Algebra Vol. I

INTERNATIONAL SERIES OF MONOGRAPHS IN

PURE AND APPLIED MATHEMATICS GENERAL EDITORS: I. N. SNEDDON and M. STARK

EXEcuTivE EDITORS: J.-P. KAHANE, A. P. ROBERTSON and S. Uz.Ai

VOLUME 91

ALGEBRA 1

ALGEBRA Volume 1 BY

L. R$DEI Maths

rtka! Institute, University of Szeged Szeged, Hungary

PERGAMON PRESS OXFORD LONDON EDINBURGH NEW YORK TORONTO SYDNEY PARIS BRAUNSCHWEIG

Pergamon Press Ltd., Headington Hill Hall, Oxford 4 & 5 Fitzroy Square, London W. 1.

Pergamon Press (Scotland) Ltd., 2 & 3 Teviot Place, Edinburgh I Pergamon Press Inc., 44 - 01 21st Street, Long Island City, New York 11101

Pergamon of Canada, Ltd., 6 Adelaide Street East, Toronto, Ontario Pergamon Press (Aust.) Pty. Ltd., 20-22 Margaret Street, Sydney, N.S.W.

Pergamon Press S.A.R.L., 24 rue des tcoles. Paris 5e Vieweg & Sohn GmbH, Burgplatz 1, Braunschweig

Copyright © 1967

AKAD$MIAI KIADO, HUNGARY

First English edition 1967

This is a translation of the original Hungarian Algebra published by AkadBmiai Kiadb, Budapest

Library of Congress Catalog Card No. 64-24548

2016/67

CONTENTS Preface to the German edition Preface to the English edition . List of Symbols . . . .

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CHAPTER I

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SET-THEORETICAL PRELIMINARIES

§1.Sets

2. Relations . . . . . . . . . . . . . . . . 3. Mappings . . . . . . . . . . . . . . . . 4. Multiplication of mappings . . . . . . . . § 5. Functions . . . . . . . . § 6. Classification of a set. Equivalence relations § 7. Natural numbers . . . . . . . . . . . . . § 8. Equipotent sets . . . . . . . . . . . . § 9. Ordered and semiordered sets . . . . . . . § 10. Well-ordered sets . . . . . . . . . . . . § 11. The lemma of Kuratowski- Zorn . . . . . § 12. The special lemma of Kuratowski-Zorn . . § 13. The lemma of Teichmuller-Tukey . . . . § 14. The theorem of Hausdorff-Birkhoff . . . . § 15. Theorem of well-ordering . . . . . . . . § 16. Transfinite induction . . . . . . . . . . § § §

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§ 17. Compositions . . . . . . . . . . . . . . . . . . . . . . § 18. Operators . . . . . . . . . . . . . . . . . . . . . . . . § 19. Structures . . . . . . . . . . . . . . . . . . . . . . . . . . § 20. Semigroups . . . . . . . . . . . . . . . . . . . . . . . § 21. Groups . . . . . . . . . . . . . . . . . . . . . . . . . . § 22. Modules . . . . . . . . . . . . . . . . . . . . § 23. Rings . . . . . . . . . . . . . . . . . . . . . . . . § 24. Skew fields . . . . . . . . . . . . . . . § 25. Substructures . . . . . . . . . . . . . . . . . § 26. Generating elements . . . . . . . . . . . . . . . . . . § 27. Some important substructures . . . . . . . . . . . § 28. [somorphisms . . . . . . . . . . . . . . . . . . . . . . § 29. Homomorphisms . . . . . . . . . . . . . . . . . . . . . § 30. Factor structures . . . . . . . . . . . . . . . . . . . . . § 31. The homomorphy theorem . . . . . . . . . . . . . . . . . . . § 32. Automorphisms. Endomorphisms. Autohomomorphisms. Meromor-

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CHAPTER It STRUCTURES .

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33

34 40 51

57 59 67 70 76 80 86

90 97 99 100

CONTENTS

vI

§ 33. Isomorphic structures with the same elements . . . . . . . . . . . . . . . § 34. Skew products . . . . . . . . § 35. Structure extensions . . . . . . . . . . . . § 36. Representation of groups by permutation groups . . . . . . . . . . . . . . § 37. Endomorphism rings . . § 38. Representation of rings by endomorphism rings . . . . . . . . . . . § 39. Anti-isomorphisms. Anti-automorphisms . . . . . . . . . . . § 40. Complexes . . . . . . . . . . . . § 41. Cosets. Residue classes . . . . . . . . . . . . . . . § 42. Normal divisors. Ideals . . . . . . . . . . § 43. Alternating groups . . . . . . . . . . . . . . . . . . . § 44. Direct products. Direct sums . . . . . . . . . . . . . . § 45. Basis . . . . . . . . . . . . . . . . . . . . . § 46. Congruences . . . . . . . . § 47. Quotient structures . . . . . . . . . . . . § 48. Difference structures . . . . . § 49. Free structures. Structures defined by equations . . . . . . . . § 50. Schreier group extensions . . . . . . . . . . . . § 51. The holomorph of a group . . . . . . . . . . . . § 52. Everett ring extensions . . . . . . . . . . § 53. Double homothetisms . . . . § 54. The holomorphs of a ring . . . . . . . . . . § 55. The two isomorphy theorems . . . . . . . . § 56. Simple factor structures . . . . . . . . . . § 57. Commutative factor structures . . . . . . . . . . . § 58. Zassenhaus's lemma . . 9 59. Schreier's main theorem and the Jordan-Holder theorem § 60. Lattices . . . . . . . . . . . . . . . . . . .

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200 205 207 207

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103 104 106

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110 113 115 117 118 122 125 134

140 152 154 157 162 163 174

184 187 194 198

CHAPTER III OPERATOR STRUCTURES . . . . . . . § 61. Operator structures . . . . . . . . § 62. Operator groups, operator modules and operator rings .

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223 228 234

§ 63. Remak-Krull-Schmidt theorem . . . . . . . . . . . . . . . § 64. Vector spaces. Double vector spaces. Algebras. Double algebras 238 § 65. Cross products . . . . . § 66. Monomial rings . . . § 67. Polynomial rings . . . § 68. Linear mappings . . . § 69. Full matrix rings . . § 70. Linear groups . . . § 71. Alternating rings . . . . . . § 72. Determinants . § 73. Cramer's rule . . . . § 74. Characteristic polynomials § 75. Norms and traces . . . § 76. Complex rings . . § 77. The quaternion group . § 78. Quaternion rings . . . .

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253 259 267 274

277 280 282 290 295 297 298 299

CONTENTS

V:i

CHAPTER IV DIVISIBILITY IN RINGS § 79. Factor decompositions and divisibility . . . . . . § 80. Ideals and divisibility . . § 81. Principal ideal rings . . . . . . . . . . . § 82. Euclidean rings . § 83. Euclid's algorithm . . . . . . . § 84. The ring of the integers . . . § 85. Szendrei's theorem . . . . . . . § 86. Polynomial rings over skew fields . § 87. The residue theorem for polynomials § 88. Gauss's theorem . . . . . . . . . . § 89. The ring of integral quaternions . . .

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369 376

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CHAPTER V FINITE ABELIAN GROUPS § 90. Cyclic groups . . . . . . . . . . . . . . . . § 91. Frobenius-Stickelberger main theorem § 92. Haj6s's main theorem § 93. The character group of finite Abelian groups § 94. The Mdbius-Delsarte inversion formula . . § 95. Zeta functions for finite Abelian groups . . § 96. The group of prime residue classes mod in

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381

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399

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CHAPTER VI OPERATOR MODULES . . . . . . . § 97. Operator modules and vector spaces § 98. Determinant divisors and elementary divisors . . . . . . § 99. The main theorem for finitely generated Abelian groups § 100. Linear dependence over skew fields . . . . . . . . . . . . § 101. Vector spaces over skew fields . . . . . . . . . . . § 102. Systems of linear equations over skew fields . . . . § 103. Kronecker's rank theorem . . . . . . § 104. Schur's lemma . . . . . . . . . . . . . . . § 105. The density theorem of Chevalley-Jacobson . . § 106. The structure theorems of Wedderburn-Artin . . . . . .

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409

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CHAPTER VII COMMUTATIVE POLYNOMIAL RINGS § 107. McCoy's theorem . . . . . . . . . § 108. Differential quotient . . . . . . . . . . . § 109. Field of rational functions . § 110. The multiple divisors of polynomials § 111. Symmetric polynomials . . . . . . § 112. The resultant of two polynomials . . § 113. The discriminant of a polynomial . . § 114. The Newton formulae . . . . . . . § 115. Waring's formula . . . . . . . . .

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325 328 329 335 337 340 342 345

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414 422 422 423 426

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433 438 440

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441

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443 450 453 454

CONTENTS

viii

. . . . . . . . . . . . . . . . § 116. Interpolation . § 117. Factor decomposition according to Kronecker's method . . . . . . . . . § 118. Eisenstein's theorem . . . . . . § 119. Hilbert's basis theorem . . . . . . . . . . . . . . . . § 120. Szekeres's theorem . . . . . . . . . . . . . . . . . . . . . . § 121. Kronecker-Hensel theorem . . . . . . . § 122. Tschirnhaus transformation of ideals . . . . § 123. Rings generated by a single element . . . . .

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CHAPTER VIII THEORY OF FIELDS . . . . . § 124. Prime fields . . . . . . § 125. Relative fields . § 126. Field extensions . . . . . . § 127. Simple field extensions . . § 128. Extension fields of finite degree § 129. Splitting field . . . . . . § 130. Steinitz's first main theorem . § 131. Normal fields . . . . . . . . § 132. Fields of prime characteristic . § 133. Finite fields . . . . . . . . . . .

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458 460 462 464 466

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471 473

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477 478 481 482 488 490 494 496 498 499 507 508 513 515 518 523 527 532

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548 551

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§ 134. Kong-Rados theorem

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§ 135. Cyclotomic polynomials . . . . . . . . § 136. Wedderburn's theorem . . . . . . . . . . § 137. Pure transcendental field extensions . § 138. Steinitz's second main theorem . . . . . § 139. Simple transcendental field extensions . . § 140. Isomorphisms of an algebraic field . . . § 141. Separable and inseparable field extensions § 142. Complete and incomplete fields . . . . § 143. Simplicity of field extensions . . . . . . § 144. Norms and traces in fields of finite degree .

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§ 145. Differents and discriminants in separable fields of finite degree 555 . . § 146. Ore polynomial rings . . . . . . . . . 558 . . . § 147. Normal bases of finite fields . . . . . . . . . . 560 .

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CHAPTER IX ORDERED STRUCTURES § 148. Ordered structures . . . . . . . . . . § 149. Archimedean and non-Archimedean orderings . § 150. Absolute value in ordered structures . . .

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CHAPTER X FIELDS WITH VALUATION § 151. Valuations . . . . . . § 152. Convergent sequences and limits § 153. Perfect hull . . . . . . . § 154. The field of real numbers . . .

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587 594 602

CONTENTS

is

. . § 155. The field of complex numbers . . . . . § 156. Really closed fields . . . . . § 157. Archimedean and non-Archimedean valuations . . . . § 158. Exponent valuations . . . . . . § 159. Discrete valuations . . . . . . . . .

§ 160. p-adic valuations

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. . . . . . . . . . . . § 161. Ostrowski's first theorem . . . . . . . . . . . . § 162. Hensel's lemma . . . . . . § 163. Extensions of real perfect valuations for field extensions of finite

degree

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. § 164. Ostrowski's second theorem . . . . § 165. Extensions of real valuations for algebraic field extensions . . . . § 166. Real valuations of number fields of finite degree § 167. Real valuations of simple transcendental field extensions . .

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CHAPTER XI GALOIS THEORY . § 168. Fundamental theorem of Galois theory . . . . § 169. Stickelberger's theorem on finite fields . . . . § 170. The quadratic reciprocity theorem . . . . . . § 171. Cyclotomic fields . . . . . . . . . . § 172. Cyclic fields . . . . . . . . . . § 173. Solvable equations . § 174. The general algebraic equation . . . . . § 175. Tschirnhaus transformation of polynomials § 176. Equations of second, third and fourth degree . § 177. The irreducible case . . . . . . . . . . . § 178. Equations of third and fourth degree over finite fields . . . . . § 179. Geometrical constructibility . . . . § 180. Remarkable points of the triangle . § 181. Determination of the Galois group of an equation § 182. Normal bases . . . . . . . . . . . .

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691

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692

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CHAPTER XII FINITE ONE-STEP NON-COMMUTATIVE STRUCTURES . . § 183. Finite one-step non-commutative groups . . § 184. Finite one-step non-commutative rings . . . . . § 185. Finite one-step non-commutative semigroups . . . .

.

Bibliography Index

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Other titles in the series

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639 644 647 648 649

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655 663 664 669 672 680 687

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610 615 617 619 627 629 633 635

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701 703

708 713 729 733

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736 753

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786

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799

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809

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821

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PREFACE TO THE GERMAN EDITION Algebra originated from the connections performable in sets of natural numbers. This surely constitutes one of the oldest collective accomplishments of mankind. With the development of civilization numerous other examples

of connections came into being before it was realized that they could be viewed from a uniform standpoint and that far-reaching generalizations could be reached, i.e. in each set one or more connections can be defined by aattching one element of it to each two elements of the same set, whereafter it can be termed a structure. Algebra deals with the investigations into such structures. Algebraic research is limited by restrictions. One of these is that of all imaginable connections only those of practical importance must be investigated. Such are associative connections wherefore the author has confined himself to this case only, as is generally done in text-books. Non-associative connections are merely considered as expedients. What is of primary importance is the modification of that notion of Algebra which springs from the realization that the nature of the elements of a structure need not be considered. The exact formulation of this idea is owed

to the genius of Ernst Steinitz who by establishing the "Principle of Isomorphy" in his epoch-making treatise "Algebraic Theory of Bodies" created therewith something similar in algebra to what Felix Klein had done

in geometry in his "Programme of Erlangen". As algebra is a relatively young branch of science, one should not be surprised at the tremendous amount of algebraic studies in recent years. Though many of its chapters were extensive before Steinitz, development remained incomplete and its parts were not sufficiently coordinated with one another as a consequence of the lack of a uniform standpoint, and at present it still falls behind other mathematical disciplines in completeness and method. This suggests the necessity for continued research. The author believes that a methodical approach is to set out from generality and to proceed to the particular. A systematization of this kind was attempted but such efforts had to be kept within bounds, as the author wanted to write a text-book. These requirements, partly contradictory, have been met by placing the general formation of concepts at the beginning of Chapters II and III, and by dealing with special cases of application either parallel to the exposition or immediately following it. In these two chapters some problems of structure are treated to satisfy the natural demands of students ai

PREFACE TO THE GERMAN EDITION

xii

who are mostly interested in the elaboration of structures. In order to prevent any further extension of Chapters II and III, which are already a little too comprehensive, applications had to be used sparingly, but further applications are dealt with in Chapters IV - XII which are devoted to various other topics. In accordance with the author's efforts at systematization much attention

has been paid to the analogies between group theory and ring theory; it would be very desirable to find more such analogies. Among other text-books not mentioned individually, the author has drawn largely from B. L. van der Waerden's "Algebra" and Pickert's "Introduction to Advanced Algebra". It must be mentioned that R. Kochen-

dorffer's recently published "Introduction to Algebra" has enabled the author to bridge some gaps in his completed manuscript. The contents of paragraphs 52, 53, 54, 85, 92, 94, 95, 120, 122, 123, 146, 178, 180, 183, 184, 185 have not previously been dealt with in any text-books

and are, for the most part, completely new. The same is largely true for paragraphs 27, 44, 49, 51, 58, 59, 64, 65, 66, 73, 82, 103, 121, 134, 148, 172, 173; in these the novelty consists merely of the fact that well-known theorems have been worded in a more general way than hitherto. More or less new methods are contained in paragraphs 11, 33, 50, 89, 91, 106, 115, 130, 136, 138, 147, 153, 169, 170, 179. The reader of this book requires little mathematical knowledge beyond a

knowledge of natural numbers; at the very most a certain aptitude for mathematical thought is required. Fundamental questions have not been taken into consideration by the author. Examples, exercises and some unsolved problems are to be found at the end of many paragraphs. Some examples have also been inserted in the text itself where they are necessary for a better understanding of the subjectmatter treated. In the exercises and examples some well-known concepts (e.g. that of complex numbers) have been assumed; these are defined in a subsequent part of the book. The reader is advised to study the examples carefully. Paragraphs marked with an asterisk require considerable effort. The reader may omit the proofs at a first reading.

For the sake of simplification the customary terminology has been changed slightly. The author says, e.g. (analogously to "coefficient group") "coefficient ring" instead of "remainder-class-ring". The "full permutation group" (quite analogous to the "full matrix-ring") means the "symmetrical permutation group". A "main-polynomial" means a polynomial with the initial coefficient of 1. Instead of using the usual "main-polynomial of an

element" the author uses "minimal polynomial of an element". Though the author talks of "normal or standardized polynomials" in the customary sense, the notion of the "main-polynomial", however, can still not be avoided.

PREFACE TO THE GERMAN EDITION

It was the author's endeavour to use as simple notations as possible. As a novelty in this field of notation, the system S - S' (a a') should be mentioned, by which the author wishes to express that a - a' is a homomorphic transformation of the structure S on structure S'. The author has made great efforts to facilitate the reader's task. This purpose is served by cross-references to serial-numbered theorems at the bottom right-hand side of odd-numbered pages. In spite of the size of this volume completeness in any direction could not be hoped for. In particular, in the theory of finite groups some interesting special themes have been treated, but only a few fundamental theorems

are given. It would have been desirable to include "General Algebra" which is becoming more and more important but this had to be renounced owing to pressure of space. Besides some text-books and monographs only such other works are included in the bibliography as have been quoted in the text. Many modifications have taken place since. the first Hungarian edition of this volume. Some paragraphs (mainly in Chapters II and III) have been rearranged, some have been split, and further on many new paragraphs have been added. Some errors to which the attention of the author was drawn, partly by his colleagues and readers, have been corrected. For this and many other services of various kinds the author owes a great debt of gratitude to Messrs. A. Adam, W. Blaschke, G. Fodor, E. Fried, L. Fuchs, L. Kalmar,

F. von Krbek, G. Pickert, G. Pollak, L. Prohaska, L. Pukanszky, S. Schwarz, E. Sperner, 0. Sternfeld, J. Szendrei. The author is deeply grateful to his late friend, T. Szele, who was of great assistance. The author's sincere acknowledgements are due to the Academic Publish-

ing Company Geest & Portig K.-G., Leipzig, and the printing office "Magnus Poser" at Jena, who have always willingly met numerous special wishes, often with difficulty. After the delivery of the manuscript in April 1956, various changes were made, so that this work was concluded only at the end of 1958. Szeged, December, 1958

The Author

PREFACE TO THE ENGLISH EDITION Compared with the original German edition this volume contains the results of more recent research which have to some extent originated from problems raised in the previous German edition. Moreover, many minor and some important modifications have been carried out. For example paragraphs 2-5 were amended and their order changed. On the advice of G. Pickert, paragraph 7 has been thoroughly revised. Many improvements originate from H. J. Weinert who, by enlisting the services of a working team of the Teachers' Training -College of Potsdam, has subjected large parts of this book to an exact and constructive review. This applies particularly to paragraphs 9, 50, 51, 60, 63, 66, 79, 92, 94, 97 and 100 and to the exercises. In this connection paragraphs 64 and 79 have had to be partly rewritten in consequence of the correction. Besides those already mentioned the author wishes to express his thanks to his colleagues for their comments and advice and to the publishers and

printers for their careful work. The Author

Szeged, July, 1964

xv

LIST OF SYMBOLS of )

E

48

0 c U n

f x'

49 50 2

2

p

2

a

2 2,252 4

na

5

E

58 61

64 66 67

57

-«

57

0+()

Q_i

6,30

R *

r

9,

(Pm)

min, max 0

10

F*

10

Q

10,154

I, j, k

19,568

{}

00 Z()

19

28 28 28

S

S' (cc -*x)

57 59

68 69,3C 76 81

82 87 88 92 92 98,12 98,15

0 S+

32,43

sx 00

37 38 38

A-Structure

38

SIO

40,226

S.

117

O-Structure

40 40 40 40

0(G:H)

122

()

130

( )l

130

sign

O()

.5!'o

17

S-.S'

37

S/@

mod

111

113

( )r

130

<"r' 0>

40

11

42,592

135 136

F*

46

141

or

47

141

xvii

xviii

LIST OF SYMBOLS

cA °G

175

182

JP -I- R A, V

188

214 216 230,231,232

inf, sup IR-

c`I.'

232 224 242

JI (H)

255

--I[]

259

F I Jv'

SIG-'S' 0(i

'Yf° [xlo f° ,.

x (au ),,,,,. (a )

x')

263 264 269

it ()

o (a (mod m)) d

FI-7 -- F'IJ F ()

[G:F] F' fGIF, NGIF

b

b(), D(

Ia ,,

282

.f, (x)

All

289

adj

289

Q8

ass

301 306

()i, H rI

308

n(), s() O*, [ ]* Qc

TGIF

551

556 557 558 583

w()

586

99-1im

587

log

602 609

9'io) (a)

610

o)

613

614 620

310

WO

325 333,585

Iv4, ( )

630

5P)

632

345

( l Ipl

665

366 [k]

.

SGIF' DGIF

II

479 482 482 509 542,547

F (x)

275

297 298

477

FP

E;j

T

451

D (f)

`r

434-6 440 444

R9)

274 275

296 , 551

13

oeklifl

12.

N(), S(

x

f', x ' ,

382,384 393

673

CHAPTER i

SET-THEORETICAL PRELIMINARIES § 1. Sets

The notion of a set, i.e. that of a collection of certain various entities, is taken to be known. A fuller treatment of set theory may be found elsewhere, e.g. in FRAENKEL (1946). The entities of which a set consists are called the elements of the set. We denote that a is an element of the set S as follows: aES a S means that a is not an element of S. Later we similarly denote by putting a stroke through a sign in this way that the assertion, which would otherwise have been expressed by the sign, is false.

The notion of the empty set, denoted by 0, is also useful. This is defined as the "set with no elements". If we give up the restriction that the elements are all different, then we use the term system (instead of set). (For further details see the end of § 8.) Two sets are called equal, when each contains the elements of the other.

We denote by the set consisting of the elements a, b, .. . Hence we have, for instance, = (b, a, c>. ("=" is the sign used to denote equality.) We often identify the set , consisting of a single element

a, with the element a. We often define a set by some property of its elements, e.g., the set of unmarried men in London who are at least thirty years old. If no element satisfies the defining property, then the set so defined is the empty set. We can also think of sets as entities. Thus we allow the existence of sets,

whose elements are also sets. If, for example, A, B.... are different sets then S = denotes the set with elements A, B, ... Such a set S is called a set of sets. Similarly we can have the set <, >.

If we have two sets S and T, we say that T is a subset of S, and we write

TCS if S contains all the elements of T. Alternatively we say that S is an overset or extension set of T. Further if TC S and T :A S, we call T a proper subset of S

and S a proper overset or proper extension set of T, and we write T c S. I

2

SET-THEORETICAL PRELIMINARIES

It is reasonable to take A z) B to mean the same as B c A. For other signs, to be introduced later, we shall interpret oppositely directed signs in a similar way.

Given sets S, T, ..., their union, denoted by S U T U ..., and their intersection (or cross-cut), denoted by s fl T fl

....

are taken to mean, respectively, the set of alI elements occurring in at least one of the given sets and the set of all elements occurring in each one of the given sets. By the difference set A - B of two sets A, B we mean. the set of all elements of A not contained in B. The sets A, B are called d i s j o i n t if A fl B = 0. The following abbreviations are sometimes used : n

U A, = A, U ...

U A,,

...

fl A

i=1

fl A; = A1. fl

i=1

.

Given a collection of sets, we say one of them is minimal or maximal, if it is not a proper overset or not a proper subset, respectively, of any one of the given collection of sets. For example, among the sets and , the first and third are mini-

mal, the first and second are maximal. Among the sets



(n=0,± 1,±2,...)

there is neither a minimal nor a maximal set.

By the product of two sets A, B we mean the set of ordered pairs (a, b)

(a E A, b E B)

and denote it by A x B. Here we have to point out that in general two such ordered pairs, (a, b) and (c, d), are considered equal only when a = c and b = d. The definition of a product is extended in the obvious way to the product A x B x ... of several sets A, B.... When St = S., = ... = S,,, the product St x Sz x ... x S is written S" and called the nth power of the set S (n = 1, 2, ...). It is well known that the so-called naive set-concept, applied above, involves contradictions. This was realized by CANTOR the founder of set theory. However these contradictions may be avoided by means of an axiomatical establishment of set theory, which implies a restriction of the general set-concept. (See e.g., FRAENKEL

(1946), or, in more modern treatment, G(DEL (1953). We shall consider only such sets as conform to the established principle of the limited axiomatical set-concept.

RELATIONS

3

§ 2. Relations There often exists a so-called relation between certain pairs x, y of the elements of a set. We have, for example, "x is the son of y", or "x is smaller than y". In this section we are going to generalize this notion.

We consider two given sets A, B and a subset R of the product set A x B. 91 is thus a collection of certain pairs (x, y) where x E A, Y E B. We introduce a sign r such that x r y means (x, y) E 91. We now say we have defined a relation between the elements of the sets A, B. In other words: The rela-

tion x r y is true or false according as the assertion (x, y) E 91 is true or false. r is called the sign of the relation. For brevity we refer to the relation denoted by r as the "relation r". In the important special case A = B we speak of a relation defined in the set A. The simplest example is the equality relation or briefly equality already used. This relation, denoted by =, is definable in every set and corresponds to the case when IR consists of all pairs (x, x) where x E A.

The signs E, c (but not the signs fl, U) introduced in § I similarly denote relations. With respect to a relation r, two elements x, y with the property x r y are usually described as relative or corresponding elements or some similar term. Relations are also linked together as follows :

a, r a,, r ... r a,, .

(2.1)

(2. 1) is called a chain (of relations) and denotes that

atra,+,

(i = 1, 2,...,n - I).

We interpret in a similar way a chain of the form

...

tr...

Occasionally more than one relation is involved. Thus. for example, a E A e B means that both of the relations a E A and A c B hold. which mean When dealing with assertions we use the signs and implies" and "if and only if" respectively. Thus given two assertions a and b, a . b denotes that a implies b and a a b denotes that a holds if and only if b holds. Clearly . and denote special relations. A relation r defined in a set A is called reflexive if a r a for all a E A,

symmetric if arb=*- b r a (a, b E A), and transitive if a r b, b r c = a r c (a, b, c E A). If r is a transitive relation, it would follow from (2.1) above that

a,ra,

(I <_i
4

SET-THEORETICAL PRELIMINARIES

EXAMPLE. In the set of real numbers the sign <, applied above to integers, denotes a transitive relation. This relation is neither reflexive nor symmetric. The reader will find the complete definition of this relation in §§ 7, 148.

§ 3. Mappings A particularly important concept, that of a so-called mapping, arises if to each element of a set we make correspond, in some prescribed manner, an element of a second set (or, in some cases, of the same set). However mappings are defined below much more explicitly as special cases of relations.

A relation -'- defined between the elements of two sets S and S' is called a mapping of the set S into the set S', if there exists for each x E S exactly one x' E S' such that x - x'. This element x' is called the image of x (or we may refer to x' as the element corresponding to the element x). Conversely,

we call x an inverse image of x'. Where x -* x we call x a fixed element of the mapping. (Each element of S has only one image. On the other hand, an element of S' has in general several inverse images, although it may have no inverse image. The latter is the case for an element of S' which is not the image of any element of S.) If each element of S' is the image of at least one element of S, we say the mapping is of the set S onto the set S'. (It is important to note the different meanings of the

prepositions "into" and "onto".) Where S = S' we speak of a mapping of the set S into itself or onto itself. For example, n -- n + 1 (n = 1, 2,.. .) is a mapping of the set of natural numbers into itself, but not onto itself.

We extend the use of the terms image and inverse image to subsets of the sets under consideration as follows: For a mapping of the set S into the set S', the image of a subset St of S is taken to mean the set of images of all the elements of S1. Similarly the inverse image of a subset Sl of S' means the set of inverse images of the elements of S'. We should note the following. If Si (s S') is the image of Sl (s S), the inverse image of Si obviously contains St but may, in fact, be more extensive

than this. On the other hand, if Si is given and has the set Sl as its inverse image, then Si is the image of S1. In this case, i.e. where SL s S and S]'. C- S' and where Si is the image of Sl and Sl is the inverse image of Si, we call SI

and Si corresponding subsets (of S and S'). We must note also the following. For a mapping x -- x' (defined as above), the element x is an inverse image of x'. However if x' is thought of as a set (consisting of the single element x'), then the inverse image of x' consists of the set of all x such

MAPPINGS

that x

5

x'. To avoid possible misunderstandings, we continue to call such x the

inverse images of x' and their set the set of inverse images of x' (instead of "the inverse image of x"').

If x -* x' is a mapping of the set S into the set S', and S (s S') denotes the set of all images of the elements of S, we shall consider that we have a mapping of S onto So, which consists of all the relations x --> x' (x E S). We may identify the latter mapping with the original one and denote it also by x -> x'. Thus we may from now on give any mapping of S into S' as a mapping of S onto a subset of S'. If x --> x' is a mapping of S onto S' and SI (C S), S'(9; S') are such that Si is the image of SI, then we consider that we are also given the mapping of S1 onto Si, which is defined by the correspondencex --> x' (x E SIX E SD. We call this mapping the induced mapping of S1. Conversely, we then say

that the mapping x --> x' (x E S) is an extension or continuation of the mapping x --* x' (x E Sl C S). More generally, we sometimes describe as a mapping and denote by x -> x' a relation between S and S' where arbitrarily many (or possibly no) elements x' correspond as images to the individual elements x E S. In this case we speak of a many-valued mapping. Hence we may call mappings, as originally defined, single-valued mappings. However, in conformity with our original definition we normally leave out the term "single-valued". In the case of many-valued mappings we may omit the term "many-valued" if the multiplicity is obvious from the context. By the inverse mapping of a mapping x -> x' of S onto S' we mean the (usually many-valued) mapping of S', consisting of all the correspondences x' -k x (x' E S'). This means that all the inverse images, with reference to the original mapping, of each element of S' are defined to be the images of that element of S' in the inverse mapping. By a one-to-one mapping of S into S' we mean a mapping of S into S', where two different elements of S always correspond to different elements

of S'. It is obvious that a mapping of S onto S' is a one-to-one mapping if and only if the inverse mapping is single-valued. The one-to-one mappings of a set S onto itself are called the permutations of S. Thus, in particular, the so-called identical mapping x - x, in which all elements are fixed elements, is a permutation. Evidently, every permutation produces, by taking the corresponding inverse mapping, again a permutation which we call the inverse permutation (of the given permutation).

Any mapping may be represented very clearly by

a b.

(a ' b'...)

[

=

fx' I, '

(3.1)

6

SET-THEORETICAL PRELIMINARIES

where immediately below each element of the first line is placed its image. This type of notation is especially favoured in the case of permutations. It is evident that (3.1) is a permutation if and only if the elements of the first and second lines are the same, apart from their order. It is to be noted that we shall sometimes for brevity speak of a mapping x' instead

of a mapping x - x'. Similarly, e.g., the permutation 1', ... , n' shall denote

Finally, in our consideration of mappings, we shall often use the so-called operator notation. This means that we denote the mapping by a, say, and the image of x by ax [or a(x)]. The mapping itself may then also be written in the form x -- ax. (We can then say that the image ax of x is obtained by "carrying out a on x".) This notation is very convenient when we are

considering several mappings o, a.... Instead of ax the notations xa and x° are sometimes used for the image element. If a is a one-to-one mapping. we denote the inverse mapping by a-'. § 4. Multiplication of Mappings

We define the multiplication of mappings by carrying them out one after another as follows: Let a be a mapping of a set A into a set B and n a mapping of the set B into a set C, then the product of the mappings o, a means the mapping oa of A into C for which

(na)x = &x)

(x E A).

(4.1)

(In order to determine (ea) one has first to carry out a and then e.) THEOREM 1. For the mappings o, a, r

(oa)r = e(ar)

(4.2)

provided that either side of (4.2) is meaningful. By the definition of the product of mappings, both sides of (4.2) exist if and only if - for some sets A, B, C, D - r, a and o symbolize the mappings of A into B, B into C and C into D, respectively. It is now only necessary to show that ((ea)r)C = (o(aT))x (x E A). Applying (4.1)

((oa)r)x = Ox) (ra) and

=

.o(a(rx))

(e(ar))x = o((aT)x) = o(a(rx)).

The theorem is now proved.

MULTIPLICATION OF MAPPINGS

7

If the image of x is denoted by xc or x° then the product er is usually defined as x(ea) = (xe) a, or x" _ (x)°°. It is obvious that Theorem I retains its validity.

The multiplication of mappings of a set S into itself forms an important special case of multiplication of mappings. If A denotes the set of these mappings, then, by definition any two elements e. a of A may be multiplied

together. Their product, too, belongs to A. It is important to note that pa 0 co may hold. For example, if S = and !, _

a b1 b a)

r=

th en

er= fI ba b

bJ

ae =

ab a aJ

ra b a

aI

o that ga 0 ae.

Two mappings e, a are called interchangeable (each to each) if the products pa and c exist and are equal. § 5. Functions "Mappings" and "Functions" are both terms for the same concept. The latter term might therefore be considered superfluous but it has some historical importance and is in current use. The notation and terminology applied to functions differ from those used with reference to mappings.

Let two sets S, S be given. Then a mapping of S into S, denoted by f, is called a function, with domain of definition S and range S. The latter is not uniquely determined, for, in addition to S every overset of S is a range for

the given function. The image of an element x (E S) determined by f is usually symbolized by f(x). In this notation the function f means the mapping x -I- f(x). If, as here, a definite letter x is used for the arbitrary element of S, then x is called the variable of the function. (This term suggests that x runs through all elements of the domain of definition S.) The image f(a) of

a particular a (E S) is called the value of f at the place x = a. It may be said that f(a) is obtained from f(x) by the substitution x = a. For convenience, the function f with the variable x and the domain of definition S is often denoted byf(x) (x E S.) This is sometimes simplified to f(x) (without the inclusion of the domain of definition). This notation will cause no misunderstanding. Occasionally many-valued functions will arise. These differ from the

above in that an arbitrary number of (possibly no) function values f(x) may be assigned to a given x. Except where specifically stated, or where

SET-THEORETICAL PRELIMINARIES

S

the context indicates the many-valued case, only single-valued functions will be considered. Nevertheless, functions of several variables are frequent, which are merely variants of single-valued functions. Namely in the case of given sets S1, ..., S (n >_ 1), S we define as a function of n variables, a function f with the domain of definition S1 x ... x S,, and with range S. Therefore such a

function may be denoted by f(xl, .

.

.,

where xi is a variable in Sl (i =

= 1, ..., n). In the important special case where SI = ... = S = S we call f(xl, ..., xj a function of n variables defined on the set S. If S = S we speak of a function of n variables defined in the set S meaning that each of the variables x1, . . ., x,, runs through all elements of S and the function values also belong to S. The idea of a function is used in various ways in Mathematics generally, and in Algebra in particular. The following general problem, which will recur subsequently in this book, concerns the -solution of equations". Given two functions with a common range, f;(xl, . . ., (i = 1, 2; Xk E Sk; k = 1, . . ., n) the problem is to decide for which particular systems

X1, ..., x the equation f1(x1, ..., x,,) = f2(.CI, ..., Xn)

(5.1)

is valid. The systems for which (5.1) is valid are the solutions or roots of the equation (5.1). In connection with this equation x1, ..., x are the unknowns.

It is emphasized that (5.1) does not imply the equality of the functions f 1, f2 occurring in it; preferably it may be called "an equation submitted for solution" and it might, for example, be recognized as such by the insertion of a question mark (?) after it. There are, however, always other forms of notation for such an equation. Should the unknown be required to satisfy several equations simultaneously, we speak of systems of equations (instead of equations). The following is particularly important. In the case of arbitrarily chosen non-empty sets A, B,.. . , by a choice function we understand a function which assigns to each set A, B, . . . one element a, b, . . . from the set itself. More precisely the function f(X), whose domain of definition is the set of the sets A, B, . . . and whose range is the union of the sets is defined by the validity of f(X) E X. The existence of a choice function is self-evident even if it is not always given. Its use is indispensable for the proof of many theorems, as ZERMELO

was the first to point out. The assumption of the existence of a choice function is known as Zermelo's postulate or the axiom of choice. The truth of this axiom is presupposed throughout this book. (Cf. the text in small type at the end of § 15.)

CLASSIFICATION OF A SET. EQUIVALENCE RELATIONS

9

§ 6. Classification of a Set. Equivalence Relations By a classification of a set S is meant its partition into pairwise disjoint subsets C, C'...., whose union is S, denoted shortly by

S = C, C". .

.

(6.1)

This symbolism is restricted, to avoid misunderstandings, to cases where classification is obviously intended. The subsets C, C',. . . themselves are called classes (of the given classification). Unless stated to the contrary, it is to be assumed that none of the classes is empty. Obviously, every element of S belongs to a definite class called the class of this element. Conversely, every element of a class is called a representative of that class. By taking a representative out of each class, we obtain a system of representatives (of that classification).

In order to designate a system of representatives a choice function is employed and therefore the axiom of choice is often used.

From (6.1) we have the set which we call the set of classes. It often plays an important role. The concept of classification leads to another concept: After a set has been classified, the elements of a particular class may be considered to be equal. This is the concept of abstraction, but instead of using this concept, we proceed to the next definition.

In the case of a classification (6.1) we call the elements x, y of the set S equivalent if they belong to the same class. That is denoted by x = y. This relation is called an equivalence relation, or more precisely, the equivalence relation belonging to (6.1). It is, obviously, reflexive, symmetric and transitive.

Conversely, we prove that every relation r defined in a set S having the three last mentioned properties is an equivalence relation.

We denote by C(a) the set of those elements x for which x r a is valid (a, x E S). We then show that the set of the different classes C(a) forms a classification of S for which the corresponding equivalence relation is identical to F. Because r is reflexive, a r a is valid for all a E S. Hence a E C(a), and further, no C(a) is empty, so that every element of S is contained in at least one C(a). We assume that C(a), C(b) contain a common element x. Then

x r a, x r b. From this it follows, because of the symmetry, that a r x, and on account of the transitivity that a r b. Consequently if y E C(a), i.e., y r a it follows from the transitivity that y r b which implies that y E C(b). Therefore C(a) c C(b). Similarly C(b) S C(a) implying that C(a) = C(b), so that C(a), C(b) are either equal or disjoint. From this it follows that C(a) is a classification of S.

SET-THEORETICAL PRELIMINARIES

10

The equivalence relation belonging to this classification is denoted by

_, so that all that remains to be proved is that r and = have the same meaning. As x = y denotes that x, y lie in the same class, there is an a(E S) such that x r a, y r a. This implies that x r y, and vice versa, proving the assertion.

Since, according to this, the three properties, reflexivity, symmetry and transitivity are together characteristic for equivalence relations, they are known as the equivalence properties or axioms of equivalence. Since the equivalence relations belonging to two different classifications are obviously not the same, we may speak of a classification belonging to an equivalence relation. The classes so formed are classes associated with the equivalence relation or, briefly, equivalence classes.

The relation = obviously has the equivalence properties, hence equality (in every set) is an equivalence relation where the corresponding classes

contain only one element each. The equivalence concept is therefore a generalization of equality. For the sake of clarity the axioms of equivalence are recapitulated:

I a = a (reflexivity)

II a = b = b = a (symmetry)

III a

b, b = c => a = c (transitivity).

Obviously, in this axiom system the axioms II and III may be replaced by the axiom

II'a=b, a=cab =c.

To prove that a given relation is an equivalence one, we often use the axioms I and 11, instead of I, II and 111. THEOREM 2. Let a set S be given a relation r. Define in S a further relation

as follows: For two elements a, b (E S), a - b if, and only if, there are in S elements cI, ..., c,,, such that in a, cz, .... c,,, b for any two neighbouring

elements x, y, at least one of the relations x = y, x r y, y r x is valid. This relation =_ is always an equivalence relation.

Since the relation = as defined in the theorem satisfies the axioms I, II and III, the theorem is true. The equivalence relation defined in Theorem 2 is called the induced equi-

valence relation from the given relation. Consider the classification in a set belonging to the equivalence relation _. If z denotes the class represented by the element x then, obviously We may, therefore, proceed from the equality of two classes to the equivalence relation between any representatives of this class and conversely. This is the transfer rule qr equivalence.

CLASSIFICATION OF A SET. EQUIVALENCE RELATIONS

II

The mapping of a set S (into any set) results in a classification by joining

elements with a common image into a class. This is the concomitant classification (to the given mapping).

§ 7. Natural Numbers Although some knowledge of the natural numbers is assumed, yet we wish to recapitulate their fundamental properties (LANDAU, 1930). 4 The natural numbers satisfy the following two axioms: I. The set,,- of the natural numbers contains an element e such that there is a one-to-one mapping n -* n' of 'r onto the difference set - e. of contains the element e and with every element II. If a subset

n (E A) its image n', then _lF = ..f''. We shall see next that conversely every set ,.' with the properties I, 11 has all the essential properties of the set of natural numbers, so that we may consider I, II as a system of axioms of the natural numbers. Henceforth we denote the element e by 1. We call n' the successor of n. From II immediately follows

THEOREM 3 (theorem of complete induction). Let a proposition A uniquely correspond to every natural number n. If A is true for n = 1 and the truth of A implies that of A is true, for every n. The assumption made in this theorem i.e. "A,,, is true for n" is called the induction assumption or induction hypothesis and the method of proof. justified by this theorem, complete induction. Since validity of Theorem 3 is ensured by axiom II, this is called the axiom of complete induction.

We prove that

1. L" (1')', ((1')')', ...

(7.1)

is the set of natural numbers. Because of II, every natural number occurs at least once in (7.1), wherefore we have only to prove that no two elements of (7.1) are equal. It is sufficient

to show that

n 0 n', (n')', ((n')')', .

. .

(7.2)

for every n. On account of I, (7.2) is true for n = 1. Since we accept the truth of (7.2) for one n, it follows, again from I, that

n' # (n')', ((n ')I)', ..

,

i.e. the truth of (7.2) for n' instead of n. According to Theorem 3, this implies the truth of the statement (7.1). From now on we denote the natural numbers, relying upon what has been proved, by 1, 2, 3, . . ., where 2 = 1', 3 = 2', etc.

SET-THEORETICAL PRELIMINARIES

12

It is possible to use the recursive definition (or the definition by complete induction) as follows: THEOREM 4. Let a be an element of a given set S and g a function of two variables n( = 1, 2, . . .) and x (E S) with the range S. Then one, and only one, function f exists with the domain of definition < 1, 2 ...> and with the range S for which .f (1) = a, .f (n) = g(n, f ( n ) )

(n = 1, 2, ...) .

Consider the product set P = <1, 2, ...> x S. By definition the assertion is identical to : there exists exactly one subset A of P with the following three properties:

to every n( = 1. 2....) there is exactly one x with (n, x) E A, (7.3) (1, a)E A ,

(7.4)

(n,x)E A-(n',g(n,x))E A.

(7.5)

Disregarding (7.3), consider all subsets A' of P with the properties (7.4),

(7.5). There are such A': e.g. P occurs among them. Their intersection, which will in future be denoted by A, also has the properties (7.4), (7.5). [We may say, that A is the smallest subset of P with the properties (7.4), (7.5).] Obviously it is sufficient to prove that A also has the property (7.3).

By induction, it follows from (7.4) and (7.5) that there is to every n (n = 1, 2, ...) at least one x with (n. x) E A. So it remains to be proved that there is to every n (n = 1, 2, . . .) at the most one x with (n, x) E A. It is true for n = 1, otherwise A contains (1, b) with b a. It then follows that the difference set A - (1, b) has the properties (7.4), (7.5), which contradicts the definition of A. Suppose that the assertion is true for a fixed n(n = 1, 2, ...) but false for n'. Let (r., x) be contained in A. According to the assumption A contains besides (n; g(n, x)) some (n', y) with y g(n, x). Then the difference set A - (n'. y) has the properties (7.4), (7.5). This contradiction proves Theorem 4. This proof is a modification of the proof of LORENZEN (1959). Cf. also PICKERTG6RKE (1958) and KALMAR (1949-50).

We apply Theorem 4 to the definition of the sum m + n and product in, n of two natural numbers m, n. For every m (m = 1, 2, ...)

m+1=m', m+n'=(m+n)+1 (n=1,2,...) and

ml =m, mn'=mn+m are valid.

(n = 1,2,...)

NATURAL NUMBERS

t3

It maybe proved that the well-known arithmetic rules for the natural numbers follow from these recursive definitions, which implies that axioms I and II in fact encompass all the essential properties of the natural numbers.

We say of the natural numbers in, n that in is less than n or n is greater than in, symbolically in < n, if there is a natural number x with in + x = n. THEOREM 5. In every non-empty set 4' of natural numbers there exists a least element.

Let" denote any set of natural numbers, in which at least one of the numbers 1, . . ., n occurs (n = 1, 2, ...). If /9 is an 4'1, then I is the least

element of /,f, therefore the theorem is true. We assume that for some n we have already proved the theorem for all and we now consider an lK= *"+1. It is sufficient to show that the theorem is true for this ,ff. If 'dl= AV,, is valid, then the proof follows from the induction assumption. If, on the other hand, ff then , GI' contains among 1, . . ., n + I. only n + 1, whence it follows that it + I is the least element of. This proves Theorem 5. THEOREM 6 (a second form of the theorem of complete induction). Let a proposition A" be in unique correspondence with every natural number n. If for every n, the truth of A,, follows from the supposition of the truth of all propositions AX (x < n), then A is true for all it.

If A were false for one n, then according to Theorem 5 there would be a least n of this nature. Since then all At (x < n) are true, the truth of A follows from the induction assumption. This contradiction provesTheorem6.

A second kind of recursive definition is contained in the following. THEOREM 7. Let a be some element of a set S and to every n (= 1, 2, ...) assign a function g,, of n variables xl, . . ., x,, (= 1, 2,. ..) with the range S. Then there exists one, and only one, function f with the domain of definition

< 1, 2, ...> and with the range S, for which

.f(l) = a. f(n + 1) = g"(f(l), .....f(n))

(n = 1, 2,...)

is valid. The uniqueness statement of this theorem follows obviously by induction.

It still remains to be established that such a function f can be given. We may assume without any essential restriction of the generality that the power sets S" (n = 1, 2, ...) are pairwise disjoint. We then put

S=SUS2U... x E S" holds for every x E S for only one n (= 1, 2,.. .). Consequently, this x has the form

x=(x1,...,x,,)

(x1,...,x,, E S).

14

SET-THEORETICAL PRELIMINARIES

So if we put for this x

h(x) = 9n(x1...... ,;) ,

(7.6)

then a function h(x) (x E S) with range S is defined. On account of Theorem 4, we define the function k(n) (n = 1, 2,. ..) with range S by

k(1) = a,

k(n + 1) = (k(n)), h(k(n))

(n = 1, 2, ...),

(7.7)

. . ., x,,,), x,,,+i) is to be interpreted as (xl,..., x,,,+i) Evidently k(n) lies in S. Finally, we define f(n+ 1) (n = 1, 2, ...) as the last "component" of k(n + 1), i.e.

where ((x1,

f(1) = a, f ( n + 1) = h(k(n))

(n = 1, 2, ...).

(7.8)

According to this and (7.7), k(n + 1) = (k(n), f(n + 1). Consequently we have

k(n)=(f(l),...,f(n))

(n= 1,2,...).

If this and (7.6) are put into (7.8), we find that f is the required function. Consequently Theorem 7 is proved. The above-mentioned axioms I, II agree with the axioms of Peano, by which he was the first to found the concept of the natural numbers axiomatically. These axioms are 1. 1 is a natural number.

2. To every natural number n there belongs uniquely a natural number n'. 3. For every n, n' 0 1. 4. From m' = n' it follows that m = n. 5. If a set .,07 contains the element I and with every element n also n'. then contains all the natural numbers.

'A//

§ 8. Equipotent Sets CANTOR calls two sets A, B equivalent or equipotent, if there exists a one-to-

one mapping of A onto B. We then write : A B. In this case we also say that A and B are of equal potency; otherwise, they are of different potency. In every set of sets A, B, C, ..., .: is indeed an equivalence relation. Since the set A is mapped one-to-one onto itself by its identical mapping, then A

A holds. Further, if A

B, i.e., there exists a one-to-one mapping a of A

onto B, then the inverse mapping a-1 is a one-to-one mapping of B onto A whence B x A. Finally, if A ,:; B, B :, C, i.e., there exist two one-to-one mappings Q, a, by which A is mapped onto B and B onto C, respectively, then A is mapped one-to-one onto C by the product ag, whence it follows that A ^' C. Consequently the assertion is proved.

EQUIPOTENT SETS

15

The empty set and the sets which are equipotent to a subset of the set of the natural numbers of the form (1, ..., n) are called finite, any other set is called infinite. Accordingly the finite sets are 0 and the sets of the form


First, we prove that it is impossible to map a finite set one-to-one onto any proper subset of itself. Since this is obviously true for the empty set, we can restrict ourselves to the sets < 1, ..., n> . In the case n = l the

statement is true. We assume its truth for any n and prove that it

is

true for n + 1. To do this we suppose that, on the contrary, the previous

statement is false, i.e., that a one-to-one mapping x -+ ox of the set <1,

.

. ., n + 1> onto a proper subset S of it exists.

If moreover

SC < 1,...,n),

(8.1)

then we conclude that because <1, ..., n> c < 1, ..., n + 1>, <1, ..., n) is mapped by a onto a proper subset of S and consequently, on account of (8.1), onto a proper subset of itself. This contradicts the induction assumption.

But if (8.1) does not hold, then n + 1 E S. Then there certainly is an a

(= 1,

.

.

., n)

not belonging to S. On the other hand, for a certain b ob = n + 1

(8.2)

holds.

We define the mapping r which differs from a only in that for it rb = a [as distinct from (8.2)]. Now, z is a one-to-one mapping of <1, ..., n + 1) and, moreover, all the image elements belong to the set <1,..., n). Consequently, we have reduced the present case to the previous case

(8.1), whereby we have proved the above statement. Later we shall prove the second part of Theorem 8. Since < 1, 2, ...> by the one-to-one mapping

(123... I` 234...

is mapped onto the proper subset <2, 3, ...>, it follows from the part of Theorem 8 already proved that the set <1, 2, ...> of all the natural numbers cannot be finite. Consequently it is infinite. The sets which are equipotent to the set <1, 2, ...) of the natural numbers, we call countably infinite. The finite and the countably infinite sets are called countable sets. The countably infinite sets are the sets of the form . 2 R-A

SET-THEORETICAL PRELIMINARIES

16

THEOREM 9. Every infinite set contains a countable infinite subset.

Let S be an infinite set and p(A) (E A) a choice function for the set of all the non-empty subsets A of S. We define recursively a function f(n) (n = 1,

2, ...) according to Theorem 7 by n-t f(i)) f (n) T(S

=

-U

(n = 1, 2, ...) .

(8.4)

!=I

(In particular, for n = 1: f(1) = qq(S).) Since S is infinite, the difference set which figures on the right-hand side of (8.4) is not empty so that f(n) is, in fact, defined by (8.4). It follows from (8.4) that f(n) differs from f(1), ...,

f(n - 1), consequently all the f(1), _f(2).... are different. They constitute a countably infinite subset of S, which proves Theorem 9. We can now complete the proof of Theorem 8. It still remains to prove that if S is an infinite set, then it is possible to map it one-to-one onto a subset of it. According to Theorem 9 we take a countably infinite subset A of S

and put B = S - A. According to (8.3) it is possible to map A one-to-one onto a proper subset of itself. Let us define an extension of this mapping such that every element of B is its own image. Thus we obtain a one-to-one mapping of S onto a proper subset. This completes the proof of Theorem 8.

It follows from Theorem 8 that every (non-empty) finite set S is equivalent to only one set of the form (1, . . ., n). The natural number n uniquely defined in this manner is called the cardinal number of the finite set considered (or the number of its elements). We extend this definition to the empty set, giving it the cardinal number zero (= 0). A corresponding concept arises for infinite sets, if we assign some object to every infinite set, which we then call the cardinal number of this infinite set so that the same cardinal number should correspond to two sets if and only if they are equipotent. THEOREM 10. Every infinite subset of a countably infinite set is countable. It is sufficient to prove that every infinite set S(c < 1, 2, ...>) is countable.

We proceed to define a function f(n) (n = 1, 2, ...) in the following manner. Letf(l) denote, on account of Theorem 5, the least element of S. If f(n) is already defined for n then let f(n + 1) be defined as the least element of

S - <1, 2, ..., f(n)>

.

(8.5)

Since this set can never be empty, the uniqueness off(n) is ensured for every

n by Theorem 4. Furthermore, since f(n + 1) > f(n) all the f(1), f(2),. .. are distinct. As they constitute a countable subset of S, it only remains to be proved that every element k of S is equal to some f(n). Since f (n) z n, there is a least n for which k is not contained in (8.5). Then k S f (n). Moreover, since k E S - <1, 2, ..., f(n - 1)>, then f(n) < k. Consequently k = f(n) and we have proved Theorem 10.

EQUIPOTENT SETS

17

We shall now define the so-called sequences in general. (They have already occurred in this book in special cases.) By a finite sequence we mean a mapping i -- ai of < 1, ..., n> into any set and indicate this sequence simply by writing

al, . . ., a ,

(8.6)

where a,, a2, ... are the terms of the sequence. According to our definition, the terms may be any, not necessarily distinct, objects. Attention must be paid to the "order" of the members of a sequence, by which we understand that

we regard the sequence (8.6) and a further sequence bI.... , b,,, as equal if and only if n = m and ai = bi (i = l , ..., n). In (8.6) n is called the number of the terms of the sequence; (8;6) itself is also called a sequence of n terms. Further ai is called the i`h term of the sequence, i itself the index of this term.

and call it an (ordered) n-tuple. For (8.6) we sometimes write (a,, ..., In the special cases n = 2, 3, 4 the terms pair (pair of elements), triple, and quadruple are also used. (Of these we have already used "pair".) If the pairs

(a, b), (b, a) are not to be distinguished, we speak of a non-ordered pair (a, b), etc. We now want to interpret an infinite sequence

a,, a2, ... or ao, ai, ...

(8.7)

or a doubly infinite sequence,

..., a_, ao, a,, ...

(8.8)

analogously to (8.6). (Thus these are in fact functions with domains of defini-

tion <1, 2, ...), <0, 1, ...), or <0, ±1, ±2, ....>.) From the finite sequences al, ..., an and b1, ..., b there arises, by juxtaposition, the sequence al, . . ., am, bI ..., b,. (8.7) and (8.8) are also known as countably infinite sequences as their

indices constitute a countably infinite set. Finite and countably infinite sequences are countable sequences.

When speaking of a sequence al, a2, ... or set may often be understood, provided that there is no possibility of misinterpretation. A non-ordered pair (a, b) is more general than the set
infinite case.

THEOREM 11 (theorem of CANTOR). For a set S with at least two elements, the set of infinite sequences

aI, a2, ...

is not countable. 2*

(ai, a2, ...E S)

(8.9)

SET-THEORETICAL PRELIMINARIES

18

CANTOR has proved this theorem by his well-known diagonal process. We assume that the set 01 of the sequences (8.9) is countable. 1J is infinitely countable, consequently we may write

Jai _ , where

al = all, a12, ..., a2=a21,a22,...I

(8.10)

are all the different sequences (8.9). For every i (= 1, 2,.. .) let bI denote an element of S different from ail. Then the sequence P = bl, b2,

. .

is different from all the sequences in (8.10), although it is included in J'1. This contradiction proves Theorem 11. According to this theorem the set of real numbers is not countable and likewise even for the numbers of the interval 0 < x:5 1, which may be written uniquely as decimal fractions 0 - a, a2. .. by an infinity of digits, of which infinitely many differ from 0.

THEOREM 12. The union of countably many countable sets is countable. First of all, let 9J2 be the union of pairwise disjoint countably infinitely many countably infinite sets

(i = 1, 2, ...).

Si =

Then we have :dli. _ ,

from which the truth of the theorem is evident for this case. Now, in the general case only those sets are in question which are subsets of one of the sets X32 just considered. Thus the truth of Theorem 12 follows from Theorem 10.

THEOREM 13. The product of finitely many countable sets is countable. First of all, let us consider the product A x B of two countable sets A, B. This consists of the pairs (a, b) with a E A, b E B. Of this the pairs with fixed a constitute a set Ba which is equipotent to B. Because

AxB= U Ba aEA

the countability of A x B follows from Theorem 12 for this case. The general case can be proved by induction.

EQUIPOTENT SETS

19

It follows from Theorems 10 and 13 that the set of rational numbers is countable

since one may write them in the usual manner as fractions,

(a, b integers, b > 0),

consequently their set is equipotent to a subset of the product bof two countable sets.

A more exact formulation of the concept "system" is as follows (cf. § 1): Let a system E be such that one takes a set S and a mapping a of S into a set of cardinal numbers ( = A 0); then we say that the (different) elements of E are the same as the elements of S, but in E an element x(E S) occurs with the multiplicity ax. This system is called finite, if the set S and all the cardinal

numbers ax(x E S) are finite. If we talk, for instance, about the system consisting of the elements a, a, a, b, c, c, then we mean the (finite) system

with S = , as = 3, ab = 1, ac = 2. Two systems S, a and S', a' are to be regarded as equal, if S = S' and a = a'. § 9. Ordered and Semiordered Sets

We call a relation < defined in a set S an ordering relation, or a semiordering relation, if it is transitive, and if, for any two elements x, y (E S), exactly one, or, at most one, respectively, of the following three propositions, is true:

x=y,x
(9.1)

If such a relation < is defined in S, S is called (with respect to this relation <) an ordered set, or semiordered set respectively. (An ordered set is also known as a totally ordered set and a semiordered set as a partially ordered set.) It follows therefore that every ordered set is also semiordered. What is generally true of semiordered sets, is also true of ordered sets. Two (different) elements x, y of a semiordered set S are called comparable (with each other), if x < y or y < x holds. If x < y we say that x is less than y and y is greater than x. An element a of S is called minimal or maximal if there

is no element x (E S) with x < a, or x > a, respectively. If the set S has only one minimal or maximal element, we denote it by writing

min S or max S.

(9.2)

If, for instance, S is the set of the inhabitants of a city and x < y denotes that x is a descendant (child, grandchild, etc.) of y, this is a semiordered set. If, on the contrary, x < y denotes that x is a child of y, there is no semiordered set owing to the absence of transitivity. In the set <1, 2, ...) of the natural numbers and also in the set of the real numbers,

then the usual relation < denotes an ordering relation. In these sets the sign < retains its former meaning. If, on the contrary, we wish to define another ordering or semiordering relation in one of these sets, then we denote it by the sign -<. This applies to other analogous cases. Similarly the terms "less" and "greater" are not used when they might give rise to a misunderstanding. So, e.g., the sign c, previ-

SET-THEORETICAL PRELIMINARIES

20

ously introduced, defines a semiordering relation in the set of all subsets of a set. However, for A c B we will not say that "A is less than B" (or "B is greater than A"), but, only that "A is a proper subset of B". In the (ordered) set <1, 2, ...> there is only one minimal element, i.e., 1. If a semi-

ordering relation is defined in it by

1-< 3-<5-< ..

;

2-<4-(6-< ...,

then I and 2 are the only minimal elements. In the set <0, ± 1, ...> (ordered according to <) there is neither a minimal nor a maximal element. A semiordered set , in which

a
element. In a finite ordered set the same holds with "exactly" instead of "at least"; on the other hand, an (infinite) ordered set has at most one minimal or maximal element. It is not at first clear that every (infinite) set may be ordered. We shall deal with this matter later.

If we write x S y, it means that either x = y or x < y, i.e. (9.3)

It is evident that this relation 5 is transitive, and that the rules

x<_x

(xES),

x
(x, y E S)

(9.4) (9.5)

hold.

One often defines a semiordering relation in S by first defining < and then <, the latter being defined by the rule

x
(x,YES).

(9.6)

For this the following is valid: The relation < is a semiordering relation corresponding to the relation < by (9.3) and (9.6) if, and only if, the relation _< is transitive and (9.4) and (9.5) hold. Therefore we call such a relation a semiordering relation too. The "if" part of the above theorem remains to be proved. If x < y, y < z are valid, then from (9.6) x < y, y < z are valid afortiori, hence x5; z. If also

x = z, then x < y and y < x, from which, according to (9.5), it follows that x = y. But since x < y, this is impossible. Consequently, we have x # z. From (9.6), x < z, so proving the transitivity of <. We have still to show that for each pair x, y (E S) at most one of the three possibilities (9.1) is true. Because of (9.6), it suffices to prove that x < y, y < x cannot hold simultaneously. This has already been demonstrated in the first part of the proof which is therefore completed.

ORDERED AND SEM[ORDERED SETS

21

If for two elements a, b of a partially ordered set a < b and no element x such that a < x < b exists, then we say that a is a lower neighbour of b, b is an upper neighbour of a and b covers a. By an extension of the concept of a sequence, we take this to mean

a mapping of any ordered set 1= < .

.

., K,

A, µ, ...> into an arbitrary

set. We denote this, in the general case, as

..., aK, aA, a., ... , where we always suppose that

...
a2, a#, ...

.

However, for the sake of simplicity, we shall use the symbolism al, a2, ..., where 1, 2, ... no longer denote the natural numbers, but are the elements of an arbitrary index set. Likewise, we shall henceforth indicate by
ones. Of course, care must be taken to avoid any misinterpretation which might arise from this notation. § 10. Well-ordered Sets A set S is called well-ordered if it is ordered and every non-empty subset of it contains a (unique) minimal element. For example, the set <1, 2, ...> is well-ordered. The same set in the "reversed" order

...-< 3-<2-< 1

is no longer well-ordered, as it has no minimal element. We shall prove later, in § 15, that every set may be well-ordered (and consequently ordered). Here we must be content with some simple statements relative to wellordered sets.

It is evident that every non-empty subset of a well-ordered set is wellordered.

In the case of an arbitrary semiordered set S, we denote by S(a) (a E S) the set of those elements of S which are less than a. These subsets of S and also S itself are called the cuts of S. THEOREM 14. The union set V of any cuts of a well-ordered set W is again

a cut of W.

SET-THEORETICAL PRELIMINARIES

22

It is sufficient to consider the case V 0 W. We put k = min (W - V). We show that V = W (k). Since the elements x (< k) of W belong to V, it

suffices to show that the x (> k) lie outside V. For x = k it is evident. If an x (> k) lies in V, then one of the given cuts would contain the element x and also k, which is impossible. This contradiction proves the theorem. In the case of a well-ordered set W, for every cut W(a), the union W(a) U a is also a cut of W, since, if W(a) U a ,-4- W, for

m=min (W-(W(a)Ua)) we clearly have W (a) U a = W (m) .

§ 11*. The Lemma of Kuratowski-Zorn Let a semiordered set S be given. A subset U of S is called bounded from above, if in S there is an element s such that for all x (E U) the condition x < s is satisfied. Then s itself is called an upper bound of U. One of the most important theorems in mathematics, of far-reaching effect especially in algebra, is as follows : THEOREM 15 (KURATOWSKI-ZORN lemma). If each ordered subset of a semiordered set S has an upper bound, then S has a maximal element.

For proof, we make use of a choice function f for the set of all non-empty subsets of S. We call the function value f(U) (E U) the chosen element of U(c= S). A well-ordered non-empty subset R of S is called regular (relative to f), if the minimal element of R is f(S) and every other element a of R is the chosen element of the set of those elements of S, which are greater than all elements of R(a). A regular subset of S is, e.g., the set which consists of the single element f(S). Since, according to this, every element a of a regular subset R of S is uniquely defined by the cut R(a), the following are valid :

(a) If RI, R2 are regular subsets of S, then R1(a1) = R2(a2) is only possible for a1 = a2.

(b) If a regular subset R of S has an upper bound lying outside of it, then there is a proper overset of R which is likewise regular. For, if the chosen element of the set of upper bounds mentioned above

is joined to the set R, this results in a regular subset of S; consequently (b) is true. (c) Of two arbitrary regular subsets R1, R2 of S, one is a cut of the other. Let V denote the union of all common cuts of R1 and R2. From Theorem

14, applied to W = R1 and W = R2, it follows that either V is equal to at least one of the R1, R2 or that there are elements a1 (E R1), a2 (E R2) with V = R1(al) = R2(a2). In the first case the statement is trivial. In the second

LEMMA OF KURATOWSKI-ZORN

23

case, it follows from (a) that a1= a2 = a, hence V U a is a common cut of R1 and R2. This contradicts the definition of V, since (VU a) c V. Thus we have proved (c). (d) If R* is the union of arbitrarily many regular subsets R1, R2, ... of S, then R.(a) = R*(a;) is valid for each Ri and at (E R,). For, if the statement is false, then there is a pair R;, aj such that for a suitable x x E R*(ai) , R;(a)

are valid. Hence it follows that

x
Re(a) = Rk(a)

But since the right-hand side contains x, there is a contradiction, which proves (d). Next let R* denote the union of all the regular subsets of S. First we prove that R* is ordered. To do this we consider two arbitrary different elements r1, r2 (E R*). Then r1 E R1, r2 E R2 holds with suitable regular subsets R1, R2 of S. By reason of (c) we may also assume that RI c R2. Since then r1, r2 E R2, r1, r2 are comparable with each other. Hence R* is in fact ordered. Secondly we prove that R* is well-ordered. Consider a non-empty subset U of R* and an element u of it. Now, u E R*, consequently u is contained in a regular subset R of S. From (d), R(u) = R* (u), further U(u) Q R* (u) so that U(u) c R(u) and also U(u)cR. Hence U(u) has a minimal element. Since U is ordered, this is at the same time the minimal element of U. Hence R* is in fact well-ordered. Thirdly we prove that R* is regular. For f(S) is the minimal element of all

regular subsets of S and consequently also of R*. We then consider an arbitrary element r of R*. Then r E R for a suitable regular subset R of S, whence it follows that r is the chosen element (by./) of the set of those elements of S, which are greater than all the elements of R(r). Since, according to (d), R(r) = R*(r), the above holds for R* instead of R, whence we have proved that R* is regular. Now, R* has, according to the premise of the theorem, an upper bound s* (in S). If moreover s* is not a maximal element of S, then there is an 2/a R - A

SET-THEORETICAL PRELIMINARIES

24

element x in S which is greater than s*. This x is an upper bound for R* outside R*. Hence according to (b), a proper overset of R* exists, which is also regular. However, as this is impossible by reason of the definition of R*, s* must be a maximal element of S, proving Theorem 15. In the literature this theorem is usually referred to as ZORN's lemma, although KURATOWSKI (1922) was the first to enunciate it. ZORN (1935) was the first to apply

it to algebra. Cf. SZELE (1949-50) as to the above proof and the following sections.

§ 12. The Special Lemma of Kuratowski - Zorn As mentioned above, c denotes in each set C5 of subsets of a set a semiordering relation. From now on any set C5 will be regarded as semiordered (according to this relation c). In particular if the set c is ordered, then it is called a chain of sets. Accordingly, this means a set of sets, where for any two different elements A, B either A c B or B c A holds. It is usual to call an ordered subset of an arbitrary semiordered set a chain.

A special case of Theorem 15, applied to the above set e, is as follows : THEOREM 16 (special lemma of KURATOWSKI -ZORN). If, in a set e of subsets of a given set, there is to each chain of sets an upper bound (in Cam), then c

has a maximal element. (This is true in particular if S contains the

union set of each of its chains of sets.)

We may briefly say that Theorem 15 refers to semiordered sets, and Theorem 16 to sets of sets. When applying them, we shall refer to each as "the KURATOWSKI-ZORN lemma" since Theorems 15 and 16 are in fact two equivalent propositions as is shown in the following paragraphs and may also be proved directly. § 13. The Lemma of Teichmiiller-Tukey THEOREM 17 (lemma of TEICHMULLER-TUKEY). Let a non-empty set G

of subsets of a set S be so constituted that a subset U of S belongs to S if, and only if, each finite subset of U belongs to G. Then c contains a maximal element.

NOTE. In general we say that a property E is of finite character, first of all

if that property E holds for the empty set and further if every set has the property E whenever all its finite subsets have the property E. According to this definition, the lemma of TEICHMULLER-TUKEY may be

expressed in the following manner: If E is a property of finite character and S is an arbitrary set, then the set of its subsets having the property E always contains a maximal element.

25

LEMMA OF TEICHMULLER-TUKEY

To prove Theorem 17 it suffices to show according to Theorem 16 that the union set V of an arbitrary chain of sets from C5 belongs likewise to G. Let be an arbitrary finite subset of V. Each a, is contained in a member Al of the chain of sets considered. Since one of the Al, . . ., A,,, let us say Al, contains the others, then


9;Al E G

From the definition of C5, it follows that
§ 14. The Theorem of Hausdorff-Birkhoff It is evident that a partially ordered set is ordered if, and only if, all its subsets consisting of two elements are ordered. Hence it follows that the property to be ordered is of finite character. Consequently Theorem 17 directly implies the following THEOREM 18 (theorem of HAUSDORFF - BIRKHOFF). Every semiordered set

contains a maximally ordered subset. This theorem is known in the literature as BIRKHOFF'S theorem, however, HAUSDORFF (1914) was the first to find it. Cf. also BIRKHOFF (1948).

§ 15. Theorem of Well-ordering THEOREM 19 (theorem Of ZERMELO or the theorem of well-ordering). Any set can be well-ordered.

Before the proof we note by way of introduction that: Given two semiordered sets A, B, we shall only then say that A is a subset of B (or B is an o verset of A), if, first of all, the set B contains the set A and any relation x < y (x, y E A) which is true in B, is also true in A. (Moreover it is assumed

that the semiordering relation is denoted in the same way in A and B.) Naturally this also applies to the concept of the cuts of a semiordered set. Now, let A, B, ... be ordered sets such that for any two of them one is a cut of the other. Obviously the union set V = A U B U

. .

. can be ordered in

only one way so that each relation x < y, in one of the sets A, B, ..., is also valid in V. The ordering relation defined in this manner in V = AUBU ... is always presupposed. If A, B.... are well-ordered, then V is well-ordered too. In the sense of the above convention one has to regard two semiordered sets as equal,

only if they are equal in the usual sense and are semiordered in the same manner. 2/a

SET-THEORETICAL PRELIMINARIES

26

To prove the theorem let us consider an arbitrary set S. C5 shall denote the set of all the different well-ordered subsets of S. (We repeat that, consequently, we have to regard "equal" subsets as different, if they are differently ordered.) If A, B are two different elements of c and A is a cut of B, then we put A < B. So S is a semiordered set and consequently, according

to Theorem 18, contains a maximal ordered subset U. Let A, B.... be all elements of U. According to the definition of < and the above-mentioned

observation, the union V = A U B U ... (9 S) is well-ordered and is therefore an element of S. Now the A. B.... are cuts of V, consequently this must be equal to one of the A, B, ..., otherwise would be an ordered subset of e, which contradicts the maximal property of U.

If now VCS, then let us add to V an element x of S - V as greatest element, which gives a well-ordered V`(9; S). Moreover A, B, ... are proper subsets and also cuts of V', consequently is an ordered subset of e. But this is a contradiction, which proves that V = S and so Theorem 19. From this theorem it follows that every set can be ordered. We remark that the axiom of choice is also a conclusion from Theorem 19. If S is a set, we use this theorem to make it well-ordered, then let the minimal element of each (non-empty) subset correspond to that (non-empty) subset of S, thus creating a choice function.

As we have seen, the six propositions, i.e., the axiom of choice, the lemma of KURATOWSKI-ZORN, the special lemma of KURATOWSKI-ZORN, the lemma of TEICHMOLLER-TUKEY, the theorem of HAUSDORFF-BIRKHOFF and the theorem of

well-ordering Of ZERMELO, are connected in such a way that each of them follows

from the previous one, and the first follows from the last. This means that all six propositions are equivalent. This is particularly surprising, since the axiom of choice itself expresses a very obvious proposition, while the same does not apply to the five other theorems. The important conclusions which one may draw from the axiom of choice have led many mathematicians to attempt to replace it by a simpler proposition. As these attempts have been unsuccessful, mathematicians have now accepted the axiom of choice.

§ 16. Transfinite Induction

By means of (complete) induction we are able to prove only such propositions as can be suitably split up into countably many component propositions. The following is more general. THEOREM 20 (theorem of transfinite induction). Let a proposition A, correspond to each element v of a well-ordered set I. If the truth of A follows

from the assumption of the truth of all propositions Ax (x < v) for each v, then all the propositions A, (v E I) are true. Suppose that one of the given propositions is false. Let I' denote the set of those indices v(E I) for which the proposition A. is false. Let v be the

TRANSFINITE INDUCTION

27

minimal element of F. Then A,, is false, while all A,, (x < v) are true. This contradicts the assumption of our theorem, so the theorem is proved. The varied applicability of transfinite induction is guaranteed by the theorem of well-ordering. If it is necessary, in the case of an arbitrary "index set", to prove certain assertions A,. (v E I), then we have first to render I well-ordered, so making the application of Theorem 20 possible. Of course, the proof depends upon the particular way in which the well-ordering of I is carried out. We can apply transfinite induction likewise in the case of countably many pro-

positions. For example, we can use, for the inductive proof of the propositions Al, As1 ..., the well-ordering

1-< 3.<5-<...-<2.<4-<... of the index set.

CHAPTER 11

STRUCTURES § 17. Compositions In the set of the natural numbers certain "compositions" are defined, for example the sum a + b and the product ab of two numbers a, b. A further important example is the product a,B of two mappings a, ,B of a set into itself, as defined in § 4. We shall generalize these and similar well-known examples below.

We say that a composition is defined in a non-empty set S = , if an element of S is uniquely assigned to each ordered pair a, j9 of elements of S in a certain way. We denote this element by a 0 j3 where 0 is the sign of

composition. This may then be called the "composition 0". Alternatively, the composition is a mapping of the product set S X S into S or a functionfloc, fi) defined in S. However, since the above concept of compositions is

too general, we shall only consider in detail compositions with certain special properties. Since in a o /I the elements a, fi are variables, we can call our compositions "com-

positions of two variables". Similarly, one may speak of "compositions of several variables", although we shall not discuss these. We shall, however, deal with "compositions of one variable".

We shall usually denote compositions by + or . The composition is then called addition or multiplication, respectively. In these cases we say that the additive, or multiplicative notation is used for the composition. We call a + j9 the sum and a 9 the product of a and f, where a, # are the

terms of the sum a + fi or the factors of the product a fl. For a fl we write simply afi, provided that the meaning is clear. (For instance, for 2.3

we do not write 23.) Certainly the kind of notation used for compositions is unimportant. It may happen that in the course of our considerations we shall vary our notation. Unless otherwise stated, we shall denote the standard compositions in the set of the complex numbers in the customary way. According to this, e.g. 2 + 3 means the number 5. In school mathematics one is accustomed to speak of "subtraction" (denoted by

a - fi) and "division" (denoted by a _ (3 or ) which are also considered to be compositions or more exactly "inverse compositions". We do not regard these as compositions, but we shall discuss them later. 28

COMPOSITIONS

29

If in the set S a composition has been defined, special notations are used

for certain elements (connected with this composition). The most important of these are "neutral element" and "inverse element". Furthermore, certain nomenclature is also introduced for special properties of a composition, e.g. "associativity", "invertibility", "regularity" and "commutativity". To these may be added "distributivity", a common property of two compositions which are simultaneously defined in S. We shall elaborate these concepts here. First we consider only one composition in S which now we write in multiplicative form. If an element s of S is such that for all a(E S)

Ea = aE = a

(17.1)

is valid, then we call s a neutral element or a unity element (or briefly unity) of S. There is in S at the most one unity element. For, if E, s' are unity elements, then from the definition it follows that s = EE' = E'. If the elements of S are denoted by small Greek letters, then a will always denote the unity element (if it exists). However, if the elements of S are denoted by small or large Roman letters, then the unity element will be denoted by e, or E. As regards multiplication of integers, the number 1 is obviously the unity element for which we retain the normal notation. We shall sometimes denote the unity element of a set by 1 even when it is not the number I provided that it will give rise to no misunderstanding. If there is such a possibility then we may use the symbol 1.

In the set <0, ±2, ±4, ...> of even numbers, let the customary multiplication be defined. (That is possible since the product of even numbers is again an even number.) Since for even numbers a, b (0 0) ab ¢ b, there is no unity element in this set.

In the set of mappings of a set into itself, obviously the identical mapping is the unity element.

If there exists a unity element s in S and there is for an element ocE( S) an element 9(E S) such that

floc =a#=E.

(17.2)

then we call # an inverse of a. (Obviously a is, at the same time, an inverse of i.) If a has an inverse, we call it an invertible element. In the set of rational numbers any element different from 0 has an inverse, but 0 itself has no inverse.

In the set of natural numbers elements other than 1 have no inverse. The use of parentheses () and brackets [) in connection with compositions will be discussed here, briefly. First, for a o fi we might have introduced the "more complete notation" (a o 9). By means of this notation the order of formation of compositions, e.g. of the form ((a o ,B) o (y o S))

is then quite clear. For this we write, omitting the outer parentheses, (a o 9) o (y o S). This example is sufficient to illustrate the general case.

STRUCTURES

30

In the following, we consider in S the previous multiplication a#. This we call associative, if, for all a, 9, y there always holds

(an)y = a(fly)

(a, jS, y E S) .

(17.3)

NOTE. In this book we shall only thoroughly investigate associative compositions, because they are the most frequent and therefore the most important. Since non-associative compositions are disregarded, henceforth, for the sake of brevity instead of "associative composition" we shall only say "composition". However, if non-associative compositions are taken into consideration, this will be mentioned. If the associativity is particularly to be emphasized we shall add the adjective "associative". As has been pointed out above, we shall investigate compositions only under certain restrictions, mainly concerning the associativity. The addition and multiplication of complex numbers are associative compositions, as is also the multiplication of mappings in conformity with Theorem 1. If we apply the "function-theoretical notation" f (a, f) to compositions, then (17.3) takes the much more complicated form f(f(a, /1), y) = f(a, f(f, y)).

We now prove that as a consequence of associativity, one element can have at most one inverse. If i and y are inverses of a, then it follows from the definitions (17.1), (17.2), (17.3) that

fi=NE=NIcY)=(a)Y=ey=Y. The inverse of a will be denoted henceforth by a-1. Accordingly,

a-1a=0Ca-1=e

(17.4)

is valid. For a one-to-one mapping a we have formerly denoted the inverse mapping by a-1. Both notations are in accordance with each other, since (r-la = aa-1 = 1 is the identical mapping.

For two given elements a, fi (E S) there arises the important problem whether the equations fl,

rja=P

(17.5)

are (in S) solvable at all, and in case of solvability how many solutions they have. If the equations (17.5) are soluble for each pair a, P, then we call the multiplication invertible. The following theorem is surprising. THEOREM 21. For an invertible multiplication, equations (17.5) have for and 97, respectively, and there exists the unity element; finally each element a has its inverse a-'-, and the solutions of (17.5) are each pair a, # only one solution each,

y

=a

rl=Cgx1.

(17.6)

If, conversely, the inverse a 1 always exists, then the multiplication is invertible.

31

COMPOSITIONS

Let us consider an invertible multiplication in S. For a fixed a there are then two elements s', e" with

s'a=a, ME'=a.

(17.7)

If fs is a further element, let us take solutions , r, of (17.5). It follows from (17.5) and (17.7) that and

fle.,=(rla)s"=si(as")=rla=fl -

With P = a" and P = E, respectively, we obtain e'e" = e" = S'. From pit follows that s" = s' these and the previous equations e'i = fl, is the unity element for which we now write s. Then we take solutions p, a of the equations

aO=e, as=e. It follows that

e=sO=(aa)ee=a(ae)=as=a,

and consequently, Lo = a is the inverse a` of a. Using the inverse we obtain from (17.5) S = ey = (CC la) = a 1(a5) = a 1N , rl = rfs = ?7 (mm-

1)

= (rl(x)a 1 = fga 1

This means that the equations (17.5) have, in fact, only the solution (17.6). Since, conversely, the equations

a$=a(a lfg)=(aa-1)N=sj9=i, rla=(jga 1)a=fg(a 1a)=fe=P follow from the existence of a-l and equations (17.6), comparison with (17.5) shows that the last assertion of Theorem 21 is also true. Accordingly, a multiplication is invertible if and only if the equations (17.5) have unique solutions. We call the multiplication regular if equations (17.5) have at most one solution $ or i, respectively, for each pair a, j3 (E S). According to this an invertible multiplication is always regular. In the set of the natural numbers multiplication is regular but not invertible. In the set of rational numbers the multiplication is not regular, since the equation Ox = 0 has more than one solution.

On the other hand, multiplication is invertible in the set of rational numbers distinct from 0. Obviously Theorem 21 is not true for non-associative multiplication.

32

STRUCTURES

Multiplication is called commutative, if it is always true that a# _ 9a .

(17.8)

We have already seen that multiplication is non-commutative in the set of mappings of a set into itself. This important example is sufficient to show that non-commutative

compositions are frequent in algebra, therefore it is important to discuss them in detail. The fact that we extend our considerations to non-commutative compositions is an essential feature of "higher" in contrast with "elementary" algebra.

Naturally, the foregoing is also applicable to additive notation of an associative composition, except that the neutral element is denoted by 0 and is called the zero element (briefly zero). The inverse of a is denoted by -a. Consequently the definition of these elements is as follows:

0+a=a+0=a,

(17.9)

(-a)+cc =a+(-a)=0.

(17.10)

The conditions of associativity and those of commutativity are expressed as follows:

(a+9)+y=a+0 +y),

(17.11)

a+,e=19+a.

(17.12)

Invertibility and regularity now mean that the equations

t7+a=j9

(17.13)

have unique solutions or at most one solution each, respectively by assuming the validity of Theorem 21 for all associative compositions). It should be stressed that according to the above statements there exists at most one zero element 0 and every a has at most one inverse -a. If necessary we distinguish the two inverses a 1 and -a by calling them the multiplicative and additive inverse of a, respectively. Although, as a rule, we denote the zero element in any set by 0 (just like the number 0) no ambiguity arises from it and this notation proves to be advantageous. If necessary, the zero element can be denoted by 0.

NOTE. In future we shall use the additive notation only for compositions which are commutative and regular. In the set of complex numbers addition has the above-mentioned properties.

Finally, if in the set S an addition and a multiplication are simultaneously

defined, and if, for any a, 9, y,

a(ig+y)=afi+ay, (ig+y)a=Aa+ya

(17.14)

COMPOSITIONS

33

are valid, then we call the multiplication distributive (with respect to the addition). Nom. The distributivity of the multiplication is always assumed. THEOREM 22. If an addition and a multiplication are defined in a set S, and S has a zero element 0, then

Oa=a0=0 is always valid.

Since 0 + 0 = 0, (0 + 0)a = 0a ,

a(0 + 0) = a0 .

(17.15)

Hence it follows from the distributivity that

Oa+Oa=Oa,

a0+a0=a0.

Since, on the other hand, Oat + 0 = Oa and a0 + 0 = a0, (17.15) follows from the regularity of the addition. We repeatedly emphasize that the type of notation used for compositions is of no importance. Accordingly we may write non-commutative and non-regular compositions in the additive notation, but not without pointing it out. Further we suppose the above definitions to be meaningful with respect to arbitrary, and arbitrarily denoted

compositions. If, e.g., V, A denote two arbitrary compositions in the set S, then the equations

acV(,BAy)=(av9)A(avy),

vac)A(yvac),

similar to the equations (17.14), express the fact that the composition v is distributive with respect to the composition A.

§ 18. Operators

We say that a set O = is an operator domain for a set S = - aat, are sometimes called operations. It is usual to call as an operator product. Since in this the operator a is on the left-hand side, we call it more exactly a left operator. Accordingly, by a right operator we mean that the operator product is written in the form aa. As a third possibility we have the exponential notation a° for operators. If T is a subset of S, then aT denotes the set of distinct elements aoc (a E T). Ta and T° are to be interpreted similarly.

If the operation is denoted by ax or a°, then the operator is sometimes called a coefficient or an exponent. Compositions and operations have a similarity of content and form, so that one s accustomed to calling them inner and outer compositions respectively. (We repeat

34

STRUCTURES

that a composition means a mapping of S X S into S; similarly an operation means a mapping of 0 x S into S.) There is nothing to prevent a similar notation being used for both compositions and operations, e.g. a o a, but the additive type of notation is rarely used for operators.

Usually a set S is assigned an operator domain O only if some compositions

have already been defined in S. Consequently O plays an "auxiliary role" with respect to S. (This auxiliary role is evidenced by the fact that as lies not in (9 but in S. Of course, this holds only "in general", since the sets 0, S need not necessarily be distinct.) We find the best example of operators in geometry. For that purpose we con-

sider expressions of the form alas + . . . +

in three-dimensional Euclidean space, where a, denotes a vector, a, a real number, + the usual addition of vectors and as the usual "multiplication" of a vector a by a "scalar" a. From its meaning as is clearly an operator product. If in particular al, as, as are three "independent"

vectors (i.e., not lying in the same plane), then the set of all alas + alas + alas exhausts, as is well known, all the vectors of the space considered, a circumstance which is evidently to be attributed to the use of the set of the real numbers as the operator domain.

Mostly two cases occur; either the sets O, S are disjoint, or 0 c S. This second case arises when we define a multiplication (and possibly

further compositions) in S and we identify the operator product as (a E O c S, a E S) with the product existing in S. If, moreover, 0 = S, then we say that S is assigned to itself as operator domain. Thus S is always an operator domain for itself in this sense, if a multiplication is defined in S. As mentioned, the application of an operator a upon the elements a of S effects a mapping of S into itself, namely the mapping a -> am. Here we

have to point out that the same mapping corresponds to two different operators a, b if as = boc for all a. It is evident, conversely, that any mapping may be considered as an operator. (This is why we have introduced from the beginning the operator

notation for mappings.) It should be noted that two different mappings are always different as operators. An operator a is called a null operator, or identical operator, if as = 0 or as = a, respectively, for all a. Of course, several null operators and identical operators may occur in O. § 19. Structures

A norx-empty set S = is called an algebraic structure, if at least one composition (or more exactly an ordered n-tuple of compositions) is defined in it. We say that S constitutes a structure with these compositions. The task of algebra is the study of the structures.

STRUCTURES

35

If a set S, after introducing some compositions, becomes a structure, its notation should differ from the set S, e.g., as S', but for the sake of convenience, we shall indicate the structure in the same way as the set of its elements. With respect to two structures S, T, we say that they are equal (S = 1) if and only if they are equal as sets and also if the compositions defined in them are the same.

It must now be made clear that the properties of compositions which are valid in a structure will be treated more fully than the nature of the elements of the structure. (For details cf. § 28.) Examples of structures are the set of complex numbers (with the usual addition and multiplication of the complex numbers), and the set of mappings of a set into itself (with the usual multiplication of the mappings).

The above definition of structures is too general for the successful study of all structures. Here, we shall confine ourselves to the most important kinds of structures. In general we shall only consider structures for which the number of defined compositions is one or two. Sometimes we denote these as structures with one composition, or structures with two compositions.

(In the latter case we always mean an ordered pair of two compositions, which are known as the first and second compositions.) In structures with one composition, we employ either the additive or the multiplicative notation, depending on whether we are referring to additive or multiplicative structures. In structures with two compositions, one of the compositions is

written in the additive, the other in the multiplicative notation, with the exception of "lattices" which will not be considered before § 60. (Lattices will be discussed not for their own sake, but with respect to other structures.)

A structure is called commutative or associative if all the compositions defined in it are commutative or associative, respectively.

We now define structures in which we shall take a deeper interest. We mean by structures, unless otherwise indicated, only these structures. 1. Multiplicative structures: 1. A group is a (non-empty) set, in which an associative invertible multiplication is defined. A commutative group is also called an Abelian group. 2. A semigroup is a (non-empty) set, in which is defined an associative

multiplication. If this multiplication is regular, we speak of a regular semigroup.

H. Additive structures

3. A module is an Abelian group, additively written. 4. A semimodule is an additively written commutative regular semigroup.

STRUCTURES

36

IIl. Structures with two compositions

5. A ring is a set which simultaneously is a module and a semigroup, for which the distributive law is valid.

6. A semiring is a set which simultaneously is a semimodule and a semigroup, for which the distributive law is valid. It should be noted that, for the zero element of a ring, i.e., of the module constituted by the elements of the ring, Theorem 22 still holds. Hence it follows for any ring with at least two elements that the semigroup constituted by its elements is not a group. The same holds for a semiring with a zero element. 7. A skew field (or s-field) is a ring, in which the elements other than the 0 constitute a group. A commutative skew field is called a field. 8. A semi-skew-field is a semiring, in which the elements other than the 0 constitute a group. A commutative semi-skew-field is called a semifield. Well-known examples of the structures t to 8, in that order, are represented by the sets consisting of the following elements: 1. the complex numbers with the abso-

lute value t or the motions of the Euclidean space; 2. the mappings of a set into itself; 3. the vectors of the Euclidean plane; 4. the complex numbers of the form a + bi (a, b > 0); 5. the integers; 6. the natural numbers; 7. the rational numbers; 8. the positive numbers. (Cf. the examples at the end of §§ 20 to 24.) For the sake of clarity we give the above definitions in tabular form as follows group (Abelian group) semigroup module semimodule ring semiring s-field (field) semi-s-field (semifield)

X

+°

+

+1

+

+`

+

X X X X ui 111

The signs + and x indicate here that addition and multiplication, respectively, are defined in the structure in question; further i indicates that the composition is invertible, and [i] indicates that it is invertible for the set of elements different from 0. (The corresponding information regarding the commutative cases is added in parentheses.)

One might denote, e.g., rings by S(+', X), whereby the terms "ring", etc., would be superfluous. However, we retain the above terms for historical reasons, though they are less systematic.

The most important of these structures are the groups, rings and skew fields (in particular fields). Accordingly, the most important chapters of algebra are group theory, ring theory and the theory of skew fields (in particular field theory). It is not easy, however, to define sharply the limits of these chapters, as they are so intermingled and interdependent and, consequently, they cannot successfully be developed as independent theories.

STRUCTURES

37

When we speak of groups, then of course modules are included, since these are themselves special groups.

Skew fields may be characterized as rings in which for each pair of elements a ria = fi are solvable. For this reason skew fields

(; 0), 9 both the equations a are often called division rings.

We have still to remark that in algebra the attributive "skew" is used to mean "commutative or non-commutative", but is often omitted, as e.g., in the terms "group" and "ring", by which we refer to both commutative and non-commutative cases. It would be more logical to say simply "field" instead of "skew field", but in that case we should have to change field into "commutative field". In the literature we find this terminology as well, but we shall keep to the above terminology. Since fields often arise, it is advantageous to have as short as possible a name for them. In certain cases the attributive "skew" will be omitted in the denomination "skew field". Thus, e.g., instead of "non-commutative skew field" we shall use the shorter "non-commutative field". Furthermore by "quaternion fields", to be introduced later on, we shall mean not fields generally, but certain special skew fields. The semigroups, semimodules, semirings and semi-skew-fields are also called

semistructures. The other four structures, groups, modules, rings and skew fields may be called full structures. In the text books, already published, only "semigroup" is used among the four semistructures, though the others are not less important. (The schoolboy becomes acquainted, in about the first four school-years, with semistructures, but not with even one full structure.) We also note that, in the theory of "ordered modules, rings, and fields" (cf. Chapter IX) semimodules, semirings and semifields play a very important role.

If S is a structure with two compositions, then we denote by S+ and S x the additive and multiplicative structures, respectively, constituted by the elements of S where, of course, the addition or the multiplication defined in S is understood. Generally we use those concepts and terms relative to S which have a meaning relative to S+ or Sx, respectively. Thus, e.g., the element 0 of S+ means, as above, the element 0 of S. Similarly, the unity element of S x (if it exists) is called the unity element of S. The structures S with two compositions, distributivity being assumed, may be tabulated as follows: S

S+ J semimodule

l module

semigroup

[group]

Bemiring

semi-skew-field

ring

skew field

where "[group]" stands as abbreviation for "group after omitting the zero element if it exists".

A structure is called finite or infinite according as the set of its elements is finite or infinite. In the case of a finite structure S the number of elements

STRUCTURES

38

is called the order of the structure and we denote it by O(S). In the case of an infinite structure S we write: O(S) = oo. If O(S) = 1 we may call S a structure with one element. If, moreover, S is a group, a module or a ring, then the single element of S must necessarily be the neutral element of S. (It is the unity element, if S is a group, and the zero element, if S is a module or a ring.) Accordingly, we call the structure with one element in the three cases: unity group, null module and null ring. We do not distinguish them from their unique element. (It should be noted that the single element of a null ring is not only the zero element but also the unity element of this ring.) Obviously, with the unity group, the null module and the null ring, we have enumerated all structures which consist of only one element. We often leave these trivial structures out of consideration, but in some cases they are important. Axioms which relate to the properties of structures are generally called structure axioms. Let A denote a system of such axioms. By an A-structure we mean a structure which fulfills this system of axioms A. For a fixed A, all the A-structures constitute a class of structures, which we call more precisely the A-structure class. According to this, the terms "group", "semigroup", etc. each define a class of structures, the axiom systems of which we have given above, under 1- 8. Similarly, "commutative ring with unity element" means a class of structures the axiom system of which is such that to the ring axioms are added further axioms which prescribe the commutativity of the multiplication and the existence of a unity element. Two systems of axioms are called equivalent, if they define the same class of structures. For groups we shall give in § 21 (Theorem 32) a second equivalent axiom system

which has been formulated in a narrower sense and therefore its use is often more convenient than that of the above axiom system 1.

The following should be noted: if a statement is made with respect to semigroups, it also relates implicitly to groups, semirings, rings, etc., i.e., to all structures in which an associative multiplication is defined, since these

may be included in the class of structures which are semigroups. The results for semigroups also hold with suitable reformulation, i.e. by changing from the multiplicative to the additive notation, for semimodules and modules. Similarly, statements valid for groups are also valid for skew fields or fields (except the zero element), and (after passing to the additive notation) even for modules and rings. Similarly, we also have to include rings and skew fields among modules and also fields among commutative rings.

If multiplication is defined in a structure, then for two elements a, t, it is always possible to form the two products a,B, fla. Of these two products one is called the dual of the other. The passing from one product to the dual

STRUCTURES

39

product is called dualization. From any concept connected with multiplication, by the dualization of all products another concept is formed, which we call the dual of the former. We can speak in a similar way of dual definitions and theorems. Those concepts (definitions, theorems) which are invariant under dualization are called self-dual. We discriminate between dual notions by using the words "left" and "right" (or "left-hand" and "righthand"). In the case of self-dual concepts, no such discrimination is necessary. Of course, in commutative structures all concepts are self-dual. Now, it is important in connection with multiplication that associativity, invertibility, regularity, commutativity and distributivity are self-dual properties.

Associativity means the validity of all equations (a9)y = a(f'y) (a, 73, y E S) ,

(19.1)

where S denotes the structure under consideration. By dualization we obtain from it: (19.2)

If we interchange the sides (and change the elements a, P, y, into y, j9, a), than we obtain equations (19.1), so that (19.1) and (19.2) are essentially identical. This shows that associativity is self-dual. Invertibility demands that the equations

as=fi,

77a=j9 (a, 9, E, 11ES)

(19.3)

are solvable for each,pair a, fl. Since the equations (19.3) after dualization (and the interchange of E and r) themselves interchange, it is clear that invertibility is self-dual. The remainder of the statements are proved in a similar way. Many concepts concerning structures are not self-dual. Examples of such concepts will be given in § 20.

It should also be noted that the zero element and the unity element are both self-dual (especially the zero element since it is independent of the multiplication). Finally, it must be stated that the proposition "a multiplication is defined in a set" is obviously self-dual. These observations show that all the structure axioms 1 to 8 are self-dual.

Hence we have the following duality principle (or principle of duality): In the above defined structures (groups, semigroups etc.), together with each theorem deducible from its axiom system, the theorem which is dual to it is also valid. The reader will not fail to notice that this "algebraic" duality principle is very similar in wording to that of projective geometry, but conceptually simpler. If we

STRUCTURES

40

should postulate in the ring axioms in place of distributivity the validity of the equations

a(fl + y) = a9 + ay, then in the structures so obtained ("rings with left-hand distributivity") the duality principle would be false (just as in affine geometry where the duality principle of projective geometry is false).

Of course, while composing our theorems, we shall make use of the possibilities of simplification offered by the duality principle in such a way that we always state only one of the two theorems : the other will then be deduced, in each case, from the duality principle. (The reader will easily recognize the "dualizable" theorems.) Likewise with the dualizable notions, we shall always define just one of the two dual concepts and regard the other as being defined implicitly. In this way we shall obtain simplifications and greater clarity in our working. (Sometimes, if it seems necessary, we shall

briefly carry out the dualization.) If operator domains are assigned to a structure S, then S is called an operator structure. If there is only one domain of operators 0 then we use the notation S 10 or the standard phraseology "S is a O-structure". In the present chapter, however, operators will rarely be taken into consideration. The names "group", "module", "ring", "field" were introduced by GALOIS, GAUSS, HILBERT and DEDEKIND, respectively.

We denote these structures usually by their initial letters G, M, R, F. Skew fields are also denoted by F. The elements of a structure or those or an operator domain are usually indicated by small Greek or Roman letters; i, j, k, 1, m, n, p, q, r generally mean integers. All

this is also valid for letters provided with indices. However, this convention is not obligatory. Semistructures, especially semimodules, semirings and semi-s-fields, will rarely

occur. We shall not emphasize them, thereby avoiding any needless strain on our attention.

We constantly use Y0 for the field of the rational numbers (the rational number field), 2' for the ring of the integers and 41for the semiring of the

natural numbers. By <J", 0> we denote the semiring of the non-negative integers. § 20. Semigroups

Let a semigroup S = be given, i.e., a non-empty set, in which is defined a multiplication, for which associativity (alg)y = a(ny)

(20.1)

holds. Everything in this paragraph relates to this structure S and its elements even when it is not stated explicitly. We shall proceed similarly in subsequent paragraphs, under similar circumstances. Since semigroups are the most general of our structures, the following concerns more or less all structures.

SEMIGROUPS

41

What has been said in §17 with respect to an associative multiplication, relates to semigroups. We shall now consider a further series of important concepts in semigroups.

All the following definitions will be independent of associativity; on the other hand, our theorems would lose their validity without the assumption of associativity.

We illustrate multiplication in F by means of the Cayley table (or the multiplication table), which has the form

.... or ..... ea

e

(20.2)

We illustrate it with examples. The multiplication table

f

y

14

a

y

p fi

I4

a

a a

fi means as = fi, aft = a, etc. But this multiplication is not associative, since (aa)a = _ Pa = fi and a(aa) = a# = a. Hence this multiplication table does not define a a

Y

a

i3

a p

a

y

a

Y

a

fl y

Y

semigroup. On the other hand y y y y

defines an associative multiplication, and consequently a semigroup as may be verified with some trouble.

We shall generally give a multiplication not by means of a multiplication table but by other, much more convenient, ways. Then the examination of a multiplication with regard to associativity (commutativity, invertibility, etc.) will mostly not be difficult.

It is very important that we extend the concept of multiplication in F with two factors to the case of arbitrarily (but finitely) many factors. For that purpose let us agree that for n >t 3 the (recursive) definition shall be

al .:. an`= (aI ...

(20.3)

For example, afly = (an)y, a f yt = ((afl)y)8. Now (20.3) is also valid when n = 2, since then the right-hand side means (al)oc2, by which we understand (omitting the "superfluous" parentheses) ala2. Furthermore if in a "product" of the form a( )f4 ..., in which "empty parentheses"

STRUCTURES

42

occur as "factors", we agree to disregard these factors, then a( )9 = afl, a( ) = a, ( )a = a. So it follows that (20.3) is valid for all n >- 1. If F has a unity element e, then, for n = 0, the left-hand side of (20.3) denotes this element e. (When n = 0 we may call (20.3) an "empty product".) We also write the left-hand side of (20.3) as jn7

llai+ i=1 for n = 0, this means the unity element a (provided it exists). For certain purposes we also define infinite products, by writing

11 ai=a1a2...

i=1

where the factors ai are equal to s with at most finitely many exceptions. This product is to be interpreted so that the factors a are simply left out of consideration and the other factors, observing the order of succession, are multiplied together.

We shall show that in the case of an associative multiplication it is all the same whether a product consisting of several factors is defined by (20.3) or by a1 ... an = a1 (a2 ... an) . (20.4) Moreover, we prove the following THEOREM 23. In every semigroup F

(0 < i < k< n).

al ... an = al ... ar(ai+l ... ak)0ki-1 ... an

(20.5)

(20.5) is called generalized associativity. First, we prove the special case (20.4) of Theorem 23. First of all, (20.4)

is true for n = 1. For n >_ 2 we assume the truth of (20.4) for n - I instead of n. From (20.1) and (20.3) al . . . an = (al/(... an-1)an = (alla2//... al((an ... an-1)an) = a11a2

an-))an =

. . .

an)

by which (20.4) has been proved. By virtue of (20.3) and (20.4) from each equation N1

Nr = Y1

Y;

there follow the equations #l ... fl Q = Y1 ... YSA , afl, ... 19, = my1 ... YS

SEMIGROUPS

43

for any (arbitrary) P and a. Consequently if we put

a1+1...ak=a, and then multiply this equation on the left-hand side by ai, ..., a2, (Xl, and on the right-hand side by ak+,, ..., cc,,, we obtain (20.5), by which we have proved Theorem 23. The next special case of Theorem 23 is often applied. The product of two products 71 ... ai , Pt ... Nk

is equal to

al.aifli.A.

We repeat that the unity element s is defined by the property

ea = ae = a (a E F) .

(20.6)

Accordingly, we call e (E F) a left unity element (or left unity) of F, if e'a = a

(a E F)

(20.7)

holds. Consequently, by dualization a right unity element e" is characterized by the property ae" = at (a E F).

THEOREM 24. If a semigroup F has a unity element, then it has no further left unity element. If F has no unity element, then either the set of left unities or the

set of right unities is empty. It follows by dualization from the first part of the theorem that if there exists a unity element then F has no further right unity element. Since a unity element is always

a left unity, it follows from the first part of the theorem that a semigroup can have at most one unity element (as is already known). This is not presupposed in the following proof of the theorem. Moreover we note that the theorem (as well as the proof)

remains valid without any supposition of associativity.

It is obvious that Theorem 24 is equivalent to the next proposition : If s' is a left unity element and e" a right unity element in F, then e' = e". The latter assertion follows from s' = E's" = s", which proves Theorem 24. An element 0(E F) is called a zero element of F, if

0a = aO = 0 (for all a E F) .

(20.8)

THEOREM 25. A semigroup contains at most one zero element.

If both 0 and 0' are zero elements in F, it follows from the definition that 0 = 00' = 0', which proves Theorem 25. For the sake of clarity the following should be noted. We have defined, with respect

to addition, the zero element 0 by the property

0+a=a+0=a;

STRUCTURES

44

to distinguish it from the zero element defined by (20.8) in semigroups, the latter might have been denoted "multiplicative zero element". This is unnecessary as no confusion can arise except in the case of a structure S with two operations; but here (by virtue of distributivity) Theorem 22 holds, so that it follows that the zero element of S x agrees with that of S+. Obviously Theorem 25, as well as the proof, is valid without associativity.

The concept of a left zero element of a semigroup is rarely used.

We have already defined the inverse a_1 of cc in a semigroup with unity

element s by a-la=aa_1=E.

(20.9)

Accordingly, we call an element a' a left inverse of a if

a'a=a.

(20.10)

THEOREM 26. If an element a in a semigroup (with unity element) has an inverse, then there is no other left inverse. If a has no inverse, then either the set of left inverses or the set of right inverses of a is empty.

This theorem is obviously equivalent to the next proposition: If a' is a left inverse and a" a right inverse of a, then a' = a". The latter assertion follows from

a' = a'(aa'") _ (a a)a" = a' which proves Theorem 26. It is easy to see the formal similarity of Theorems 24 and 26, which also appears in the proofs. Similar observations made with respect to Theorem 24 are also valid for Theorem 26. The only essential difference is that Theorem 26 is false without associativity, as can easily be shown.

With respect to inverses, the next two rules are important:

(a_1)_l = a

(20.11)

and 1

Mn-,

-1 oci

(20.12)

if a-1 and ai 1, ... , an 1, exist. (In words : If a has an inverse a-1, then a-1 also has an inverse and this is equal to a. If all the factors of a product have

inverses, then the product also has an inverse and this is obtained by multiplying the inverses of the factors in reverse order.)

(20.11) follows from (20.9) and from the uniqueness of the inverse. Further, if we multiply the right-hand side of (20.12) either from the right or from the left by a1 ... a,,, then after a simple application of Theorem 23 a is obtained in both cases. Hence, again on account of the uniqueness of the inverse, (20.12) follows.

45

SEMIGROUPS

If there is an element j9(O 0) with

afS=0

(20.13)

for an element a(# 0), then we call a a left zero divisor. (It is evident that in the above fi is simultaneously a right zero divisor.) Of course, zero divisor means an element that is both a left and right zero divisor.

By a left annihilator (of F) we mean an element a for which

a$ = 0 (for all

(:_ F) .

(20.14)

The left annihilators different from 0 are obviously left zero divisors. An element a is called left regular (in F) if for each element y the equation

e ==y has at most one solution . It is important that the definition of left regularity has

(20.15)

another formulation. First of all we say: The left-hand cancellation law is valid for an element e, if from ' ea = efl it follows that a = fi. Expressed in formulae :

ea = e/9 = a = i4

(a, f E F) .

(20.16)

This can be worded as follows: An element e is left regular if, and only if, the left-hand cancellation law holds for e. The condition that equation (20.15) has for each y at most one solution is equivalent to the fact that the conditions ea = eis, a P are incompatible. This is equivalent to condition (20.16), which proves our assertion. Of course, an element a shall be called regular if, and only if, it is left and right regular, i.e., if for all elements y both equations e$ = y, rle = y have at most one solution and t7, respectively. Consequently a semigroup

is regular if and only if all its elements are regular. Moreover, from the above, it follows that in any semigroup an element a is regular if, and only if, the (both-sided) cancellation law holds for e. We say that the left-hand cancellation law holds in a semigroup, if it holds for each element. Then the former may be stated (on account of the principle of duality) as follows : A semigroup is regular if, and only if, the cancellation law holds in it. Expressed in formulae, the regularity of a semigroup F means

a=P ace =Per.ac =9

(e, a,9EF).

A nonregular element of a semigroup is called singular. Note that left regular elements may very well occur among singular elements too. The zero element and left zero divisors are singular elements.

STRUCTURES

46

THEOREM 27. A regular element of a semigroup either has an inverse or. has no left inverses. The product of two regular elements is regular. All the regular elements of a semigroup F, providing such exist, form a semigroup F*.

In a regular semigroup it follows from each equation sa = a that a is the unity element.

The semigroup F* of this theorem will always be denoted by an asterisk and called the regular semigroup of F. (Under certain circumstances F may contain regular semigroups, which contain F* as a proper subsemigroup.) In order to prove the theorem we first note that its first part may be stated as follows : in a semigroup F a left inverse of a regular element must be at the same time the inverse of this element. For this it suffices to show that if for the elements o, P of a semigroup with unity element s the equation

P e = e holds and moreover a is regular, then ee' = s. It now follows on account of the supposition that

ee'e=e

-

On the other hand, so = o. From these two equations and the regularity of o it follows that oo' = s, which proves the first part of the theorem. As to the second part, by virtue of the principle of duality, it suffices to prove that the product pa of two regular elements o, a is left regular, i.e., for pa the left-hand cancellation law is valid. Since this holds individually

for e and a, eaa = pap

as = afl

e(aa) = p. (aj9)

a = fl

(a, /9 E F)

so that the truth of the assertion is evident. Since, according to this, the product of elements of F* still belongs to F*,

i.e., multiplication may be carried out within F*, then, on account of the associativity of the multiplication in F, which holds a fortiori in F*, the third assertion of Theorem 27 is valid. The proof of the last part results from the following:

em =a=osa= oa=). os=

ea=sa=a.

Two elements a, /9 of a semigroup are called permutable (or interchangeable) with each other if aj9 _ /9a. A semigroup is obviously commutative if and only if its elements are pairwise permutable. THEOREM 28. If the elements a.1, ..., a of a semigroup are pairwise permutable, then the product al ... a is independent of the order of the factors.

It follows from the generalized associativity and the assumption that a1 ... an = a1 . . . ai-1 (ai ai+i)aa+2

. .

. an = a1 ... ai- i (0C1+1 ai) ai+2 .

.

.a.

47

SEMLGROUPS

After cancelling the parentheses on the right-hand side, we see that our product al ... a" remains unaltered if two arbitrary neighbouring factors are interchanged. Since by repeating such interchanges all the orders of the factors are obtainable, the truth of the theorem follows. We define the nth power a" of an element a by

a"=al... an

(a1=...=an=a;

nE d'_),

(20.17)

where a is called the basis and n the exponent of this power. For instance al = a, a2 = ma, a3 = aaa. The special powers a2, a3 are called: square, cube of a. If a unity element s exists in F, we define

a°=s.

(20.18)

(It should be noted that according to this 0° = s, if there is a zero in F; here the basis of the power is the zero element, and the exponent is the number 0.) Finally, if the inverse a-1 exists we define

a-"=(cc I)"

(nE

(20.19)

By the definition of the power a", the semiring .A'' of natural numbers was assigned to each semigroup F as a domain of operators. If, furthermore, F has a unity element, then we may regard the semiring <.,f', 0> as a domain of operators of F. Since according to Theorem 21 each element in a group

has an inverse, the ring .7 of integers is a domain of operators for each group. For this reason we sometimes call the structures 4'', <.-f'', 0>, .7 natural operator domains. Their elements (considered as operators) may be called natural operators. For powers the rules aman = am+n, (am)n = amn ,

(20.20)

are valid where the exponents are integers, but the number 0 and the negative integers are only permitted if a unity element exists in the semigroup and the element a has an inverse. The validity of (20.20) results from (20.17) to (20.19) together with (20.5) and (20.9). A more detailed proof is unnecessary. (See also the recent note REDEI 1965.)

As a special case of (20.20.,) we have

(an) -1 = a-" . This is a generalization of (20.11).

(20.21)

According to (20.201) the powers of a fixed element x serve as a simple example of pairwise permutable elements.

3 R-A

STRUCTURES

48

We say that an element a of a semigroup is of finite order or infinite order, respectively, according as finitely many or infinitely many different elements occur among the powers a, a2, ... . In the first case we denote by o(a) the number of different elements among the above powers, in the second case

we put o(a) = 0. In both cases o(a) (= 0, 1, 2, ...) is called the order of the element a. It is apparently illogical to define an element a of infinite order by o(a) = 0; it would seem to be more logical in this case to write o(a) = oo. We shall see later that our convention "o(a) = 0" is, for several reasons, more suitable than "o(a) = oo".

We prove that, in the case o(a) = 0, the sequence a, a2,

. consists entirely

of distinct terms and in the case o(a) = n( = 1, 2, ...) its first n terms a, ..., a" are distinct and every term in the sequence is equal to one of these n terms. If a, a2, ... are not all different then there is a unique natural number n such that a, . . ., a" are all different and ant' is equal to some am+l (0 5 m < < n). Hence it follows that an+t = am+t

(t= 1,2.... ).

The repeated application of this equation shows that every term of the sequence is equal to one of the first n terms. Since the latter are different, it follows that we have o(a) = n, which completes the proof. We gather from the proof that the sequence a, a2, ... in the case of an a of finite order is always of the form a,

. .

.,

am.

0.m+1, .

.

., an ., am+t

(0 5 m < n = o (a))

.

. ., an.,... (20.22)

,

where a, ..., a" are all different. We can call a, ..., a' the aperiodic part and am+',

., a" the period of the sequence a, a2,

...

.

An element a is called idempotent, if a2 = M. Then a = a2 = as = .. . and o(a) = 1. The unity and the zero element are idempotent. THEOREM 29. In a semigroup the left regular idempotent elements are identical with the left unity elements. A regular idempotent element is necessarily the unity element. A finite semigroup with at least one regular element always has a unity.

If a is an idempotent, then a2 = a. Consequently, for each element o of the semigroup ate = at). If, moreover, a is also left regular, then it follows after cancelling that aO = Co. This means that a is a left unity.

SEMIGROUPS

49

If a is also regular, it follows from the principle of duality and Theorem 24 that a is the unity element. Conversely, if e' is a left unity element, then we can write for every equation

s'Q = e'a simply e = a. This means that the left-hand cancellation law holds for s', i.e. s' is left regular. As we have seen above a' is also idempotent. Since, furthermore, the unity element is obviously regular and idempotent, only the last assertion of the theorem remains to be proved.

For this purpose let a denote a regular element of a semigroup of finite order. Then a is also of finite order. Consequently, a = an(n-m)

is idempotent, where m, n have the same meaning as in (20.22). Since a' is also regular, by Theorem 27, from the second assertion of Theorem 29 the truth of the last assertion of this theorem follows. An element a (of a semigroup with zero) is called nilpotent, if there is a

natural number n with a" = 0. Then a" = a" +I = .. = 0. If moreover n is minimal, then obviously o(a) = n. Therefore we call the order of a nilpotent element also its degree of nilpotence. The zero element is nilpotent. The rest of the nilpotent elements are zero divisors.

Except for the zero element there are no elements simultaneously nilpotent and idempotent.

For an a (E F) and a natural number n we denote by the equation

the solution of

to = a

and call it an nth radical of a. According to this, is in general a manyvalued symbol. If the equation has no solution, we say that " a is non-existent. When n = 1, then-Va = a, therefore we usually assume n > 2..Ja is also called the nth root of a and in the cases n = 2, 3 a square root, or cube root, of a respectively. Here n is called the radical (or root) exponent. For V/a we write simply .,fi-c. The many-valuedness of Ja may often be reduced by assigning to /a one of its possible values and considering this value consistently. In cases where we have chosen once and for all a value for ./a , we usually denote this value by ait" . Some-

times we use .,/a to denote the set of all solutions of the above equation, but this must be pointed out whenever used. An equation

a=al...a"

(or the right-hand side of it) is called a factor decomposition of a where we mostly suppose n z 2.

STRUCTURES

50

If for two elements a, ft the equation

aE = P is solvable, then we say that a is a left divisor of 9 or that 9 is a left multiple

of a, and we write oc 9. We also say that fi is left divisible by a. (For right divisibility, we introduce no notation. In a commutative semigroup alfl always means that a is a divisor of 9.) Divisibility will be examined further only for certain rings (cf. Chapter IV).

For integers a, b the notation alb always means that this divisibility holds in the semigroup of the integers (or, what comes to the same thing, in the

ring 7). Those natural numbers p > 1 which are divisible by no natural number except 1 and p are called prime numbers. A natural number which is neither 1 nor a prime number, is called a composite number. The composite num-

bers are the products ab (a, b = 2, 3, 4, . . .). Each composite number is divisible by at least one prime number, and the least among its divisors d (> 1) is certainly a prime number. The next remark is of fundamental importance. According to G. SzAsz (1953-54) there may be defined in each set S, consisting of at least four elements, a non-associative

multiplication so that the equations (an)y = a(ny)

(a, 8, y E S)

hold for all but a single arbitrarily chosen one. This means that the conditions of associativity in the case of at least four elements constitute an independent system of axioms. This shows the sharpness of the condition of associativity for compositions. The above consideration means in practice that the equations, which together constitute associativity, have to be re-examined individually, to confirm the associ-

ativity of a multiplication. Fortunately, as already stated, other means are at our disposal in the most important cases. THEOREM 30. I f i n a set S i s d e f i n e d a (not necessarily associative) multipli-

cation such that (20.1) h o l d s f o r all a, 9 (E S) and certain y = yl, y2, ... (E S),

then (20.1) holds for those a, f9, y (E S) for which y is the product of some Yi, Y2,

..

.

Let ar denote a product consisting of n factors which all occur among the Yi, Y21 ... . We have to prove that all the equations (aft-r. = ac(rt,,)

(a, b' E S)

are valid. It is true, if n = 1. We assume the assertion for some n. Since n,,+i is equal to nny;, it follows that i = (aft) (nnYs) _ (((cP)n.) Y = W#70) Yi = a070Yr) _ (0 (n,,yl)) = oc(97en+i) This proves the theorem.

SEMIGROUPS

51

Semigroups will frequently be mentioned in this book, usually in connection with other investigations. The general theory of semigroups has developed into an important discipline especially as a result of the monographs of LYAPIN (1960) and CLIFFORD -PRESTON (1961). It should be noted that

this theory is far from complete. A detailed treatment of finitely generated commutative semigroups (cf. § 26) is to be found in RUDE! (1965). EXAMPLE 1. A "< is a commutative regular semigroup with unity element. EXAMPLE 2. By the Cayley table

is defined a non-commutative non-regular semigroup of order three, in which is the zero element, a a left unity element and i4 a left annihilator. (One has to show that from these three facts associativity follows.) EXAMPLE 3. The set F of mappings of a set S into itself, S consisting of at least two elements, is a non-regular non-commutative semigroup. For instance, for S = < 1, 2 (1 2211

1

(2 2) (1

1) -(2 1) (1 1)[

is not right regular.

hence (

1)

EXAMPLE 4. One could regard a° as an operation in _r, which on account of 38' 96 (33)3 is neither associative nor commutative. EXERCISE 1. If in a semigroup with unit element the element a is a left inverse of

the element fl, then a" is a left inverse of f" (n = 1, 2,. ..). EXERCISE 2. If a is an element of finite order in a semigroup, then there occurs only one idempotent element among the powers a, as, ... This is equal to a'tu-"'°, where n = o(a), a"+' = 8m+1 (0 < m < n) and k is the least natural number with k(n - m) > m. [Cf. (20.22).] Moreover the elements of the period a'"+', ..., a" form a group. EXERCISE 3. If a semigroup contains only one left zero element, then it is the zero element.

EXERCISE 4. A semigroup without unity element, but with at least one left unity element, is without right regular elements.

§ 21. Groups

Let a group G = (a, 9, ...> be given. This means that an associative, invertible multiplication is defined in G. By Theorem 21 this is equivalent to saying that in G an associative multiplication is defined such that there exists a unity s and each element a has an inverse a-1. We emphasize again that from Theorem 21 and its proof it follows that if the equations 17oc=P

(21.1)

STRUCTURES

52

are multiplied by a-i from the left and right, respectively, then there arise, on account of associativity, the equations

e=a-lp, 77 =Pa-1.

(21.2)

If, conversely, these are multiplied by a from the left and right, respectively,

then we recover equations (21.1). So that, in a group, equations (21.21), (21.22) are merely another form of equations (21.11) and (21.12). Similarly, we may write, e.g.,

aEr1 yf-a=6,

ai9Y=a, afla 1 =Y, &-I ft =asfla

e=

Ysf = e,

in the form : flat- 2 (1Y3-1,

ft

=«

,

a-5 = YE.

As further examples: (a

'M_a

= ((a I#)- 1)a = (fl- 'a)' = fl- 'a#- larl a,

(af(X- ')$ = OC#a-lava ' = ate'.

Take a fixed a and let $ run through all the elements of G. From the above, each element 9 of G then appears exactly once in the form at. This proves the following THEOREM 31. For each element a of a group G, the mapping -->

(4 E G)

(21.3)

is a permutation of G.

On account of the principle of duality, the mapping - sac is also a permutation of G. It is evident, conversely, that each semigroup G must be a group, if the mappings (21.3) and their duals are permutations of G.

The next remark is equally important: In a group, on account of the uniqueness of the solution of (21.12), it follows from the single equation am- a that s is the unity of the group. (It also follows from Theorem 27.) It is both of theoretical and practical importance that the group axioms (cf. § 19) are equivalent to some weaker axioms. THEOREM 32. A set G is a group if and only if the following axioms I to IV hold: I. A multiplication aq is defined in G. II. For any three elements a, 9, y of G, (a /3) y = a (fly) holds.

III. There is an element s' in G such that s'a = a for all elements of G. IV. To each element a of G there is an element al of G such that ala = e'. We have already seen that axioms I to IV are valid for a group G.

GROUPS

53

We now assume the axioms I to IV in order to prove that G is then a group. By IV for each a there exists an a1 and an ap such that aia = E', a2a1 = e' .

Then it follows from I, II and III that aa1 = e'(aa1) _ (e'(X)0C1 = ((a2a1)a)ai = (a2(0C10C))a1 = (0C2- _')0C1

=

= a2(E a1) = a2a1 = E'

Furthermore, we have

as' = a(ala) =

e'a = a .

From this and axioms 1, II we have a (aift) = (0Ca1)N = 8 'fl = 9, 001)a = 9(a10) = 9e' = /1 for all elements fi (E G).

We have shown that in G the equations a = (l, qa = P are solvable, i.e., multiplication is invertible. This together with II means that G is a group, and so Theorem 32 is proved.

NoTE. We see that axiom III postulates the existence of a left unity and axiom IV requires something less than the existence of a unity, namely that of left inverses. Thus a set G is a group whenever an associative multiplication is defined in it, for which G has a unity and each element of G has a left inverse. In most text-books the group is defined immediately by axioms I to IV. (However, cf. KuROSCH (1953a).) It is noteworthy that the axiom system I to IV is not self-dual, although it defines a self-dual structure.

It is needless to say particularly that all facts valid in semigroups are, a fortiori, valid for groups, where, however, several simplifications take place, as we have already

seen. Because of the regularity of multiplication in a group ( e), there is no zero element, nor are there left zero divisors or nilpotent elements. Further, it has one and only one idempotent element, i.e., the unity element and each element is divisible by every other element, as a consequence of which it is unusual to speak of divisibility in a group.

We show that for the powers a (i = 0, ± 1, ...) of a group element a only the two following cases are possible: these powers are either all different

and then o(a) = 0, or a°, ..., a"-1 are all the different powers of a and

then o(a) = n. It is sufficient to consider the case where not all the a' (i = 0, ± 1, ...)

are different. Then there are integers r, s such that r > s, a' = a' and, moreover, r - s = n is minimal. It follows that a''s = s, i.e., a" = e. . . ., a"-i. These must be different, otherwise one would contradict the minimal property of n. From this it follows that o(a) = n, proving the assertion. It follows also from the proof that : When o(a) = n > 0, then n is the least natural number with a" = s.

Therefore every element a` occurs among a0,

STRUCTURES

54

Hence follows the useful rule : If o(a) = ab, where a, b are natural numbers, then o(a°) = b. Since, moreover, any natural number other than I is divisible by at least one prime number, we obtain from the latter: Among the powers of a group element of finite order, different from e, there is an element of prime number order.

The simplest groups are those which consist of the powers of a single element. These are called cyclic groups. From the above there are but two possibilities for a cyclic group G, as follows:

G=<...,a-i,s,a,a2,...), O(G)=oo,o(a)=O, G =<e,a,...,an'>, O(G)=o(a)=n. For instance, the numbers 2' (i = 0, ±1, ...) constitute an infinite cyclic group and the nth complex roots of unity a (finite) cyclic group of order is.

Two elements a, P of a group G are called conjugate, if there is an element 0 of G with j9 = oao-1. We prove that this "being conjugate" is an equivalence relation.

Since a = sas-1, a is conjugate to itself. Moreover, if fi = oao-1 and y =

= aaa-1 then

= oao-1 =

e(a_1

Ya)o_i = oa_1 yaP_i =

Thus we have proved the assertion. The corresponding equivalence classes, which consist of conjugate elements, are called the Frobenius classes (of the group). These play an important role in many questions. We prove two simple theorems. THEOREM 33. Each regular finite semigroup is a group.

Let F denote a regular semigroup. For each a (E F),

- al: is a one-to-

one mapping of F into itself. If F is finite, this mapping must be a permutation

of the set F. Accordingly the equation aE = P has one, and only one, solution in F for each pair a, fl. Hence and from the dual the above theorem follows.

THEoREM 34. Let F denote a semigroup with unity and F the subset of those elements which have an inverse. Then F' is a group.

To prove this, we denote two elements of F by a, f. Since the inverse = fl-1 a-1 exists, afl E P. Thus multiplication is closed in F'. The unity of F is its own inverse, and so belongs to F' of which it is the unity. Finally, because (a-1)-1 = a, a-1 E P. Hence, from Theorem 32, (aP)-1

Theorem 34 follows.

GROUPS

55

A group whose elements are permutations of a given set S is called a permutation group (of S). From our previous statements relative to permu-

tations and from Theorem 32 it follows in particular that the set of all permutations of S constitutes a group. We call this the full permutation group

of S and denote it by .93 (S). Full permutation groups are commonly called "symmetric". We prefer, however, to use the expression "full" because of the analogy with the "full matrix ring" to be introduced later.

On account of the great importance of permutation groups we wish to deal with permutations in more detail and in a rather simple form. The simplest permutations of all are those of the form

a_1 ao a1 ... ... a o

a1

a2 .. .

a1 ... a,, -1 an

a2 ... a

a1

which we call cyclic permutations and denote by

(... a_ 1 a0 aj ...)

(21.4)

(a1... a,.) ,

(21.5)

or by

respectively. They are also called an infinite cycle or a (finite) cycle of length

n respectively. We can also give (21.4) as al -* ai+1 (i = 0, ± 1, ...) and similarly (21.5) as ai -+ ai+1 (1 = 1, ..., n), where, however, we have to a1. It is obvious that (21.4) and (21.5) (conceived as group interpret elements) are of infinite order and of order n, respectively. A cycle (a) of length means, of course, the identical mapping a --> a. The cycles (a b) of length 1 two are called transpositions. Note that the notation (21.4) is unique, and that, on the other hand, (21.5) may be written in n different forms (ak ... a a1... ak_1) (k = 1, ..., ii). In particular (a b) = (b a). The ai in (21.4) and (21.5) are called the elements of the cycle. Of course, these must always be different. We call two cycles disjoint if they contain no common elements. If 1, y21 ... are (finitely or infinitely many) pairwise disjoint cycles, then we define their product

7r =512...

(21.6)

as follows : if Si denotes the set of the elements of i, then n shall mean the mapping of the union set S = S1U S2U ... into itself for which nx = fix (x E Si). It is evident that v is a permutation of S, independent of the order of the factors and for finitely many factors, is equal to the product defined previously.

THEOREM 35. Every permutation may be uniquely factorized (to within the order of the factors) into a product ofpairwise disjoint cyclic permutations.

3/a R-A

STRUCTURES

56

In these factorizations one usually disregards the one-term factors, whereby the uniqueness is not altered.

The assertion of uniqueness of Theorem 35 is trivial. In order to prove the

possibility of the desired factorization, we consider a permutation n of a set S. We divide S into classes so that two elements a, b belong to one class

if and only if there exists an i such that a = n b. It is obvious that this classification is possible. Now the class represented by an a (E S) consists of all the nka (k = 0, f I , ...). If these are different, we form the cycle

(... n-1a a na ...) . If, on the other hand, they are not all different, then those that are different

can certainly be given in the form a, na, ..., n"'1 a. In this case we form the cycle (a Ira . . . 7rn-1 a) .

Since in either case the elements of the cycle exactly make up the considered class, we see that the product of all these cycles is equal to n. Consequently Theorem 35 is proved. We usually write permutations in the form determined by this theorem and also compute the product of permutations in this form, e.g., 1 2 3 4 5 61 7 8

62485 17

3) = (1 6)(2)(3 4 8)(5)(7) = (1 6)(3 4 8),

(1 2 7)(4 6) (2 4 6)(3 5 7) = (1 2 6 7 3 5), (1 2 3 4 5)(1 3 4 5 2) = (1 4)(3 5). The powers of a cycle constitute a (cyclic) permutation group .9'. For 4 = (1 2 3 4) for example, .0 consists of the permutations

(1 2 3 4), (1 3)(2 4), (1 4 3 2),

1,

where I denotes the identity permutation. The full permutation group of (1, 2, 3> consists of the permutations 1,

(1 2), (1 3), (2 3),

(1 2 3), (1 3 2).

This group (of order six) is non-commutative.

For Abelian groups the following useful rules hold: i, a1 ... anin. a1kl ... an kn= Ocii,+k, ... Mnin+kn,

i,

ink

(ai ... Sc:)-'

ilk

ink

= IX i` ... OCn in .

The simplest examples of Abelian groups are the cyclic groups.

GROUPS

57

NOTE. In every group, by virtue of the unique solvability of equations (21.1),

it is possible to introduce the concept of an inverse composition. First, we consider the case of an Abelian group. Now both the equations (21.1) have the common solution a-1 ,q

This is sometimes written as

=

(21.7)

fla-1 .

or occasionally as fi = a, and is called a

fraction or a quotient. (Accordingly we can then use the well-known terms

"numerator, denominator, fraction line, dividend, divisor and division sign".) The inverse composition (of multiplication) defined by (21.7) is called division. However, it is normally not associative, so that it is not a composition in our sense. The case for non-commutative groups is less simple since now the solutions (21.2) of the equations (21.1) are in general different so that we must talk of "left or right division". But we do not define any inverse composition in the non-commutative case. It would, in fact, be unnecessary, since by the help of the inverse a-1, the solutions (21.2) of (21.1) can be obtained directly and without difficulty. It is somewhat inconsistent to use the term "invertible composition", but not "inverse composition" in non-commutative groups; however, we confine ourselves to the concept of the "inverse of an element" which replaces it. EXERCISE 1. Determine the permutations interchangeable with a cyclic permutation. EXERCISE 2. In the set of linear polynomials f(x) = ax + b with rational coefficients

a (# 0), b we define the composition f(x) o g(x) = f(g(x)). Prove that this yields a

non-commutative group in which the inverse of f(x) is equal to 1 x - b a

a

§ 22. Modules

Since modules are essentially Abelian groups and differ from the latter only in the additive way of writing the composition, it will be sufficient at present to become acquainted with the usual additive notation. Let a module M = be given. We already know that in M the composition is ordinarily denoted by +, and for a + fi the conditions for associativity and commutativity are as follows:

(a+fl)+y=a+(fl+ y), ac +fl=fi+a. Invertibility means that for any a, fi the equation a + = fi is uniquely solvable. The zero element (neutral element) and the inverse of a are denoted

by 0 and -a, respectively. A further difference from the multiplicative notation is that we use the notation n a instead of an where n denotes coefficients (not exponents). (na is not a product, but an operator product.)

sraucruaes

58

This briefly explains what "the transition from an Abelian group to a module" implies. This is now elaborated as follows. We repeat that 0 and -a are defined by:

a+0=a, a+(-a)=0. Further the definition of na in complete detail is:

na=a1+...+an

(a1=...=an=(x; nE-4),

(-n)a = n(-(x) Oz= 0.

(n E

(Here "0" on the left-hand side means the number 0 and on the right-hand side the zero element of M.) For this reason we should write the zero element of M, say, as 0, but it is unnecessary

unless there is a possibility of misunderstanding.

The cancellation law in modules becomes

a+y=P+y=a=j9. Additionally for the inverse - (na) of na we simply write - na, thus: Hence

- (na) = - na

(n E J) .

(-n)a=-na

(nE.7).

Consequently, we write, e.g., for (- ka) + 1# + (- my) just - ka +

+I#-my.

For sums with many terms we also use the notation n

=1

ai=a1+... +an.

This means 0, if n = 0. We also use infinite sums

Yai=a1+(X2+..

i=1

with only finitely many terms different from 0. The solution of the equation a + _ fi is fi + (- cc), i.e.,

Ii - a. This is called the difference of fi and a. This defines the inverse composition of addition, and is called subtraction. In order to avoid possible misunderstandings we stress that subtraction is not regarded as a "composition in the narrow sense".

59

MODULES

(Occasionally, in commutative structures, addition, subtraction, multiplication and division are called the four fundamental compositions.) In a module we denote the order of an element a by o+(a), which we call (in order to distinguish it from the previously defined "multiplicative order"

o(a)), the additive order of a. According to this, o+ (a) means, for an a of finite order, the number of different elements ka (k = 0, ± 1, . . .); if, on the other hand, these elements are all different, i.e., a is of infinite order, then o+(a) = 0. If o+(a) = n > 0, then 0, a, . . ., (n - 1) a are all the different elements of the form ka. Here o+ (a) = 1 if and only if a = 0. By a cyclic module is meant a cyclic group written in additive notation. The above concepts also apply with slight modifications to semimodules. EXAMPLE 1. S+ is an important cyclic module, the module of integers. Observe that here the operator product na agrees with the ordinary product of the integers. EXAMPLE 2. Another important cyclic module is

<..., -2m, -m, 0, m, 2m, 3m.. . .>

(m E .'1l,

which consists of all the multiples of m. GAUSS was the first to discover the meaning

of this module in the elementary theory of numbers and to introduce the notation "module". We shall denote this module by (m). EXAMPLE 3. The vectors of a Euclidean plane consitute a non-cyclic module. EXAMPLE 4. f-+ is a semimodule without zero.

EXAMPLE 5. The position vectors of the points lying in the first quadrant of a Euclidean plane constitute a semimodule.

§ 23. Rings

We repeat that we call a structure R = a ring if it is simultaneously a module and a semigroup, and the distributive law

a((i+y)=afl+ ay, (j3+y)a=floc +ya

(23.1)

holds. In particular, by a ring is meant a set in which two associative composi-

tions are defined, addition, a + fl, and multiplication, afl, so that the addition is commutative and invertible and the distributive law holds. (Hence (23.11) is called left distributivity.) The structures R+, Rx are called the module and semigroup of the ring R,

respectively; further we call the regular subsemigroup of Rx the regular semigroup of R and we denote this by R*. (If Rx has no regular elements, then R* is to be regarded as non-existent.) In due course we shall see that in a ring addition is "predominant" compared with multiplication. Of course, we can explain this by the fact that R+ is a "more complete" structure, viz., an additive group while Rx is "merely" a semigroup. The same fact

STRUCTURES

60

also explains why we frequently conceive the ring R to be a module R+ provided with R as a left and right operator domain, so that the operator products are identical with the products defined in R. We shall return to this later. In conformity with what has been said, it often happens in practice that a module M is changed into a ring in such a way that we define in it, afterwards, an associative and distributive multiplication relative to addition. (Cf. BEAUMONT (1948), REDEc (1953-54)."

We also relate the concepts introduced in modules and semigroups to rings, too. In addition to that we observe that the zero element of R is the common zero element 0 of R+ and Rx and that it has the properties as

a+0=a, 0a=a0=0,

(23.2)

was already mentioned above. The notions left regular, singular, idempotent

nilpotent and left unity, left inverse, left zero divisor, left divisor, (and the corresponding dual and self-dual concepts) are to be interpreted in R

as in Rx. On the other hand, -a in a ring shall be expressly called the additive inverse of a. Finally, we note that in rings with multiplicative order

o(a) and additive order o+(a) the latter is of much more frequent use so that we mean by the order of a ring element a simply its additive order o+ (6c).

From (23.1), generalized distributivity easily follows by induction: Q

r

l

Q

(al + ... + ar) (01 + ... + Ns) = IE ail ( #.,j = L, ai Nk . k-1 i=1 k=1 1=1

(23.3)

Further

19)= -aj9

(23.4)

where we mean by the right-hand side the additive inverse -(a19) of afl. This follows from

0=(a-a)19=00 +(-(X)fl, 0=a(j9-j9)=GO +a(-19). More generally,

(ra)(sj3) = rsaq (r, s E -7),

(23.5)

where the right-hand side is to be understood as (rs) (aj3) . By virtue of (23.4) it suffices to prove (23.5) for r z 0, s >--_ 0. Obviously,

(23.5) is trivial for rs = 0. When r > 0, s > 0, we obtain (23.5) so that in (23.3) we insert for all a; and Yk the element a or 19, respectively. Notice that the usual rule ((X + P)2 = as + 2a,8 + #z holds in general only for commutative rings. On the other hand, by virtue of (23.3) we always have

(a+#)E=a'+aP+#a+ fi$.

RINGS

61

For commutative rings we prove the so-called polynomial theorem n!

(a1 + ... +

i mill

... a; (n

1, r z 1), (23.6)

O

where the customary notations

0!=1, k!=1.2...k

(23.7)

are used. (Of course factors of the form a° are disregarded

in (23.6).)

Before we prove the theorem, we remark that the polynomial coefficients occurring in (23.6)

n1)

n!

i1 ... i,! !

(23.8)

are integers, which could easily be proved directly, but the integral property of (23.8) will be shown in the following proof of (23.6). In order to prove this we denote the polynomial coefficient (23.8) by (n, il, . . ., i,) when this exists and the number 0 otherwise. Then we write (23.6) as follows :

(a1 + ... + a.)" _

(n, i1i ..., i,) cc" ... ar (n ? 1, r

1)

,

(23.9)

where we now have to sum over all r-tuples i1, ..., i, which consist of integers. It suffices to prove (23.9). For n = 1 (23.9) is trivial. For n Z 2 we suppose the truth of (23.9) for

n - 1 instead of n. Hence there follows after multiplying by al + ... + a., (and using (23.3)) : r (a1+...+a,)"(a-1, i1, .. ., i t=1

i,)ai...arOct.

For this we may write (a1 + ... + (x,)" _

E E (n - 1, i1) ..., i,)t a ... a; , i t=1

(23.10)

where (n - 1, i1, ..., i,)t indicates that i, in (n-1, i1, ... , i.) is to be replaced by it - 1. Obviously

(n-

t n

(n,i1

STRUCTURES

62

thus r

t=1

11 + ... + i,

(n -

(n,

n

We can omit the first factor on the right-hand side since for i1 + the second factor vanishes. Since, according to this we have t=1

(n - 1, i1, ..., i,)t = (n, i1, . . ., ir) ,

... +

it 96 n

(23.11)

(23.9) follows from (23.10), whereby we have proved (23.6). (The integral property of (23.8) follows likewise from (23.11). ) The special case

(a + fly =

n!

'

r=o i! (n - f)!

fln-i

(n >_ 1)

(23.12)

of (23.6) is called the binomial theorem (for commutative rings) and the n

_

_ n(n-1)...(n-i+1)

n

(n - i)! -

i!

05iSn),

(n '?=

(23.13)

occurring in (23.12), are called the binomial coefficients which are likewise

integers. The notation (23.13) is retained for use later and we complete it (for integers i, n) with the conventions

t01= 1, (n)=0 (n>0; i<0 or i>n). Let us notice that from (23.11) follows the recursive formula

nj

1I+ (n

(n

i

Z1)

(23.14)

and that we may also write the binomial theorem (23.13) in the form

(a +

p)tt

(n =E i=o

l

at /fin-i

i

THEOREM 36. In any ring an element different from 0 is left regular if, and only if, it is not a left zero divisor. Consequently, the only singular elements of a ring are the zero element and the left and right zero divisors.

If a is left regular, then it follows from every equation acf = 0 , i.e.,

a# = a0, that iS = 0 , therefore a is not a left zero divisor. If, on the contrary, m (A 0) is not left regular, then there is an equation

RINGS

63

a/i = ay with f # y. Since, according to this, we have a(fB - y) = 0, fi - y # 0, then a is now a left zero divisor. Consequently the theorem has been proved. COROLLARY. From this and Theorem 29 it follows that in a ring without

unity element all left unity elements are right zero divisors. THEOREM 37. In any ring R the following propositions hold. If R has only

one left unity element, then this is the unity element. If an element a has only one left inverse, then this is the inverse of a. If a has two left inverses, then it has infinitely many. If R is finite and contains at least one regular element, then R has a unity element and every regular element has an inverse. Assume that a is the only left unity element in R. For each pair a, j9 (E R)

(e+as -a)fl =a#+as/4-ap =/i+afl-aq=j9. From this follows e + as - a = s, thus as = a. Since this holds for all a, s is in fact the unity element of R. We now assume that R has a unity element s and j9 is a left inverse of cc. Then

9a=s.

From this it follows that

(/i + afl - s) a = fla + a/ia - a = e + as - a = e . If, consequently, fl is the only left inverse of a, it follows that i9+afl -e = fl, a/i = e. Accordingly, /i is now the inverse of a. Further, we have still to consider the case where a has left inverses other than /1. We have to prove that then there exist infinitely many left inverses

of a. We put ei=a'(aj9 - s)

(i = 0, 1,...). From (afl - e)a = a/ia - a = me - a = 0 it follows that Qja = 0, thus (/i + o )a = s. Since, according to this, all /i + t); are left inverses of a, we have only to consider the possibility that not all the Q; are different. It suffices

to deduce a contradiction from this. We take two i, k such that P,_I-Ok

Then

(i>k).

AMP - e) = ak(afl - s) . If we multiply on the left by #', then because we have

e and /i(a/i - s) = 0

a/i-e=0.

Thus we have verified that /i is a right inverse and consequently the inverse of a. This is the required contradiction, which proves the assertion.

STRUCTURES

64

In order to prove the last assertion of the theorem, let us assume that R is finite and has at least one regular element. According to Theorem 29 the semigroup R X then has a unity element e. This is at the same time the unity element of R. Since now the regular semigroup R* of R exists, it follows from Theorem 33 that each element of R*, i.e., each regular element of R has an inverse. This completes the proof of Theorem 37. This theorem originates from BAER (1942) and KAPLANSKY (1949).

A ring is called zero-divisor-free (or zero-divisorless) if it contains neither left zero divisor nor right zero divisor. From Theorem 36 a ring is zero-divisor-free if, and only if, its non-zero elements are regular. Commutative zero-divisor-free rings with unity element are called integral

domains (or integrity domains). An important example of an integral domain is the ring 7 of the integers. A ring in which all elements are nilpotent is called a nil ring. A ring is called a zero ring if all products afl in it are equal to 0. (A zero ring is always a nil ring.) Every module can easily be made a zero ring by defining in it the "trivial" multiplica-

tion xf = 0. This is why zero rings are often called trivial rings.

It is appropriate to define here the characteristic of a structure S. Let S be,

first of all, a module (ring, skew field or field). Then, if o+(a) = 0 for all a 0 in S, we denote the characteristic of S by the number 0. On the other hand, if there is a natural number n such that na = 0, for all a then we call the least such number n the characteristic of S. In any other case we say that S has the characteristic o. Finally if S is a pure multiplicative structure (i.e., group or semigroup), then the characteristic of S is interpreted as the multiplicative analogue of the previous concept. According as the characteristic is a number n (= 1, 2, . . .) or co, we call S a structure of finite (positive), or infinite characteristic, respectively; the third case is where S has the characteristic 0. Obviously all finite structures are of finite characteristic.

THEOREM 38. The characteristic of a zero-divisor-free ring with at least two elements is 0 or a prime number.

If suffices to prove the theorem for the case where the characteristic is different from 0. Then there is an element a ( 0) of finite order. Because a # 0, o+(a) > 1, whence it follows that there also exists an a with o+(a) = P where p is a prime number. Then, for an arbitrary element fi c .pfl

PLO=Pa ft = 09=0.

RINGS

65

Since a # 0, it follows that pfl = 0; this means that p is the characteristic of the ring, proving the theorem. EXAMPLE 1. We shall deal with so-called matrices more generally in §68. Here we consider only the special case of matrices of the type 2 x 2. These are denoted by A =

(ai asl

B =

,

as a

bi b2 be b4)

where a,, a2, ..., bl, ... are the elements of a ring R. (Matrices of type 2 x 2 are essentially four term sequences with terms from R arranged in a square.) In the set of all such matrices we define addition and multiplication : A -h B =

AB =

rai + bl

a2 + bs

Ias+b$ a4+b ' aibi + albs

aib2 + a2b4 asb2 + a4b4 )

a3bi + a4b8

which, clearly, defines a ring, which we denote by R2 and call the full matrix ring of rank four (over R). The zero element of R2 is 00

-0.

We see that R2 ordinarily is not commutative, even when R is commutative. (More exactly: R. is commutative if, and only if, R is a zero ring.) If R 0, then R. is not zero-divisor-free, it even contains nilpotent elements, as we see from the examples:

a

(0

b 00)

(0 0) (-b

- 0'

0)2 = 0.

If R has the unity element e then ( e 0 is the unity element of R2, moreover 111

1

(e0

0)

(p 0) are idempotents. EXAMPLE 2. Let m be a natural number. In the set

R=(0,l,...,m- 1> we define an "addition" and "multiplication" (different from the usual), as follows. For a, b (E R) we form, first of all, a + b and ab, then we subtract suitable elements

mr, ms, chosen from the module (m) _ (..., -m, 0, m, 2m, ...>, from each, so that the numbers

a+b-mr, ab-ms

obtained in this way fall into R. These two numbers are denoted by a + b, and a X b, m

m

respectively, and we call the compositions, hereby defined, addition mod m and multiplication mod in, respectively. It is easy to see that R constitutes a ring with respect to these compositions. Obviously R is commutative, moreover O(R) = m,

STRUCTURES

66

so that R is our first example of a finite ring. When m = 4 the Cayley tables for R run as follows:

+

0

1

2

3

X

0

1

2

3

0

0

1

2

3

0

0

0

0

0

1

1

2

3

0

1

0

1

2

3

2

2

3

0

1

2

2

0

2

3

3

0

1

2

3

0 0

3

2

1

(mod 4).

This shows that the element 2 is a zero divisor and even nilpotent. We stress that we

have constructed, both in this and in the previous example, a new structure from a given structure. (Such constructions will occur frequently later in a great variety.) EXAMPLE 3. Let us construct a module M having the property that every ring R with R+ = M is necessarily a zero ring. To do this, we take a prime number p. The set of all pkth complex roots of unity (k = 0, 1, ...) evidently constitutes a group G.

We denote by ak (k = 1, 2....) the root with amplitude

2k

.

(ak is a so-called

primitive pkth root of unity.) Obviously

a = ail ... a." (ak = 0, ..., p - 1; k = 1, .., n; n > 0) are all the elements of G. (It follows from Theorems 12, 13 that G is countably infinite.)

By putting ap = 1 (this is the unity element of G), we get (k, ! = 0, l , ...), and especially

(k = 0, 1, ...). We turn now to the additive notation, by which a module denoted by M arises from G. However, to avoid misunderstandings, we simultaneously write Ilk for ak. Then all the elements of M are

(ak =0,...,p- 1; k= 1,....n; n>0), and we have pk Ilk +1 = Ill , PkIlk = 0

(k,1 = 0, 1, ...).

We show that M has the required property. Consider an arbitrary ring R with R+ = M. In this case necessarily k

Ilk Ill = Ilk 'P Ilk+I = Pk Ilk !lk+l = 0 Elk+1 = 0. On account of distributivity it follows from this that the product of any two elements of R likewise vanishes, i.e., that R is a zero ring, as asserted. The above infinite Abelian

group (whether in multiplicative or additive form) is called the Prefer group. It is important for various reasons. (Cf. SZELE (1949- 50b).) We denote the Prefer group by (p"). According as we mean (p°°) in the additive or multiplicative notation, we refer to it as the "module (p°°)", or "group (p°°)", respectively. EXERCISE 1. Prove that for the group (p°°) the radical ./a is never the empty set. EXERCISE 2. For which m (> 1) is the ring, mentioned in Example 2, zero-divisor-

free? Show that R is then a (finite!) field. EXERCISE 3. In the system of axioms for a "ring with unity element" the commuta-

tivity of addition is a consequence of the other axioms (even without the axiom of associativity of multiplication).

67

SKEW FIELDS

§ 24. Skew Fields We may repeat the definition of the skew fields in the following form: a ring F is called a skew field if the elements of F different from 0 constitute a group. T noREm 39. Every skew field is zero-divisor-free. COROLLARY. The characteristic of a skew field is 0 or a prime number.

Since the elements different from 0 of a skew field constitute a group, so the product aj9 of two such elements a, fi is also different from 0. This proves the theorem. The corollary is true by virtue of Theorem 38. We can express Theorem 39 by saying that the elements different from 0 of a skew field are regular. On the other hand, we recall that for a ring R we generally denote the semigroup of the regular elements of R by R*. If R is a skew field F, it follows from above that this semigroup is a group. This will henceforth be called the group of the skew field F and denoted by F*.

We repeat that F* consists of the non-zero elements of F. The unity element e of F*, because e0 = Or = 0, is at the same time the unity element of F. In F each element a ( 0) has an inverse a-1, which agrees with that

in F*. (On the other hand, 0-1 does not exist since O5 = e is insoluble.) It should be noted that in F with given a, fi the equations

7a=fi

(a#0)

are uniquely solvable. (In particular, for 9 = 0,

_ 17 = 0.) The truth

of the assertion follows, with respect to Theorem 39, from the fact that F* is a group. If F is a field, we often indicate the solution of a (i.e., of

a = fl) by writing

P

a

(= a-1 9 = fi a-1).

TimOREM 40. Every zero-divisor-free finite ring with at least two elements is

a skew field. For, if R is such a ring, then the regular semigroup R* of R consists of all elements of R other than 0. On the other hand, because of Theorem 33 R* is a group whence the truth of Theorem 40 follows. This theorem will be extended later by Theorem 318 which says that each finite skew field is necessarily a field. EXAMPLE 1. In a finite field with two elements these must be the zero element and the unity element. According to this the Cayley tables of such a field must have the following form: 0 S

I

0

'-

0

e

0

0

0

8

0

8

0

e

STRUCTURES

68

Conversely, it is obvious that these Cayley tables define a field. It should be noted that the compositions mod m, defined in Example 2 of the previous paragraph, if m = 2, lead to the following Cayley tables: 0

1

x

0

1

0

0

1

0

0

0

1

1

0

1

0

1

(mod 2).

We see that these correspond to the above Cayley tables (to within the notation for the elements). (Cf. also Exercise 2 of the previous paragraph.) EXAMPLE 2. The so-called real rational functions of one variable constitute a (noncountably) infinite field. EXAMPLE 3. We want to construct a non-commutative skew field. For that purpose

let F = (a, i , ...> denote the field of complex numbers. From the full matrix ring F$ we take into consideration only matrices of the form

B=

A

where e' denotes the complex conjugate of e. We denote the set of these matrices by Q. Since a

A l B

AB=

+Y j9+a ( - 6' a' + Y'1 = - (fta ++ Y6)'

-'y-

t4+6 1 (a + Y)')

ay- 0,

fl' 6 + a' y'

a6+RY' - (a8 + fly')' (ay - #g')'

we see that addition and multiplication are compositions in Q. Further the matrix

0 (as the special case at = = 0 of A) lies in Q and together with A its additive inverse

also belongs to Q. Since addition and multiplication are associative in F$ they are so, a fortiori, in Q. Finally, addition is commutative and the distributive law holds so that it follows that Q is a ring. We show that Q is a skew field. Since the unity 1)

belongs to Q, it suffices to show that in Q each element other than 0 element (0 has a right inverse. According to the above formula for the product A B the assertion is equivalent to the following: if the complex numbers a, ,B do not both vanish, then there are complex numbers y, d such that ay - Pb' = 1,

These are (since aa' +

as + fly' = 0.

0) as follows: a'

E_-

# aa'+j9fl,

SKEW FIELDS

69

Consequently we have proved that Q is a skew field. This skew field Q is called a quaternion field. Other "quatemion fields" will be introduced later. The discoverer of quaternion fields was HAMILTON. EXERCISE 1. To the matrix ring Rs (over a ring R) one usually assigns R as operator

domain by the definition

r

ab c d)

_

ra

rb

rc

rd

(a, b, c, d, r E R).

(24.1)

The elements of R itself (conceived as operators) are then called the scalars, and we say that r on the left-hand side of (24.1) is a scalar factor. Further, the operator product (24.1) is also called a scalar product. Show that one may now give the a elements (a, j9 complex numbers) of the quaternion field Q by

(,fl,)

aI

l +b

r

(p 10)

+ c (- 0 p) + d (o p (a, b, c, d real numbers). )

Denote the four matrices occurring here byE,1, J, K, respectively (E is the unity element

of Q). Then the elements of Q appear in the form

aE + bl + cJ + dK.

(24.2)

Show that

12=J2=K2=-E, IJ=-JI=K, JK=-KJ=1, KI=-IK=J. We write (24.2) somewhat simpler by replacing the unity element E by the number 1

(later this will be carried out in a logically more rigorous manner) and by writing simultaneously i, j, k, instead of I, J, K. Then (24.2) becomes

a + bi + cj + dk

(a, b, c, d real numbers),

(24.3)

moreover

i2=j8=kEij=-ji=k, jk=-kj=i, ki=-ik=j.

(24.4)

Show that, after interchanging the elements of Q, the sum and the product of two elements become

(a + bi + cj + dk) + (r + si + tj + uk) = _ (a + r) + (b + s)i + (c + t)j + (d + u)k,

(24.5)

(a+bi+cj+dk)(r+si+tj+uk)= (ar - bs - ct - du) + (as + br + cu - dt)i + + (at - bu + cr + ds)j + (au + bt - cs + dr)k.

(24.6)

One is accustomed to writing an element of the quaternion field Q in the form (24.3). We may then regard (24.5) and (24.6) as defining the compositions in Q. (It is unnecessary, however, to remember (24.6), as (24.4) is sufficient to be able to form (24.6).) The elements (24.3) of Q are called quaternions. EXERCISE 2. For quaternions we have the rule

(a+bi+cj+dk)(a-bi-cj-dk)=as+b2-I-ca+d8.

STRUCTURES

70

The factors of the left-hand side are called conjugate quaternions. Their product is consequently a real, non-negative number. EXERCISE 3. In the quaternion field Q the equation x2 = -1 has infinitely many solutions. How can we give all these solutions? EXERCISE 4. In a zero-divisor-free commutative ring R (and especially in a field R = F) each equation axe+bx+c=0 (a,b,cE R) can have at most two solutions. EXERCISE 5. The eight quaternions ±1, ±i, ±j, ±k constitute a non-commutative group (of order eight). This noteworthy group is denoted by Q8 and we call it the quaternion group. (This will be given in § 77 in another form.) The

i'j'

(r=0,1,2,3; s=0,1)

comprise all the different elements of Q,,.

§ 25. Substructures We have thus far met cases where a subset of a structure again constitutes

a structure. For example, in the ring 7 the set of even numbers also constitutes a ring, the numbers 1, -1 a group. In the semigroup of mappings of a set into itself the permutations constitute a group. We now want to deal generally with such eventualities. As a preliminary, we consider a structure S and a composition 0 in it. A subset T ( 0) of S is called closed relative to the composition 0 if all aOl9 (a, fi E T) lie in T, in which case we may say that the composition 0 can be carried out in T. Similar expressions may be used with respect to the

inverse compositions or the formation of inverses. E.g., if we say that a subset T (# 0) of a field is closed relative to subtraction or the forming of the multiplicative inverses, then it means that all a - ig (a, # E T), or all a-1 (a E T, a 0 0), respectively, lie in T. A subset of a structure S, with respect to a composition defined in S, constitutes a structure if, and only if, it is closed with respect to that composition.

By a substructure of a structure S we mean any subset T ( 0) of S, which is closed with respect to at least one of the compositions defined in S. We then say that the structure S contains the structure T (as a substructure). If T is also, with respect to the above compositions, an Astructure, where A signifies a system of structure axioms, then we say that T is an A-substructure of S. Especially when A is the system of axioms for a group, semigroup, module, ring, skew field or field, we call T a subgroup, subsemigroup, submodule, subring, sub-skew-field or subfield, respectively. We sometimes denote the fact that T is a substructure of S by the set-theoretical

notation T c S, but only if the context reveals what kind of substructure is intended.

SUBSTRUCTURES

71

If S, T are structures and T C S (in the set-theoretical sense) it does not necessarily mean that T is a substructure of S. For instance, two groups G, H are defined by means of the multiplication tables: E

a

E

a

V# E

a

where H is not a subgroup of G, since a2 has different meanings in G and in H. We have already given examples for substructures. Further simple examples are as follows: If a is an element of a ring R, then the elements ma (m E 7) constitute

a submodule of R. If, moreover, a is idempotent, then, because ma na = mna, this submodule is also a subring.

We have already met with such important substructures of a ring R as the module R+, the semigroup R" and the regular semigroup R*. Instead of "T is a substructure of S" we also say that S is an overstructure (or extension structure) of T. If, moreover, S constitutes an A-structure for

a structure axiom system A, then we call it an A-overstructure, and in special cases we say, overgroup (or extension group), extension semigroup, overmodule (or extension module), overring (or extension ring), extension skew field and overfield (or extension field). The subgroups of a group, or the subrings of a ring, are called the substructures of the same kind as the given structure, furthermore we speak in the same way of the overstructures of the same kind of a structure. This short manner of speaking is not quite correct, yet it leads to no misunderstanding. Henceforth substructures will frequently be discussed. On the one hand, we shall consider a structure as sufficiently known only if its substructures are also known; on the other hand, substructures supply, as will be seen, a powerful aid in the investigation of a given structure. The following simple theorem is clear without proof. THEOREM 41. With respect to any kind of compositions in a structure S, associativity, regularity, commutativity and distributivity are properties which hold good even if one passes over from S to a subset of S closed with respect to the compositions in question. On the other hand, invertibility has not the same behaviour. E.g., the infinite cyclic

group (..., a 1, E, a, ...> contains the subsemigroup in which multiplication is not invertible.

We insert here the following simple statements of a kind similar to Theorem 41. If H is a subgroup of a group G, then the unity element of G is also that of H; further, the inverse a-1 of an element a of H, formed in G, lies likewise in H and is the inverse of a also in H. The first assertion follows from the fact that the unity element 8 of H is idempotent and G contains,

72

STRUCTURES

apart from e, no other idempotent element. If, furthermore, a' is the inverse of a in H, then a'a = s = a la, from which, after cancelling a, we obtain a' = a-t. The corresponding statements hold for modules and rings with respect to the zero element and the additive inverse. For skew fields we have the following: if G is a sub-skew-field of a skew field F, then the zero 0 and the unity a of F are also those of G. It holds for 0 by the above and since, according to this, G* is a subgroup of F*, it follows for s also. It is otherwise with semigroups. The multiplication table

E

a

1

I

E

of

E

a

a

a

defines a semigroup with the unity element a and the further idempotent element a, each of them constituting by itself a subgroup the unity element of which is a or a respectively. (By the way a is at the same time the zero element of the given semigroup.)

The remark about the unity element of a skew field has also no analogue with respect to rings with a unity element. Consider, e.g., the full matrix ring 02 over S. The unity element of 3$ is (0 1) , but on the other hand, the elements of the form (0 0)

constitute a subring of .92 with the unity element

I O'(1 0)] We extend the notion of a "proper subset" to substructures in a somewhat modified sense and call a substructure T of the structure S proper, if

T is a proper subset of S and - in case S is a group, module or ring consists of at least two elements. Thus, from the above remark, the proper subgroups of a group G are those subgroups of it, which are different from G and the unity element. Similarly, the proper submodules of a module M are those submodules of M, which differ from M and 0. Further, the proper subrings of a ring R are those subrings of R, which differ from R and 0. We complete this by defining the proper submodules of a ring R to be the proper submodules of R+. If, in a set a of substructures of a structure, we talk of the maximal or minimal elements of C5, this should be interpreted in the usual set-theoretical sense. If, on the other hand, we mention the maximal subgroups of a group G, then we mean the maximal elements of the set of subgroups of G different from G. In other words : a maximal subgroup of G is any subgroup H (0 G)

of G such that there is no subgroup F of G with H c Fe G. We similarly interpret such terms as: maximal subsemigroup, maximal submodule,

maximal subring, maximal sub-skew-field and maximal subfield. When we

SUBSTRUCTURES

73

speak of the minimal subgroups of a group G, we mean similarly the minimal

elements of the set of subgroups of G different from the unity group. We interpret a minimal submodule or a minimal subring in a similar way. So, e.g., all the maximal submodules of the ring 7 are those modules (m) for which m is a prime number. On the other hand it is obvious that 5' has no minimal submodules, and the Prefer group (p'°) has no maximal subgroups. We now enunciate some simple criteria for substructures. First it follows immediately from Theorem 41 that a subset T of a semigroup S constitutes a subsemigroup of S whenever T is closed with respect to the multiplication. THEOREM 42. A non-empty subset H of a group G is a subgroup of G if H is closed with respect to the multiplication and the forming of inverses. For a finite G the first condition is sufficient. In order to prove the first part of the theorem, we denote by e the unity element of G. For each a (E H), ax -1= s, consequently, s necessarily lies in H. Since, further, a is necessarily the unity of H, the assertion follows from Theorems 32, 41. The second part of Theorem 42 now follows from Theorem 33. THEOREM 43. A non-empty subset N of a module M is a submodule of M if N is closed with respect to subtraction.

For any two elements a, f of N it follows that the elements

0=a-oc, -3=0-9, a+ 3=a-(-3) belong to N. From this and from Theorem 42 Theorem 43 follows. THEOREM 44. A non-empty subset S of a ring R is a subring of R if S is closed with respect to subtraction and multiplication.

The truth of this results from Theorems 41, 43. THEOREM 45. A subset S consisting of at least two elements of a skew field F is a sub-skew-field of F if S is closed with respect to subtraction, multiplication and formation of multiplicative inverses. By virtue of Theorem 44, S is a ring. Further the difference set S - 0 is a group on account of Theorem 42, which proves Theorem 45. The following table A

Sg.

Sg.

B

G.

M. R.

SF.

G.

M.

R.

SF.

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74

shows by dots " " all the combinations A, B in which an A-structure can contain a B-structure where both A and B signify the axioms for semigroups (= Sg.), groups

(= G.), modules (= M.), rings (= R.) or skew fields (= SF.).

THEOREM 46. Let S denote an A-structure and S1, S2, ... denote B-substructures of S where A, B is a suitable pair of systems of structure axioms for semigroups, groups, modules, rings or skew fields. If the intersection

D=s1nS2n... is not empty, this is also a B-substructure of S, except in the case when A denotes

the ring axioms, B the skew field axioms and D is the zero element of S.

We formulate the special case A = B, which is frequently applied, as: COROLLARY. The intersection of arbitrary subgroups of a group is a subgroup of the given group. The same applies if for "group" we read "module", "ring", "skew field", or "semigroup". In the latter case, however, it must be assumed that the intersection is not empty. In the proof of the theorem we may restrict ourselves to the case where there are only two substructures S1, S2, for the proof is entirely analogous for the general case. First we consider the case where B is the axiom system for semigroups, i.e., S1, S2 are semigroups. We now have to prove that D is likewise a semigroup. For that purpose we consider two elements a, 8 of D. These belong both to S1 and S2 and the same holds also for their product aq. Accordingly a,9 E D. Since, consequently, D is closed with respect to multiplication, it follows from Theorem 42 that D constitutes a semigroup. This completes

the proof for this case. The next (and most difficult) case is where B denotes the axiom system for groups, i.e., S,, S2 are groups. From the above we know that in this case D is always a semigroup. Therefore it is sufficient to show that D contains a unity

element and the inverse of each a (E D). This is simple if S is a group, but since S is arbitrary, we have to proceed as follows. We denote the unity

element of Si by a,(i = 1, 2), and (since a E S1i S2) the inverse of a in Si by ai (i = 1, 2). Then the equations E1a=ae1 =a,

E2a = ae2 = a ,

ala=aal=El,

a20C =aa2=E2

hold in S1, and S.., respectively, and consequently in S. It suffices to prove

that el = e2 and a1 = a2, for then el (= e2) belongs to D and is its unity element; furthermore, al (= a.,) similarly belongs to D, where it is the inverse of a. We now have Et = ala = al(E2a) = al(aa2)a = (a101) (a20) = E182 .

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75

By symmetry we obtain 82 = 8281. Hence it follows by virtue of the duality principle that e2 = E1E2. Consequently el = 82. Further we obtain al = E1(h = E20C1 = (c 2o) al = 02(7-al) = a2e1 = a282 = a2.

This completes the proof for this case. The case where B is the system of axioms for modules is essentially no different from the previous case, since modules are in fact (additive) groups. For the case where B denotes the system of axioms for rings, the theorem

is easy to prove, for in compliance with the foregoing D is, at the same time, a module and a semigroup; further, because of Theorem 41 distributivity holds in D. Finally, we have the case where B is the system of axioms for skew fields, i.e., S1, S2 are skew fields, and D contains at least one element different from the zero element 0 (of S). From the above D is a subring of S. Further

0 E S1, S2 and D. It still remains to prove that the difference set D - 0 is a group. This follows from the above, since D - 0 is the intersection of the groups S, (= S, - 0) (i = 1, 2). Consequently Theorem 46 has been proved.

A counterpart to Theorem 46 is THEOREM 47. If S11 S..,, ... are arbitrarily many A-structures where A is the system of axioms for groups, semigroups, modules, rings or skew fields,

and moreover if for each pair Si, Sj, one of Si and Sj is a substructure of the other, then S11 S2, ... have only one common A-overstructure S with the property

S=S1US2U....

(25.1)

It suffices to prove the case where A signifies the group axioms, since the other cases are proved similarly. We have to prove that in the set S defined by (25.1) multiplication may be defined in one, and in only one, way so that S becomes a common extension group of all the groups S1, S2, ... . We consider two arbitrary elements a, P (E S). Then there exist, because of (25.1), an Si and an S j with a E Si, 9 E S j. Since S, S Si or S j 9 Si, there exists

also an Sk such that a, # E Sk .

(25.2)

(X/ = ), (E Sk)

(25.3)

In this Sk the product

exists. Since for two arbitrary Sr, SS one is a subgroup of the other, the element y in (25.3) is independent of the group Sk in (25.2), but is uniquely defined by the elements a, P. Accordingly, we can define a multiplication in the set S by means of (25.3).

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76

This multiplication is associative. For if 0, a, ,r are arbitrary elements of S, then they all lie in a suitable Sr. Since the multiplication in S, is associa-

tive, it also follows for S. Multiplication in S is also invertible, as on account of (25.2), e.g., the equation a = P is solvable in Sk, and consequently also in S. From this, S is a group, and, obviously, an extension group of SI, S2, ... . Since this cannot be the case if another multiplication is defined in S, the uniqueness of the group S follows. Theorem 47 is therefore proved. The structure S defined in this theorem is called the union of the structures ... and is denoted as in (25.1). (In special cases it is called the union group, etc.) SI, S2,

Theorem 47 is trivial, if there are only finitely many structures Sts S2, ..., since then one of them is an overstructure of the others. (For generalizations of Theorem 47 cf. KLROSH (1953a), p. 66.)

THEOREM 48. Let A, B each denote a system of axioms for groups, semigroups, modules, rings or skew fields and let an A-structure S and two mutually disjoint subsets K, L (L 0 0) of S be given. If the set Z of those B-substructures

of S which contain K and contain no element of L, is not empty, then has a maximal element. Consider any elements Si, S2, ... of Z with the property

SICS2c.... It follows at once from Theorem 47 that the union of these elements SI, S21

... is an element of Z. Hence Theorem 48 follows from the special

lemma of KURATOWSKI-ZORN (Theorem 16). EXERCISE 1. If the intersection of all subsemigroups of a regular semigroup S is non-empty, then S is a group all of whose elements are of finite order. (Hint: H contains no element of infinite order; H contains one and only one idempotent element; this is a unity element; each element of H has its inverse.) EXERCISE 2. The elements of finite order of an Abelian group constitute a subgroup. a EXERCISE 3. For every natural number n, the fractions of the form - (a E :I) conn

stitute a cyclic module M. Prove that the union module Ml U M2 U ... (consisting of all rational numbers) is non-cyclic.

§ 26. Generating Elements Theorem 46 makes the following important and very general concept possible. Let A, B each denote a system of axioms for semigroups, groups, modules, rings or skew fields; further let S denote an A-structure and S a non-empty subset of S which is contained in at least one B-substructure of S. Then on account of Theorem 46 the intersection of the B-substructures of S containing S is a uniquely defined B-substructure of S which we denote by {St and call the B-structure of S generated by the elements of S or by S.

GENERATING ELEMENTS

77

We now somewhat generalize this concept by giving up the requirement that all the elements of S should be distinct i.e., we admit for S (non-empty) systems of elements of S and define {S} as {S'}, where S' indicates the set of all distinct elements of S. It suffices to consider the previous case; however, the statements concern also the latter case. For {S} we also write {a, i4, ...}, where a, 9, ... denote the elements of

S. If, furthermore, S is the union of the sets S1i S2, ..., then {S1, S.,, ...} is interpreted as {S}. The phrase "the B-substructure { S} of S generated by S" is seldom used in this form but often in such versions as: "the subgroup {S) of the group G generated by S" ("the subgroup {S} of G") or "the subring {S) of the field F generated by S" ("the subring {S} of F"), etc. E.g., the subgroup {2, 3} of the field .7-0 consists of the elements 2' 3k (i, k = 0,

±1,...).

If a is an element of infinite order of a group, then the semigroup {a} consists

of the elements a, a',...

For a structure S and a subset S of S, if the symbol {S) is used without fuller explanation, then this will denote the substructure of the same kind (as S) generated by S. Thus, e.g., {a, f9} denotes for the elements of a given

group G - unless otherwise indicated - the subgroup of G generated by a, P. We now consider, for a given S, all the {S}. Among these S itself also occurs, since S = {S}. A (non-empty) subsystem S of S is called a generating system of S, if S = {S}. The elements of S are then called generating elements

or generators of S. However, this terminology may also be related to the general case {S} c S, and then S, for an arbitrary substructure {S} of S, is a generating system of {S). Accordingly, we may then call the elements of {S} the elements (of S) generated by the system S.

Henceforth we shall often consider a structure to be of the form {S} where S is a generating system for that structure. If we speak, e.g., of the group {a, #}, this means the group with the generators a, P. The important question arises as to how we can compute the elements of a structure {S} from the elements of S. To answer this question some preparation is needed. For that purpose, we shall again summarize the five structure classes in question by the help of the following table :

a+f semigroup group . module ring skew field

-a

aj

I

a_1

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78

In this table the dots indicate that in the relevant structures the above compositions are defined or that the corresponding inverses exist. In particular [.] indicates the restriction required for a-1. Now we wish to consider the forming of the inverses -a, a-1 in a structure as "compositions", which we call compositions with one variable. If we also include in a structure those compositions with one variable, then we speak of the compositions in the wider sense (i.w.s.). Compositions i.w.s. are,

e.g., in rings and skew fields the compositions a + 9, ap, -a, and a + fi afl, -a, a-1, respectively; but in a skew field a-1 is understood to apply only for a 0. In a structure S, an (algebraic) expression of certain elements a, i4, y, .. . of S means first of all each of these elements, and then all those elements (of S) which may be obtained from them by a finite number of repetitions of the compositions i.w.s., defined in S. The expressions of the form a,

a + 9, afl, -a, a-1 are called the elementary expressions (of a, P.... ). Thus every expression of certain elements may be obtained (mostly in several ways) by forming elementary expressions from the given elements finitely many times. Examples of expressions in the elements cc, fi, y of a ring are

x = -2x3 + xfBx,

2x-5y2, xi+fly +y«. In the case of a field, we have the example: (x + f9) -' - («

z

-

(« + fl) (« -

y)3

if x+j9.0, x-yOO. The following is now almost trivial. THEOREM 49. A structure S = {S} consists of expressions of the elements

of the generating system S. Let G be the set of those substructures of S of the same kind as S, which contain S. In particular the set T of expressions of the elements of S constitutes a structure belonging to G and T is also a substructure of all structures in S. Thus the intersection of these is equal to T, but by definition it is equal to {S}. Hence it follows that T = {S), which proves the theorem. A question to be discussed later, is how all the different elements of S among the expressions mentioned in Theorem 49 may be distinguished. Here we briefly consider the possible expressions, according to the different structure classes. In a semigroup we can write all expressions of the elements of a subset S

of this semigroup in the form

IM2 ...Pk

kz 1).

(26.1)

GENERATING ELEMENTS

79

Similarly, in a group we have (26.2)

el 1e2 1 ... ek I

instead of (26.1). In a ring we have to consider finite sums of the form Ecet...,Qk e1... eK1

(26.3)

where the coefficients co.,..., Q, are integers. We may restrict ourselves in (26.3) to the case where every product el ... ek occurs at most once and the coefficients differ from 0, but then we have to permit the "empty" sum (with the value 0). The order of the terms in (26.3) is of no consequence. In a skew field the situation is much more complicated. For although every expression may be written as a "parenthesis expression", e.g., as 3y_1)2

+

_

My-IL /9)2

+ 2y,

in general we cannot remove the parentheses. However, under special circumstances, this may be done, e.g., we can give the elements of the quaternion field Q - as we have seen - in the form

a+bi+cj+dk, where a, b, c, d are real numbers. The real numbers and i, j, k are now generators.

In a field we may write all expressions in the form

AB-1=

B

where A, B (B 0 0) are of the form (26.3). In a module we have expressions of the form

I cee

(e E S) .

(26.4)

(This is similar to (26.3) with k = 1 for each term.) For Abelian groups, of course, we have the multiplicative analogue of (26.4). Likewise, in (26.1) and (26.3) the corresponding simplification takes place in the commutative case.

In a structure S we often denote an expression of the elements a, , .. . by f(a, i4, ...). Even if there may be infinitely many a, fl.... only finitely many of them may appear in an floc, /9,.. ....), for which reason it is sufficient to consider f(a, ..., v) in order to cover each possible case. In an expression floc, /9.... ) we may regard the elements a, fl.... or some of them as variables (in S), in which case they are often denoted by letters taken from the end of the alphabet. We retain the term "expression" also for such functions. We shall frequently deal with equations of the form A = B, where A, B are two expressions in a structure S. If in the latter variables also occur, 4 R-A

STRUCTURES

80

and if we consider A = B as an equation submitted for solution, then we call the variables (as in § 2) the unknowns. If such an equation holds for all values (in S) of the variables, then it is called an identical equation or identity. The case where no variables occur (and the equation is true) is included. Knowledge of such identities is of great importance for the determination of the different elements of a structure given in the form S = {S} (i.e., by

generators). We have still to note that we can write an equation A=B in a group in the form AB-1 = e, and in a module, ring or skew field in the form A - B = 0, so that we often assume an equation in these structures to be in the form A = e, or A = 0, respectively. Finally we introduce, with respect to generating systems, the following important concepts. A generating system S of a structure is called minimal if no proper subsystem of S is a generating system of the given structure. A minimal generating system always consists of distinct elements, i.e., it is a set. A finite generating system after cancelling some superfluous elements may always be reduced to a minimal one. A structure with at least one finite generating system is called finitely generated. Consequently, the finitely generated structures always have at least one minimal generating system. EXAMPLE 1. The module .$o has the generating system

(1, 2 , 3 , ... ), but

it has no minimal generating system and a fortiori it is not `finitely generated. EXAMPLE 2. If Pi. pz, ... are all the distinct prime numbers, then for the sub1 module I PI P2 erating system.

. of. o , the elements in the brackets constitute a minimal gen-

y

11J

EXERCISE 1. The numbers a + b. J2 + c,1-3+ d1/ 6 (a, ..., d E .9) constitute a ring with the minimal generating system (v/ 2, J Y>. EXERCISE 2. The only minimal generating system of the semigroup -V x consists of 1 and all the prime numbers. EXERCISE 3. For the full permutation group of the set <1, . . ., n) (n > 2) the transpositions (1 2), (2 3), . . ., (n - 1 n) constitute a minimal generating system.

EXERCISE 4. For two elements a, j of a ring the equation a" - fl" _ (a - f) (a"-` + a"-° S + ... + fl"-') (n > 2) holds if and only if of and fl are permutable. EXERCISE 5. Every semigroup with at least two elements contains at least one proper sub-semigroup. PROBLEM. The newly discovered and noteworthy theorem (of Isbell): The multiplicative semigroup of an infinite commutative ring can never be finitely generated. Let us examine whether this is true in the non-commutative case. Try to prove Isbell's theorem more simply. (Cf. ISBELL (1959), REDEI and STEINFELD (1952), and JOHNSON (1958).)

§ 27. Some Important Substructures In this section we shall define some substructures of a structure S, viz., the "Frattini substructure", the "centre" and the "commutator structure" (of S). As a preliminary, it is necessary to discuss certain semigroups.

SOME IMPORTANT SUBSTRUCTURES

81

A semigroup is called an 1-semigroup if each element in it is a left uni-

ty, an r-semigroup is one in which each element is a right unity. It is evident that in every non-empty set we can define a multiplication in only one way such that it becomes an 1-semigroup. If all non-empty subsets of a semigroup are subsemigroups, then we call the semigroup breakable. The next theorem holds for these very special semigroups, which, however, we shall only prove at the end of this paragraph (in small type). T FmoREm 50. A semigroup S is breakable if, and only if, it can be partitioned into classes and the set of the classes can be ordered in such a way that every class constitutes an 1- or an r-semigroup and for the elements a (E C), j9

(E C') of two different classes C, C' with C < C', aq = floc = P. We now introduce the following terminology. For a structure S let ch(S) be the set of those elements which may be omitted from each generating

system (containing them) of S. We prove that - apart from some exceptions - P(S) constitutes a substructure of the same kind as S, and we call O(S) the Frattini substructure of S. In special cases depending on S, we refer to the Frattini subgroup, Frattini subsemigroup, Frattini submodule, Frattini subring, Frattini sub-skew-field and Frattini subfield (of S). However,

there is an exception when (b(S) is the empty set; this is always the case when S is a breakable semigroup, the unity group, the zero module or the zero ring. Moreover we also have to exclude the case where S is a skew field and O(S) is equal to 0. (In § 124 we shall see that this exceptional case occurs if, and only if, S is a so-called prime field.) Apart from semigroups there are no other exceptions. In the exceptional cases the Frattini substructure O(S) is said to be nonexistent. First we prove our assertions for O(S) where S is a semigroup. If this is breakable, i.e. every non-empty subset of it is a semigroup, then it has only one generating system, viz., S itself whence it immediately follows

that O(S) _ 0. We notice that besides the breakable semigroups there are other semigroups S with cP(S) = o (cf. H. J. WEINERT 1964).

It still remains to be proved that a non-empty O(S) is a semigroup, i.e it is closed with respect to multiplication. For that purpose we conside two elements a, f (E O(S)). If S = {af, S) for some S (c S), then a fortior

S = {af,a, f,S}= {a, f,S}. On the right-hand side, because of the assumption, we may cancel first a, then f. From this we obtain {af, S} = {S}, so it follows that of E (b(S), proving the assertions concerning O(S). We now consider O(G) for a group G. O(G) is empty if G = a. If G Z) e then 8 E O(G). This, together with the above, means that O(G) is a semigroup

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82

with the unity element s. It suffices to prove that from a E O(G), it follows that a-1 E 0(G). This is obviously true since {a, S} = {a-1, S} (S (-- G), so proving the assertions about 0(G). From the assertions proved so far the corresponding ones for modules and rings follow immediately. The same also holds for a skew field F except in the case where 0(F) is equal to 0. These are all the assertions about 0(S). THEOREM 51. The Frattini substructure 0(S) of a structure S is the intersection of S and its maximal substructures of the same kind, except in those cases, where S contains only one element or where S is a skew field with

0(S) = 0. If D denotes the specified intersection, we have to prove that for all ccES

a ff D a a 10(S) .

(27.1)

Let Sm denote one of the maximal substructures, mentioned in the theorem, if such exists.

If the left-hand side of (27.1) holds, then there is an S. which does not contain a. Hence S = {a, Sm} 0 {Sm}, consequently a ff 0(S). Conversely let us assume that a ff 0(S). Then apart from the excepted

cases, there is a non-empty subset S of S with S = {a, S} 0 {S}. In the set of those substructures of the same kind as S which contain S and do not

contain a, there is, on account of Theorem 48, a maximal substructure which we denote by S'. Every proper overstructure of S' of the same kind as S contains S and a and consequently also S = {a, S}. From this, S' is an Sm.

Since a is not contained in this Sm, the left-hand side of (27.1) follows. Theorem 51 is therefore proved. THEOREM 52. If every proper substructure of the same kind as a structure S

is contained in a maximal one of the same kind and a subset S of S together with (P(S) generates S, then S alone generates S. If S # {S}, {S} would be contained in one of the maximal substructures Sm of S. This S. contains 0(S) and S = {S, 0(S)}. This contradiction proves Theorem 52.

Now let S be a group, a ring or a skew field. Then we denote by Z(S) the set of those elements e of S, for which pa = me

(for all a E S) ,

(27.2)

and we prove that Z(S) constitutes a commutative substructure of the same kind as S. We call Z(S) the centre and its elements the central elements of S. S may be a semigroup too, on the understanding that then the centre

can be empty. Finally, if S is a module, then we have Z(S) = S; for this case our assertion is trivial.

SOME IMPORTANT SUBSTRUCTURES

83

It suffices to prove the assertion for a skew field S, since the rest of cases are easier. We have to show that Z(S) is a field. For the sake of brevity we put Z = Z(S), The letters e, a denote elements of Z and a an element of S. On account of oa = c , as = as we have

(e-a)a=em -aa=ace-aa=x(ee-a), (ea)a = e(a0) = e(aa) = (eo)a = (oP)a = a(ea) .

Further by virtue of (27.2) we also have aN-1 = e-1a

From these we have

e - a,

(e

0)

.

o-'EZ

aa,

(provided 9:00 for 0 '). It follows from Theorem 45 that Z is a skew field. But since, by virtue of (27.2), Loa = ae, Z is a field, which proves our assertion.

From two arbitrary elements a, fi of a group G or of a ring R we can

form the expressions afa-1 #-', and afl - j9a, respectively, which we call in both cases the commutator of the elements a, ft. (This terminology is

explained by the fact that for K = aj9a-' f-' and A = aj9 - /9a we have KIa = ajI and A + fJa = ocfl respectively.) We also call ala-1 f-' and

a# - 8a the multiplicative and additive commutator, respectively. The substructure of the same kind, generated by all commutators aft a-' 9-1 and ocfl - #a, thus : [(X# a-1

-], ...}

{a fl - #a, ... }

(cc, fl, ... E G)

,

(a, f9, ... E R)

of G and R are called the commutator group of G, and the commutator ring of R, respectively.

In a skew field we can form both kinds of commutators a/I a-' fl-1 0), a/I - fa. The next theorem holds for these, so that we do not

(a, ft

define the notion, as above, of the "commutator skew field". THEOREM 53. In a non-commutative skew field F both the multiplicative commutators and the additive commutators generate F itself. For the proof we introduce the notations (a, /I) =

afla-lf-l

[a, #] = a/I - floc

(a, f # 0) ,

.

We denote the skew field generated by the (a, j9), and [a, ,#], (a, /I E F) by F1 and F2, respectively. We have to prove that F1= F2 = F. It suffices to

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84

show that each element a (( Z(F)) of F lies both in Fl and F2. Since then

for each y(E Z(F), # 0), y = a-1 ay(a-1, ay ( Z(F)) and so lies in F, (i = 1, 2).

We take an element i4 (E F) not interchangeable with a. Such a fg exists since a is not a central element. Then (0c, P)

e, 1a,fl)#0

where a is the unity element of F. First we show that a E Fl. For this we consider the commutators

x=(a,fg), 2=(a+f3-',f9), µ=(a,fg+e). These exist by virtue of a + f-1 # 0, fg + e 96 0. Then icjIoc = a# ,

A(19a+e)=0j9+e,

µ(fa+a)=afl+a. Since K ,-E e, then A

is and u # E. From the first two of the last three

equations it follows that (A - x)IBa = e - A. Since K, A E F1 and A - K 0 0 we have floc E F1,

afg E Fl .

Hence, from s E Fl and the third of the above equations, ,% E Fl follows. Secondly, we show that a E F2. To do this we consider the commutators P = La, f'3] ,

We have

a = [a, affil

or =a2jI-afla=MLO.

Hence from LO, a E F21 e # 0, a E F2 follows. Consequently Theorem 53 is proved.

We call a group, or ring, centre free (or without a centre) if the centre of the group, or ring, consists only of the unity element, or the zero element, respectively. A ring with unity element, especially a skew field, is never centre-free. A semigroup is called centre-free when its centre is empty. PRooF OF THEOREM 50. The assertion "if" is trivial. In order to prove the assertion

"only if" we assume that S = is a breakable semigroup. Then each a and each pair a, f4, constitute semigroups, so that we have a$ = a

(a E S),

(27.3)

and

a# = a or a# = f3

(a, f4 E S) .

(27.4)

SOME IMPORTANT SUBSTRUCTURES

85

We define a relation = in S by

aas

afl=fl, fa=a

(27.5)

and show that this is an equivalence relation. Reflexivity holds because of (27.3) if furthermore a = 14 and a = y, then [

t

afl = fl, fla = a, ay = y, ya = x. Hence it follows that

fly = fl(ay) = (fla)y = ay = y, and by symmetry also yfl = 14. These imply that 14 = y, which proves the assertion.

We denote the corresponding classification of S by e,. Because of definition (27.5) each class in e, is a maximal l-semigroup. Similarly let e, denote the classification of S into maximal r-semigroups. We show that if any two classes of e, and e, intersect then one of them consists of only one element. For otherwise S would contain three different elements a, fl, y with {a, y} an 1-semigroup, and {14, y} an r-semigroup. Then

(fla)y = fl(ay) _ fly = fl,

y(ea) = (yfl)a = y a = a.

But because of (27.4) flx is equal to a or f, so that one of these equations contradicts (27.4).

From what has been proved, by taking the classes, consisting of at least two elements, of e, and e and all the other elements of S together as a class, then we obtain a third classification a of S. We denote the corresponding equivalence relation by = containing a by a. It follows from the definition of a and from (27.4) that for any two elements at, rg from different classes:

a#=floc =a or af=floc =fl.

(27.6)

We show that for any elements el, 22, a of S

e1=es*a= either 1a =a, = a or e,a=ae,=e,

(27.7)

with i = 1, 2. Since 01 = e2, the elements e1, e2 constitute an I- or r-semigroup. It suffices to consider the first case. Then we have 2122 = 22,

0201 = 21

Further, because a * el, 02, from (27.6) we have 2117 = aer

(i = 1.2).

Consequently if the assertion is false, then because of (27.4) with e1, ea suitably ordered,

Hence we obtain

ela = ael = e1, 2217 = ant = a. at = e1a = 01(a22) _ (21x)22 = 2122 = 22:

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86

however, (27.7) is true for el = e2 because of (27.4). This contradiction proves (27.7).

Now we define in the set of classes a an ordering relation < for which two unequal classes a, P are such that

a<

(27.8)

Hereby < is uniquely defined (i.e. independently of the representatives a, 14), since

for a - a', /1 = #' it follows from the right-hand side of (27.8) because of (27.7) (applied to ei = a, e2 = a', v = #, and eI = A e2 = 1-', o = a, respectively) that

a ' fi = f a ' _ f and a#, = fl ' a = P', respectively. Further it follows from (27.6) that for a =E fi either a < A or # < as < y. From always holds. In order to prove the transitivity of <, we assume a < the definition (27.8) it follows that

ay=a(,Y)=(afl)y=fly=Y and, by dualizing, ya = y, so a < 'y. This completes the proof of Theorem 50. EXERCISE 1. The full permutation group of <1, 2, 3> has no centre.

EXERCISE 2. The centre of the full matrix ring I2 over .1 consists of the matrices (a 0) 0a

(aE:I).

EXERCISE 3. The only subgroup of second order of the quaternion group Q8 is the centre, which is the commutator group as well as the Frattini subgroup of Q8. EXERCISE 4. By the Cayley tables

0

1

M

fl Y

1

fl

y

X

fl

y j9

0

a

0

a

a

0

y

f3

y

Y

fi

0 a

0

a

0

0

0

a

0

a

(3

0

0

y

0

0 a a

fl

0 0

Y

0 0

,B

y

fl

y

is defined a centre-free ring. This ring of order four, interesting in several respects.

is called the four-ring. EXERCISE 5. The Prefer group (pGO) is its own Frattini subgroup.

§ 28. Isomorphisms We have already remarked that with respect to a structure we shall attach less importance to the nature of the elements as to the manner of composition. Now we wish to give an exact meaning to this "idea". This will lead as to a certain revision of the concept of structure and algebra. In order to avoid beginning directly with definitions we first explain a simple but fundamental construction. Let a structure S -
ISOMORPHISMS

87

tions corresponding compositions. Here (and also later, unless otherwise indicated) we denote the corresponding compositions in S and S' by the same composition sign. The definition of the compositions in S' itself shall be as follows: we consider a composition in S which we write, e.g., in a multiplicative way and agree that if an equation of the form xi3 = K

(28.1)

holds in S then for the images a', j9', x' of x, f3, K the equation

holds in S'. By virtue of (28.1) h' _ (x#)' so that the definition formulated in (28.1) and (28.2) may also be written in the (shorter) form (213)' = x'(3' .

(28.3)

We emphasize that on the left- and right-hand side the products o c#, a'#' are to be understood in S, and S', respectively, and that (28.3) with suitably changed notation should relate to each pair of corresponding compositions. (All this means, briefly, that we have replaced the elements of S by new elements, but we treat them exactly the same as the former elements.) After these preliminary remarks the following definitions are obvious. We consider two structures S, S' and a composition in each, both of which we write in a multiplicative way. If for a mapping ; -* E' of S into S' the condition (28.3) has been fulfilled, then we say that this mapping is homomorphic for the compositions considered. If this holds for all compositions defined

in S and S', after they are properly put in pairs, then we say that E -- ' is a homomorphic mapping of S into S. The compositions assigned to each

other in a homomorphic mapping are called (as above) corresponding compositions. In the case of a homomorphic mapping of a structure into another structure unless otherwise stated it is to be assumed that similarly denoted compositions correspond. We shall deal in general with the homomorphic mappings in § 29. Here only a certain special case will be discussed.

A one-to-one homomorphic mapping is called an isomorphic mapping. An isomorphic mapping of a structure S onto a structure S' is called an isomorphism. If there exists such a mapping then we call S, S' isomorphic structures. Hereby we have defined a relation between structures, which we call isomorphy and denote by So if we write S 4/a R - A

S'

,

(28.4)

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88

it means that S is isomorphic with (or to) S'. If we write it more fully as

S' (a -> a) ,

S

(28.5)

then it means that a ---> a' is an isomorphism of S onto S'. In the construction given above we have seen how we can give to any structure S the structures S' isomorphic with it. There are always infinitely many such structures. Since isomorphic structures are equipotent, it is consistent to denote the isomorphy of structures just as we denote the equivalence of sets, i.e., by

We prove that isomorphy is an equivalence relation. Firstly, S x S (a -> a), i.e. the identity mapping of a structure onto itself is an isomorphism. Therefore isomorphy is reflexive. In order to prove symmetry, we consider an isomorphism

S z T(a sa).

(28.6)

Since this mapping s is homomorphic, by virtue of (28.3) we have s(af3) = (SO) (sf3)

(28.7)

.

Further, since s is one-to-one, the inverse mapping s-' which maps T onto S also exists. We show that S (a'

T

s-' a') ,

(28.8)

whence follows the symmetry of isomorphy. We still need to show that the mapping denoted in (28.8) is homomorphic. From (28.7) it follows that

s-1(sa sf3) = 0 = (s-1(sa)) (s -1 (SP

(28.9)

.

Since sa and sf3 run through all the elements of T, (28.9) proves the assertion. Finally, in order to prove the transitivity of the isomorphy, we consider the two isomorphisms

S z S'(a-->sot)

S' z S"(a'-.sa).

,

(28.10)

It suffices to show that this implies S

S" (a --> s'sa)

(28.11)

.

First of all S is mapped one-to-one onto S" by s's. Further it follows from (28.10) that (s's)(a(3) = s'(s(a,8)) = s'((sa) (sf3)) = (s'(sa)) (s'(sf3) = (s'sa) (s'sfi) (a, f3 E S) .

(28.12)

ISOMORPHISMS

89

These two facts together prove (28.11). Consequently we have proved that isomorphy is an equivalence relation. Now we can formulate the fundamental isomorphy principle of algebra in the following manner : Isomorphic structures are not regarded as essentially different.

We also call isomorphic structures "essentially equal" or "equal in an abstract sense". At the same time the concept of algebra will be subjected to a revision,

its task being henceforth the search for structures which are not isomorphic with each other. When interpreted in such a way, algebra is usually called abstract algebra. (However, in future the term "algebra" will be used in this sense.) If we speak of an abstract structure S, then we mean that S is not specifically defined but appears as representative of the structures isomorphic to

it, where the elements may therefore be mere symbols and we consider only the diversity of their elements and the kind of compositions which hold between them.

Although this work will generally deal with abstract structures, yet various isomorphic structures will frequently be considered. This will always

be the case, for instance, if a structure contains several isomorphic substructures.

A concept, connected with a structure, which remains invariant under isomorphism, is called an isomorphic invariant (briefly invariant) of this structure. (Such invariants are, e.g., the order and the characteristic of a structure.) Another important task of algebra is the determination of the invariants of structures since these are regarded, by the isomorphy principle,

as the only "algebraic properties" of structures. A system of invariants of a structure is called a complete invariant system of this structure, if by this system the structure is uniquely determined in the sense of abstract algebra (i.e., up to isomorphy). We call certain invariants independent if none of them may be derived from the others. Although the concept of isomorphy arised in GALOIS' papers, it was STEINrrZ (1910) who first formulated the isomorphy principle, for which reason it is also called the Steinitz principle. In this way STEINITZ has become the founder of "modern" algebra.

Briefly we point to the similarity with the well-known "Erlanger Programm" of FEr ix KLEIN. According to this the task of geometry consists of the exploration of the geometrical properties of different spaces. So the properties which are invariant under the customary mappings in geometry (collineations, affinities, motions, etc.) are called the projective, affine, metric invariants, etc. This comparison proves all the

more appropriate as in algebra further mappings and invariants, besides the isomorphisms, will also be considered, although only in an auxiliary role.

In order to give a simple example of the notion of isomorphy, we consider a cyclic group G = {a}. We want to define all the groups isomorphic with G,

STRUCTURES

90

which must, of course, also be cyclic groups. Let G, be such a group. Then O(G) = O(G). But this is already sufficient, since if G1 = with O(G1) = O(G) then G1(a`--Mfg'),

G

where i runs through the numbers 0,

1,

.

. .,

O(G) - 1, or 0, ± 1, ....

respectively, according as O(G) is finite or infinite. Accordingly, two cyclic groups are isomorphic if, and only if, they are of the same order. In other words: O(G) is for a cyclic group G a complete invariant system. Therefore

we can speak of "the cyclic group of order n" (n = 1, 2, ...) and of "the infinite cyclic group", since these expressions denote well defined abstract groups. EXAMPLE 1. Define in the set of pairs (a, b) of real numbers a, b the addition (a, b) + (c, d) = (a + c, b + d). This gives rise to a module M. If on the other hand, V denotes the module of vectors in a Euclidean plane, then we have the isomorphism V ((a, b) -- ae, + be_),

M

where el, es are two arbitrarily chosen non-parallel vectors in this plane. EXAMPLE 2. Among the quaternions a + bi + cj + dk (a, b, c, d real numbers) the quaternions a + bi constitute a subfield of the quaternion field Q, which is isomorphic with the field of the complex numbers. EXERCISE 1. There is (to within isomorphism) only one non-cyclic group of order four. This is called the (Klein) four-group. It is Abelian and is given by the Cayley table F

e

a

T

E

O

a

T

O

U

F

T

a

a

or

T

F

Q

T

T

(T

O

F

The full permutation group of four elements contains exactly four four-groups. Which? EXERCISE 2. The six values of the cross ratio (A B C D) arising from the permutation

of the elements A, B, C, D are, as is well known: ).' I - i.,

A

I-

1

1

L

Denote them by f (i) (i = 1, ..., 6). The set of these functions (of %) constitutes according to the composition (f, o fk) (),) = f, (f, (A)) a group, which in geometry is called the (Klein) anharmonic group. This is isomorphic with the full permutation group of three elements. EXERCISE 3. Determine all infinite cyclic subgroups and submod ules of the rational number field.F-0. How may we map them isomorphically onto each other? (Here we meet a case where the compositions are denoted differently in isomorphic structures.)

§ 29. Homomorphisms In the previous paragraph in connection with isomorphisms we have also

incidentally defined homomorphic mappings. Here we want to examine this concept. All statements about homomorphic mappings, of course.

HOMOMORPHISMS

91

apply, in particular, to isomorphisms so that the following discussions will at the same time give a clearer picture of the notion of isomorphy. We consider a homomorphic mapping

s =r 5'

( E S)

(29.1)

of a structure S into a structure T, which means (from § 28) a mapping of S into T with the properties (aj9)' = a.'f1'

,

(x + fi)' = a' + #'(a, f1 E S).

(29.2)

Moreover, we assume again that the corresponding compositions in S and T are denoted alike (i.e., either in the additive, or multiplicative notation); if structures with only one composition are considered then we have to regard only the relevant part of (29.2). Note that the condition (29.2) may also be expressed as follows : for a

homomorphic mapping the succession of the composition and mapping is interchangeable. It is, e.g., according to (29.2) indifferent whether we compute from a, f1 first the product a/i and pass from this to the image (a/i)', or whether we take first the images a', /3' and compute from these the product a'/J'.

By reiterated application of (29.2) we obtain : (al9y)' _ ((afl)y)' = (0j9)')' = (a fi')3" = ash';''

Similarly we obtain from (29.2) for rings, (a + 4h + yc)' = x' + 4h' + + y' (y a'). Further, by the help of complete induction it is easy to see the following: iff (a, h, ...) is an expression in S which has been constituted from the elements a, h, ... by addition and multiplication, then it follows from (29.2) that

(f(a, h, ...) )' = f(x', h', ...) ,

(29.3)

where on the right-hand side the compositions are understood to be in T. We generalize the rule later to the case where inverses also occur in f. If one assumes the homomorphic mapping in the form 5 - s$, then (29.2) becomes

s(xh) = sac sh, s(x + 3) = sa + sh

.

(29.4)

(We have already met this form of (29.2) in the previous paragraph.) We now continue the consideration of the above case (29.1). We denote the set of images a' (a E S) by S'(9 T) and call S' the homomorphic image of S (in T). If we write (29.2) in the form

x'h'=(xh)', a'+h'=(a+h)',

STRUCTURES

92

then we see that S' is a subset of T closed with respect to the composition under consideration. Accordingly S' is a substructure of T, which we shall examine more explicitly below. Note that S is now mapped homomorphically onto this S' (9 T) by virtue of (29.1). Thus the most general case originally considered may always be reduced to the special case T = S', where the structure S is homomorphically mapped onto a structure S'. Though in principle it will always suffice to consider this simplified case, it is often necessary to return to the above case, e.g., if S is homomorphically mapped into T, and if we consider at the same time several such mappings where we have different homomorphic images S' of S (in T). Of course the statements obtained for the above simplified case will also relate to the most general case.

By a homomorphism we understand a homomorphic mapping of a structure S onto a structure S'. Given the existence of such a homomorphism we call S, S' homomorphic structures, or, alternatively, we can say that S' is a homomorphic image of S. The relation between structures defined in this way is called homomorphy and is denoted by ''. Accordingly

S - S'

(29.5)

signifies that S' is a homomorphic image of S (or is homomorphic with S). Occasionally, we also denote a homomorphism of S with S' in the form (29.5). If we write (29.6) 5') , S ^' S' (E

then it means that - ' is a homomorphic mapping of S onto S', i.e. a homomorphism of S onto S'. The sides of (29.5) are not interchangeable, since a homomorphism, in general, is not a one-to-one mapping. Consider, e.g., two cyclic groups G = is}, Gl = { B} of sixth and third order. respectively. These are homomorphic and G -r Gl (0°, a3 - ,3°; a, a' -- ,3;

a2, a' -- #=).

Accordingly, in this example, there is a homomorphism in which two different elements

have a common image. Thus this homomorphism is not one-to-one. Isomorphisms are (according to § 28) one-to-one homomorphisms. This is the reason why some text-books introduce first the notion of a homomorphism, from which they obtain isomorphisms as special cases, a method which has many advantages.

By putting the notion of isomorphy at the very beginning we wish to emphasize its priority in principle, which consists of the fact that isomorphisms constitute the means by which a complete definition of the notion of structures (and of algebra) can be given.

Of course, homomorphisms constitute a powerful tool in research, as we shall see later.

We begin the examination of homomorphisms by observing that homomorphy is a transitive relation. In fact the product s's of a homomorphism

HOMOMORPHISMS

93

s of S onto S' and that of a homomorphism s' of S' onto S" is a homomorphism of S onto S". The proof follows as in § 28 from (29.10) to (29.12).

If a property of a structure is maintained under all homomorphisms, we call this property a homomorphic invariant (of this structure). THEOREM 54. The associativity, distributivity, invertibility, commutativily

of the compositions in a structure, the property «w is a neutral element" and the relation "e is an inverse of the element a" are all homomorphic invariants. The last two assertions are more explicit as follows: a homomorphism maps the unity element, the zero element and two elements, one of which is the (multiplicative or additive) inverse of the other, onto elements of exactly the same kind.

COROLLARY. The homomorphic images of semigroups, groups, modules

and rings are again semigroups, groups, modules and rings, respectively. The assertions of this theorem are deductions from the following propo-

sitions and those dual to them: for a homomorphic mapping - ' of a structure S = (a, j3, y.... > the following holds: (7-3)Y = a(fy) = (x fly) Y' = a'(j9'Y'),

x(f+y)=«fl+«y=>«'(j3'+y')=+a'f .

(29.7) (29.8)

(29.9)

'4) = fla - a',3' = j"a'

,

E'a'=a'; 0+a= a=*, 0'+a'=7',

(a-1)'=a, " (-a)'= -a'.

(29.10) (29.11) (29.12)

Consequently it suffices to prove (29.7) to (29.12). We shall only prove (29.8), (29.111) and (29.121) as the other assertions may be proved similarly. It follows from (29.2) and the left-hand side of (29.8) that

«V'+y')=a'(`N'+y)'=(a(f +"'))'=(cq3+ay)'= _ (cc/9)' + (Oct,)' = x'13' + M'Y',

by which we have proved (29.8). From (29.2) and the left-hand side of (29.11 I)

it follows that 8,ot = (Ea)' = 7.'

.

therefore (29. 111) is true. From this the image s' ofthe unit element e (E S) is the unity element of the homomorphic image of S and it follows from a-Ia=

= as -I = s, because of (29.21), that (a I)'z' = a'(a-1)' = s', implying

STRUCTURES

94

(29.121). This proves Theorem 54. From this the corollary follows immediately. From the last assertion of Theorem 54 it also follows that (29.3) holds for all expressions f(a, is, ...) in S. In Theorem 54 no mention was made of regularity. This is, in fact, not a homomorphic invariant as the following example shows. We have seen (§ 23, example 2) that the elements 0, 1, 2, 3 with respect to addition and multiplication mod 4 constitute a ring R with zero divisors. Moreover R is a homomorphic image of the zero-divisorfree (!) ring .7 because a homomorphic mapping of J onto R is obtained by assigning to each element 4n + i (n = 0, ±1,. ..) of 9, the element i of R (i = 0, 1, 2, 3). In the above corollary no mention was made of skew fields. Cf. in this respect, Theorem 57.

As with mappings in general, so in the case of a homomorphism (29.6), two subsets S, S' of 5, and S', respectively, are called corresponding, if S is the inverse image of S'. THEOREM 55. Let us consider a homomorphism S - S'

two corre-

sponding subsets S (S S), S' (c S') and two corresponding compositions in S, 5', both written multiplicatively. If S' is closed with respect to multiplication, so too is S. If the multiplication is invertible in both S' and S, then it is

in S, too. oc'fl'. Thus Let a, i be two arbitrary elements of S. By (29.21) if S' is closed with respect to multiplication, then it follows that (aj9)' E S' and a# E S, by which the first assertion of the theorem has been proved. In order to prove the second assertion we assume that S' (as well as S) is closed with respect to multiplication and that multiplication is invertible in S' and S. It suffices to show that ocE = fi has a solution $ in S. Because 9'. Since, of the assumption there is a solution in S. Then by Theorem 54, multiplication is invertible in S' the latter equation has no solution in S' except 1:'. Because of the invertibility of the multiplication in S', ' must fall within S' whence it follows that $ E S. This completes the proof of Theorem 55. Theorem 54, its corollary and Theorem 55 result in the following: THEOREM 56. Let a homomorphism of S onto S' be given, where S and S' are either semigroups, groups, modules or rings. If T' (c S') is a substructure of the same kind as S', its inverse image T (C S) is also a substructure of the same kind as S.

The substructures T, T' in this theorem are called corresponding substructures of S and S'. We have to notice that T' runs through all the substructures

of the same kind as S', while T need only run through some of the substructures of the same kind as S. Now we consider the three most important cases of Theorem 56, i.e., where S and S' are groups, modules and rings. In these cases S' has only

HOMOMORPHISMS

95

one substructure of the same kind consisting of one element. The corresponding substructure of S is called the kernel of the homomorphism. So in case of a homomorphism of a group G onto a group G', the kernel of this homomorphism is the subgroup of G which consists of the inverse images

of the unity element of G'. The same holds for modules and rings where of course the zero element replaces the unity element. From § 31, the kernel of a homomorphism will play an important part. THEOREM 57. Every homomorphism of a skew field is an isomorphism unless

the homomorphic image is the null ring.

We consider a homomorphism F - F (E - ') where F is a skew field, so that F is, from the corollary to Theorem 54, at least a ring. It is sufficient to examine the case in which this homomorphism is not an isomorphism. Then there are two distinct elements z, i3 of F with a' = i3'. On the other hand, 0' is, by Theorem 54, the zero element of F. Consequently

(a-f3)'=a'-(3'=0'. Now if y is an arbitrary element of F, then because x - fl element a in F such that

0 there is an

7, =(a- f3);. Hence

y'=(a - f3)'e' =0'E,'=0'. Consequently each element of F is mapped onto 0', i.e., F = 0'. This proves Theorem 57. A homomorphism of a structure S onto a structure which consists of one

element, is called a trivial homomorphism of S. It is evident that every structure has trivial homomorphisms. The isomorphisms of a structure are other examples of homomorphisms of a structure which always exist. We call a structure simple if all its homomorphisms are either trivial or

isomorphisms. This definition is one of the most important in algebra. From Theorem 57 all skew fields are simple. Furthermore all structures with at most two elements are simple, as are groups, modules and rings without proper substructures of the same kind. Later we shall see (cf. corollary of Theorem 69) that a group of prime

order has no proper subgroups whatever, so that it is simple, and that among the finite groups of compound even order there are infinitely many non-isomorphic simple groups. A homomorphic mapping -* ' of a structure S with generators

ell e2' ... is uniquely determined by the images a (i = 1 , 2, ...) of the generators. Now each element of S may be written as an expression = f(911 e21 ...) in the generators and the image of f takes the form

96

STRUCTURES

f(oi, o', ...). Therefore we shall often give a homomorphic mapping of a structure with the generators o,, o.,, . . . thus

of - et

(29.13)

by giving the images of the generators. If we write (29.14) S^'S'(21-91,02-'os,...), then it means that el, o2, ... are generating elements of S and (29.13)

yields a homomorphism of S onto S'.

For instance, let .4' denote the full permutation group of the elements a, b, c, further-

more denote by G the group of order two, which consists of the unit element e and a further element a. Then (a b) - a). . .. G ((a b c) - e, Find how the elements of .9' must be mapped and show that this is, in fact, a homomorphism. Show also that W7 '.. G ((a b c) -- (x,

(a b) - e)

leads to a contradiction.

Under certain circumstances a homomorphic mapping of a structure S into a structure T is called a representation of the structure S by the structure

T. If the mapping is an isomorphism, then we speak of a true or faithful representation. This terminology is most frequent if T is a well-known structure. The problem of finding all possible representations of a structure S by a structure T is called, in general, a representation problem. Further important problems are, to find all structures, non-isomorphic to each other, among all substructures and homomorphic images, respectively, of a structure S. We call these two problems the substructure, and homomorphy problem, respectively (of S). Later we shall repeatedly meet these problems.

Suppose that, starting from a structure S1, finite sequences S1, .

.

.,

S

can be constructed such that every Si (i = 2, ..., n) is a homomorphic image of Si_1 or a substructure of Si_, of the same kind. Every structure S,, arising in this way is called a derived structure (or derived) of S1.

THEOREM 58. All derived structures of semigroups, groups, modules and

rings arise as homomorphic images of substructures (of the same kind). B is meant here that B is a substructure of the same kind as By A A. This relation is transitive. Hence and from the transitivity of homomorphy it follows, by induction, that it suffices to prove that if S is one of the structures in the theorem and

S - S' ;? T' then there is a T such that

S? T-. P. Since this follows from Theorem 56, Theorem 58 is true.

HOMOMORPHISMS

97

As one of the most brilliant results of the whole algebra, FELT and THOMPSON (1963)

have recently proved the old suggestion of Burnside, that among the finite groups of an odd order only those of a prime order are simple. EXAMPLE 1. The group of complex numbers different from 0 is homomorphically

mapped by E -- 14: I, because Str/ 1 = I 1 711, onto the group of positive real numbers. (I I denotes the absolute value of The kernel of this homomorphism is the group of the complex numbers Sk with I I = 1. With respect to addition the mapping E I of the complex numbers is not a homomorphism, since we only I

1

I

andnot

have

EXAMPLE 2. The group of Example 1 is mapped homomorphically by 5 - sgn 5 = 1-1, because sgn r! = sgn E sgn q, onto the group of complex numbers E _ I

with I = 1. The kernel of this homomorphism is the group of the positive real numbers. I

EXAMPLE 3. The group of positive real numbers and the module of real numbers

are isomorphic structures. A suitable isomorphism is x -- log x because log xy = = log x -I- logy.

§ 30. Factor Structures In this section we shall use a method by the help of which we are able to construct new structures from a structure S, which are all homomorphic images of S. In § 31 it will be shown that in this way we obtain all the homomorphic images of S (to within isomorphism). Let a structure S = be given. We consider a classification L .

of S into non-empty classes; the class represented by a and the set of all classes are denoted by a and S, respectively. We consider an arbitrary composition in S which we write in the multiplicative notation. We try defining a multiplication of classes in the set S by

a fl= a .

(30.1)

(This means that, in order to multiply two classes, we take arbitrary representatives a, j9 one from each class, form their product afl and pass over to its class But since the classes a, fi cannot be represented by the elements a, 9 alone, the right-hand side of (30.1) is in general not always uniquely defined by the left-hand side, so that a multiplication is not defined by (30.1). If the right-hand side of (30.1) is uniquely defined for arbitrary a, fi by the left-hand side, then the multiplication defined by (30.1) in the set S is called class multiplication. Class addition and class compositions are to be similarly interpreted. (A composition and the corresponding -class composition are always denoted alike.) We have still to examine what is the necessary and sufficient condition that the right-hand side of (30.1) should be uniquely determined by the .left-hand side. This condition may be written as

a=y, =bra#=yb

((x,j3,y,b,S),

(30.2)

'Which we can show to be identical to

x= =>pK=OR,rcQ=1.n

(30.3)

STRUCTURES

98

For the two special cases of (30.2), a = y = o, ,B = K, b = i, and a = K_ y = A, 8 = b = o, together form (30.3). While if (30.3) holds, from the left-hand side of (30.2) we get of = ab = 5 f . Accordingly (30.3) is the necessary and sufficient condition for the multiplication (30.1) to be uniquely defined.

If (30.3) is satisfied then we say the classification e of S is compatible with

the multiplication in S. We have to interpret in the same manner a classification of S compatible with addition or any composition in S. The last result may now be expressed as follows : the class composition corresponding

to a composition in S exists if, and only if, the considered classification of S is compatible with that composition. By a compatible classification C of S we mean a classification of S which is compatible with all the compositions in S. These classes themselves are also called the classes mod e. If C is a compatible classification of the structure S, then (and only then) the set S of the classes (relative to the corresponding class compositions) is a structure which we denote by (S =) S/C and call the factor structure of S belonging to or with respect to e (or mod 0-). The homomorphism S ~ S/6?

($ - 5)

(30.4)

follows immediately from (30.1) applied to all class compositions. We call this the natural homomorphism of S onto S/F? which consequently consists of assigning to each element of S the class (as image in S/Cl) represented by this element. Part of (30.4) is stated as follows: THEOREM 59. Every factor structure is a homomorphic image of the given structure.

Because of its importance we summarize what has been said above. The compatible classifications e of a structure S yield the factor structures S/c). The elements of S/e are the classes themselves. Their composition, e.g., multiplication, is defined by (30.1). We see that the solution of the problem of determination of factor structures necessarily leads to the notion of homomorphism. The attributive "mod d" in the terms "classes mode", "factor structure of S mod Ca" will be explained in § 42.

According to the transfer rule of equivalence, the condition (30.3) for compatibility may also be expressed in the language of equivalence relations. (See § 46.)

From Theorem 59 and the corollary to Theorem 54, the factor structures of semigroups, groups, modules and rings are structures of the same kind. Therefore for these factor structures we use the terminology factor semigroup, factor group, factor module and factor ring, respectively. As regards the factor structures F/p of a skew field F note that they are, by virtue of Theorem 59, homomorphic images of F and consequently, by Theorem 57, either isomorphic with F or null rings. Since both cases are

FACTOR STRUCTURES

99

trivial, we never speak, in a sense similar to the above, of the "factor skew fields of a skew field". On the other hand, if a factor ring is a skew field, then we call it a factor skew field (in the commutative case a factor field) of the considered ring. Similarly a factor semigroup may be called a factor group of the given

semigroup, if it is a group. In the literature factor rings are often called residue class rings.

Two compatible classifications (21, e., of a structure S are called related. if the factor structures S/CI (i = 1, 2) are isomorphic. Because isomorphism is an equivalence relation, the relationship of compatible classifications is an equivalence relation too.

For an arbitrary structure S with at least two elements, the following two classifications @ may be constructed. For the first, C consists of only one class (equal to S), and for the second, each element of S constitutes a class by itself. These two classifications are obviously compatible. We call

them the trivial compatible classifications of S. For these the factor structure S/tv is in the first case the structure consisting of one element; in the second case it is isomorphic with S. EXERCISE 1. The semigroup defined by the Cayley table I«

i

y

«I y has exactly two non-trivial compatible classifications. How is multiplication defined in the factor structures? EXERCISE 2. The four-group has exactly three non-trivial compatible classifications.

All three are related to one another.

§ 31. The Homomorphy Theorem THEOREM 60 (general homomorphy theorem). Each homomatphic image

of a structure is isomorphic with a factor structure of the given structure. More precisely: If S _, S,

(;

;')

(31.1)

and C(x') denotes the set oj' inverse images of x' ('-Z S'). then

'C(x) , C(li') , .. .

(31.2)

constitute a compatible classification F> of S with

S' ,: S/Q (x' - Q%'))

(31.3)

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100

(Since C(, according to the terminology introduced in § 6, is the concomitant classification of S to the homomorphism (31.1), so by (31.3) the image of a structure by a homomorphism is isomorphic with the factor structure, obtained from the concomitant classification of the structure.) In order to prove the theorem we have to show first of all that C is compatible. It is sufficient to prove the compatibility for multiplication. We introduce the notation x = C(a') . Because of the duality principle it is sufficient to show that

(a,fl,oES). This assertion is identical to (2m), = (AL

'-

Since from (31.1) the right-hand side of this may be written as o'a' = o'#'. we have proved compatibility. As regards (31.3), it is evident that the mapping oc' -> C(a') is one-to-one and maps S' onto S/?. Consequently it is sufficient to show that it is homomorphic with respect to multiplication. This means that C(a) C0') = C(a'' f ') .

This equation is true because oc'f3' _ (oc3)' and because of the definition of class multiplication. Consequently the proof is complete. Nom. By Theorems 59, 60, the problem of determination of the homo-

morphic images of a structure has been reduced to the determination of its compatible classifications. We shall deal in detail with this problem for groups, modules and rings in § 42. A skew field has (according to Theorem 57) no compatible classifications other than trivial ones. Of course the homomorphy problem of a structure formulated in § 29 is equivalent to determining the mutually inequivalent compatible classifications of this structure.

§ 32. Automorphisms. Endomorphisms. Autohomomorphisms. Meromorphisms

When considering isomorphic mappings of a structure S onto or into a structure T, of course we do not exclude the case S = T, which is in many ways an interesting case. An isomorphic or homomorphic mapping of a structure onto itself is called an automorphism or autohomomorphism, respectively, of this structure. An isomorphic or homomorphic mapping, of a structure into itself is called a meromorphism or endomorphism, respectively, of this structure.

AUTOMORPHISMS

lot

The table below is a useful mnemonic iso-

homo-

onto autoautohomo-

into

mero- j morphism. endo- j

As the automorphisms of a structure S signify its isomorphisms onto itself, we can denote them in the form

S;z S(;-* s').

(32.1)

Every structure has at least one automorphism, namely the identity mapping of S which we may therefore call the identity automorphism of S.

If s, t are automorphisms of S then obviously so are st and s-'. Consequently the automorphisms of S constitute a group, which we call the full automorphism group of S and denote byve(S). Every subgroup of ' (S) is called an automorphism group of S. The automorphisms of a structure indicate the "symmetry properties" of this structure. These are analogous to the "symmetries" of a geometric figure which are mappings of a geometric space such that they map the considered figure onto itself. These generally constitute a group.

Some automorphisms for a group G can easily be given. If n is a fixed element of G, then, as will be shown, the mapping

a -> gao-'

(,c E G)

(32.2)

is an automorphism of G which we call an inner automorphism (induced by 0) of G. Otherwise our assertion may be expressed in the concise form

G z G(a-oao-'). First of all, G is mapped onto itself by (32.2), since every equation eo e ' = i4 0 E G) has the solution a = o-' i3O. As there can be no further solution, this mapping is one-to-one. Because

oaf o - ' = eoc ' efle-'

(32.3)

the mapping is a homomorphism. This proves the above statements.

We denote the inner automorphism, induced by o, by [o]. From emc(ee)' = e(a(xa-1) o-' it follows that [go,] _ [o] [a] .

(32.4)

Hence the product of two inner automorphisms is again an inner automorphism. We show that the inner automorphisms of G constitute a subgroup of vg(G) which we call the inner automorphism group of G and denote by ,6 1) (G).

102

STRUCTURES

In any case, according to the previous statement f<'>(G) constitutes a semigroup. Since, from (32.4)

G ",t°(G) (2 - [o])

(32.5)

is a homomorphism it follows, from the corollary to Theorem 54, that ,,,e(i) (G) is a group, and that it is necessarily a subgroup of . ((G). In addition to this, (32.5) shows that vf('I(G) is a homomorphic image of G. We now ask what is the kernel of this homomorphism. This kernel consists of the elements o of G for which the automorphism [o] is the identity automorphism. According to (32.2) this is so if. and only if, P is a central element of G. Consequently we have the following THEOREM 61. The inner automorphism group

()(G) of a group G is a

homomorphic image of G, and the homomorphism of G onto i 0(G) arises by assigning to each element of G the inner automorphism induced by this element. The kernel of this homomorphism is the centre Z(G) of G. If F is a skew field then for o E F* (32.2) defines a mapping of F onto itself. Since, as in (32.3),

+ efle"

o(a +

it follows that the mappings F ^, F (a -> oao-`)

(for all o E F, # 0)

are all automorphisms, which we call the inner automorphisms of the skew field F. These constitute a subgroup ofv((F) which we call the inner automorphism group of the skew field F and denote by vet'°(F). By Theorem 61, this is a homomorphic image of F*. THEOREM 62 (theorem of HUA). If a proper sub-skew field L of a skew field

F is mapped by every inner automorphism of F into itself then L lies in the centre of F. We prove first that an= %a

(aEF-L;AEL).

(32.6)

Since this is true for A = 0, we can restrict ourselves to the case A Because a 0 0 and a + 7. 0 0, the elements

xAa-1=o,

0.

(a+AA)A(a+

both lie in L. From these equations we obtain

aA=ox,

aA+A'=as+a.'..

thus (o - a)a = (a - ),)A. Hence, because a

L and A 0 0, we must have

n- a-- 0, and a-7.=0. Hence o= 1. and (32.6) follows.

ENDO-, AUTOHOMO-, MEROMORPHISMS

103

We now show that

µ; = )p

Q,,u E L) .

(32.7)

If a is as in (32.6), then a + p E F - L. The application of (32.6) gives

(a+µ))=.a@+U). Hence and from (32.6) follows (32.7). Both together imply Theorem 62. As regards the other mappings defined in this section, we only remark for the present, that both the endomorphisms and meromorphisms, as well as the autohomomorphisms, of a structure S constitute a semigroup of which

the first contains the two latter and the automorphism group of S as substructures. EXAMPLE 1. For every Abelian group G and every integer n, the mapping il" is, because (Ej)" = if, an endomorphism of G. For modules this endo-

morphism takes the form - n . EXAMPLE 2. For an infinite cyclic group {a) and an integer n different from 0

(a) - (a")

(a - a")

is a meromorphism, since {a"} is infinite and consequently isomorphic with (a}. This meromorphism is an automorphism if, and only if, n is equal to I or -1. EXAMPLE 3. Since the Prefer group (pm) is isomorphic with the group of those complex roots of unity, which are of (multiplicative) order 1, p, p"...., so (p..) ,.. (p°°)(s - P) is an autohomomorphism (but not an automorphism), the kernel of which consists of the p' complex roots of unity. EXERCISE 1. For the full permutation group R-` of the set we have .9s x d(.9') = c4°' (.93). EXERCISE 2. Theorem 62 is even valid if L is a proper subset of F.

§ 33. Isomorphic Structures with the Same Elements

We often have to deal with structures S, S' which consist of the same elements and consequently differ from each other only in the nature of their compositions. In such cases the compositions in S and S' must be indicated in a different way. It is simple but important to decide whether S, S' are isomorphic. We solve this problem for the case when two compositions are defined in S and S', which is quite sufficient.

THEOREM 63. Given a structure S = with two compositions, addition and multiplication, all the isomorphic structures with the same elements are obtained by taking a permutation (4 -)1 7Ia

(33.1)

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104

of the set and defining in S' the "new" addition and multiplication by

_

a -1-

so that

i(n-1

a+

n(n-la n-1(3) S'(a > n a).

S

(33.2) (33.3)

First, we prove that the isomorphism (33.3) follows from (33.2). Since the permutation (33.1) is a one-to-one mapping of S onto S', we have to show that n(a + j9) = na + nj9, n ((Xfl) = a a x n(3 . (33.4)

If we apply (33.2) with na, n9 instead of a, fi and interchange the sides, then we get (33.4), proving the assertion. Conversely, let us assume that a structure S' consists of the same elements as S and S -_ S'. An isomorphism of S onto S' is then a permutation, which

we write in the form (33.1). If the compositions in S' are denoted by -i-, and x , respectively, then (33.4) holds. Hence (33.2) follows by applying (33.4) to a -a, n-1# instead of a, P, so proving the theorem. 0), s are rational numbers, then x - rx + s yields a perX-S mutation of the set of rational numbers. The inverse permutation is x -. EXAMPLE 1. If r (

r Consequently, if in the above set "sum" and "product" of a and b are defined by

l and

(r

r(a - s+ 6 I

r

rr

a-s b-s +s r

+s=J a+b - s,

s(a+b)+s(r+s)

r

r

respectively, then a field isomorphic to Jr-, is formed. The zero element and the unity element are s and r + s, respectively. EXAMPLE 2. In a group the following are permutations

awa ', a - ea,

a- eaP ',

where e denotes an arbitrary fixed element. If Theorem 63 is applied to these three permutations, then in the first two cases we find that the resultant group is isomorphic with the given group, where the product of the elements a, 14 (instead of a#) is defined as Sa and o - 1 14, respectively. In the third case the multiplication remains unchanged. (For the reason, cf. the following exercise.) EXERCISE. For which permutations (33.1) are the structures S, S' equal in Theorem 63?

§ 34. Skew Products Let a finite or infinite sequence Sl, S21

... of (not necessarily distinct)

structures be given and let T denote the product set SI x S2 x ..., i.e., the set of sequences (xl, x_, ...)

(a; E S1)

(34.1)

SKEW PRODUCTS

105

or, more generally, a subset (s 0) of this set. We can make the set T a structure by defining one or more compositions in it. We call every such structure T a skew product of S1, S21 ... and denote it here briefly by

.... For S1... S" with S1 = ... = S" = S we write S". This gives us a very general method for "construction of new struc-

S1 S2

tures from given structures", which is widely applied in algebra. We can it Hamilton's principle, as HAMILTON was the first to apply it to obtain a

logically incontestable definition of complex numbers (cf. Exercise 1). In the term "skew product" the adjective "skew" refers to the fact that the skew product of commutative structures is frequently non-commutative (cf. Exercise 1 and Examples 2, 3). Some important skew products dealt with later will be termed "direct product", "direct sum", "Schreier's product", "cross product", etc. Special notations will be introduced for them.

In a skew product multiplication takes the form (a1, a2, ...) 01, N2, ...) = (fi,f2, ...)

V. E S) ,

where fl, f2i ... are functions of the al, a2, ..., A.... As usual functions are formed by applying compositions in S1, S2, ... .

these

Henceforth in a skew product T = Sl S2 ... of finitely or infinitely many

structures - if nothing else is stated - we tacitly assume that T consists of all the sequences (34.1). The other case where T is a proper subset of the set of the sequences (34.1), will rarely occur. EXAMPLE 1. A skew product R' may be defined for every ring R by ((X, fg) + (y, 6) = (cc + y, fl + 6),

(MA (y, 6) = (sy + 06, Ay + 0).

It is easy to prove for R2 that the ring axioms (the associativity of both compositions, the invertibility and commutativity of addition, and distributivity) are satisfied, i.e. R2 is a ring. Because (y, 6) be, f3) = (;'z + yf3,

dx + V)

R2 need not be commutative even if R itself is commutative. R2 is commutative if, and only if, R is a zero ring and then R2 is also a zero ring. The elements of the form (y, -y) for y 0 0, are always right zero divisors in R2, even if R is zero-divisor-free. EXAMPLE 2. The simplest, non-trivial method of constructing a skew product RS from two rings R, S is to define the compositions

(cc,P)+(y,6) =(a+:', f3+ 6), (a,8)(y,6) =(ay, 6) We recognize at once that RS is a ring with the zero element (0,0). This is called the direct sum of the rings R, S and we denote it by R n S. In contrast with Example 1, R e S is commutative if R and S are commutative. We can similarly define the direct sum of the modules M, N and the direct product of the groups G, H, by retaining from

(34.2) only (34.21), and (34.2.), respectively. These are denoted by M n N, and G (D H, respectively. Of course these are again modules and groups, respectively. If neither of the rings R, S is a null ring, then because (x, 0) (0, B) = (0, 0),

STRUCTURES

106

R e S always has zero divisors. We shall deal in detail with direct sums and products in § 44.

EXAMPLE 3. A skew product G2 can be defined for every group G by (a, 1g) (Y, b) = («Y, Y ' 3Yb)

We show that G2 is a group. For the elements A = (x, /1), B

C = (a, v) of G.

(AB)C = (cy, Y '/3Yb) (p, v) = (avtu,,u-'Y l/3ybuv), A(BC) = (a, /3) (YR, u 'bµ v) = (ar/i, (Yfr)-'f3(YR) f 16P v) = (aYP, k 'Y-'f3Yblwv),

wherefore multiplication in G2 is associative. (e, e) is the unity element and finally the inverse of (a, /3) is (a-', a#-la 1) because (O t-

af3-'a ') (a, 3) = (e, E).

We return to this group G2 in § 51, Exercise 1. EXERCISE 1. For a ring R the skew product R2 defined by (a, f3) + (y, b) = (« +

f3 + 6), (a, f3) (Y, b) = (ay - fad, ab + fly)

is also a ring. If R is the field of real numbers, then R2 is isomorphic with the field of complex numbers. If R itself is the field of complex numbers, then R2 is not a field, and is not even zero-divisor-free. EXERCISE 2. For a ring R we define a skew product R4 where we define the sum and product of two elements

A = (a, b, c, d), by

B = (p, q, r, s)

A+B=(a+p,b+q,c+r,d+s), AB = (ap + br, aq + bs, cp + dr, cq + ds).

Then R4 is isomorphic to the full matrix ring R2 of rank 4 over R. EXERCISE 3. Change the definition in the above exercise so that multiplication in R4 is defined by:

AB = (ap - bq - cr - ds, aq + by + cs - dr,

ar-bs+cp+dq,as+br-cq+dp). Then R4 is again a ring. If R is the field of real numbers, then R4 is isomorphic with the quaternion field (over the field of real numbers).

§ 35. Structure Extensions

Frequently for a given structure S an overstructure T of S is constructed in some way. In such cases we speak, in general, of a structure extension or extension of S, or we say that S is extended to T. The study of a given structure is very often facilitated by forming an extension of it.

STRUCTURE EXTENSIONS

107

In this respect the most important cases are those in which we pass from a given structure by an extension to another which satisfies a larger system of structure axioms than the given structure, e.g., the extension of a ring to a ring with unity element, to which we shall return below.

If T is an extension structure of S and

T={S,T}

(35.1)

for a subset T of T i.e., if S U T is a generating system for T, then we say that T arises from S by the adjunction of T (or of the elements of T). The elements of T itself are then called the adjoined elements. Note that later for the right-hand side of (35.1) in some cases we shall write, in order to bring into prominence the adjunction, S(T), or S[T]. It is always trivially possible

to give extension structures of S by adjunction if an overstructure Sl of S is already known, since then the symbol {S, T} has a meaning for all T 9 S1. There is an extremely important possibility, essentially different from the above, for the formation of an extension structure of the structure S, if we have a structure T', which contains a substructure S' isomorphic with S:

T' ; S'S.

(35.2)

Additionally we assume that

(T'-s')ns=o,

(35.3)

a condition which may easily be satisfied by replacing in T' the possible common elements of T' - S' and S by any elements lying outside S and S'.

(The case where T' n s = o occurs most frequently in applications.) It is easy to construct, with the aid of conditions (35.2) and (35.3), an extension structure T of S, which is isomorphic with T':

T'zT =-;? S.

(35.4)

To do this we define an isomorphism

S' sr S ( '

(35.5)

We consider the set T which is equipotent with T', [because of (35.2) and (35.3)], defined by

T=(T'-S')US,

(35.6)

which is so obtained from T' that its subset S' is replaced by the equipotent set S. We make this set T a structure as follows: denote by a', j9', . . . the elements of S', by a, fl, . . . the corresponding elements of S [cf. (35.5)] and

by a, b, ... the elements of T' - S' (= T - S). [See Fig. 1.] We now consider all equations of the form

AB=C, D+F=G....

(A,B,... ET'),

108

STRUCTURES

by which the compositions in T' are defined, where A, B, ... are just elements a , fi', ..., a, b.... In all these equations we substitute for the a', P', ... the corresponding a, j9, ... (this means that we omit all signs ""') and we take the equations, so formed, as defining the compositions in T. It is evident that T will be a structure isomorphic to T', which also contains [by (35.5)] S as a substructure. Thus the above problem is solved. We say that this structure T is obtained from the structure T' by embedding. More precisely: T is obtained from T' by "embedding" the structure S in T

in place of the substructure S', by way of the isomorphism (35.5).

Fir. I.

If we define the one-to-one mapping A of T' onto T where A is an ex-

tension of the isomorphism (35.5) and the a, b.... (E T' - S' = T - S) are fixed elements for A, then we have the isomorphism

T' ;., T (X -> A X).

(35.7)

In the applications it is customary to use the same notation for the structure, before and after embedding. The definition of the compositions in T could have been obtained by writing down

the Cayley tables for all compositions inT'and replacing all the a', jl', ... by the corresponding a, P....

As a simple but very important application we prove THEOREM 64. Every ring can be extended to a ring with unity element. Let R be a given ring. From this and the ring 7 of integers we construct a skew product Rl = LYR with the compositions

(a, a) + (b,

(a + b, a + fl),

(a, a) (b,

(ab, b(z + afl + afi) . (35.8)

[The terms ba, a# on the right-hand side of (35.82) are meaningful, since 7 is an operator domain for all rings.] First we show that Rl is a ring.

STRUCTURE EXTENSIONS

109

From (35.81) it is obvious that R1 is a module. From (35.82) it follows that

((a, a) (b, j9)) (c, y) _ (abc, bca + acp + cap + aby + bay + a/iy + aj9y) _

(a, a) (bc, cj + by + fly) = (a, a) ((b, 9) (c, y)). Thus R; is a semigroup. We obtain e.g., left-hand distributivity with the aid of (35.8) from

(a, a) ((b, f) + (c, y)) = (a, a) (b + c, fl + y) _

=(ab+ac, ba+ca+a#+ay +afl+ay)= = (a, a) (b, fi) + (a, a) (c, y) . Thus R1 is in fact a ring. As special cases of (35.82) we have

(1, 0) (b, fl) = (b, fl), (a, a) (1, 0) _ (a, a), from which we see that R1 has the unity element (1, 0). Further by (35.8)

(0, a) + (0, f) = (0, a + I4),

(0, a) (0, f) = (0, afl) .

(35.9)

Hence the elements of the form (0, ) of R1 constitute a subring R' of R1. The isomorphism R' R ((0, ) -* E) (35.10) follows from (35.9). Accordingly R may be embedded in R1 in lieu of R'. Let the unity element (1, 0) of the ring R1, constructed by the help of (35.8), be denoted by s. Then an arbitrary element of R1 may be written as

(a, a) = (a, 0) + (0, a) = a(], 0) + (0, a) = as + (0, a) . After embedding, this result appears in the form as + a. Writing the elements of R1 in this form, (35.8) takes the simpler form

(as+a) +(be+ fl)=as+be +a+ fl (as+(x)(be+l4)=abs+ba+afi+aj9. Also, R1 (after embedding) can be written in the form {R, s}, i.e. it is obtained from R by the adjunction of the unity element e of R1. We must necessarily add "of R1" as there is the possibility that R itself already has a unity element which is then, of course, different from s. In this case R1 has the zero divisor s - so where so denotes the unity element of R. (Show this.) Further o+(e) = 0, thus R1 is always an infinite ring even if R is finite. (Later we shall see that a finite or zero-divisor-free ring may be

STRUCTURES

110

extended to a ring of the same kind with a unity element; cf. § 85.) Part of the above is implicit in SUPPLEMENT (Doxxox's theorem). Every ring R has an extension ring RI with unity element s, the elements of which may be uniquely written as

[whence o+(e) = 0]. This ring Ri (uniquely defined to within isomorphism) is called the Dorroh extension ring of R (cf. DoRROH (1932)). EXAMPLE 1. Every semigroup S by the addition of only one element e may be extended to a semigroup S1, where multiplication by e is defined by e2 = e,

ea = OW =a

((x E S).

This e is then the unity element of S, (= S U e). We briefly say that this semigroup S, is obtained from the semigroup S by the adjunction of a unity element. EXAMPLE 2. Every semigroup S by the addition of only one element 0 may be extended to a semigroup So, where multiplication by 0 is defined by

0z=0,

OM =a0=0

(aES).

This 0 is then the zero element of So (= S U 0). We briefly say that this semigroup So is obtained from the semigroup S by the adjunction of a zero element. EXERCISE. Embed R in the rings constructed in Exercises 1, 2 and 3 of § 34.

§ 36. Representation of Groups by Permutation Groups

As an important application of the concept of isomorphism we show that every group can be faithfully represented by the help of a permutation group. Namely we shall find that any group is isomorphic with a subgroup of the full permutation group of the set of elements of the given group. Moreover we shall prove that the full permutation group .9(S) of a set S essentially depends only on the cardinal number of the set S (and not on the nature of the elements of S). It must be understood that the full permutation groups of equivalent sets are isomorphic. First, we prove the last assertion. Let S, S' be two equipotent sets, which we may write in the form

so that

S =
-+ '

S)

is a one-to-one mapping of S onto S'. To the arbitrary element

pdoc

I

(36.1)

REPRESENTATION OF GROUPS BY PERMUTATION GROUPS

III

of 9 (S) we assign the element p'-Ix

µ' v' #' ...

I

of .9 (S'). We show that we have the following isomorphism

9 (S)

9= (S') (p - p').

Since the mapping is evidently one-to-one we have only to show that for p, p' and a further similar pair q. q' of permutations the equation holds. Since in (36.1) It, v,

(qp)' =tlp' .

. .

are all the different elements of S, we may assume

q to be in the form

v...

q= Oa...l whence q,

= ['U' I"

..

Since

9'P the assertion is proved. Henceforth for given n we denote the full permutation group of a finite set <x1, . . . , x,> by 93 whenever the nature of the elements xI, . . ., x,, is a matter of no consequence. We take the elements, unless otherwise indicated, to be the numbers 1, ... , n. We call S0 the full permutation group of degree n and its subgroups simply the permutation groups of degree n.

It can be proved by induction that the set <1, ... , n> may be ordered in exactly n! ways. Hence it follows that n!

The full permutation group of an infinite set is not countably infinite as can be proved by a method similar to that of Theorem 11. Quite generally in an operator structure S 10 , where O =
f rfJ...)-Irk) e

Ira

of S into itself. This general case will be dealt with in detail in Chapter III. Here and in § 37 the special case where multiplication is defined in S will be discussed, i.e.

5 R-A

STRUCTURES

112

where S may be considered as a left-hand operator domain for itself. This "principle"

suggests the examination of the mappings assigned to the element P of S, i.e., (

)

(s E S) of S into itself.

THEOREM 65. Every group G is isomorphic with a subgroup of the full permutation group ,9'(G) of the set G ; moreover

EG)

(36.2)

is an isomorphic mapping of G into 9 (G). From Theorem 31 we know that the right-hand side of (36.2) is in fact an element of 93 (G); therefore (36.2) is a mapping of G into 9 (G). This mapping is one-to-one, since from al:

we have e = a$, and e = a follows. So for arbitrary Q, a (E G) (

S

_

eat

((o. l (b ea$ I a$ l

I

ll

e

S

ob

Consequently Theorem 65 is proved. The faithful representation of G in 99(G) furnished by this theorem is called the regular representation of the group G. In particular it follows from Theorem 65 that every group of order n is isomorphic with one subgroup of the full permutation group 9; . Though according to Theorem 65 all groups are, to within isomorphism, permutation groups, it must not be supposed that the study of permutation groups alone is sufficient for a complete study of group theory. We shall deal later with various other means for the examination of groups, most of which are much more efficacious than their representation by permutations.

It is left to the reader to set up the analogue of Theorem 65 for semigroups. A faithful representation may be found not only for regular semigroups and for those with a unity element, but also for many other semigroups. Of course for modules the additive analogue is valid, though it is seldom applied. (For that purpose it is more usual to study Abelian groups rather than modules.)

As a simple application we consider the cyclic group G = {a} of order n. To the element a belongs the (cyclic) permutation

a' OEM

Consequently {a}

p} (a -p)

113

REPRESENTATION OF' GROUPS BY PERMUTATION GROUPS

is the regular representation of {a}. In the infinite case the same holds with the difference that the infinite cycle p = (... a-1 s a (X2 . . .) occurs. (The fact that in the regular representation of the group {a) a cyclic permutation corresponds to the generating element (z, furnishes an additional explanation for calling groups generated by one element cyclic groups.) EXAMPLE 1. Theorem 65 explains why the four-group (cf. § 28, Exercise 1) is isomorphic with the subgroup of '904 consisting of the elements 1, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3).

EXAMPLE 2. If we apply Theorem 65 to .9D$ we see that this group is isomorphic,

with the subgroup of .93 generated by the elements (1 2 3) (4 5 6), (1 6)(2 5)(3 4).

§ 37. Endomorphism Rings We have seen in Theorem 65 that groups maybe represented by mappings

(i.e., by permutations). The problem of obtaining an analogous result for rings confronts us for the moment with the problem that for the mappings of a set S into itself no addition but only multiplication has been defined. In general, defining the required addition is scarcely possible in a "reasonable" way, but it is possible if there is already a composition in

S itself. We carry it out here for the endomorphisms of a module M. After this important preparation we shall deal in the next paragraph with the representation of rings. We define in the following way the full endomorphism ring of a module M =
are all the endomorphisms of M, which we denote by r, s, ...

.

By

denoting by ra the image of a, under the endomorphism r, we have that r(a + fi) = roc + rig.

(37.1)

We define the addition in F(M) by

(r + s) a = ra + soc ;

(37.2)

further, multiplication is to be defined (as elsewhere for mappings) by rsa = r(sa) .

(37.3)

This completes the definition of '(M), but we have to prove that F(M) is in

fact a ring. Since, according to (37.2) and (37.1),

(r+s)(a+ fl)=r(a+i) +s(a+ (9) =ra+rfi+sa+s(3=

=(ra+sa)+(r,9+si)=(r+s)a+(r+s)#

STRUCTURES

114

r + s is an endomorphism of M. Accordingly, we have actually defined an addition in 8'(M) by (37.2). The corresponding fact concerning multiplication, defined by (37.3), is already known from previous discussions. We easily see from (37.3) that because of the commutativity of addition

in M, the similar fact also holds in FP(M). In exactly the same way the associativity of addition can be shown. The associativity of multiplication in F(M) is already known from previous discussions. In order to prove the invertibility of the addition we have to show that every equation

x+r=s

(37.4)

has a solution x in Y'(M). We show that the mapping x of M into itself defined by

xa=sot - rot

(37.5)

is a solution of (37.4). From (37.5) and (37.1) it follows that

x(a+ fl)=s(a+ ff -r(a+P =sot+sp-roc -rj9= =(sot-rot)+(sfi-rp)=xa+xj9, wherefore x belongs to F(M). From (37.2) and (37.5) it follows that

(x+r)a=xa+roc =sot. This means that (37.4) holds and, consequently, proves the invertibility of addition in 87(M). Finally we obtain for r, s, t (E F(M)) from (37.1), (37.2), (37.3) r(s + t)a = r((s + t)a) = r(sa + tot) = r(soc) + r(toc)

_

= rsa + rta = (rs + rt)o: , (s + t)ra = (s + t) (rot) = S(roc) + t(r(Z) = sra + trot =

_ (Sr + tr)a , by which distributivity holds in F(M). Thus we have proved that F(M) is a ring. Notice that 8'(M) always has a unity element, namely the identical mapping of M, which is, therefore, also called the identical endomorphism of M. The zero element of F(M) is called the zero endomorphism of M. This maps every element of M onto 0.

Of course the above also relates to an Abelian group G =
ENDOMORPHISM RINGS

115

(37.3) run as follows :

r(aq) = ra r{3

,

(r + s)a = ra sa ,

rsec = r(sa) .

(37.6)

We retain the notation F(G) for this case, too, and also the terminology full endomorphism ring and endomorphism rings of G. On the other hand, one cannot speak of endomorphism rings of non-commutative groups as the above proof is essentially based upon the commutativity of M. If we apply the above definition to a semimodule or a commutative semigroup, then the endomorphisms constitute a structure which is in general not a ring but a semiring.

If one takes a ring instead of a module then the equations r(a + /1) = ra + rfl,

r(af) = ra rfJ

replace (37.1) as the definition of an endomorphism. But these together with (37.2) and (37.3) do not define, in general, a ring in the set of endomorphisms. On the other hand, for a ring R, the full endomorphism ring i;(R+) of the module R+ of the ring plays an important part, as will be seen in the next section. EXAMPLE 1. For any module M, every mapping a - na (n E 3) is an endomor-

phism which we will denote here by [n]. Then [n]a = na. If M has the positive characteristic c, then, because ca = 0, [c] is the zero endomorphism of M; further, [m], [n] are obviously equal if, and only if, c I m - n. If, on the other hand, M has the characteristic 0 or co, then all [n] are different. In each case the distinct [n] constitute a subring of ie(M), i.e., an endomorphism ring of M which we denote here by [9]. (More precisely one should write [3]M .) It is easy to see that .? - [.7]. If, and only if, M has characteristic 0 or co, then [9] ;t;:1 . EXAMPLE 2. We show that the full endomorphism ring if(M) of an infinite cyclic

module M = {a} is isomorphic with 3. For, if - ' is an arbitrary endomorphism of M, then a' = na for some integer n, whence it follows in general that (i a)' = = ia' = ina = n(ia) (i E 9). According to this, the given endomorphism is just -- n%. Because o+ (a) = 0, this is the zero endomorphism if, and only if, n = 0. The assertion follows from Example 1. (The result may be expressed in the concise form ; (9+) , J.)

§ M. Representation of Rings by Endomorphism Rings

The analogue of Theorem 65 runs : THEOREM 66. For a ring R

e

l

jQ0,

E R)

(38.1)

is a homomorphism of R onto a subring of the full endomorphism ring F(R+)

of the module R+ of R. This homomorphism is an isomorphism if, and

STRUCTURES

116

only if, 0 is the only left annihilator in R. Thus for rings with a unity element isomorphism always holds. Because

e( +n)=eE+ en

(e,$,-q E R)

the right-hand side of (38.1) is in fact an element of.F(R+). For two arbitrary e, a (E R) (from the definition of compositions in W(R+))

I(e+a)E -

e

E

+a )= feEl+

so that the mapping (38.1) is homomorphic. Hence the first assertion of the theorem has been proved. In order to prove the second assertion we have to examine the conditions under which the endomorphisms, assigned by (38.1) to the elements e, a(E R), are equal, i.e.

This equation is equivalent to e = off, i.e., to (e - v)E = 0 (E E R). Thus the required condition is that a and a differ from each other by a left annihilator of R. This completes the proof of Theorem 66. This theorem is of great importance since it follows that every ring with a unity element may be faithfully represented by an endomorphism ring of its module. This result, modified to some slight extent, may be used for any ring R. For that purpose we take, according to Theorem 64, an extension

ring Rl of R with unity element. On the one hand, Rl may be faithfully represented by a subring S of ro (Rl ). Since, on the other hand, R is a subring of R1, it follows that R may be likewise faithfully represented by a subring of S and so also of 8'(R1 ). This proves the following COROLLARY. Every ring is isomorphic with an endomorphism ring. EXAMPLE. Let R = (a, i9, ...> be a zero ring. The endomorphisms on the righthand side of (38.1) are now all equal to the zero endomorphism of R+. Take the extension ring R, = { R, e} with unity element e, the distinct elements of which are all the is + (i E 7, y E R). The mapping

of R into ie(R;) is then a faithful representation of R.

ANTI-ISOMORPHISMS. ANTI-AuTOMORPHISMS

117

§ 39. Anti-isomorphisms. Anti-automorphisms Let S = be a structure in which the product ac9, and possibly also the sum a + 9, of the elements a, j9 is defined. S° shall denote the structure which differs from S only in that in it the product is defined by (39.1) aOP _ (3a . Then we call S° the opposed structure to S. If the multiplication is commutative in S, then of course S and S° are equal. If, in particular, S is an A-structure with a self-dual axiom system A, then S° is obviously an A-structure. (Thus, e.g., the structure opposed to a ring is a ring, the structure opposed to a semigroup is again a semigroup, etc.) If, furthermore, S° is opposed to S, then S is also opposed to S°. This is

why it is meaningful to speak of opposed semigroups, groups, rings and skew fields.

Opposed structures suggest the following definitions. We understand by an anti-isomorphism of a structure S onto a structure S' a one-to-one mapping - E' of S onto S' with the property (39.2) (a + is)' = x' + j9', (x#)' = f3'cc' (a, 3 E S). If there exists such a mapping, then we say that S' is anti-isomorphic with S. (Here, as is usual in connection with isomorphisms, we have for convenience

presupposed the same notations for the corresponding compositions in S and S'. If only multiplication is defined in S, S', then (39.2L) is omitted.) We have to interpret more generally an anti-isomorphic mapping of a structure S into a structure T. An anti-isomorphism of a structure S onto itself is called an anti-automorphism of S. In a corresponding way we may speak of anti-homomorphisms and anti-homomorphic mappings. They are seldom used, since they may always be reduced to a homomorphic mapping followed by an anti-automorphism.

Strictly speaking any anti-isomorphism may be similarly composed of an isomorphism and an anti-automorphism; for that reason the concept of anti-isomorphism could be spared, but this would often cause inconvenience.

Opposed structures are always anti-isomorphic, e.g. if S° is opposed to S, then the identity mapping of S is an anti-isomorphism of S onto S° as can be seen from (39.1). Can opposed structures also be isomorphic? We may pose this question

in the following way: Can anti-isomorphic structures be isomorphic? THEOREM 67. Anti-isomorphic groups are isomorphic.

It suffices to show that two opposed groups G, G° are always isomorphic. This is in fact the case, since the isomorphism G

G° ( -* E_I)

holds because of (aj9)-I = 8-I a-I and (39.1).

STRUCTURES

118

We likewise see that for every group G the mapping automorphism of G.

-- ;-i ( E G) is an anti-

For semigroups and rings (skew fields) Theorem 67 is not valid (cf. the example below). Nom. Theorems 66, 67 deal with the representation of groups and rings by mappings of the form (39.3)

Q --+

If, instead of this, we put

then the above theorems remain valid, with anti-isomorphism, antihomomorphism and right annihilators instead of isomorphism, homomorphism and left annihilators respectively. On the other hand, however, Theorem 65 is valid for

in place of (39.3). EXAMPLE. Let R = (# 0) denote a zero-divisorless commutative ring and S the set of the elements of the form (b 0) from the full matrix ring R2. Because

(a 0)

b0

- (c 0j = is - c 0j d0

'b-d0

is 0) (c O I

b0 ldO)

_

ac 0 bc0

it follows from Theorem 44 that S is a subring of R2. We see that S and the ring S° opposed to it are not isomorphic. The last equation shows that S contains the right annihilators (d 0) , but, on the other hand, contains only one left annihilator

00 0

O) = 0. Since by passing over to S° the left and right annihilators are interchanged

and, on the other hand, they pass over, for every isomorphism, into elements of exactly the same type, it follows in fact that S and S° are not isomorphic. For R2 and the ring Rv opposed to it, over an arbitrary commutative ring R, it is easily seen that the following isomorphism holds: abj ac)1 R2

R°1

cd

,

bd

J

§ 40. Complexes

We observe by way of introduction that if S denotes a non-empty set and S the set of all subsets of S, then the intersection A fl B and the union set A U B of two elements A, B of S belong to G. Hence in 5 two compo-

COMPLEXES

119

sitions are defined by fl, u, which we call the set compositions. Because

(Af1B)f1C=Afl(Bf1C), (AUB)UC=AU(BUC), Af1B=BflA, AUB=BUA,

(40.1)

(40.2)

both are associative and commutative, and further, because

Afl (BUC)= (Af1 B)U(Af1 C),

AU(BUC)=(AUB)fl (AUC)

(40.3)

each of these compositions is distributive for the other. Moreover, because

AUO=A, AnS=A, Aflo=O, AUS=S

(40.4)

the elements 0 and S are the neutral elements of S with respect to the com-

positions U and fl, respectively, and at the same time the zero element with respect to fl and U, respectively. On the other hand, neither of the compositions n , u is invertible or regular. Accordingly, S forms a structure (with two compositions) which, however, is not a ring. We shall deal with this structure and its generalizations more explicitly in § 60.

The subsets of a given structure S = are called the complexes of S. We usually denote these by A, B, ... . We often tacitly disregard the empty complex O. The name "complexes" seems a tautology since it means the non-empty subsets of S; however, this peculiar terminology is justified because we always regard the compositions defined in S as being defined also for the elements of a complex. Of course complexes are in general not closed with respect to these compositions. The complexes closed with respect

to at least one of the compositions (in S) are just the substructures of S. Accordingly, complexes are a generalization of the substructures. Complexes may be used in various ways for the examination of structures. If we denote the set of complexes of the above structure S by Cam, then in G

are defined the set compositions fl, U. If, further, addition or multiplication is defined in S then we define the sum A + B and the product AB of the complexes A, B as that complex which consists of the distinct a + 19 and aj9, respectively, (a E A,19 E B). If, finally, S is a ring, then we understand by the combined product of the complexes A, B the submodule of S

generated by the ai (a E A, 1 E B). Since in rings we shall deal almost exclusively with the combined product, we denote it similarly by AB, just like the "ordinary" product of A, B. Especially, if one of the complexes

A, B is a submodule of a ring S, we always mean by AB the combined product. 5/a R.-A

STRUCTURES

120

We even allow "mixed cases" like (AB)C, where A and B are complexes of S, while C is a submodule of S. Then (AB)C may again be interpreted as the combined product of the "ordinary" product AB and of C. For the present we disregard the combined product. It is evident that both

compositions AB, A + B are associative and the latter is (because of the commutativity of the addition in S) also commutative. Consequently with respect to multiplication C constitutes a semigroup and with respect to addition a commutative semigroup (written additively). It is obvious

that none of these semigroups is regular. The product and the sum of finitely many complexes is defined, in accordance with our general conventions for semigroups, by

A, ... A. _ (A, ... An-1)An,

Al+...+ An(A1+...+An_)+A,,

(40.5)

(40.6)

Then Al ... A. and A, + ... + A. obviously consist of the al ... an and al + ... + an, respectively, (a; E A). If S contains a unity element e, then the product Al A2 ... of infinitely many complexes A1, A2, ... (e EA1, A2 ...) means the set of those products

al a2 ...

(a; E Aj, i = 1, 2, ...),

in which only finitely many at are different from s. We have to interpret similarly the sum Al + A2 + ... of infinitely many complexes A,, A2, ... (0 EA1,A2...). If every a (E A) has an inverse a-1, then we denote by A-1 the complex of all such a-1 and call A-1 the inverse complex of A. (However, let us notice that A-1 A is not in general, a unity element in G, consequently A -1 does not mean the customary inverse of A in C5.)

The power An of a complex A is defined, for a natural number n, as the product A, ... A. with Al = ... = An = A (if multiplication is defined in S). If A-1 exists, then let A-" = (An)-1. (It is not customary to use an additive analogue of A'.) The rule AmA" = Am+n

evidently holds, if m and n are both positive or both negative. A product AB, or a sum A + B of the complexes A, B is called schlicht, if we can write all its elements uniquely in the form acfi or a + fi, (a E A, E B).

If, for two complexes A, B, AB = BA, then we call them permutable or interchangeable. If moreover aj9 = 9a (a E A, j9 E B), then we say that the complexes A, B are elementwise permutable or interchangeable.

COMPLEXES

121

At last we come to the combined product AB. The associativity

(AB)C = A(BC)

(40.7)

remains true. One has only to realize that both sides of (40.7) consist of the submodule of S generated by all the aj9y (a E A, 9 E B, Y E C). Likewise the distributivity

A(B + C) = AB + AC, (B + C) A = BA + CA

(40.8)

,

holds if we suppose that B and C contain the zero of S. It is sufficient to prove, e.g., (40.81). For that purpose we show that on both sides we have the submodule of S generated by all the aq, ay (a E A, j9 E B, y E Q. It is evident that this module contains both sides of (40.81). It is therefore sufficient to show that for instance ap is contained in both sides of (40.8). This follows immediately from the fact that both C and AB contain the element 0. (Observe that (40.8) for the "ordinary" product of complexes

holds in general only with S in place of =.) We apply the concept of conjugate elements in a group to complexes by calling two complexes A, B of a group G conjugate if B is the image of A under an inner automorphism, i.e., if there exists a e (E G)withB = e, 4e-'. We then say that B is a conjugate of A. Of course this generalized "conjugacy" is an equivalence relation (in the set of all complexes of G). As every auto-

morphism of a group carries each subgroup into an isomorphic subgroup, it follows that the conjugates of a subgroup are also subgroups of the given group. They are called conjugate subgroups. A complex A of a group is, according to Theorem 42, a group if, and only if, A2, A-1 S A. If this is the case then A2 = A-1 = A. If, for two subgroups A, B of a group, BA 9 AB, then BA = AB. For, B_1 A-1, it follows by hypothesis that (AB) , i.e., i.e., AB c BA. (BA)-'

A-'B-'

THEOREM 68. Let A, B be subgroups of a group. AB is a group if, and only if, AB = BA. The product is simple if, and only if, A fl B is the unity element.

Both assertions "only if" are trivial. If AB = BA, then (AB)2 = ARAB = A2B2 = AB, (AB)-1 = B-1A-1 = BA

,

which means that AB is a group. If A fl B = e, then it follows from each equation oc# = alf, (a, al E A; , P1 E B) that ai'a = N1N-1 and, (because ai'a E A, #I#-' E B) that al 'a = Y'N-' = s, i.e., a1 = ac, Nl = 3. This means precisely that the product AB is schlicht. EXAMPLE 1. If A = (a, f> is a complex of a group, then A2 = (a$, aft, flee, f> except for the case where among the elements a2, aj9, fl a, #3 two are equal. EXAMPLE 2. For the cyclic group {a} of order six we have the product decomposition {cc} = <e, a3> (e, aQ, a') = {a3} {a$}.

122

STRUCTURES

§ 41. Cosets. Residue Classes Here we shall deal with some important classifications of a given structure

S = (a, P, ...>, which should be a semigroup with unity element or in particular a group, module, ring, or skew field. These classifications will be formed by the help of a subgroup H or submodule to of S. First, we consider the case where multiplication is defined in S and S has the unity element e. H will indicate a subgroup of S wherein we assume that it also has s as unity element. (If S is a group or a skew field, then this condition is automatically satisfied.) We call each product 1:H ( E S) a left coset

of H (or by H, or mod H) in S. Accordingly each H is a right coset of H.

(The notation "left" or "right" is omitted whenever the type of cosets is clear; "class" may be used instead of "°coset" where the meaning is clear.)

Since eH = He = H, H occurs among both the left and right cosets of H. Therefore this class H is called the principal class. Since in the commutative

case H =

we may then speak of the cosets of H. We may do so even when S is not commutative but H is so constituted that H = = H always holds. (We shall deal with this more explicitly in § 42.) The are called "classes", is shown by the following:

reason why the product H (and

THEOREM 69 (LAGRANGE'S theorem). In a group S all the left cosets of a subgroup H constitute a classification of S. The theorem is also true when S is a skew field, ring or semigroup provided that in the last two cases it is presupposed that S has a unity element and that it is the unity element of H. Since the unity element of H is also that of S, E E H for all E S. Accordingly, each element of S is contained in at least one left coset of H. So we have only to show that two classes ocH, PH with at least one common element are necessarily equal. According to our assumption there exists a o (E H)

with ae E 13H. Hence it follows that oceH C PH2.

But since H is a group, eH = H, H2 = H, consequently we have aH S PH. Similarly i4H S ocH, consequently ocH = RH. This proves the theorem. If S consists of finitely many left cosets of H, then we denote their cardinal number by O(S : H), otherwise we put O(S : H) = oo. We call O(S : H) the index of H in S. The case where S is a group is important. Since each class H is then a schlicht product of 1' and H, it follows from this that all left cosets of H are equipotent. Consequently Theorem 69 yields the following. COROLLARY. For every group G and each subgroup H of it

O(G : H) O(H) = O(G) holds.

(41.1)

COSETS. RESIDUE CLASSES

123

In particular, if G is finite, we have O(H) I O(G),

(41.2)

whence by application to H = {a} it follows that o (a) 10 (G), ((x E G). A system of representatives of the classification mentioned in Theorem 69 is called a left representative system of S by H (or mod H). If

S=

(41.3)

is such a system of representatives, then because a; E a; H, S = a1H, a2H

, ...,

(41.4)

is that classification. The converse is trivial: if (41.4) is the classification of S into left cosets by H, then (41.3) is a left representative system of S by H. We certainly have also

S=SH.

(41.5)

If, moreover, S is a group, then (41.5) is a decomposition of S into a schlicht

product of the complexes S, H. The converse of this is likewise trivial: if (41.5) is a schlicht product, then this implies the classification of S into left cosets by H; the classes arise from (41.5) after multiplication of H by the elements of S and at the same time the factor Sin (41.5) is a left representative system of S by H. We prove that the left and right cosets, respectively, by H of a group S

constitute two equipotent sets, whence it fellows that (for a group G) the above-defined "left-hand side index" O(S : H) is simultaneously the "right-hand side index". Moreover we show that if S is a left representative system of the group S by the subgroup H, then the inverse complex S-1 is a right representative system of S by H. it follows immediately from According to the rule (ap)-1 (41.5) that

S-t = H-1 S-1

.

(41.6)

It is evident, too, that the products in (41.5) and (41.6) are simultaneously schlicht. Since, finally, S and H are groups, we may write (41.6) as

S=HS-'. This is the proof of the assertion. We now want to apply these facts to the "additive case". We have to consider the cases where S is a module, ring or skew field and H is replaced by a submodule m of S. However, instead of "coset" and of "representative system", in the additive case we refer to the term "residue class" and "residue system"; further the distinctions "left" and "right" are no longer to apply

STRUCTURES

124

because of the commutativity of addition. We note that S in these cases always constitutes a group (more precisely a module) with respect to addition. If S is a module, ring or skew field and m a submodule of S, then we call the (necessarily schlicht) sums + m the residue classes of S mod m (4 E S). The principal class m is also called (because 0 E m) the zero class (or the class 0). The classification of S into the residue classes E + m takes the form (41.7) S = zi + m, a2 + m ,....

The system of representatives
(...,r- m, r, r + m, r + 2m,...> which is, according to our general terminology, just the residue class mod (m) represented by r, where (m) denotes, as above, the submodule {m} of 9. Instead of "mod (m)" we shall write briefly "mod m". The residue class considered may then be denoted as the "residue class r (mod m)". In the multiplicative case, where H is a group, we denote the left cosets given in (41.4) as "mod H", merely for the sake of uniformity. EXAMPLE 1. All the residue classes mod m of 9 are the r (mod m) (r, = 0, ..., m - 1). According to this O(9 : (m)) = m.

EXAMPLE 2. Let 1' be a subgroup of order 2 of the full permutation group 608. We shall see that the left and right cosets of .Ji' are not identical. We may take = {e} with e = (1 2). If we put a = (1 2 3), then, as we have already seen, 68 = . = (e, a, a8, e, ae, x2e>. These classes are as follows:

Jr = ,

aa%° = (a, a e>,

a8 `le _ (a2. x'e>,

_° = (E, e>, ira = (a. xEe>,

i°aa = (a'. xe>.

whence we see that the left and right cosets are distinct. Nevertheless e, a, a2 is a common left and right system of representatives of Ae in tee. The existence of such a common system of representatives is not accidental on account of the following noteworthy theorem of HALL: If H is a subgroup of finite index of the group G, then

there exists a common system of representatives for the left and right cosets of H in G. (Cf. ZASSENHAUS (1958); p. 11.)

EXAMPLE 3. In the (commutative) ring 3 the cosets of the subgroup <1, - 1> are

the following: (0>, (1, -1>, <2, -2), ... . EXAMPLE 4. A cyclic group of prime number order has according to the corollary of Theorem 69 no proper subgroups. EXAMPLE 5. If the subgroups H, K of a group G are of finite index, then

O(G :(HnK));: O(G :H)O(G: K).

125

COSETS. RESIDUE CLASSES

For, if <ei, e$, ...) is a left representative system of H mod (H n K), then the classes eiK, e2K, . . . are different, thus O(H : (H n K)) S O(G : K).

The assertion follows from this and from (41.1).

§ 42. Normal Divisors. Ideals

According to Theorems 59, 60 the search for all factor structures of a structure S, and thus also that of the homomorphic images of S, is equivalent

to determining all compatible classifications of S. We shall solve this problem for groups (thus also for modules) and for rings with a surprisingly simple general result.

Let G =
o).

(42.1)

We already know that the kernel of this homomorphism is a subgroup N of G, which consists of the elements mapped onto the unity element s of G. Because s E N,

e=N.

(42.2)

Then we consider an arbitrary class . This consists of all the

such that (42.3)

In order to find these we assume them to be of the form = ev. The condition (42.3) then states that 0 = O, for which we may write, by virtue of (42.1), Vii' = O. This is equivalent to v = e, i.e., to v E N. Consequently we have found that the required are the set of Lov (v E N), i.e., P = eN. Because of the duality principle, N p. From these equations it follows that

oN=Ne

(eEG)

(42.4)

(e E G).

(42.5)

or in another form

ioNe` = N

This suggests the following important definition : By a normal subgroup

(self-conjugate subgroup or briefly a normal divisor) of a group G we

126

STRUCTURES

mean each subgroup N of G which satisfies condition (42.4) [or (42.5)]. This condition means that a subgroup is normal if, and only if, its left and right cosets are identical, i.e. if it coincides with all its conjugate subgroups. The above result may now be expressed by saying that every compatible

classification of G necessarily consists of the cosets of a normal divisor. Conversely, we assume that

P=PN=NP

(PEG),

(42.6)

where N is a normal divisor of G. We show that the sets N constitute a compatible classification of G. The necessary condition for this is: P=6=*. re,ra,

for all rEG.

Pr=or,

By virtue of (42.6) this condition may also be written in the form

Noz=Nar, which, however, is trivially satisfied. Consequently the following theorem is valid :

THEOREM 70. The compatible classifications of a group are just the classifications into cosets with respect to a normal divisor.

More precisely this theorem means that the compatible classifications of G with respect to a normal divisor N of G may be given in the form

C = N, PN, aN,

...,

(42.7)

where all the different cosets of N stand on the right-hand side. Since moreover C is uniquely defined by the principal class N, we introduce the simpler notation G/N for the corresponding factor group G/@. We call GIN the factor group of G with respect to N (or by N or mod N). According to the above the following theorem holds: THEOREM 71. All the factor groups of a group G are of the form GIN, where N runs through all the normal divisors of G. The elements of G/N are all the cosets of G with respect to N. The product of two elements of G/N is computed according to the rule

aN 19N = ajN

,

(42.8)

and, in particular, the principal class N is the unity element of GIN. It is to be noted that (42.8) remains valid even if the left-hand side is interpreted as the product aN flN of the complexes aN, flN, since, from (42.4),

aN,BN = aBN$ = ON.

127

NORMAL DIVISORS. IDEALS

Conversely, if, for a subgroup H of G, aHfi H = aflH

always holds, then H is normal. For, it follows by hypothesis that H9H = PH, thus.

HP S PH.

If this is applied with fl-1 instead of fl, then it follows that Hi4-3 S fl-1H and so PH S H fl; consequently H fl _ flH, i.e., H is normal. From this we have obtained the proposition that the set of left cosets mod H of a group G, with respect to multiplication of complexes, constitutes a semigroup if, and only if, the subgroup H is normal, and that then this semigroup is just the factor group G/H.

The natural homomorphism of G onto G/N takes the form

G - G/N (e - eN).

(42.9)

N is the kernel of this homomorphism. Now G always has the two trivial normal divisors s, G. For the corresponding factor groups we have G/E

G ,

G/G

1,

where I signifies the group consisting of the unity element alone. The case

N = e, G are thus those in which (42.9) is either an isomorphism or a homomorphism onto the unity group. Since by (42.9) and on account of the homomorphism theorem (Theorem 60) all homomorphic mappings of G have been covered, we have the following THEOREM 72. A group is simple if, and only if, it contains no proper normal divisor.

It is evident from what has been said that the concept of normal divisor is of great importance. We now prove two simple theorems: THEOREM 73. Every subgroup of index 2 of a group is normal.

THEOREM 74. The intersection and the product of two normal divisors of a group is again a normal divisor of the group.

If H is a subgroup of the group G with O(G : H) = 2 and a(E G) is an arbitrary element outside H, then both the cosets ocH, Ha must be equal to the difference set G - H, whence ocH = Hoc. Since this equation also holds for a E H, Theorem 73 is proved. In order to prove Theorem 74 we consider two normal divisors A, B of G.

By Theorems 46 and 68, A fl B and AB are subgroups of G. If e is an arbitrary element of G, then, by virtue of (42.4),

o(AfB)=oAfleB=AoflBo=(Afl'B)o, This proves Theorem 74.

oAB=AoB=ABe.

STRUCTURES

128

The above also applies to a module t =
now take on the role of normal divisors. Accordingly, the compatible classifications of M agree with the classifications of M into residue classes with respect to a submodule m of M. The corresponding factor module is accordingly denoted by M/m. This has for its elements all the different residue classes mod M, so that we may write

M/m=<m,o+in,a +m,...>,

(42.10)

where <0, o, o,.. .... > is a residue system of M mod in. The addition of these elements is defined by

(a+m)+(fl+m)=a+fl +m.

(42.11)

The zero element of M/m is the zero class m, which we denote, as an element of M/m, by 0, provided the meaning is unambiguous. A module is simple if, and only if, it has no proper submodule, i.e., if its order is 1 or a prime number.

Finally, we consider a ring R =
C=m,o+m,a+m,...

(42.12)

be this classification. We have only to find for which m the classification C is also compatible with the multiplication in R. For this, a necessary and sufficient condition is:

ae+m=,Bo+m. This is equivalent to i.e., to

a-

andae- PeEm. andpeEm,

for which we may also write

eµ, yeEm

(AEm, oER).

(42.13)

By an ideal of a ring R we mean a submodule m of R with the property (42.13), i.e.,

Rm, mR 9 m

.

(42.14)

The historical background of this peculiar term "ideal" is that KUMMER had called certain concepts in number theory "ideal numbers". DEDEKIND later replaced them by ideals.

NORMAL DIVISORS. IDEALS

129

From each of the two conditions (42.14) it follows that m2 S in. This means that in is closed with respect to multiplication. According to this an ideal of R is always a subring of R. Since, on the other hand, "normal divisor of a group" and "ideal of a ring" are analogous notions, we sometimes call ideals normal subrings. From the above we obtain the following THEOREM 75. The compatible classifications of a ring are the classifications

into residue classes with respect to an ideal. If a is an ideal of the ring R and C the corresponding compatible classifi-

cation of R, we denote the factor ring, R/C, by R/a. Then the following theorem holds. THEOREM 76. All the factor rings ofa ring are the R/a, where a runs through

all the ideals of R. The elements of R/a are the residue classes mod a in R. The sum and the product of two elements of R/a are computed according to the rules:

(a+a)+(f+a)=a+# +a, (a+a)(,e+a)=a,8+a. (42.15) The left-hand side of (42.152) may not be regarded as a product of complexes, otherwise it is in general only a subset of the right-hand side.

Both trivial subrings 0, R of R are, according to (42.14), always ideals. In the same manner as Theorem 72, we obtain THEOREM 77. A ring is simple if, and only if, it contains no proper ideals. Extending (42.14), we call a submodule in of R with the property

RmSm a left ideal of R. A right ideal of R means, correspondingly, a submodule in

with mR S in. The left and right ideals are sometimes called one-sided ideals. Of course these are a fortiori subrings. One-sided ideals are important

in the examination of ideals. It should be noted that all the two-sided ideals occur among one-sided ideals. THEOREM 78. The intersection, the sum and the product of two ideals is again an ideal. More generally the intersection, the sum and the product of two left ideals is a left ideal, furthermore this holds for the product even if the right-hand factor is an arbitrary complex. COROLLARY. If a is a left ideal, b a right ideal and C a complex, then ab and aCb are ideals. It suffices to prove the second part of the theorem. We denote by a, b two left ideals and by C a complex of R. It suffices to prove that

aflb, a + b, aC

130

STRUCTURES

are left ideals of R. All three are submodules of R, moreover

R(aflb)9 Ra fl Rb9aflb,

R(a+b)=Ra+Rbca+b, RaC = (Ra)C c aC.

Hence the theorem is proved. From this and the dual theorem follows the corollary. From Theorems 57 and 77 it follows that a skew field contains no proper

ideals. We even prove the following THEOREM 79. A skew field has no proper left ideals.

Assume that a (# 0) is a left ideal of the skew field F, then Fa = F and Fa S a whence a = F and the theorem follows. As an analogue of the symbol { }, which generally denotes the substructure (of an arbitrary structure) generated by the elements in brackets, we denote ideals, left ideals and right ideals of a ring by the symbols (), (),, ()r, respectively, which we define as follows. Let A denote an arbitrary complex of the ring R. We consider all the ideals, left ideals and right ideals of R, which contain A. (R occurs among them in all three cases.) Their intersection constitutes, according to Theorem 78, an ideal, left ideal and right ideal, respectively (containing A). These we denote by (A), (A),, (A)r and call the ideal, left ideal and right ideal of R, generated by A, respectively. If further A, B, . . . are arbitrary complexes of R, then we write . . .) = (A U B U ...), etc. We prove the important formulae:

(A, B,

(A) = {A}+ + RA + AR + RAR (A), _ {A}+ + RA , (A),. = {A}+ + AR .

,

(42.16) (42.17)

(42.18)

where {A}+ denotes the submodule of R generated by A. Moreover we prove that for a ring R with unity element the simpler formulae (A) = RAR , (A), = RA , (A),= A R

(42.19)

(42.20) (42.21)

are valid. First of all we prove (42.17). The right-hand side is a submodule of R. Furthermore :

R({A}+ + RA) = R{A}+ + R2A 9 RA + RA = RA.

[31

NORMAL DIVISORS. IDEALS

Consequently the right-hand side of (42.17) is a left ideal. This contains A, therefore

(A)1S{A}++ RA. On the other hand evidently

{A}++RA9(A)1. Hence follows (42.17). By dualization we obtain (42.18). It can be seen in the same way that the right-hand side of (42.16) is a left and a right ideal, i.e. an ideal. A is contained in this, therefore (42.16) is true with S instead

of =. However, (42.16) is trivially true with ? instead of =. Thus we have proved (42.16). If R contains a unity element, then

{A}+S RA,AR9RAR. Then (42.16), (42.17), (42.18) become (42.19), (42.20), (42.21) in that order. If a is an ideal of R, then a = (a). Hence all ideals of R may be written in the form (A). This also holds for left and right ideals. For an element a, we shall call (a), (a),, (a), the principal ideal, principal left ideal and principal right ideal generated by a, respectively. By (42.16)

(a) = {a}+ + Ra + aR + RaR. Accordingly the principal ideal (a) consists of the elements not + Boa + as + Z e1oa1

(n E 7;

o'; E R) ,

(42.22)

where the sum is finite. Likewise :

(a)1 = {a}+ + Ra,

(a)r = {a}+ + a R .

These consist of the elements na + @cc

and na + ago

(n E 7; N E R),

(42.23)

respectively. We see from this that the "generation" of the elements of the two-sided ideals is much more complicated than that of the one-sided ideals.

Because of (42.16),

(A, A. . .) = (A) + (B) + ... (A, B.... c R) . In particular

(a, A ...) = (a) + (,B) + ...

(a, /1.... E R) .

Accordingly, every ideal is a sum of principal ideals. This formula, together

132

STRUCTURES

with (42.22), makes it possible to write explicitly all the elements of an ideal (A) (in terms of the elements of A and R) which is left to the reader. The same holds for one-sided ideals. From (42.17), (42.18) it follows that

(A), (B),. = ({A}+ + RA) ({B)+ + BR) = {AB}+ + RAB + ABR + RABR.

Accordingly we have the formula (A), (B). = (AB) .

(42.24)

(Cf. this with the corollary of Theorem 78.) Likewise it is easy to prove that (A) (B) = (AB, A RB).

(42.25)

In a commutative ring R, (42.16) is simplified to

(A) = {A}+ + RA

.

(42.26)

(This results from (42.17), since (A) and (A), now mean the same.) If R is a commutative ring with unity element, then

(A) = RA

.

(42.27)

According to our terminology two compatible classifications are called related, if the connected factor structures are isomorphic. Correspondingly we call two normal divisors of a group, or two ideals of a ring, related, if the corresponding factor groups, or factor rings, are isomorphic. The above considerations serve as a basis for the following definition. Let S denote a structure in which multiplication is defined, and let A denote a complex of S. By the normalizer and the idealizer of A (in S) we understand the set of those elements a of S, for which

aA = Aa

(42.28)

aA, Aa S A ,

(42.29)

and

respectively. Finally we call the set of elements of S elementwise permutable with A, the centralizer of A. [This, according to (42.28), is contained in the normalizer of A and coincides with it if A = a (E S).] It is obvious that the normalizer, idealizer and centralizer of A are subsemigroups of S. Tm;o1M 80. The normalizer N(A) of a complex A of a group G is a subgroup of G, which consists of the elements v (E G) with the property vAv-1 = A .

(42.30)

NORMAL DIVISORS. IDEALS

133

If A is, in particular, a subgroup of G, then A is a normal divisor of N(A), and N(A) contains all the subgroups of G which contain A as normal divisor. THEOREM 81. The idealizer 1(A) of a submodule A of a ring R is a subring of R. If A is, in particular, a subring of R, then A is an ideal of I(A), and I(A) contains all the subrings of R which contain A as an ideal. As the proofs of these theorems are easy, they are left to the reader. In accordance with the given definition of the normalizer of a complex, we understand by the normalizer of a ring R the set of all a (E R) with the property aR = Ra. (42.31)

(Note that the normalizer of R is in general only a subsemigroup and not a

subring of R.) From (42.17), (42.18) we have: If a is an element of the normalizer of R, then (a), = (a),. For finite groups G the following so-called class equation O(G) = Z O(G : Z2)

(42.32)

holds, where a runs through a system of representatives of the Frobenius classes and Z. is the centralizer of a. That is, the class of a consists of the

eae_', where e runs through a left representative system of G mod Z. We understand by an ideal of a semigroup H a complex a of H with the property Ha, aH c a. Evidently every ideal of H is a semigroup. The ideals of a semigroup are important although they have little to do with the factor semigroups. For further information on idealizers, see FUCHS (1950). EXAMPLE 1. Because 9

(m) = (m), the submodule (m) = {m} (m > 1) is an

ideal of 9, the principal ideal generated by m. The factor ring 9/(m) is also called the residue class ring mod m. This is isomorphic with the ring considered in § 23, Example 2. O(9/(m)) = m (cf. § 41, Example 1). We shall prove in § 84 that the (m) (m > 0) are all the ideals of S. EXAMPLE 2. The product of a subgroup and a normal subgroup of a group is a group.

EXAMPLE 3. Among the four proper subgroups of .9 there is a normal subgroup (of rank three) and the three others (of rank two) are conjugate. EXAMPLE 4. In a ring R each Ra (9; (a),) is a left ideal and each RaR (9 (a)) an ideal ((x E R).

EXAMPLE 5. In the full matrix ring R2 of rank four over an arbitrary ring R the

matrices of the form (e 0) constitute a left ideal. EXAMPLE 6. Let a submodule a of a ring R be called a quasi ideal of R (STEINFELD

(1955), if Ra n aR s a. Then a2

Ran aR

a,

therefore each quasi ideal is a subring. One-sided and two-sided ideals are always quasi ideals. In the above ring R2 the matrices ( 01 constitute a quasi ideal.

STRUCTURES

134

EXERCISE 1. The full matrix ring of rank four of a skew field is simple (cf. Theorem 255).

EXERCISE 2. Construct a ring R where there are elements a with (a), = (a),, and

Ra0aR.

§ 43. Alternating Groups

Alternating groups are, in more than one respect, important (finite) permutation groups. Before defining them, by way of preparation, we establish some statements which are interesting in themselves. For (finite) cyclic permutations we have the formula

(1 ... k) = (1...1)(i...k) Hence

(I

(1
(43.1)

(k >_ 2) .

(43.2)

... k) = (1 2)(2 3) ... (k - 1 k)

From (43.1) we have (2 ... k)(k 1) = (I ... k), thus

(I k) _ (k ... 32)(1 ... k)

(k >_ 3).

(43.3)

THEOREM 82. Let

p=( x

,I,

a = fax)

(x E S)

(43.4)

be a permutation of a set S and a mapping of S into itself. Then

pap-1=

Vt x) '1

.

(43.5)

COROLLARY. If' two permutations q, q' are each aecomposed into a product of pairwise disjoint cycles:

q=c1...c,, q'=ci...c;,

(43.6)

then q, q' are conjugate if, and only if, r = s and in some suitable order of the factors (43.7) o(c,) = o(ct) (i = 1, ..., r).

The theorem follows from pap-1

- Ix'XX 1 xlX' Ixl - lz 1 lax/ = I(x)'J 1

The statement "only if" of the corollary follows immediately from the theorem. In order to prove the "if" part we have only to consider the case

ALTERNATING GROUPS

135

where the decompositions (43.6) are of the form

q =(a, ...ak)...(m,...m,) (a', ...ak)...(mi...ml). q' Then

P=

a,...m, a,. .ml

is a permutation, for which, by Theorem 82, Two simple examples: For p

we have

Pap .For

(31254)'

__

124155)

(31 254 _ 12345

15344)

53144)

q=(16254)(78)

p=(356)(128),

we have

a

pqp ' = (2 3 8 6 4) (7 1).

For a permutation

_ 1 ... nn

we call a pair of elements j'. (For this we also say that in p the elements i', j' constitute an inversion.) We call the permutation even or odd according as the number of its inversions is even or odd, and we write accordingly: sign p = 1, or sign p = - l For instance P

.

_(1234 24 1 3)

contains only the inversions: <2, 1)', <4,1>, <4, 3>. Accordingly sign p = -1.

We now consider the full permutation group n >- 2. We take the product

P=

17

(a, - as)

where we assume (43.8)

15r<s$n where a1, ..., a,, are any different integers. If we carry out a permutation p (E JO J on the indices 1, . . ., n of a,, . . ., a,,, then P becomes

STRUCTURES

136

another product, which we denote by P". Clearly

Pp = ± P.

(43.9)

Further, it is evident that P°4 = (P4)°

(43.10)

(p, q E

From this it follows that the p such that P°= P constitute a subsemigroup of .9 , which, by Theorem 42, must be a group. We denote this group

byve and call it the alternating group of degree n. Since P can only take is at most two. But since two values, it follows that the index for a transposition of the form (i i+1) p(i+I)

_ -p,

(43.11)

this index is equal to two. Accordingly O

n!

(43.12)

,

and it follows from Theorem 73 that .,e is a normal divisor of 9,,. THEOREM 83. The alternating group Pf consists of the even permutations of

,9, These are the permutations which may be written as the product of an even number of transpositions. Consequently the

(ij) (kl)

(i#j, k# 1;i,j,k,I=1,...,n)

(43.13)

constitute a generating system of %,,. Another generating system is (i j k)

(i 0j, i # k, j # k; i, j, k = 1, ..., n).

(43.14)

COROLLARY. For arbitrary permutations p, q (E

sign pq = sign p sign q . For the proof we consider a permutation p number of its inversions. Then sign p

(43.15)

and denote by u the (43.16)

The first assertion of the theorem is equivalent to

P" = (-1)"`P .

(43.17)

ALTERNATING GROUPS

137

If p = 0, p is the identity permutation, therefore (43.17) is true. If p > 0 we assume (43.17) with ,u - 1 instead of ,u. Let p be written in the form n

P= 1'...n'j

(43.18)

If p had no inversion of the form <1', (i + 1)'> (i = 1, . . ., n - 1), then 1' < 2' < ... < n' and consequently p would have no inversions at all which is impossible because p > 0. Accordingly p has at least one inversion

of the form . With this i we form the product of the permutationsp and (i i+1):

q=p(ii+l).

(43.19)

Because of (43.18)

_

i+1 i+2 ...n i 1)'(i + 1)' i' (i+2)'...n'J

1 ... i-1

q-11'...(1-

According to this the inversions of q are the same as those of p, except for , consequently q has exactly p - 1 inversions. By (43.17) and the induction assumption,

P°= (-l)N-I P. From this and from (43.10), (43.11) and (43.19), (43.17) follows for all µ' With respect to the second assertion of the theorem we first observe that because of (43.2) each element of .51,, may be written as a product of transpositions. Let v denote the number of factors in such a decomposition of p.

We have to prove that

21 v - p .

(43.20)

By virtue of (43.2) and (43.3) each transposition may be written as a product of transpositions of the form (i i + 1) in which the number of factors is odd.

Consequently p may be written as a product of v + 21 of such transpositions where t is some integer. On account of (43.10) and (43.11) P°=(-1)v+"p=(-1)VP.

From this and from (43.17) we obtain (43.20). This proves the second assertion of the theorem. Since the assertion in (43.13) follows from this, it is sufficient for the complete proof of the theorem to show that the same group is generated by (43.13) and (43.14). This follows from the fact that for distinct i, j, k, i,

(ij)(k1)=(ik1)(ij1), (ij)(ik)_(ikj).

STRUCTURES

138

Since by (43.16), (43.17)

P° = sign p P, the corollary follows from (43.10). THEOREM 84. The alternating group ve is simple for n g0 4. If n = 4 it has exactly one proper normal divisor: this is the four-group consisting of the permutations 1, (1 2) (3 4), (1 3) (2 4), (1 4) (2 3).

Since from (43.12) O(yez) = 1, O(, 1g) = 3, the assertion is true for n = 2, 3. Now, we have only to consider the case n z 4. We assume that

N is a normal divisor (0 1) of,,. 1. N contains at least one permutation (a b) (c d) with distinct a, b, c, d (= 1, ..., n). Then there is in N a permutation v for which the decomposition into a product of pairwise disjoint cycles is one of the following four cases :

v=(ab)(cd)..., (abc), (abcd...)..., (abc)(de...)... (The dots indicate that there may be further elements or cycles.) Corresponding to these four cases, we have recourse to a permutation a (Eve,) where

a = (a b c), (a b d), (a b c), (a b d)

in the four cases respectively. Compute in all four cases the following permutations :

vav-1 = (bad), (b c d), (b c d), (b c e),

a-1 = (b a c), (bad), (b a c), (bad), vav-1 a-1 = (a c) (b d), (a b) (c d), (a d b), (a d c e b)

.

Since vav-la-1 =

c(ava-1)-1 belongs to N, we have proved the above assertion, in the first two cases directly, in the third case by reduction to the second case and in the fourth case by reduction to the third case.

2. N contains all the permutations (a b) (c d) with distinct a, b, c, d (= 1, . . ., n). Because of 1. we can assume that (1 2)(3 4) E N. We take a permutation

(l234...n1 flab

cd...

J1

from 95,,. Either a or a' = a(I 2) belongs to all 2)(3 4)a--1 = a'(1 2)(3

,,. 4)a'-1

Consequently

= (a b)(c d)

belongs to N, which is what we wanted to show.

ALTERNATING GROUPS

139

3. If n ? 5, N contains all the permutations (a b c) (E 95J. To show this take two distinct elements x, y (= 1, ..., n), which are also distinct from a, b, c. Since

the assertion follows from 2. If n >- 5, according to 3. and the latter part of Theorem 83 we have N = (,,, by which we have proved the first part of Theorem 84. If n = 4, it follows from 2. that N contains the group of order four mentioned in Theorem 84, which we denote by G4. Since it follows that 4 O(N) and, on the other hand, O(N)1 12 (= 0(vf4)), N can only be G4 or vf4. Since G4 is normal in "f4 (even in 934), the proof of Theorem 84 is completed. Regarding this proof cf. BAUER (1932-34) and REDEI (1951b). Another short proof by induction is given by POLLAK (1955).

THEOREM 85. The full permutation group 93 has, when n = 3 or n >--_ 5,

no proper normal divisor other than ve,,. 934 has, apart from 1'f 4i only one proper normal divisor, this being the proper normal divisor of vfr4 mentioned in Theorem 84. Since the theorem is true for n = 3, we may assume n 4. Let N (A 1) denote a normal divisor of 93,,. First we consider the case where there is a permutation v in N, whose decomposition into a product of pairwise disjoint cycles has the following form:

v=(abc...).... If a, b are interchanged here, then the permutation obtained

µ=(bac...) .. belongs to N because of the corollary to Theorem 82. The same then holds for ,u v -1 = (a b c), and so, because of the same corollary, for all threec N, the theorem is true. cycles (a' b' c'). Since, according to this, In the case not yet dealt with, each permutation v (0 1) of N is a product of pairwise disjoint transpositions: v = (albs) ... (akbk)

(k >- 1) .

(43.21)

If 2k < n, then N contains, besides (43.21), _ (c bl) (a2b2) ... (ak bk) ,

where c is different from all at, b,. Since then vee = (c al bl) is contained in

N, this case has been discussed already. Consequently 2k = n, whence k z 2. In addition to (43.21) a = (alb2) (albs) . . . (ak -1 bk) (akbl) ,

STRUCTURES

140

and so also

va = (a, ... ak) (b1 ... bk)-1

belong to N. Consequently k = 2, n = 4 unless we are again going to deal with the previous case. Since further there are now only three permutations

of the form (43.21) and these together with the identical permutation constitute a normal divisor of .J (= ,94), we have proved Theorem 85. The group c 41, is also called the icosahedron group, since it is isomorphic with the

group of motions of Euclidean space which map an icosahedron onto itself. (Cf. SPEISER (1956).) REDEI (1951a) and SUZUKI (1957) have proved that except for the

icosahedron group every non-cyclic finite simple group contains at least one noncommutative second-maximal subgroup; a subgroup which is a maximal subgroup of a maximal subgroup is called second-maximal. EXERCISE. The maximal subgroup of c l5 is not commutative, the second-maximal ones are commutative.

§ 44. Direct Products. Direct Sums So far, we have hardly used skew products of structures (§ 34). Here we shall deal with their simplest application, the so-called direct composition of structures. Let a sequence S1, S.,, ... of arbitrarily many (not necessarily distinct)

structures be given. The elements of S; are denoted by a; (i = 1, 2, ...) and e; denotes the unity element of S;. For the present we denote by S the set of all sequences

a = (a1, a2, ...)

(44.1)

(a; E Si) .

(Later, however, we shall take for S certain subsets of that and denote these again by S.) If in all S, is defined a (similarly denoted) composition 0, then we define in S the composition 0 by (%10

We then say that compositions are directly defined in S. In particular for multiplication and addition we can make the following definitions: (a1, a2, ...) 0i, #2, . . .) = (001, a3 F'2, .

.

.)

,

((Xl, a.,, ...) + 011 fl 2l ...) = (al + N1, a2 + N21 ...) .

(44.2)

(44.3)

It is obvious that for the direct definition of compositions, associativity, commutativity, invertibility and distributivity are all invariant properties,

DIRECT PRODUCTS. DIRECT SUMS

141

i.e., if, for instance, in all the Si there is an associative multiplication, then

the multiplication (44.2) is also associative, etc. The concepts "neutral element" and "inverse" likewise show an invariant behaviour, i.e. denote the unity element, zero element, multiplicative and additive inverse [of the element (44.1)] in S, respectively, provided that the corresponding elements exist in all the Si. Henceforth all the Si will be semigroups with a unity element, groups, modules, rings or skew fields, respectively. Further the sequences (44.1) will

now be restricted by the "finiteness condition", by which only a finite number of the ai (i = 1, 2, ...) may be distinct from the neutral element of Si (namely ei or 0 according as the Si ale semigroups, groups, modules, rings or skew fields, respectively. We retain the previous notation S for the set of all such sequences (44.1). The compositions should then be

defined in S directly, i.e., by (44.2) and (44.3). No difficulty arises, since, because e2 = e,, and 0 + 0 = 0' = 0, the right-hand side of (44.2) or (44.3) always lies in S, if the same holds for the factors or terms on the left-hand side. The above invariance properties also apply to S. If, there-

fore, the Si are all A-structures, where A denotes the axioms for semigroups with a unity element, groups, modules or rings then S is also an Astructure. This structure S is called the direct composition of the structures S1, S2, . . ., for which, however, we introduce the terms direct product of semigroups (with unity element), direct product of groups, direct sum of mod-

ules and direct sum of rings, respectively. As a special case of the direct sum of rings, the direct sum of skew fields is thus defined; however, if there are at least two Si, S is not a skew field, but only a ring, as it is easy to see. We denote the direct product and the direct sum by and

S=SI®S..,®....

(44.4)

S=S1®S.2®...,

(44.5)

respectively. Correspondingly we call the structures S1, S, . . . the direct factors, or the direct summands, or in both cases the direct components of the structure S. (We say that Si is the it' component of S.) It should be noted that all the above statements remain meaningful even without the finiteness condition. In this case we call S a complete direct product or a complete direct sum. Since in the case of finitely many S, the finiteness condition is automatically

fulfilled, in this case "complete direct" means the same as "direct". For infinitely many S,, the direct composition is a substructure of the complete direct composition, however, complete direct compositions differ considerably in type from direct ones. They play an important role in recent researches on Abelian groups (modules). See

For "complete direct" and "direct" some authors say "direct" and "discrete direct". SZELE-SZENDREI (1951), FUCHS (1958).

STRUCTURES

142

Of course, the components S1, S21 ... are not substructures of S, which is

often inconvenient in applications, but we shall avoid this by means of embedding.

Before doing so we submit the above to a slight generalization which is based on the abstract concept of structures. If S', Si, S2, . . . are structures isomorphic to the above S, S11 S21 ..., we call S' a direct composition (a direct product or a direct sum) of Si, S2, ... In this more general case we also use the notations (44.4), (44.5) and call the S; the direct components (i.e. the direct factors or summands) of S'. Now we consider for an arbitrary i (= 1, 2, ...) an element (44.1) of S, in which at (E St) is arbitrary and each ock (k rA i) is the neutral element of Sk, i.e., ak = Ek or ock = 0 according as we are dealing with a direct product or a direct sum. We denote this special element of S by al, for which

at = (..., e, .1, ate Et+i.... ) or

(44.6)

We consider still another, similarly constituted element of S : or

flt = (..., 0, N!, 0, ...) .

(44.7)

From (44.2) it then holds for semigroups (and groups) or rings, respectively, that a'#i' = (..., --j-1, aiate El+le ...)

or at Y! =

l..., 0, alNie 0, ...)

,

(44.8)

and from (44.3) in the case of modules and rings that

a; + i 9 = (..., 0, at +

Nt,

0, ...) .

(44.9)

Hence we see that in all cases with fixed i all the a; constitute a substructure S; of S, for which the isomorphism (44.10)

S, x s,

holds. This means, according to the above generalization of the concept of a direct composition, that in the cases (44.4) and (45.5)

S=Si0Sa®... and

S=SiED S2ED

....

(44.11) (44.12)

respectively.

S is already a (common) extension structure of the Si, S', ..., which are, however, no longer the original S1, S2, but by (44.10) are only isomorphic to them. In order to embed Sx, S2, ... in place of Si, Sz in S, we note first of

DIRECT PRODUCTS. DIRECT SUMS

143

all that Si, S2, . . ., taken pairwise, have only the neutral element (s1, E2, ...) or (0, 0, ...) of S in common. Therefore we assume that the neutral element of SI, S2, ... is also common, which we denote for semigroups (groups) by s and for modules (rings) by 0, and that the S11 S21 ... contain no other common element; it is evident that there is no objection to these assumptions. Finally, by way of (44.10), we may (simultaneously) embed the S1, 5.,, ... in

S, in place of Si, S2, .... Thereby in S each at is replaced by a; (ai E S;; i = 1, 2, ...). After the embedding (44.11), (44.12) again assume the form (44.4) and (44.5), respectively, where, however, the S11 S21

... are now sub-

structures of S. In order to discriminate this case from the more general one, we say that S is the direct composition (i.e., the direct product or the direct sum) of the substructures SI1 S2, ... . Instead of this, in the four cases which we have considered, we use expressions similar to those used at the beginning of the Theorems 86, 87, 88, 89 (see below), which give a simple criterion for S to be the direct composition

of the substructures S11 S2, .... We shall then regard these criteria as the "final definition" of this notion. We further examine the case where S is the direct product or sum of the substructures S1, S2, ..., and prove that S = {S19 S2, ...}

(44.13)

,

i.e., S is generated by the elements of S11 S2, ..., and in the two cases the elements of S may be uniquely written in the form

a = a1a2.... and O= a1 + a2 + ..,

(a; E Si)

(44.14)

respectively, where among the act, a2, ... only finitely many are distinct from s or 0, respectively. [Without this condition (44.14) would be meaning-

less, so that we always assume it, either explicitly or implicitly, in connection with (44.14).] We call (44.14) the (unique) component representation of the elements of S, the a1, a2, ... the (multiplicative or additive) components of a, and a; the 1`h component or even the S,-component (of a).

It is sufficient to prove assertion (44.14) since from it follows (44.13). We consider, for instance, the case of the direct product. Before embedding, the elements of S might be uniquely written in the form (44.1) so that there are only finitely many a, different from E. By (44.2) and (44.61)

(al,a2i...)=a1,a2,...I

(44.15)

where the ai with finitely many exceptions are equal to the unity element of S.

This means that, after embedding, (44.1) becomes (44.141). We can prove (44.142) similarly. 6 R.-A.

STRUCTURES

144

For finitely many S1, . . ., Sn, (44.14) signifies that

S=S1...Sn or S=SI+...+SA, where the product or sum of complexes on the right-hand side is schlicht. (The same

holds for infinitely many S1, S$, ..., the exact formulation of which is left to the reader.) Comparison with (44.4) and (44.5) shows that the direct product and the direct sum of substructures is, in the general sense, the product or sum of these substructures. (This is dealt with more precisely below.)

For a complete direct composition S of infinitely many S1, S2, , .., the above embedding may be carried out in exactly the same way and the S1, S2, ... will be substructures of S after embedding. Neither (44.13) nor (44.14) is then valid, since then the right-hand side of (44.13) is only the direct composition of S1, S8, ... We see from this the importance of the above finiteness condition.

We have still to examine how the composition in S may be expressed by means of the component representation (44.14) of the elements. This leads to a surprising result and to the new definition of direct composition. First we consider the case where the S. are semigroups with unity element. Take two elements of S in the component representation (44.141): a = ala2 ..., 9 = P1P2 ...

(ar, #t E S1) .

(44.16)

In order to compute the product ao, we first assume a, fi (without embedding) to be in the form (44.1) : (X = (01,cc2,...), N = (,9,#22

...).

Then by virtue of (44.2) aN = (001, a2,2, .

. .),

and by (44.15) ofQ/

//

RR

//

,qq

= (ccj,9)' (MAY

...,

whence (after embedding) we get the component representation a,9 = 21#1 a2,9.5 ... .

(44.17)

This is the required rule for multiplication in S which we can formulate as: In the direct product S the elements may be multiplied componentwise, more precisely, the ith component of the product of two elements of S is the product of the it' components of the factors. Writing this rule as a formula we get S

S, ( a -* at)

(i = 1, 2, ...) ,

(44.18)

i.e., we get an endomorphism of the direct product onto its ith component,

if we assign to each element its 1 h component. This endomorphism is

145

DIRECT PRODUCTS. DIRECT SUMS

called the projection of S onto S,. This is also valid for groups, modules and rings, since (44.18) remains valid in these cases, as will be shown. An important inference can be drawn from (44.17). To do this we apply (44.17) to 1a: Na = N101 . /920'2

... .

By comparison with (44.17) we see that afi = #a if for each i (= 1, 2, ...) at least one of a,, /9, is equal to e. This condition is trivially satisfied if a = 0'r (E S) ,

k).

fl _ /9k (E Sk)

Consequently we have proved that the components of S are elementwise interchangeable. This proves the "only if" part in the following theorem. THEOREM 86. A semigroup S with unity element is the direct product of its

subsemigroups S, S2, ... if, and only if, (i) S1, S2. ..... contain the unity element of S, (ii) S1, S21 ... are elementwise interchangeable, (iii) the elements of S may be uniquely written in the form (44.141).

The "if" of this theorem remains to be proved. To do this we assume that the conditions are satisfied for S1, S2,... . We consider two arbitrary elements of S in their unique representations: a = 0(10'2 ..., N = /911'2...

(0',, N; E Si).

/k0', (i # k), it follows that

Since, according to the assumption, 0'N = al /91 "42 .

... .

If we compare this with (44.2) we see that the isomorphism S ,. S® ® S., ®

...

(a -- (0'1,0'2,...))

holds. (By this we mean S1 ® S2 ® ... without embedding.) This proves Theorem 86. THEOREM 87. A group S is the direct product of its subgroups S1, S2, .. . if, and only if, the elements of S may be uniquely written in the form (44.14) and the S1, S2 ... are elementwise interchangeable. This last condition is satisfied if, and only if, the subgroups S1, S2, ... are normal. COROLLARY. If the intersection of two normal divisors A, B of a group is the unity element, then AB = A ®B B. (44.19)

AB is then a normal divisor of the given group.

STRUCTURES

146

The first part of Theorem 87 follows from Theorem 86. In order to prove the second part we first assume that S = S1 ® S2 0 .... We have to show that the S, are normal in S. Since Sk, for k # i, is elementwise interchangeable with Si, it follows from (44.13) that Si is normal in S. We now assume, conversely, that S1, S21 ... are normal divisors of the group S and that the elements of S maybe uniquely written in the form (44.141).

Since Sl, S21 ... are normal, of kal 1 ak 1 = ai - akaI lxk 1

(a; E SI, ak E Sk e

i # k)

belongs to both subgroups S;, Sk. Since, because of the unique representability of the elements of S in the form (44.141), these have only the unity element a of S in common, it follows that

a,akai ]. ak 1 = a

,

i.e., aak = akG'i. This proves Theorem 87. We prove the corollary as follows: It follows from the assumptions, by Theorems 68 and 74, that AB is a normal divisor of the given group and at the same time the schlicht product of A and B. Consequently we may write its elements uniquely in the form ab

(a E A, b E B).

By Theorem 87 the truth of the corollary follows from this. We now consider the case of modules. It follows from the above that if the module S is the direct sum of the submodules Si, S2, ..., then two elements of S have by (44.142) the component representations a=a1+a2+...,

#=#I+fg2+...

(ai,f3iES)

(44.20)

so that their sum has the component representation

GE +P=((X1+PI) +(a2+P2)+....

(44.21)

The additive analogue of (44.18) is similarly valid. Because of commutativity the following simpler theorem can replace Theorem 87. THEOREM 88-4 module is the direct sum of its submodules S1 S21 ... if, and only if, the elements of S may be uniquely written in the form (44.142). Only rings remain to be considered. Let S be the direct sum of the subrings S1, S2, ... . As in the above case the component representation of the elements of S may be expressed by (44.142); further, by virtue of the definition (44.3) of addition, the module S+ is the direct sum of the submodules St. S t . . . . . Hence the sum a + P of two elements a, fi of S is computed according to the rule formulated by (44.20) and (44.21). We have still to .

DIRECT PRODUCTS. DIRECT SUMS

147

compute the product a#. For that purpose we shall conclude as in (44.16) and (44.17), but with an essentially different result. We have now to proceed from (44.20) [instead of (44.16)], for which we write first

a=(x1,a2,...)Y= 01,N2,...). Hence follows by (44.2)

MP =(aif1,a212, hence, on account of (44.3) anRRd`` (44./62),

000= (a1N' + 12)' + ...,

and finally (on account of the embedding) (44.22)

all = °1191 + a2fl2 + ....

Consequently the sum and the product of two elements (44.20) of S are computed according to the rules (44.21), (44.22). Of course, (44.18) also follows for this case. We again draw an important inference from (44.22) [as above from (44.17)]. We apply (44.22) to two elements

a = a; (E S), P = Rk (E Sk)

(i 0 k) .

Since a1 = 0 (1 96 i), F', = 0 (10 k), it follows from (44.22) that a;Yk = 0, i.e., S;Sk = 0 (i # k). (44.23) THEOREM 89. A ring S is the direct sum of its subrings S11 S21 ... if, and only if, it holds when they are considered as modules, i.e. its elements

may be uniquely written in the form (44.142), and (44.23) holds. Here (44.23) may be replaced by the condition that the subrings SI, S21 ... are ideals.

We have already proved above the assertion "only if". In order to prove the assertion "if" we assume that the conditions stated are satisfied. By Theorem 88, S+ is then a direct sum of the submodules Sl , S2 , . . ., thus the sum of two elements a, fi of S, given in the form (44.20), is computed according to the formula (44.21). For the product of the same elements formula (44.22) follows from (44.20) and (44.23). Both together mean that S is the direct sum of the subrings Sl, S21 ... .

For the proof of the second part of the theorem we first assume that S is the direct sum of the subrings S1, S2..... From (44.13) and (44.23) it then follows that SS,=S,S=S?

(i= 1,2,...).

STRUCTURES

148

Since, moreover, S, is a ring, Sic S,. This means, together with the former, that all S11 S21 ... are ideals of S. Conversely, if the elements of S may be uniquely written in the form (44.142), i.e., S+ is the direct sum of the submodules Si , SZ , ... , and if S1, S21 ... are ideals of S, then

S,f1Sk=0, S.Sk c S,f1Sk

(ilk).

Hence (44.23) follows, by which Theorem 89 is proved. If a structure S is the direct composition of the substructures S11 S2, (as in Theorems 86 to 89), then we say the substructures S1, S2, . . . constitute a direct decomposition of S. Corresponding to the special cases

S=S1®S2®..., S=S1ED S2@...

(44.24)

we call (44.241) a direct product-decomposition and (44.242) a direct sum... . We might have defined direct decomposition immediately by the properties defined in Theorems decomposition of S into the substructures S1, S2,

86 to 89 and from this attained the above general definitions of direct composition. However, this way would not have been shorter, and the uniformity of the point of view would not have been so prominently displayed.

We emphasize once more that a direct composition (by embedding) can always become a direct decomposition, but this does not necessarily happen. However, if either (44.241) or (44.242) holds and there is no direct decomposition, then the following symbolisms may replace (44.241,2)

S,:;S1®S20..., S;ZeSI®S20..

.

(44.25)

This form is always used when the Si are substructures of S and (44.24) is their direct composition (in the general sense), provided that no direct decomposition is involved. (Concerning this matter, the following theorem provides an interesting example.) TuaoI uM 90. Let

S;:t; StED ...®S

(44.26)

be a direct decomposition of a ring S with unity element s into finitely many ... , S and let

subrings S1,

a = al + ... + a,,

(44.27)

be the associated component representation of the elements a of S, where a; (E S;) is the ith additive component of a. Let Sx, S; denote the semigroups of S and S,, respectively, where S; is a subsemigroup of SX and has

DIRECT PRODUCTS. DIRECT SUMS

149

s, as unity element. Further, put

&;=...+eI-I+cci+E;+I+... (=E-E;+a;).

(44.28)

a = &I ... &,, .

(44.29)

Then

For a fixed i, the a; constitute a subsemigroup Sl of S X with unity element s. Then

SxSi ®...

(44.30)

Sx=Si ®... ®Sn,

(44.31)

Si

(44.32)

Si (a; -* &;)

.

The component representation of the elements associated with the direct decomposition (44.31) is given by (44.29), so that &; is the i`h multiplicative component of a [in the decomposition (44.31)].

(Note that (44.30) is not true with = instead of , since according to (44.26) Sl Sk = 0 for i k.) Coxol.LnxY. For the regular semigroups S*, S*,

of the above semigroups

Sx, S19 S1 S*

(44.33)

S* = Si ®... ®Sn ,

(44.34)

S' (a; - &;)

(44.35)

S*

hold.

Here S*, St consist of the components a; or &;, respectively, of the regular elements cc of S.

The proof of the theorem is very easy. Because a = sa = as, by (44.27) and (44.22),

Hence, again by (44.27), it follows that a; = e;a; = a;e,, therefore e, is the unity element (of S' and so also) of S;. Hence and from (44.22), (44.28) it follows that

ee-I + ;/3i + EI+1 + ... (= E - E; + ;3j .

(44.36)

Because a;fl; E S;, &;fl; E Sl. This shows that Sl is a semigroup. It follows from (44.28), (44.36) that a (= e;) is the unity element of S; . Because of (44.22), (44.28)

+a,,.

STRUCTURES

150

Since, on the other hand, by (44.28) ai, &i determine each other uniquely, it follows from the uniqueness of (44.27) that (44.29) is likewise a unique representation of the elements (of S and so also) of Sx. From (44.22), (44.28)

E - Si - ek + ai + F'k = kak

((xi E Si , F'k E Sk , 1 9 6k ),

i.e., the Si , Sr , ... are elementwise permutable. The above proves on account of Theorem 86 that (44.31) is the direct decomposition of S x into the subsemigroups S; , ..., Si .

Since xi, ai determine each other uniquely, the isomorphism (44.32) follows from (44.36). Hence and from (44.31) follows (44.30). This proves Theorem 90. Because of the isomorphism (44.32) cc, is regular in S; if, and only if, &i is regular in S x, whence (44.35) follows. By (44.29) a is regular (in S ") if, and only if, every component &i is regular in S; (i = 1, . . ., n). Accordingly (44.34) is a consequence of (44.31). From (44.34) and (44.35) we get (44.33). Consequently the corollary is proved. Theorem 90 and the corollary would be false if in (44.26) there were infinitely many components S1, SZ, ... instead of S1, ..., S,,. We see from (44.22) that the direct sum of infinitely many rings can have no unity element. On the other hand, the complete direct sum of arbitrarily many rings with unity elements evidently has a unity element.

In cases (44.4), (44.5) we call the homomorphic image Ti (9 Si) of a

substructure T of S, obtained from the projection S - Si (a -' ai), the i`s or Si-component of T or the component of T in Si. We now give some simple properties of direct decompositions. From Theorems 86 to 89 it follows that (44.4) and (44.5) remain true if the components S11 S21 ... are arbitrarily permuted.

If

S=S1®S2®..., S,=T1®T2®..., then evidently

A similar "replacement rule" is valid for direct sums. If S decomposes into the direct product

S=S1®S2®.. and T, is a subsemigroup of Si with unity element (1= 1, 2, ...), then T = T1 T2

...

DIRECT PRODUCTS. DIRECT SUMS

151

is a subsemigroup of S with unity element, and

T=T1®T.9®.... The same holds for direct sums. A similar result holds for groups. Nevertheless all substructures of the same kind as the considered structure S do not always arise in this way. The following simple example illustrates this: Let a group G be decomposed into the product of two infinite cyclic subgroups:

G = {a) a {/1} . The elements arfik (2 1 i + k) constitute a subgroup H of G ; however, it is clearly impossible to have

H = A a B, A s {a}, B q (13} . We add to this the following definition. if G = G1 a G, a . is the direct decomposition of a group G into the subgroups G1, G:, ..., and H is a subgroup of G with the property that for each i (= 1, 2, ...) all the elements of G, occur among the ith components of the elements of H, then H is called a subdirect product of G1, G81 ... This definition also holds when H and G1, G2, ... are replaced by arbitrary isomorphic groups. The subdirect sum of modules and of rings are similarly defined. For subdirect compositions cf. McCoy (1948) and FUCHS (1952b).

A structure (in particular a group, a module or a ring) with more than one element is called (direct) indecomposable if it may not be directly decomposed

into at least two substructures (of the same kind) with more than one element. It is evident that simple groups, modules and rings with more than one element and, in particular, all skew fields, are indecomposable. The same holds for rings with more than one element without zero divisors. One of the above structures is called completely reducible, if it can be directly

decomposed into finitely many simple substructures (of the same kind) (cf. § 63, Theorem 146). For direct composition cf. KERTESZ (1951- 52), (1952).

EXAMPLE 1. The four-group is the direct product of two groups of order two. EXAMPLE 2. The infinite cyclic group, though not simple, is indecomposable. For,

if {e} is such a group and A, B are two subgroups of it, then take an element ea, eb (96 e) from each of these; since nab (A e) is a common element of A and B, {a} _ = A e B is impossible. EXAMPLE 3. The module of the rational numbers is indecomposable. (Prod' as in Example 2.) EXAMPLE 4. For the cyclic group {e} of order six

{e} _ {e=} a {n3} . EXAMPLE 5. 3/(6) = 9/(2) a J/(3). To show this, let a denote the unity element of J/(6), i.e., the residue class 1 (mod 6). Then 0, e, ..., 5e are all the elements of 3/(6). The sets <0, 3e>, <0, 2e, 4e> each constitute a subring of 3/(6), which is even an ideal. Then we have the direct decomposition J/(6) = <0, 3e> a <0, 2e, 4e>. On the other hand, <0, 3e) .9/(2), <0, 2e, 4e) J/(3). EXAMPLE 6. The zero divisors of a direct sum of skew fields are those elements, of which at least one component vanishes. The direct sum of infinitely many skew fields consists entirely of zero divisors.

6/a R.- A.

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EXAMPLE 7. If a group G has the direct decomposition G = A ® B and H is a subgroup of G which contains A, then H =A ® (B n H).

(44.37)

Since the factors on the right-hand side contain only the unity element in common,

their product is direct. Evidently (44.37) holds with ? instead of =, so that we need only show that (44.37) holds with s instead of =. Every element of H may be written in the form a# (a E A, /1 E B). Because a, afJ E H, fi E H, so that fi E B n H. Consequently the assertion is proved.

§ 45. Basis

It is obvious that a direct composition is commutative if, and only if, its components are commutative. In particular, the direct product of cyclic groups is always an Abelian group. If an Abelian group G ( e) has the direct decomposition G = {coi} ® {cot} 0

...

(45.1)

into cyclic subgroups (wl}, {co.,}, ... ( e), then we call the elements wl, w2, ... (or the set of these elements) a basis of G. By the help of this basis we may write the (unique) component representation of the elements of G in the form

a=co'wz'...

(al,a2,... E.2).

(45.2)

The factors co r, uniquely defined by a, are just the components of a. Conversely, if for wl, W2, ... (E G, e) the elements of G (to within the order of succession) can be uniquely written in the form (45.2), then wl, w2, ... is evidently a basis of G. Then we call (45.2) the (unique) basis representation of the elements of G. We see from (45.2) that a basis is a minimal generating system of G. Any (finitely or infinitely many) elements wl, co., . . ., different from the unity element e, of an Abelian group are called independent (of one another), if from every equation (45.3) wi'w28 ... = e it follows that all the factors m,°1 are equal to E. Evidently the independent elements are always distinct. Further we see that a set of elements from G is independent if, and only if, the same holds for the finite subsets of this

set. This means that independence is a property of finite character. It is trivial that the basis elements must always be independent. THEOREM 91. A system of generators of an Abelian group G (0 e) is a basis if, and only if, it consists of independent elements.

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BASIS

We have still to prove the "if" part. Assume that wl, c02, ... are independent generators of G. In any case every element of G may be written in the form (45.2), so we have only to prove that the factors of the right-hand

side are uniquely defined therein by a. This is the case, in fact, since if

2 ... = wl w2 .. . b,

wa,wa, 1

b,

then coal -b'w2'' b' .. = s , whence it follows that

,,at-bt=E,

Coal oCt)bl

(i = 1, 2, ..).

Since modules are Abelian groups, all the previous arguments, after changing to the additive notation, are equally applicable to modules. Further we understand by a basis of a ring R (0 0) every basis of its module R+. Of course, this definition also applies to skew fields. According to this, a basis

of a module (ring, skew field) different from 0 means a set of elements wl, w2, ... such that all its elements may be written in the form a = al wl + a2 w2 + ...

(al, a2, ... E a/)

(45.4)

where the terms at w, are uniquely defined by a. We call (45.4) the basis representation of the elements. The elements wl, w2, ... (; 0) of a module (ring, skew field) are called independent, if

alwl + a2w2 + ... = 0

(45.5)

implies that al wl = a2 w2 = ... = 0. We also say that w1, w2,

. are linearly independent. (In skew fields we shall later define an independence different from linear independence. We omit "linear" when the meaning is clear.) The above considerations, especially Theorem 91, remain valid in the present case after the appropriate modifications. Finally we understand by a basis of a commutative semigroup H ( e) with unity element s a set of elements wl, w2, ... (# E), such that we may write all elements of H with uniquely defined factors co" in the form

a = w1`w2' ...

(al, a2i ... z 0).

(45.6)

We call (45.6) the basis representation of the elements of H.

We call co" or afol the i`h component of a in the basis representations (45.2), (45.4), (45.6). If an Abelian group has a basis and

are the basis representations of two elements of the group, then aY R = fl war+bt 11

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154

is the basis representation of the product. We can express this briefly by saying that the components of the basis representation are multiplied when the elements are multiplied. The same applies to the other cases. If a ring R has a basis Col, W21 ... and if the basis representations WiWk = Y_ Ctkrwr 1

are known for the products of the basis elements, then, because of the distributivity, we can compute the product of elements of R according to the formula Yaico) (Y bkwk) = Z aibkCiklW1 1

k

i,k.1

A similar result holds for the case where the w; do not constitute a basis, but an arbitrary generating system of R+, although the terms cikl col are then not uniquely defined. (Cf. RiDEI (1953).) EXAMPLE 1. The infinite cyclic group {a} has the basis a. Another basis is a-'. There are no other bases. A finite cyclic group {a} of order three or more has the two bases a, a 1 and other bases if the order is at least five; if the order is, e.g., six, then a3, a3 is also a basis. EXAMPLE 2. If the group G is the direct product of two cyclic subgroups {a}, (,'E e), then G has the basis a, j9. More generally a#', 11 is a basis if and only if o(p) I o(a). EXAMPLE 3. The semigroup r* has only one basis, which consists of all prime num-

bers. (This will be proved later.) EXAMPLE 4. The field . 0 has no two linearly independent elements, consequently it does not have a basis. EXAMPLE 5. The ring of complex numbers of the form a + bi (a, b E 9) (i.e., the so-called Gaussian integers) has the basis 1, i. EXERCISE. Let M be the direct sum of the (cyclic) submodules {cwt} . . ., (co") (96 0).

There is one, and only one, ring R with R+ = M, in which the product of the basis elements is defined by wi wk = (01,if,k) (i, k = 1, ..., n). § 46. Congruences

We are already acquainted with the importance of the compatible classifications of a structure. We also know that a classification of a set and the corresponding equivalence determine each other uniquely. It is high time

to avail ourselves of the possibility inherent in this for the purpose of the compatible classifications, which we could have done before now. The equivalence relation belonging to a compatible classification C of a structure S is called a congruence relation or briefly a congruence (with respect to @ or mod C) and we denote it by writing

a - /3 (mod -0)

(a, /3 E S)

.

(46.1)

This means then that a, /3 represent the sameclass (mod 6?). If (46. 1) is satisfied

by a, f9, then we call a, /3 congruent elements (mod C). We also say that a is

CONGRUENCES

155

congruent to j9. C itself is called the module of the congruence (46.1). If there can be no misunderstanding, instead of (46.1) we may write

a - #.

(46.2)

If, in particular, S is a group, a module or a ring, then instead of (46.1) we write

a m fl (mod H) ,

(46.3)

where H denotes the principal class, namely the normal divisor, submodule or the ideal of S, the cosets of which constitute the classification C. Then H (instead of @) is called the module or kernel of the congruence. If S is a module or ring and H is given as a submodule {,u, v... } or an ideal (,u, v...),

respectively, then we write for (46.3) a - j9 (mod p, v...). Congruences in semigroups S have no kernels. However, if S is commutative and finitely generated, then a congruence in S may be characterized by the help of a "kernel function". (Cf. RsrE1 (1963).) From the rule (30.3) and the transfer rule of equivalence we derive the following criterion : For a ring S, an equivalence relation -, defined in S, is a congruence if, and only if,

KmA=K+ eMI + v, PK= eA, K0=AP (K, .I, Q E S),

(46.4)

(where we have taken into consideration the commutativity of addition). For other structures, we have to alter the right-hand side of (46.4) correspondingly. We sometimes use the notation (46.1) in cases where the classification e is not compatible, but then (46.1) means the associated equivalence relation.

Nom. Clearly if mod

s/I' S/('',

are two compatible classifications of the structure S and every class is the union of certain classes mod C. [In particular

G/N-.G/N'(N

N'), R/a- R/a'(aca'),

if G is a group, R a ring, N, N' are normal divisors of G, and a, a' are ideals of R.] This useful rule can be worded in the language of congruences as follows: S/e - S/c', if all congruences mod C are also true mod C'. We now take into consideration certain congruences

a, - # i

(i = 1, 2, ...).

(46.5)

On account of rule (46.4) and the axioms of equivalence we may obtain from (46.5) further congruences a = fi, which we call the congruences resulting

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156

from (46.5). The congruences (46.5) are, of course, included. If a = ft is a congruence resulting from (46.5), we can say that a = B follows from (46.5).

By the help of the concept of resulting congruences, we can define a congruence relation in a structure S in the following way: We give in S arbitrary "formal" congruences (46.5) so that the a;, fe are quite arbitrary elements of S. A congruence relation is defined in S by the set of all congruences resulting from (46.5). We call this congruence the congruence relation induced by (46.5) or briefly induced congruence. This concept corresponds

to that of induced equivalence; the justification of the definition for induced congruence closely follows from the proof of Theorem 2. It suffices to illustrate this by an example. Consider in a group G the two congruences

a = A.

y = B.

(46.6)

As resulting congruences, we have, first and foremost, all the "identical congruences"

eme

(QEG).

(46.7)

By (46.4) we obtain ay = fly, fly = fl8, so that

ay=A6.

(46.8)

(Thus we have the following rule: The multiplication of two congruences results in a congruence.) From (46.6) it follows that #-'a = e, ,B-' IS a-', consequently

M-1 _ f'.

(46.9)

By repeated application of (46.7), (46.8), (46.9) we obtain e.g., (in view of equivalence properties) the resulting congruence

efl2ag-i« s =

z

If{-s

(), a arbitrary in G).

All computations with the elements of the factor structure S/6? (i.e. with the classes mod C) may of course be replaced by computations with the congruences mod

existing between the elements of S, i.e., by computations

mod e. EXAMPLE 1. If we make the computation with the elements of the ring 9 mod 7, then we obtain, among others, the congruences 3 + 6 = 2, 3. 6 - 4 (mod 7). These may be interpreted as saying that in the ring 9(7) the sum and the product of the residue classes 3, 6 (mod 7) are equal to the residue classes 2 and 4 (mod 7) respectively.

EXAMPLE 2. In a group the congruence a = fi is equivalent to a#-' = e (and to f' a = e). According to this it is sufficient in groups to consider the congruences of the form Q e. In modules and rings it is sufficient to consider the congruences of the form 8 = 0. EXERCISE 1. The congruence induced by certain relations

arm#I

(i = 1,2,...)

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157

in an associative structure S agrees with the equivalence induced by the relations

at=A , Boa;m B6A, ato0mA o0, ooatoam Bofltoa (i = 1,2,...; o,aE S), where o indicates any composition e in S. EXERCISE 2. Prove the theorem: All (compatible) classifications e' related to a compatible classification e of a structure S, and only these, result when we take all the mappings a of S into itself such that

a(ooa)=aooaa(mod )

(o, a E S)

(where o indicates any composition in S), for which there is a further mapping b of S into itself such that

abo = o (mod ) and define e' by B

a (mod C') a at) = as (mod C).

Then

b(e o a) = be o ba (mod e'), bao - B (mod

B m a (mod e) p be = ba (mod e').

§ 47. Quotient Structures Here and in § 48 we shall deal with some structure extensions, where the new structure (under certain conditions) satisfies certain new axioms. As a

simple special case, we shall obtain the extension of the semiring ,f of natural numbers, first to the semifield of positive rational numbers and then to the rational number field Y0(just as one is accustomed to in schools).

First of all we require extension groups of semigroups. The regularity of the given semigroup is then a trivial, necessary condition, which, however, according to MALCEV (1937) is not sufficient. On the other hand, as we will see, every commutative regular semigroup may be extended to an (Abelian) group.

Let S = denote a given semigroup or semiring, in particular a ring, and H the set of regular central elements of S, which we suppose to be non-empty. Then H is a regular commutative semigroup. By a quotient structure of S we mean an overstructure T (9 S) of the same kind as S, which

is so constituted that T has a unity element, each element B (E H) has an inverse B-' (E T), which lies in the centre Z(T) and each element of T may be written in the form B-Ia (47.1) (a E S, e E H) . (Here B-I means the inverse of o in T.) It will be shown that T always

exists and is (to within isomorphism) uniquely determined. According to the context, we call T the quotient semigroup of the semigroup S or the quotient

semiringof the semiring S or the quotient ring of the ring S. If in particular T

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158

is a group, a semi-skew-field or a skew field then we refer to the quotient group, quotient semi-skew-field or quotient skew field (of S), respectively. In the commutative case the latter two are called the quotient semifield and quotient field (of S). THEOREM 92. Every semigroup S with at least one regular central element

has (to within isomorphism) a uniquely defined quotient semigroup T. If S is commutative and regular, then T is an Abelian group (the quotient group of S). Let H denote, as above, the semigroup of regular central elements of S. We denote by o, a, r the elements of H. We define a skew product U = S H by the multiplication

(a, 00, a) = (af, ea) .

(47.2)

(The notation for the skew product has nothing to do with the product of complexes. By the way, if S has a unity element, then U is isomorphic with S ® H.) Associativity is evidently satisfied, so that U is a semigroup. We define in U an equivalence relation by (a, e)

(f, a) a as = e8 .

(47.3)

This is possible, since first of all clearly (a, p) __ (a, O). If, furthermore,

(a, e)= (f, a), (a, e)= (Y, r) then by (47.3), as = ef, roc = Oy, hence Orj9 = rofl = raa = ara = aeey = Oay,

and consequently r# = ay, i.e. (,B, a) - (y, r), again by (47.3). We prove that this equivalence relation is a congruence. On account of the duality principle it is sufficient if we show that (a, P) =

a) = (Y, r) (a, P) = (Y, r)

a)

Since by (47.2) the right-hand side may be written as (ya, rp) (47.4) is equivalent to

(47.4)

(yf, -ca),

as=Qfl=rays=rvyfl. Since this is true, __ does in fact denote a congruence in U.

Denote the corresponding compatible classification of U by C. We put

T' = U/? .

(47.5)

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159

We see from (47.3) that all the (e, &(e E H) constitute a class mod We denote this class by 1. Because

(e, e) (a, a) = (a, a) (e, e) = (ea, ear) = (a, a) 1 is the unity element of T'. For fixed a (E S) and e (E H) we consider all the solutions (fl, a) of the congruence (j3+ a) = (ea, e)

This congruence is, on account of (47.3), equivalent to e9 = aea, i. e., to fi = aoc. This means that for a fixed a (E S) all the (aa, a) (a E H) constitute a class mod Q. We denote this class by a'. We show that the different a' constitute a subsemigroup S'ofT' and that the isomorphism S

S' (a -> a)

(47.6)

holds. First of all this mapping a -* a' is one-to-one, since from a' _ fl' for any a (E H) we obtain the congruence (ea, e) = (ej9, e). Consequently, by (47.3), the equation a = f4 follows. From (47.2) we obtain

(ea, e) (efl,

_ (e2aq, e2)

(a, fl E S; e E H) ,

thus a'#' Consequently (47.6) holds. For a fixed a (E H) all the (a, ae) (a E H) likewise constitute a class mod @, which may be proved just as the above assertion concerning a'. We show that this class is the inverse of /e' in P. Since (a, ae) (a0, a) = (ao, a) (a, ea) = (a2e, ate)

belongs to the class 1, the assertion is true. We denote the class (a, e) (mod C) by [a, e]. Since (a, e) = (a2a,a2e) = (a, ae) (aa, a) = (aa, a) (a, ae),

it follows that

[a, e] = e'_la' =

a'e'-1

.

(47.7)

Now we embed the semigroup S in T', by way of the isomorphism (47.6), in place of the subsemigroup S' and denote the semigroup so formed from T' by T. We show that T is a quotient semigroup of S. Since T T', T has a unity element. Since in T' each N has an inverse -I, so in T each a has an inverse e-1 and this must be the inverse image

of o'-' under the isomorphism (47.6). Thus after embedding, by (47.7), the arbitrary element [a, e] of T' passes into e-Ic (= ae-1), so that 0-1 lies in Z(T) and every element of T appears in the form (47.1), as was required.

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160

In order to prove the uniqueness requirement of Theorem 92, we assume that, besides T, also T1 is a quotient semigroup of S. We have to show that T T1. By denoting the inverse of a in T1 by e(-1), we show that

T ., Tl

(e-la -- e(-11a) .

(47.8)

First we prove that this mapping is one-to-one. This assertion is equivalent to

e-1 a= a-1# r-s e(-1>a =

(47.9)

From the left-hand side of (47.9) it follows that eao-la = eaa-'fl, as = e(-')a(-1)ofl, coni.e., as = ofJ. Furthermore (in TI) sequently e(-')a = a(-')#. This proves the "r*" part of (47.9). The °° " part follows similarly, which proves the one-to-one property of the mapping (47.8). We next consider two arbitrary elements e-la, a--'# of T. Their product (ea)-1 ai4. Since this is mapped onto (ea)(-1)afl = e(-1)aa (-1)#, the is proof of (47.8) is complete. In order to prove the last assertion of the theorem, we consider the special case where S is commutative and regular. Then H = S and each element of S now has an inverse in T. Accordingly by (47.1) T is a group generated by S and so an Abelian group. This proves Theorem 92. THEOREM 93. Every semiring S with at least one regular central element has (to within isomorphism) a uniquely determined quotient semiring T. If S is zero-divisorless and commutative, then T is a semifield (the quotient semifield of S). The semiring S and its semigroup SX have the same regularcentralelements.

Hence it follows that for every quotient semiring T of S its semigroup T" must be the quotient semigroup of SX. Since this, according to Theorem 92, is (to within isomorphism) uniquely determined, it is sufficient, for the

first part of Theorem 93, to prove the following: We can define in the afore-mentioned semigroup T" an addition in exactly one way, and that in

such a manner that T" becomes a semiring T containing S as a subsemiring.

In order to prove this, we retain the above notations H, o, a, r. Next we assume that we have already defined in T" the required addition. Then

for two arbitrary elements e-1a, a-'# of T

e-"z +a-'#='t-ly

(47.10)

with suitable y, -r. On account of the distributivity in T we get, after multiplication by ea, the following equation

as + ei = ea'r-ly

QUOTIENT STRUCTURES

161

whence r-'y = q_Ia-1(aa + ei9). This, inserted in (47.10), results in o-1a + a-1j9 = o-1a-1(aa + Pfl).

(47.11)

Conversely, let us assume (47.11) as a definition of the addition in T. An easy computation will show that this addition is commutative and regular

and that distributivity is valid, thus T is a semiring. If, furthermore, both terms on the left-hand side of (47.11) lie in S, i.e., Q`a = y; a-Ij3 = = S (y, S E S) then a = ey, 9= orb. Therefore the right-hand side of (47.11)

is y + b (namely, the sum of y and b in S). Hence it follows, since S" is a subsemigroup of T, that T contains the semiring S as a subsemiring. So the first part of Theorem 93 is proved.

We now prove the second part of the theorem. The elements different from 0 in T constitute, by (47.1), the quotient semigroup of the semigroup of elements different from 0 in S. However, if the semiring S is zero-divisorless

and commutative, then the last-named semigroup is regular and commutative, consequently, by the second part of Theorem 92, the elements different from 0 in T constitute an Abelian group. This means that the quotient semiring T is a semifield. So Theorem 93 is proved. THEOREM 94. Every ring S with at least one regular central element has (to within isomorphism) a uniquely determined quotient ring T. If S is zerodivisorless and commutative, then T is afield (the quotient field of S). A quotient ring T of S must be afortiori a quotient semiring of S. Since this, by Theorem 93, is uniquely determined, we have only to show that it

is a ring. This follows simply from the fact that in it, by (47.11), the element a-a has the additive inverse Thus we have proved the first part of Theorem 94. The proof of the second part of Theorem 94 follows from this together with the second part of Theorem 93. NOTE. Theorems 92, 93, 94 leave two problems unresolved. The first is how to distinguish different elements of the quotient structure T of S, in other words, to find the necessary and sufficient condition for two elements o-1a, a-19 (a, 9 E S ; o, a E H) of T to be equal. This condition is expressed

in the form as = e fl, so that this question can always be posed within S. (In general we cannot state any more.) The second problem is how the elements of T can be multiplied and, if T is a semiring or ring, how they can be added. This is done according to the trivial rules

Ia . a-1# = (eo)_Ia#, ,g_1a +

a_t _ (Oa)-1(aa +

already mentioned. A generalization of the first part of Theorem 94 may be found in ASANO (1949) Also cf. FRIED (1953-54) and WEINERT (1962, 1963, 1964).

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162

§ 48. Difference Structures

Let S = denote a given semimodule or semiring. By the difference structure T of S we understand an overmodule or an overring of S such that all elements of T may be written in the form

a - fl

(a, 9 E S).

(48.1)

(Here -P means the additive inverse of j3 in T.) Corresponding to the previous two cases we then call T the difference module or difference ring, respectively, of S. It will be shown that T always exists and is (to within isomorphism) uniquely determined. If T is, in particular, a skew field or field, then we use the more significant terminology difference skew field or difference field, respectively (of S). THEOREM 95. Every semimodule has (to within isomorphism) a uniquely determined difference module.

Since a semimodule is a commutative regular semigroup in additive notation, Theorem 95 follows from Theorem 92. THEOREM 96. Every semiring S has (to within isomorphism) a uniquely determined difference ring T. Since S+ is a semimodule, so T+ is (by Theorem 95) the uniquely determined difference module of S+. Consequently if we show that in this difference module T+ a multiplication may be defined in exactly one way so that T+ becomes an overring T of S, then we shall have proved Theorem 96. This multiplication can only be (on account of distributivity in rings)

(a-i)(Y- h)=(ay+f8)-(aS+fly)

(a, f, 7,6ES).

(48.2)

Conversely, let us take (48.2) as the definition of multiplication in T. One can easily show that this multiplication is associative and that distributivity holds. Consequently T is a ring. If in particular the factors of the left-hand

side of (48.2) lie in S, i.e., a - = o, y - S = a (o, a E S), then a = f + o, y = S + a. Thus the right-hand side of (48.2) is just ea (i.e., the product of o, a in S). Hence it follows, since S+ is a subsemimodule of T, that T contains the semiring S as a subsemiring. This proves Theorem 96. THEOREM 97. If S is a semi-skew-field and for any two different elements

a, f (E S) at least one of the equations

,B+ a=«,

a+e=fl

(48.3)

with e E S is solvable, then the difference ring T of S is a skew field (the difference skew field of S).

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163

Since T is a module, it follows from the assumption about (48.3) that a - 8 is equal to 0 or - o (o E S). On account of (48.1) this means that every element of T appears among the o, - o (o E S). Since, on the other hand, S is a semi-skew-field, the equation o = a

has for all o, a (E S, the equations

0) a solution

(48.4)

(E S). Since, moreover, T is a ring,

(-o) = -a, o(-E) = -a, (-o) (-5) = a follow from (48.4). All this means, from the duality principle, that the ring T is a skew field, which proves the theorem. We apply Theorem 93 to the semiring..Y' of natural numbers. Accordingly ,f has (to within isomorphism) a uniquely determined quotient semifield. It is easy to see that Theorem 97 may be applied to this semifield, so that this semifield has a difference field. This is called the rational number field, which we now denote as before by Y. Its elements are called the rational

numbers. We may also define .7 so that, by Theorem 96, we form the difference ring r,.7 of..r and then by Theorem 94, the quotient field -9'0 of 7. A detailed and generalized treatment of these questions is to be found in WEINERT (1962, 1963, 1964); for example we note that the above-mentioned facts about ..if" are not valid for every regular commutative semiring and that the condition formulated

in Theorem 97 in (49.3) is not necessary for the difference ring of S to be a skew field.

EXAMPLE 1. The difference ring of the subsemiring {3, 5) _ <3, 5, 6, 8, 9, 10, 11, ...>

of J' is isomorphic with .f. EXAMPLE 2. The condition formulated in Theorem 97 in (48.3) may not be omitted (not even in the commutative case). Let S denote the semifield of the rational functions

r(x) with non-negative rational coefficients. (Cf. § 109.) For S the condition is not satisfied in Theorem 97, nor is the difference ring of S a field. (The quotient ring of this difference ring is a field, viz., the field of all rational functions r(x) with rational coefficients.)

EXERCISE 1. Every extension group of a commutative regular semigroup H con-

tains a subgroup (? H) isomorphic with the quotient group of H. ("Minimal property" of the quotient group.) EXERCISE 2. Every extension field of a commutative zero-divisorless ring R contains

a subfield (2 R) isomorphic with the quotient field of R. ("Minimal property" of the quotient field.)

§ 49. Free Structures. Structures Defined by Equations Let an arbitrary non-empty finite or infinite set

A=(a,1,...)

(49.1)

be given, which we call subsequently the alphabet A, and its elements

STRUCTURES

164

a, #.... the letters (of A). (The nature of the a, 9, ... is a matter of no consequence; they may be arbitrary different "symbols".) The "formal products"

al ... ak

(al...., ak E A; k ? 1)

(49.2)

are called words. (Thus they are the finite sequences formed from the elements

of A. Examples for words: a, j9, afl, aa, /9aaya.) We denote by (A) the set of all words. In this set we define the multiplication

al ... ak #1 ... Nl = al ... Mk fl ... #,.

(49.3)

Accordingly multiplication in (A) means simply the juxtaposition of elements. It is trivial that this multiplication is associative, i.e., (A) is a semigroup. Since this is obviously generated by A, we call it the free semigroup generated by the alphabet A. We have to notice that in this the word (49.2), which was above called a "formal product", is now an actual product. All the different words shall be denoted by cpl, cp2, ... The "formal sums"

'c;4'i =C1T1+c29%2+...

(c,E.7),

(49.4)

with only finitely many c, different from 0, are called word sums. We agree to

disregard in (49.4) the "terms" c,q', with c, = 0. Accordingly, the word sums are now all finite (formal) sums. In addition, we agree to disregard in (49.4) the order of succession of the "terms" of the word sums, and also the coefficients c, = 1. Hence all words and, in particular, all letters occur among the word sums. If all the c, are equal to 0, then we call (49.4) the empty word sum, which we denote by 0. The set of all word sums is denoted by (EA). In this set we define addition and multiplication, respectively, as follows :

Ec;9,i + Id,q, = E(c, + dd)9p, (Ec,9q,) (Ed;991) _ Ic,di9vi9?i .

(49.5)

(49.6)

[The reader may keep in mind that the right-hand side of (49.6) is meaningful only by (49.5). On the right-hand side of (49.5) and (49.6) there are only finitely many c, + d, and c, d1 distinct from 0, therefore (49.5) and (49.6) do in fact define compositions in (ZA).] Evidently both compositions (49.5), (49.6) are associative, addition is commutative and invertible and distributivity holds. Thus (EA) is a ring. Since this is generated by A, we call it the free ring generated by the alphabet A. We have to note that in this the word sum (49.4), which was above called a "formal sum", is

an actual sum of its terms c,gv, and that the empty sum 0 is the zero element.

FREE STRUCTURES

165

In addition to the alphabet A given in (49.1), we now consider a second alphabet equipotent to it

A' = (a', fl', ... i ,

(49.7)

for which we assume that the intersection A fl A' is empty. We have chosen

the notations such that e P is a one-to-one mapping of A onto A. The union set A U A' is also an alphabet. A word formed from this set, a1 .

.

. ak

(a1, ..., ak E A U A';

k > 0)

(49.8)

is called a reduced word, if in this two letters of the form e, eo' (A E A) are never adjacent. The case k = 0 is also permitted and then we call (49.8) the empty word, which we denote by E. We now denote by (A, A') the set of reduced words and define a multiplication in it. The product uv of two reduced words

u=a1...ak, v=#1...#t

(at,tltEAUA')

(49.9)

shall be the reduced word belonging to the word a1... ak #1... #t, which is obtained from this word by cancelling as often as possible elements of the form e, Lo' (Lo E A) which are adjacent. [For instance, for three different letters a, fl, y (E A), the product of the reduced words a.,9y' and ya is equal

to a99.] It is possible to prove that this multiplication is associative, but this involves unexpectedly tedious calculation. If W = K1... Ki (K1, ..., Kt E A U A'; i 1) is a reduced word, then we put *W = K2 ... Kt, W* = K1 ... Kt-1, *W,x = K.` ... Kt--1 (provided, in the last case, that i >_ 2). If two letters u, v (E A U A') constitute

a pair of the form e, t? or e', e (e E A), we write µ T v, otherwise we write ,a 4. v. By Theorem 30, it is sufficient to prove that (uv) v = u (vy) ,

(49.10)

where u, v are two reduced words, which we take in the form (49.9), and y E AUA'. We may assume k, 1 >_ 1, further we assume (49.10) as true for smaller values of 1. On account of this induction assumption and the definition of the product of reduced words, we prove (49.10) by direct computation, distinguishing different cases : If ak t N1, 9, T y then for l > 2 (uv) y = (u* v) y = u* (*vy) = u* *v* _ = uv* = u (vy); and for l = 1 (ak = y and) (uv)y = u*y = u = uc =u (vy). If ak T fli, #Rt 4. y then (uv) y = (u* *v) y = u* (*vy) = u(vy)

If ak 4. fl, Y, T y then (uv)y = uv* = u(vy).

STRUCTURES

166

If oak. #1, #e y y, then (49.10) is trivial.

Hence, the set (A, A') of the reduced words is a semigroup. The empty word s is evidently its unity element. Since an arbitrary reduced word

at ... ak has the left inverse Yk ... #I, where for every i (= 1, . . ., k) either 9; = ai(a; E A) or fl; = a; (a; E A'), (A, A) is a group. Since, in particular, the product oo' of the reduced words o, o'(o E A) is equal to s,

o'=o_i

(oEA).

(49.11)

Hence it follows that the alphabet A is a generating system of the group (A, A). Therefore we call this the free group generated by the alphabet A. We call the structures defined above free structures. If S is a free structure (semigroup, group, ring) generated by the alphabet
,)/l ...

n

(Ni E A, a, > 1; i = 1, ..., n).

(49.12)

0;+1 (i = 1, ..., n - 1). In free groups the only alteration is that the exponents may be negative or zero. In free rings the We can attain uniqueness if we prescribe O;

words occurring in the word sums may likewise be written in the form (49.12). It should

be noted that free semigroups and rings have no unity element.

Looking back to the definition (49.3) of multiplication in the free semigroup H generated by the alphabet (49.1), we can now see that all equations

f = g(f, g E H) become valid equations if the letters a, i, ... are replaced in them by arbitrary elements of an arbitrary semigroup. Exactly the same holds with respect to the definitions in (49.9) and (49.5), (49.6), for groups and rings, respectively, (instead of semigroups). This gives a "universal character" to free structures, which is manifested in the following three important theorems. THEOREM 98. Every semigroup is the homomorphic image of a free semigroup.

THEOREM 99. Every group is the homomorphic image of a free group. THEOREM 100. Every ring is the homomorphic image of a free ring.

For the proof let H be a given semigroup, further let S =

equipotent to S and make the a, 1, ... and a, b, ... correspond to each other in a one-to-one manner. In each element of the free semigroup H gen-

erated by the alphabet A, by replacing the letters a, P.... by the corresponding a, b,..., there is defined a unique and even homomorphic mapping of H onto H. Theorems 99 and 100 are proved in an essentially similar manner.

FREE STRUCTURES

167

NOTE. According to Theorem 98 every group is the homomorphic image of a free semigroup. Nevertheless Theorem 99 is not superfluous,

since a group may usually be obtained more conveniently as the homomorphic image of a free group than that of a free semigroup. In what follows we shall deal with a certain generalization of Theorems 98, 99, 100. Let A = be an alphabet and F a free structure (semigroup,

group or ring) generated by A. Also let (finitely or infinitely many) "formal" equations A X1, 12, ...) = g,(a1, 021 ...)

(i = 1, 2, ...)

(49.13)

be given, where f o r each i both sides are arbitrary expressions in a a2, ... , which are not required to be equal. We consider the congruence induced by

A(al, a2, ...) m g,(al, a2, ...)

(i = 1, 2, ...)

(49.14)

and denote by C the corresponding compatible classification of F and call the factor structure S = F/C

(49.15)

(or each structure isomorphic with this) the structure defined by the equation

(49.13) (or, more precisely, the structure defined by the alphabet A and the above relation). In special cases we speak of the semigroups, groups or rings defined by certain equations. If, furthermore, T is a structure (semigroup,

group or ring, respectively), in which may be given generating elements el, Q2' ... (not necessarily distinct) such that Mk = Qk (k = 1, 2, ...) is a solution of the equation system (49.13), then we call T a structure belonging

to the equations (49.13). Since the classes ak (mod @) are generators of S and evidently constitute a solution of (49.13), so S itself is one such T. Further every homomorphic image of such a T, in particular of S, is again one of these structures. THEOREM 101. The homomorphic images of a structure defined by equations

are those structures which belong to the same equations. ("Structure" means here in both cases either semigroups or groups or rings.) For the proof we retain the above notations. We have only to prove the

assertion S - T. We even show that S ,., T ( f ( a l

, a2, ...) (mod (2) -.f(Ql, Q2, ...)) ,

(49.16)

where f(a1, a21 ...) runs through all expressions in al, a2, ... and hence .f(Nl, e2, ...) arises by the replacement ak = ek (k = 1, 2,. ..). In fact (49.16) defines a mapping of S onto T. We have now to show that this mapping is unique, for the homomorphic property is obvious. For this

168

STRUCTURES

purpose we denote the class ak (mod @) by ak and consider a valid equation

of the form f(al,a2,...)

9(«1«29...

(49.17)

It is sufficient if we show that

f(el, e2, ...) = 901, P2, ...) .

(49.18)

f(al, a2, ...) m 9(a1, a2, ...) .

(49.19)

According to (49.17)

This must be a congruence resulting from (49.14), therefore f(al, a2, ...) = g(a1 (Z2, ...)

(49.20)

is a consequence of the relations (49.13). But since these are true for ak = Qk (k = 1, 2, ...), (49.18) is also true. Consequently Theorem 101 is proved. COROLLARY (DYCK'S theorem). If S is a structure defined by equations and

we add further equations to the defining equations, which all together define a structure T, then T is a homomorphic image of S. Since T is a structure associated with the originally given equations, the corollary follows from Theorem 101.

As a matter of fact the structure S defined by the equations (49.13) is scarcely ever investigated directly on account of (49.15), as the investigation of the compatible classification CU by the help of the congruences (49.14) is in general a difficult task. Instead, it is more usual to utilize Theorem 101 or its corollary. We proceed according to the following scheme. We consider all structures T (i.e., semigroups, groups or rings), which are generated by the elements al, a29 ... under the conditions (49.13), i.e., which may be given in the form {a,, a2, ...}. (These are to within isomorphism all those structures, which belong to the equations (49.13).) If we then find among

these T an S which cannot be obtained from any of the other T's, as a homomorphic image, then it follows from Theorem 101 (and from the corollary) that this S is the required structure. For instance, we consider the group defined by the equations as = e,

Y1 = e,

YcN-1 = 'x-1

(49.21)

Here e denotes the unity element, so that only a and 19 are to be regarded as generators.

Properly, we should include in (49.21) the equations ea = me = a, e19 = PC = 19, which guarantee that e is the unity element; this would mean that all three elements e, a, 19 are considered as generators. Instead of this, one could replace (49.211,1) by the equivalent equations a' = a, #3 =,8, which are, however, less convenient. [Had we meant by (49.21) a group which is generated, say, by the elements a, 19, y, . . ., v, then this

STRUCTURES DEFINED BY EQUATIONS

169

would have been expressly stated, or in (49.21) the identical equations y = y, ..., v = v, would have been inserted in order to call attention to the further generators y, ..., v.] We denote the group under discussion by G. Then we may also put G = = {a,fl). Since, according to (49.213), 9a = a ',B, the elements of G may be written in the form 4 '`. By (49.211,2) we may restrict ourselves to i = 0, 1, 2 and k = 0, 1 so that in any case O(G) < 6.

(49.22)

[Or we argue thus: because of (49.21) {a} is a normal divisor of G with 0 ({a}) < 3, O(G : {a}) < 2, therefore (49.22) follows.] On the other hand, .9`3 = ((12 3), (1 2)} belongs to the equations (49.21), since these are satisfied for a = (1 2 3), # = (1 2). Consequently the homomorphism G - .9 3 follows from Theorem 101. But since O(93) = 6, by (49.22) G 6P3, which determines G. As a still easier example we consider the group G = {a} defined by the relation (49.23)

Since, by (49.23), o (a) G 6, it follows that 0 (G) < 6. By Theorem 101 G must consequently be the cyclic group of order six. As a last example we consider the ring R = {a, 14} defined by the equations

a2 = a, fJ2 = j9, a(f = t°a = ,9, 2,9 = 0.

(49.24)

From (49.24) it follows that the elements of R may be written in the form as + bf9

(a E d ; b = 0, 1).

(49.25)

We first examine the possibility that the elements (49.25) are all different. Then R4 is the direct sum of the infinite cyclic module {a} and of the cyclic module (fl) of order two, so that R+ is infinite and non-cyclic. Hence and by (49.24) R is uniquely determined, since the elements may be multiplied according to the rule (aa + b fl) (ca + dfl) = aca + (ad + be + bd) fl .

(It is easy to see that associativity and distributivity hold.) If, as the other possibility, the elements (49.25) are not all distinct, then the module R+ is either finite or cyclic. The homomorphic image of such a module can only be finite or cyclic, whence it follows, from Theorem 101, that R is the ring discussed above.

As an interesting application we consider the groups defined by the equations

a2=8, N2=E.

(49.26)

For the sake of convenience we combine this task with the determination of all noncommutative groups generated by two elements of order two, which we call dihedral groups. It will be seen that the order of these groups is either -or an even number Q;6) and that they are (to within isomorphism) uniquely

determined by their order. Correspondingly we call them the infinite dihedral group or the finite dihedral group of order n (n = 6, 8, . . .). It will be seen, too, that the group defined by (49.26) is the infinite dihedral group.

170

STRUCTURES

We proceed heuristically by supposing first the existence of an infinite

dihedral group. This we take as G _ {a, P}, where a, 9 satisfy the equations (49.26). We put e = fla .

(49.27)

G- = {a, pl.

(49.28)

Because {a, j9} = {a, Pa} then

Since, moreover, according to (49.27) P = em-1, it follows from (49.26)

that (ea-1)2 = s, ea-lea-1 = e, aea-1= a-1. Accordingly equations (49.26) assume for the new generators the form xz

=s, aea- = e

(49.29)

From (49.292) it follows that (aea-1)-' = e`, ae- a-' = e for which we may also write

o`a=ae-`

(iE-7).

(49.30)

Hence and from (49.29) it follows that

(1 E 7)

(49.31)

exhausts all the elements of G.,. The elements (49.31) must also be distinct otherwise G- [because of (49.291)] would not be infinite. From (49.30) we obtain for G- the Cayley table xek

0k e`+k

ae-t+k

aet I ae`+k

a-1+k

e`

I

(i, k E 7).

(49.32)

On account of Theorem 65 we define the faithful representation

G .9 (a -> a, a - r)

(49.33)

of G by a permutation group

S' _ {a, r}

(49.34)

with

a=lal

r

=I)

(BEG.,).

(49.35)

From (49.29), (49.33) it then follows that a2 = 1 ,

ara-1

= r-1.

(49.36)

STRUCTURES DEFINED BY EQUATIONS

171

By introducing the notations xk = ok, A = aok we obtain from (49.32) and (49.35)

... xk ... yk ...

a

yk .xk .

1

1I

I = (... xk ... yk ... (

.xk+1. ..Yk-1

(k

,7)

(49.37)

J

We now disregard the previous meaning of the xk, yk and denote by them any distinct elements constituting a set S. Then (49.37) determines two elements a, r of the full permutation group ,.93(S). For these a, r, {a, r} becomes a well-defined infinite subgroup _0 of 93(S). Since, from (49.37) and Theorem 82 equations (49.36) follow immediately, 93 is an infinite dihedral group. Since we have shown above that there can be (to within isomorphism) at most one infinite dihedral group (which is that determined by the Cayley table (49.32), we have proved the existence and uniqueness of the infinite dihedral group. Now we can prove that the infinite dihedral group is defined by the equations [(49.26) or] (49.29). For, if (49.29) holds for G = {a,o} and G is not the infinite dihedral group, then the elements (49.31) are not all distinct and G is finite. Hence on account of Theorem 101 the assertion follows. Now the finite dihedral groups may be easily obtained. Contrary to what has been said above o must now be of finite order. Since, from (49.29), if o2 = r then mg = ooc, so

o(o)=n>_3. Therefore

o', ao`

..., iI -

(i = 0,

1)

(49.38)

are all the different elements of the group. So the order of the group is 2n, as was asserted above. Since the Cayley table (49.32) is still valid, the finite dihedral group is defined by its order. We still have to demonstrate the existence of these groups. The infinite dihedral group (49.28) has, according to (49.30), the normal divisor {o"} of index 2n (n >_ 1). The factor group G_/{o"} is of the order 2n and is, for n > 3, by (49.29), obviously noncommutative. So we have proved all the assertions. It happens that the dihedral group of order 2n (n >t 3) is defined by the equations a2 = E, aoa-1 = o-1, o" = e.

(49.39)

To avoid possible misunderstanding, it should be noted that if we speak of a structure (a, /1, ...} defined by equations, then it does not mean that the generators a, P.... are all distinct. If, e.g., the equation a2 = all occurs among the defining equations of a group, then a = /. For semigroups this equality does not follow.

On the other hand, in the semigroup defined by the equations

a2=V2,

Za=a

a=i42a=a2x=a3=fl

172

STRUCTURES

hold, so that we just have the semigroup {x} defined by a° = a. The only distinct elements in it are a, as.

As special cases among the structures defined by equations are the free structures where the system of the defining equations is empty. (This is why they are called "free structures".) If we want to define a commutative structure {al, a2, ...} by equations, then we add to the defining equations all the equations a; ak = ak a; (i, k = = 1 , 2, ...). Alternatively we can talk explicitly of a commutative structure (defined by equations). Hence, in particular, the notions commutative free semigroup, free Abelian group, commutative free ring are meaningful. Accordingly, a free module is a free Abelian group written in the additive notation. It is obvious that a free Abelian group (a free module) {a1, a2,... }

is just the direct product (the direct sum) of the infinite cyclic groups (modules), {a1}, {a2}, .... We see that Theorem 101 together with its corollary remains true for commutative structures. Skew fields present entirely different circumstances. Since, by Theorem 57, every homomorphic image of a skew field is either isomorphic with it or is a zero ring, the notion of free structures and that of structures defined by arbitrary equations cannot be carried over to skew fields so that Theorem 101 remains valid. Therefore we give the following definition: the free

field generated by an alphabet A =
This simple way of defining a structure is the most perfect one imaginable, the practicability of which is, however, faced with insurmountable difficulties, whereas the -definition of S by equations" has the advantage that we have only to give the generators of S and some of the valid equations between them. (E.g., in the elementary school the teacher does not work with the full Cayiey tables for . r, which would be impossible, but merely with the generators 0, 1, . . ., 10 and suitable defining equations,

1 x ] = 1, ..., I-1-1 = 2,. ..). The difficult problem of expressing in an obvious form all the different elements of S (if possible uniquely) in terms of the generators and of indicating the detailed composition rules in S still remains. We have seen some examples of this above, and we shall consider further ones. The problem sketched above is the so-called word problem. Cf. MARKOV (I947a, 1947b), PosT (1947), KALMAR (1952), Novixov (1955).

One may apply the notion of the definition of structures by equations in very different ways. The following is an example, to which we will allude but briefly. Let G1, G2, ... be groups for which we assume, without loss of generality, that they have the same unity element, yet no other common element when considered pairwise. By the free product of the groups G1,

STRUCTURES DEFINED BY EQUATIONS

173

G2, ... we understand that group whose generators are the elements of G1, G2, ... such that the equations, which hold in each individual GI, hold as defining equations. We similarly interpret the free sum of the rings R1, R2, ..., for which we assume that they have a common zero element, but no other common element besides that. For other foundations of the free groups, see ARTIN (1947), van der WAERDEN (1948), KUROSH (1953a).

EXAMPLE 1. The free product of two groups of order two is the infinite dihedral group. EXAMPLE 2. The ring defined by a2 = a is isomorphic with J.

EXAMPLE 3. For a natural number in, the ring defined by a2 = a, ma = 0 is isomorphic with 9/(m). EXAMPLE 4. For given u, v (E J) the ring R = {e, a) defined by

e2=e, ea=aE=a, a2 =us+ va ke+la (k,IE9),

consists of the elements

which are all distinct. Moreover, R is commutative and the unity element of R is e. The rules of addition and multiplication are

(ke + la) + (k'e + I'm) = (k + k') e + (I + 1')a, (ke + la) (k's + 1'a) = (kk' + ull') e + (kl' + k'1 + vll')a. EXAMPLE 5. The ring R = {e, a) defined by

E2=E, Ea=a, ae=0, a2=O is non-commutative. a is a left unity element and a a left annihilator. The elements may be given exactly as in the previous Example 4, with the same rule for addition, but with multiplication as follows:

(ke + Ia) (k's + I'a) = We + kl'a. Since all the la are left annihilators, all the a + la are left unities. EXAMPLE 6. If we add to the above defining equations the further equation 2e = 0, then the four-ring is obtained. (Cf. § 27, Exercise 4.) EXAMPLE 7. The Priifer group (p') (cf. § 23, Example 3) may be defined by ai = e, 042 = aU 09 = a2s ...

.

(The commutativity is evident.) EXAMPLE 8. The semigroup defined by

a2=aN = P2 ,,qq

elements

at, fJa, fl (i = 1, 2, . . .). The Cayley

is non-commutative and consists of the table is: ml

q

Y

a

a

ak

attl

al+k

a3

a2}k al+k

as

la

p

at+2

aitI

a4

a2

a3

a2

(I = 1 , 2, .. ; k = 2, 3, .

174

STRUCTURES

EXAMPLE 9. For a prime number p the ring R = (a} defined by

pa = 0

(49.40)

consists of the elements

Ea,a'

(a,=0,...,p- 1; 1=1,2,...),

where there are only finitely many a, distinct from 0. These elements are then all different. This ring is of characteristic p. [Equation (49.40) holds for all the fields F = {a) of characteristic p. There are infinitely many non-isomorphic such fields F = {a}, as we shall see later. Since none of them can be the homomorphic image of another, none of these fields may actually be regarded as the field defined by (49.40).] EXERCISE 1. Give the proofs of the statements in the previous examples and define

in each case the composition rules. EXERCISE 2. The finite dihedral group of order 2n is isomorphic with the group of motions of the three-dimensional Euclidean space, which map a regular n-sided polygon onto itself. The dihedral group of order six, the group ,9 and the anharmonic group are isomorphic. EXERCISE 3. Determine the free product of two groups of order two and three, respectively. (Cf. KUROSH, vol. II. (1956), pp. 16-17.) EXERCISE 4. Determine the free sum of two fields of order two considered as rings. EXERCISE 5. In the ring defined by the equations

cell=£, fa=aE=a, r9=9e= 9 the element # has infinitely many left inverses.

EXERCISE 6. For an integer c let G be the group defined by ao-l = o` where we disregard the trivial cases c = 0, 1. If c = - 1, oxa' are all the different elements of G. If I c I > 2 all the elements of G occur among the axo' a= and axo'a° = ax'o'wa" I. X + z = X' + Z', CxY = cxye .

Therefore the ax and ax o' az (c x y) are all the distinct elements of G. On the other hand, let R be the ring of rational numbers of the form ucv

(u, v E J).

(For c = -1, R = J.) We define the skew product S, of a and R, by

(i, r) (k, s) = (i + k; kcr + s)

(i, k E J; r, s E R).

Prove that S is a group isomorphic with G. Give one such isomorphism. For the case c = 2 cf. HIGMAN (1951). See also KALOUJNINE (1939), KRASNER (1951), REDEI (1950).

EXERCISE 7. In the semigroup H defined by a = a$9, P = a#2, there are elements such that off = H. However, H has no left unity elements. Cf. REDEI (1960b).

§ 50. Schreier Group Extensions By a Schreier extension group (or Schreier's extension) of a group G by a group 4 we understand any extension group CS of G with the property that G is normal in (53 and the isomorphism 03/G

c/&

(50.1)

SCHREIER GROUP EXTENSIONS

175

holds. The so-called Schreier's extension problem for groups consists in determining all Ci for given G, c4. For rings we may formulate an analogous problem, which we shall deal with in § 52. In order to solve the problem, we replace (50.1) by the somewhat more general conditions Ci/G1

cl7 ,

G1

G

,

(50.2)

which state that G1 is a group isomorphic with G and Ci is a Schreier extension group of G1 by 4. This generalization is unessential, since from (50.2) we can obtain (50.1) by embedding G in Oi in place of GI. Therefore, we also call any Ci satisfying the conditions (50.2) a Schreier's extension

of G by dt. We assume the two groups given in the form

G=
(50.3)

where at, fi, ... and a, b.... denote the elements of G and (4, respectively. In particular the unity elements of these groups are denoted by 8 and e, respectively.

We shall obtain the solutions of Schreier's problem as special cases of certain skew products of (4 and G, which we define by the multiplication (a, a) (b, 9) = (ab, ababj9)

(a, b E (4; at, j9 E G) ,

(50.4)

where ab, xb (E G)

(50.5)

are two functions, subjected to the "initial conditions"

ae=s, eb=8, ae=0C, 8b=8

(50.6)

Every such skew product is denoted by (OG and we call it a Schreier product of (4 and G. We call (50.5) the function pair belonging to the Schreier product (40 G. These always determine each other uniquely, for, on the one hand, a multiplication (50.4) is given by (50.5) and, on the other, if (50.4) is given,

then [because of (50.6)] (a, e) (b, e) = (ab, ab) ,

(e, a) (b, e) = (b, ab) ,

which uniquely defines the function pair (50.5). The first term ab of the function pair (50.5) is called the factor system (of d 40G). In the literature the factor system a' is usually denoted by flay (or Ca,a, etc.). We shall apply this notation, when it is expedient to do so.

7 R.-A.

STRUCTURES

176

Since, by (50.4), (50.6), (e, (x) (e, j9) = (e, aj9)

the elements (e, a) (a r- G) constitute a subgroup of c40G isomorphic with G, i.e., (e, a) -> a is an isomorphism of the above subgroup with G. By way of this isomorphism we embed G in cPOG. On account of this embedding 40G is an extension structure of G. However, we shall also retain the "old" notation (e, a) for the embedded elements, a, of c00G. It should be noted that, because of (50.4), (50.6), (e, e) (= e) is the unity element of c40G. Since the multiplication (50.4) is evidently not always associative, cPOG need not be a semigroup, much less a group. THEOREM 102 (SCHREIER'S fundamental theorem on group extensions). All the Schreier extensions of a group G by a group c.P are (to within isomorph-

ism) those Schreier products 4 0 G which are groups, and they may be characterized by the conditions

(aj3)` = 4` ,

(50.7)

abebe = bc(xb)` ,

(50.8)

ab`b` _ (ab)`(ab)`

(50.9)

from which it also follows that

G ^ G(a-±ab).

(50.10)

For these groups (c40 G)iG

4((a,e)G -±a).

(50.11)

Since, when cPOG is a group, (50.10) means that, for every b, a --> (X 1' is an automorphism of G, we call the second term ab of the function pair (50.5) in this case the automorphism system (of c4 0G). We shall prove the theorem together with the following, which is an important supplement and, in part, a converse of it. THEOREM 103. Let a group 03 with the normal divisor G be given. Put c& = W/G

(50.12)

and denote the elements of G and c4 as above [in (50.3)], as a consequence of which the a, b, ... (E cP) now mean the cosets of 0 mod G. In particular, e is the principal class G. Further, take a choice function

f(a) E a

(a E cg-; f(e) = e)

(50.13)

and put

ab =flab) -'f(a)f(b) (E G) ,

ab =f(b)-1 af(b) (E G)

(a, b E c; a E G). (50.14)

177

SCHREIER GROUP EXTENSIONS

For this function pair the conditions (50.6), (50.7), (50.8), (50.9) are fulfilled, and for the corresponding Schreier extension group COG

COG . 1.i ((a, a) -> f (a)a) .

(50.15)

First of all we prove that the conditions (50.7), (50.8), (50.9) are necessary and sufficient for COG to be a group. Since COG always has a unity element and by (50.4) every element (a, a) in COG obviously has a right inverse, it follows from Theorem 32 that COG is a group if, and only if, the multiplication (50.4) is associative. The condition for this is (ab. abab9) (c, y) = (a, a) (bc, b`9r')

i.e.,

,

/

R (abc, (ab)c(ababfJ)cy) = (abc, al+cabrbcfcy) ,

(ab)c(abab 9)c = abcab. be /3c

(50.16)

.

If we first substitute b = e, then a = e, fi = e and finally a = fi = e, then by (50.6) we successively obtain (50.7), (50.8), (50.9). Conversely, if these equations hold, (ab)r (abab/i)c =

(ab)c(ab)c(ab)c,gc = abcbr(ab)c%lc

= abcabcbefic

i.e., (50.16). Thus the statement is proved. Further we prove the statement in (50.10). If (50.7), (50.8), (50.9) hold, then by previous statements COG is a group. So if we take a fixed (b, j9), and (a, a) runs through all the different elements of COG then the right-hand side of (50.4) must likewise run through all

the different elements of COG. Hence it easily follows that at - ab is a permutation of G. This, together with (50.7), proves (50.10). In order to prove (50.11) we consider a group COG. Because of (50.4)

c4oG ^' C ((a, a) - a).

The kernel of this homomorphism is G. Since, on the other hand, (50.4) and (50.6) give (a, a) = (a, e) (e, a), the (a, e)G (a E C) are all the different cosets mod G. From these [on account of Theorem 60 (31.3)] the truth of (50.11) follows. Accordingly every group COG is a Schreier extension of G by C. Since the converse of this is a consequence of Theorem 103, we need only prove this. For the function pair (50.14), since f(e) = e, condition (50.6) is trivially satisfied. Because of the definition of f(a)

f(a)a

(a E (4, a E G)

(50.17)

178

STRUCTURES

are all the distinct elements of 0. For the product of two arbitrary elements of th we obtain from (50.14) the rule .f(a)a . f(b)fl =.f(a)f(b)ab8 =.f(ab)abab9

Comparison with (50.4) establishes, by (50.17), the isomorphism (50.15). Hence '4OG is a group and it follows from the part of Theorem 102 already proved that the function pair (50.14) satisfies the conditions (50.7), (50.8), (50.9). Consequently Theorem 103 is proved and the proof of Theorem 102 s completed. Since, according to the first part of Theorem 102, all the Schreier extensions of G by d4 are essentially given by the groups 'OG, we shall take the Schreier extensions mainly in the form' 0G. This has the advantage that for given G, cp all the c.4OG consist of the same elements; thus they differ from each other only in the multiplication of the elements [i.e., in the function pairs (50.5)]. In order to give the Schreier extension d4oG it is sufficient according to Theorem 102 to determine the function pairs ab, ab, which constitute a solution of (50.7), (50.8), (50.9). Two Schreier extensions cf 0G of G by '4 are called equivalent, if there is an isomorphism between them, in which the elements of G are fixed and

each class mod G is mapped onto itself. It is evident that this is indeed an equivalence. The function pairs belonging to equivalent extensions are called associated. Two factor systems are also called associated, if they may be completed by two suitable automorphism systems for associated function pairs. THEOREM 104. If ab, ab is an arbitrary function pair, satisfying the conditions (50.6) to (50.9), then the formulae (ab)._1 abalbb', b'-labb'

(50.18,)

give all the function pairs (possibly each one several times) associated with it where a --> a' denotes every mapping of 4 into G with e' = S. Between the two (equivalent) Schreier extensions (A, (SS', belonging to the function pair ab, ab and the function pair (50.18), respectively, we have the isomorphism

0 ;z W' ((a, a) -* (a, a'-' ac)).

(50.19)

COROLLARY. For all the choice functions (50.18), (50.14) runs through a full system of (not necessarily distinct) associated function pairs.

In order to prove the theorem, we take the Schreier extension 05, belonging to the given function pair ab, a, in the form CS = 40G. The multiplication (50.4) then holds. Furthermore 05' will denote the Schreier extension of G by cp, belonging to a further function pair (a), (ab).

SCHREIER GROUP EXTENSIONS

179

We can assume that W consists of the same elements as Ci and that multipli-

cation is carried out as follows: (a, a) x (b, j9) = (ab, (ab)(a)P) .

(50.20)

Ci, 03' are equivalent if, and only if, there exists an isomorphism

0

0i' ((a, a) --- (a, a)°)

(50.21)

such that (a, a)° is of the form (a, a) (a1 E G) and (e, a)° _ (e, a). If this is the case, we put (a, e)° = (a, a'-) with a' E G, e' = B. Then we have (a, a)° = ((a, e) (e, a))° = (a, E)° x (e, a)° = (a,

a'-') x (e, a) = (a, a'_la) .

Therefore (50.21) now becomes (50.19). Conversely, if (50.19) and a' E G, e' = e, then CS is, according to the above, equivalent to W. Now we know from Theorem 63 that (50.19) is equivalent to (a, a) x (b, 9) = n (n-1(a, a), il-1(b, 9)) , where n denotes the permutation (a, a) --* (a, a'-' (x). This equation becomes, after the computation of the right-hand side,

(a, a) x (b, 9) = n ((a, a'a) (b, b'f)) = n (ab, ab(a'a)bb'#) _ = (ab,(ab)'-1ab (a'a)bb'1) .

(50.22)

The combination of (50.20) with (50.22) results in (ab) (ab) = (ab)'-1 ab(a'a)bb'

.

(50.23)

If we substitute first a = e and then a = e in (50.23) we obtain (ab) _ (ab)'-Iaba'bb', (ocb) = bi-labb'.

These are the function pairs in (50.18), whereby we have proved Theorem 104

In order to prove the corollary we have to consider that all the choice functions (50.13) may be given in the formf(a)a', where a' means the same as

in (50.18). If we replace f(a) by f(a)a', then after an easy computation (50.14) becomes (50.18). Hence the corollary follows from Theorems '103, 104. If we want to obtain the Schreier extensions of G by C4, or, what is the same thing, the corresponding function pairs, then Theorem 103 can render

a good service. Because of the corollary to Theorem 104 it is sufficient to substitute in (50.14) only one choice function (50.13). The method sketched here will be elucidated in the proof of the following highly specialized theorem, which is, however, capable of generalization.

STRUCTURES

180

THEOREM 105. Let G be an arbitrary group and c4 = {g} a cyclic group. If, moreover, c4 is finite of order n, then the function pairs (without regard to

associated function pairs) belonging to the Schreier extension groups cJ0G, are given by

ab =

Ie(i +kn), (50.24)

where A, v denote an automorphism and an element of G, which are subject to the conditions A v = v , Mac = v -la v ((z E G). (50.25) If c4 is infinite, then (50.24) must be replaced by

ab = e , a° = Aka

(b = gk; k E 7)

(50.26)

and (50.25) has to be omitted. For the proof we assume t to be a Schreier extension of G by cQ and consider &j = {g} to be identified with 03/G, so that g is now a generating class mod G for c0. We take an arbitrary element Q of g and consider first the case of a finite c4. Then a choice function (50.13) may be given by f(a) = 92`

(a = g'; i = 0, ..., n - 1).

(50.27)

St"= v(EG),

(50.28)

If we put then (50.141) passes over into (50.241). We denote the automorphism a -+ Q-lcQ of G by A :

Aa = o-last .

(50.29)

Because of (50.27), (50.29), (50.14`) passes over into (50.242). (50.25) follows from (50.28), (50.29). Conversely, we can easily show that the function pair (50.24) satisfies the conditions (50.5) to (50.9). Hence the proof for finite cP

is complete. If c4 is infinite, then a choice function may again be taken in the form (50.27), where i has now to run through all integers. Otherwise the proof is similar to the previous one, but much simpler. We see that (with fixed G, n) the function pairs (50.24) and the pairs (A, v) determine each other uniquely. Of course, among all the function pairs (50.24), associated ones may still occur, as will be evident from the proof of the theorem, since for 0 one might have taken any Stw (w E G). Certain special Schreier extensions cAOG are of particular importance. Referring to this we note primarily that the trivial case ab = e, ab = a of

(50.5) means that (50.4) becomes (a, a) (b, j) = (ab, a#), i.e. d oG is equal to the direct product cd ® G. This implies that at least one Schreier

SCHREIER GROUP EXTENSIONS

181

extension of G by d4 always exists, namely the direct product of these groups. By applying (50.18) to ab=8, ab = a. we see that the Schreier group extension c4oG is equivalent to cPf & G, if, and only if, a function pair

of the form

(ab)'-la'b' , b'-lab'

(50.30)

belongs to it. (In § 51 we shall deal with groups which occur as a direct factor in all their Schreier extensions.) The function pairs ab, ab of the form e, ab and also the Schreier extensions associated with them are called factor free. From Theorem 102 we immediately obtain the following: THEOREM 106. In a factor free Schreier extension group 4toG of G by 4t the elements are multiplied according to (50.31)

(a, a) (b, j9) = (ab, ah(3) ;

furthermore the automorphism system ab is subject only to the conditions that the a - ab are automorphisms of G and ab` _ (ab)`.

(50.32)

In each such group 4OG the elements (a, E) constitute a subgroup 4t' with

d4oG = (fi' G ,

4' : 4

.

(50.33)

It follows from (50.33) that the classes mod G may now be represented

by the elements of a subgroup, i.e., 4OG "splits" into the schlicht product of the normal divisor G and another subgroup. (Conversely, every group "splitting" in this way may be written as a factor-free Schreier extension.) The equivalent Schreier extensions also have this property, so that

we call them and the function pairs belonging to them splitting. We apply the same term to the factor systems, which together with a suitable automorphism system constitute a splitting function pair. As a special case of Theorem 104 (ab)'-Ia'bb' , b'-labb' (50.34) are all the splitting function pairs, where ab means the same as inTheorem 106. The function pairs ab, ab of the form ab, a and also the Schreier extensions

belonging to them are called automorphism free. As a special case of Theorem 102, we have directly the following THEOREM 107. In an automorphism-free Schreier extension group 440G of G by 4 the elements are multiplied according to the rule (a, a) (b, t4) = (ab, abaq) .

(50.35)

STRUCTURES

182

Furthermore, the factor system ab, apart from the requirement that eb = a` = E, is still only subject to the conditions that all the ab belong to the centre of G and ab`'b` = (ab)`ab .

(50.36)

Because of this theorem, since the factor system of an automorphism-free Schreier group extension c OG consists entirely of central elements of G, these and the Schreier extensions equivalent to them, and also the function pairs belonging to them, are called central. We use the same term for those factor systems which together with a suitable automorphism system constitute a central function pair. As a special case of Theorem 104

(ab)'-' aba'b',

b'-'ab'

(50.37)

are all the central function pairs. These may be characterized also by saying that the second members of them consist of inner automorphisms of G. On the one hand, this property evidently belongs to the function pair (50.37). On

the other hand, in Theorem 104, if in a function pair a', ab all the automorphisms a-->ab are inner, then the mapping a --> a' may be chosen so that the second member of (50.18) always equals a. Hence the assertion is proved. Nom 1. We shall consider a function pair ab, ab from Theorem 102 with properties (50.5) to (50.9). If we write (50.8) in the form

ab` = be(ab)`(b`)-'

,

(50.38)

then we see that the automorphism a -, ab` is equal to the product of the three automorphisms a > b`a(b)-l, ocb. If this function pair is factor-free, i.e., if ab = E, then according to Theorem 106 the simpler rule (50.32) holds. This rule holds more generally if, and only if, all ab are central elements of G; this follows from (50.8) since (ocb)`, with arbitrary fixed b, c, runs through all elements of G. (The reader will see from this note that the

above terms "central extension" and "central function pair", which we retain only because of their historical background, are not completely accurate.)

Nom 2. If two structures T, T' contain a common substructure S, then we call every isomorphism a -> a' of T onto T', for which all the elements of S are fixed, a relative isomorphism over S, for which we write

T

S

T' (a > a) .

(50.39)

If there is at least one such isomorphism, then we say that the structures T, T' are in a wider sense (i. w. s.) equivalent extensions of S. Since we have defined a narrower notion of equivalence for the Schreier group extensions above, we should call them "equivalent in a narrow sense"

(i. n. s.) instead of "equivalent". In both cases we say briefly "equi-

SCHREIER GROUP EXTENSIONS

183

valent", but when we mean, in regard to Schreier extensions, an equivalence i. w. s., then we shall emphasize it. We see at once that, if (50.39) holds and S, T, T' are groups, then by a -> a' each left coset mod S (in T) is mapped onto a left coset mod S (in T'). NoTE 3. We now denote by 0 a semigroup with unity element and by G a subgroup of it, the unity element of which is equal to that of 03. We also permit the existence of a zero element in 0, which we denote by 0. In Theorem 69 we have seen that the distinct left cosets aG constitute a classification of CAS. The same holds for the right cosets Ga. (Of course, the classes OG, GO consist only of the element 0.) If all the products aG, Goc

(a # 0) are schlicht and

aG = Ga for all a (E G) , then we call the subgroup G a normal subgroup (or a normal divisor) of the semigroup 0. It is evident that in this way the notion of a normal divisor of a group has been generalized. Of course, these classes now constitute a compatible classification, which is uniquely determined by G, therefore we denote the corresponding factor semigroups by W /G. The result of this is that the definition in (50.1) remains meaningful, if we take G to be a group, while for c0,t we take an arbitrary semigroup with a unity element and also for Cj a semigroup with a unity element. (G and Chi must always have the same unity element.) Thus we have defined the notion of the Schreier extension semigroup of a group by a semigroup with unity element. Schreier's extension theory above may, with a slight change, be generalized accordingly. This is left to the reader. [The only essential change is that, in the fundamental theorem, we have to replace (50.7) by the stronger condition (50.10), which cannot now be inferred from (50.7), (50.8), (50.9). Further, if co contains

the zero element 0, then the definition of the Schreier product has to be changed so that we identify all (0, a) with the zero element of 0. We add to the condition (50.6) that 06 = a° = fi, a° = a, and we postulate (50.8), (50.9), (50.10) only for be 0 0, abc 0, and b # 0, respectively.] But if we talk of Schreier extensions of group without comment, then we mean Schreier extension groups. As regards the Schreier extension theory cf. first of all SCHREIER (1926a, 1926b), and then ZASSENHAUS (1958), KUROSH (1953a), REDET (1952a, 1956b).

EXAMPLE 1. The semigroup 3X has the normal divisor <1, -1>.

EXAMPLE 2. If .7$ is the full matrix ring of rank four over a field F, then the 0

) (p E

elements

0) constitute a normal divisor of the semigroup .721,

0

(although the latter is not zero-divisor-free). EXAMPLE 3. Let G = (a, 11, ...> be a cyclic group with O(G) # 1,2 and dt = (e, g> a (cyclic) group of order two. Define a factor-free function pair (50.5) by a' = e, ae = a,

a' = a-'. The associated factor-free Schreier extension cIIOG is a dihedral group. These groups are to within isomorphism all the dihedral groups.

7/a R.-A.

STRUCTURES

184

EXERCISE 1. In a Schreier extension group 4 o G (with notation as in Theorem 102)

we have (a, a)-1 = (a-1, (aa- )-1 (a°-')-I)

EXERCISE 2. Which of the function pairs (50.24) and (50.26) are associated ? EXERCISE 3. For the associated function pair i. w. s. Theorem 104 holds with the difference that one takes (ab)1-1(Aa)Aba,Abb. b,-lEAbb, instead of (50.18) and

(SS ti & ((a,a) -- (A_ 1a, (A _ 'a)'-1a))

instead of (50.19), where A is an arbitrary automorphism of 4.

§ 51. The Holomorph of a Group THEOREM 108. The inner automorphisms of all the Schreier extensions of a group G induce all the automorphisms of G. First of all let 0 be an arbitrary Schreier extension of G. Each inner automorphism of 0 induces an isomorphism of each subgroup of 0. Thereby, the normal divisors are mapped onto themselves. Hence the induced mapping

of G is, in fact, an automorphism of it. In order to prove the converse, we note the following: let 4 denote an arbitrary automorphism group of G. For automorphisms we use the exponential notation and define the product ab of two automorphisms a, b (E 4) by

scab = ( a)b

(a E G) .

(51.1)

Then according to Theorem 106 we may form the factor-free Schreier extension 44OG. In this [according to (50.4), (50.6)] (a,

E)_'

(e, a) (a, s) _ (a_l, E) (a, a°) _ (e, a°)

(a E 4,

a E G) .

If we suppose (as usual) G to be embedded in d O G, then we see that the automorphism a is induced by an inner automorphism of d' 0 G. Since we can take the full automorphism group of G, for 4, Theorem 108 is proved. A subgroup of a group G is called characteristic, if it is mapped onto itself by every automorphism of G. The equivalence of the following "alternative definition" follows immediately from Theorem 108: A subgroup of'a group G is characteristic if, and only if, it is normal in all the Schreier extensions

of G.

Note that a characteristic subgroup of G must be, a fortiori, a normal divisor of G. If, further, H is a characteristic subgroup of G, then every automorphism of G induces an automorphism of H. By the holomorph of a group G we mean the factor-free Schreier extension

ve 0 G of G by the full automorphism group vg of G, in which the pro-

185

HOLOMORPH OF A GROUP

duct of the elements is defined by (a, a) (b, 9) = (ab, ab9)

(a, b E

;

(x, fi E G) ;

(51.2)

of course ab denotes here the application of the automorphism b on a, and (51.1) gives the relation (50.32) of Theorem 106.

Since the holomorph ,fOG is the special case 4 = of of the 44 OG above, the above proof of Theorem 108 gives the following: THEOREM 109. The inner automorphisms of the holomorph of a group G induce all the automorphisms of G. Hence and from the definition of the characteristic subgroups we obtain the following: THEOREM 110. A subgroup of a group G is characteristic if, and only if, it is normal in the holomorph of G. A group is called complete if it has only inner automorphisms but no centre. According to Theorem 61, every complete group is isomorphic with its inner automorphism group. Evidently, all normal subgroups of a complete group are characteristic. THEOREM 111. A group is complete if, and only if, it is a direct factor in all its Schreier extensions. For the proof let G be a complete group and 03 a Schreier extension of it.

We consider an arbitrary element a of 0. In consequence of the assumption we have for this a only one element a' of G such that a-1 em

=

a'_1 ea'

(e E G).

Since we may write this equation in the form eax -1 = aa'-le, the above means that in each coset of (3 mod G there is only one element which is elementwise permutable with G. These elements form a subgroup H of 0, which is at the same time a system of representatives of 0 mod G. Hence li has the direct decomposition 03 = H ® G, whence we have proved the "only if" part of the theorem. In order to prove the "if" part, we assume that a group G is a direct factor in all its Schreier extensions. Then it is, in particular, a direct factor of its holomorph.7eoG, where -e is the full automorphism group of G. According to (50.30) and (51.2) there is then a mapping

a -* a' of

(e' = e)

(51.3)

into G, for which

(ab)'-1 a'b' = e, a'-lea' = ea

(a, b E ,e; e E G) .

(51.4)

From (51.42) it follows that ,e consists entirely of inner automorphisms

186

STRUCTURES

of G. Therefore it remains only to prove that the centre Z of G consists of the single element e.

From Theorem 61 and the homomorphism theorem (Theorem 60) it follows that v( G/Z. From (51.42) it follows that the mapping (51.3) is one-to-one. Furthermore, if we write (51.41) as (ab)' = a'b', we see that ve is mapped by (51.3) isomorphically onto a subgroup G' of G, which,

because of (51.42), contains no element 0 E of Z. From all this we have

G/Z : G', G' rl z= F, hence ( i has the direct decomposition

G=G'®Z.

(51.5)

It is now evident that in a direct product-decomposition of a group every automorphism of one direct factor may be extended to an automorphism of the whole group. Since G has only inner automorphisms, the same follows

easily from the above by (51.5) for both the factors G', Z. But since Z is Abelian, it follows that Z has no non-identical automorphism at all, consequently 0 (Z) < 2. For 0 (Z) = 1 the proof is complete. Only the case 0 (Z) = 2 remains to be considered. We extend Z to a cyclic group Z1 of order four and form the direct product

G, = G' ® Z,

,

(51.6)

where we can assume that G, is an overgroup of G. Since G1 is a Schreier extension of G, it must contain G as direct factor. Hence, by (51.5), 0 has the direct decomposition

G, = G' ®Z ®A ,

(51.7)

where A is a subgroup of G1. From (51.6) and (51.7) it follows that G1/G' is isomorphic both with Z1 and Z ® A. Thus, the last two groups are isomorphic with each other. But this is impossible, since a cyclic group of order four is obviously indecomposable. By this contradiction we have proved Theorem 111. As regards Theorem 111 cf. BAER (1946) and REDEJ (1954c). As regards the holomorphs of groups and of the complete groups see also SPEISER (1954), ZASSENHAUS (1958).

EXERCISE 1. For every group G we have the factor-free Schreier extension G o G (of G by itself), in which multiplication is defined by (x, 1g) (y, b) _ (xy, Y 1 Yyb)

If G is complete, G o G is isomorphic with the holomorph of G, and so must contain G as a direct factor. This holds even for an arbitrary G, i.e., G o G is the direct product of two subgroups isomorphic with G. Show this from (50.30) or by Theorem 63.

HOLOMORPH OF A GROUP

187

EXERCISE 2. The full permutation group of a countable set is, with the exception of fflg and 9J, a complete group (cf. HOLDER (1895), SCHREIER-ULAM (1937)).

EXERCISE 3. Let G be an arbitrary group, c4 its full automorphism group and 99(G) the full permutation group of the set G. The permutations of the form Q') and

()(a E c4;

s, tI E G)

constitute subgroups ca'(-- alt) and G'( G) of .9'(G), respectively, which have only the unity element in common. Now c g'G' = G'c' ', therefore c &G, is a subgroup of ,1i(G). This contains G' as a normal divisor and is isomorphic with the holomorphc4 o G of G. (In former text-books c-4' is called the holomorph of G.)

PROBLEM. Must a group G be complete, if it is a direct factor of its holomorph? (By the above proof of Theorem l ll, G is either complete or the direct product of a complete group and of a group of order two.)

§ 52. Everett Ring Extensions We shall deal here with EVERETT'S (1942) solution of Schreier's extension

problem for rings. The analogy with § 50 permits it to be somewhat shortened.

By an Everett extension ring of a ring R by a ring V we understand any extension ring 91 of R with the properties that R is an ideal of 91 and the isomorphism

R/RxV

(52.1)

holds. We briefly call R'1 an Everett extension.

In order to be able to obtain the solutions, we generalize the previous definition by replacing (52.1) by the conditions

N/RI;ze R, RI^ R.

(52.2)

We assume the given rings in the form

R= .

(52.3)

We denote the zero elements of R and A by 0 and o, respectively. We define certain skew products of 2 and R by the compositions

(a, a) + (b, fi) = (a + b, [a, b] + (x + P) , (a, b E -V; a, l4 E R) , (52.4) (a, a) (b, j9) = (a b, + ab + al4 + aq) where

[a, b], , ab. a#

(E R)

(52.5)

STRUCTURES

188

are four functions, which are subject to the "initial conditions"

[o, b] = [a, o]= = =Ob=ao=ofi=a0=0.

(52.6)

[Apart from (52.3) <> will only be applied in these paragraphs to denote

the function (52.52).] We denote every such skew product by J9 + R and call it an Everett sum of V and R. We call (52.5) the function quadruple corresponding to the Everett sum .+ R. This correspondence is one-to-one, for from (52.4) [on account of (52.6)] we have

(a, 0) + (b, 0) = (a + b, [a, b]), (a, 0) (b, 0) = (ab, ) , (o, a) (b, 0) = (o, ab)

,

(a, 0) (o, 3) = (o, a#)

,

which uniquely determine (52.5). The first two terms [a, b ], of the function quadruple (52.5) are called the additive or multiplicative factor systems,

respectively (of J9 + R). Since by (52.4) and (52.6)

(o, a) + (o, 3) = (o, a + A (o, a) (o, 9) = (o, cc ),

the elements (o, a) (a E R) constitute a subring of R + R isomorphic with

R, i.e., (o, a) -+ a is an isomorphism of this subring onto R. By way of this isomorphism we embed R in 2 + R and henceforth regard J9 + R as an extension structure of R. For convenience we retain also the "old" notation (o, a) for the (embedded) elements a of LR + R. It should be noted that by (52.41) and (52.6) (o, 0) (= 0) is the additive zero element of V + R. Of course V + R is not, in general, necessarily a ring. THEOREM 112. (EVERETT'S fundamental theorem on ring extensions). All the Everett extensions of a ring R by a ring. are (to within isomorphism) those

Everett sumsJI + R, which are rings, and may be characterized by the conditions

a(f+y)=aj9+ay, (a+j9)c=ac+3c, a(fly) = (a9)y,,

(aj9)c = a(rc).

(aj9)c = a(rc) ((xb)y = a(by)

(52.8) (52.9)

, ,

(a + b)y + [a, b]y = ay + by, a(b + c) + a[b, c] = ab + ac, (ab)y + y = a (by),

(52.7)

a(bc) + a _ ((xb)c,

[a, b] = [b, a] ,

[a, b] + [a + b, c] = [a, b + c] + [b, c],

(52.10) (52.11) (52.12)

(52.13) (52.14)

EVERETT RING EXTENSIONS

189

+
(52.15)

a[b, c] + .

(52.16)

For these rings

(,* + R)/R ; J9 ((a, 0) + R

a).

(52.17)

Since, according to (52.7), both a -> ab and f -± a# are endomorphisms

of the module R+, provided 59 + R is a ring, in this case we call the last two terms ab, a/i of the function quadruple (52.5) the two endomorphism systems of * + R. We shall prove Theorem 112 together with Theorem 113, which is an important supplement and, in part, a converse of it. THEOREM 113. Let a ring 91 together with an ideal R be given. Put = 91/R

(52.18)

and denote the elements of R and,* as above [in (52.3)], as a consequence

of which the a, b ... (E JP) now mean the residue classes of 91 mod R, in particular, o is the principal class R. Further, take a choice function f(a)Ea

(aE -

; f(o) = 0)

(52.19)

and put [a, b] _ .f (a) + .f (b) - .f (a + b),

ab = af(b), a/ = f(a)f3


(a, b E'; a, f E R).

(52.20)

For this function quadruple the conditions (52.6) to (52.16) are fulfilled, and for the corresponding Everett extension ring JP -[- R,

,P + R ,:: R ((a, a) -> f (a) + ac) .

(52.21)

First we prove that the conditions (52.7) to (52.16) are necessary and sufficient for R + R to be a ring. Primarily it follows from Theorem 107 that the conditions (52.13), (52.14) are necessary and sufficient for V+ R to be a module with respect to the addition (52.41). Therefore we suppose (52.13), (52.14) to be satisfied henceforward. Then we still have to prove that (52.7) to (52.12) and (52.15), (52.16) are necessary and sufficient that the multiplication (52.42) is associative and, with respect to the addition (52.41), distributive.

The condition for distributivity is expressed according to (52.4) by the following two equations

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STRUCTURES

(a,a)(b+c, [b,c]+il+y)= (ab,+ab+afi+(xq)+(ac,+ac+ay +ay), (a + b, [a, b] +a+ f3) (c, y) = = (ac, + ac + ay + ay) + (be, + is + by + fly) . These become, again by (52.4),

+a(b+c)+a([b,c]+fl +y)+a[b,c]= = [ab, ac] + + + ab + ac + ail + ay,

(52.22)

ca+b,c> +(a+b)y+([a,b]+a+ fl)c+ [a, b]y = = [ac, bc] + + + ay + by + ac + 9c .

(52.23)

We show that these two conditions are equivalent to (52.7), (52.11), (52.16).

From these (52.22), (52.23) follow immediately. Conversely, if we sub-

stitute in (52.22), (52.23) b = c = o and a = b = o, respectively, then, by (52.6), we obtain equation (52.7) and then (52.22), (52.23) are reduced to

+a(b+c)+a[b,c]+a[b,c]= = [ab, ac] + + + ab + ac,

+ (a + b)y + [a, b]c + [a,b]y= = [ac, be] + + + ay + by. From these, for a = o and c = o, respectively, equations (52.11) and so also equations (52.16) follow. We have proved that distributivity follows from the conditions (52.7), (52.11), (52.16), and conversely. In the following we still suppose also the last three conditions to be satis-

fied. We have then only to prove that the conditions (52.8) to (52.12) and (52.15) are necessary and sufficient for multiplication (52.42) to be associative.

The conditions for this are (a b, + zb + a(i + aft) (c, y) = = (a, a) (be, + fic + by + fly) .

By (52.4.,) and (52.7) this equation is equivalent to

+ (ab)y + c + (ab)c + (afl)c + (afl)c + + y + (ocb)y + (afl)y = + a + a(re) + + a(by) + a(fly) + a(bc) + a + a(f c) + a(by) . (52.24) On the one hand, this equation obviously follows from (52.8) to (52.12)

EVERETT RING EXTENSIONS

191

and (52.15). On the other hand, all these equations are obtained from equation (52.24) if we identify - always in a suitable manner - three of the six variables a, b, c, a, f3, y with the zero element o or 0 and also take (52.6) into consideration. Hence we have proved that '2 + R is a ring if, and only if, (52.7) to (52.16) are satisfied. In order to prove (52.17), we consider a ring JP + R. According to (52.4)

J'i + R . JP

((a, a.) -- a).

The kernel of this homomorphism is R. Since, on the other hand, according to (52.41) and (52.6), (a, a) = (a, 0) + (o, a), so the (a, 0) + R (a E,59) are

all the different residue classes mod R. From both [by Theorem 60, (52.3)] the validity of (52.17) follows. Hence, every ring J? + R is an Everett extension of R by 2. Since the converse of this is a consequence of Theorem 113, we have only to prove this theorem. For the function quadruple (52.20) the condition (52.6) is trivially fulfilled on account of f(o) = 0. From the definition of f(a) the

f(a) + a

(a E

'; a E R)

(52.25)

are all the distinct elements of 91. For the sum and the product of two arbitrary elements of N we obtain on account of (52.20) the formulae

(f(a)+a)+(f(b)+ j9)=f(a+ b)+ [a,b]+a+fi, (f (a) + a) (f (b) + ,B)

= f(ab) + + ab + afl + aj3 .

Comparison with (52.4) proves, by (52.25), the isomorphism (52.21). Since, by this, JP + R is a ring, it follows from the part of Theorem 112 already proved that the function quadruple (52.20) satisfies the conditions (52.7) to (52.16). Thus Theorem 113 is proved and the proof of Theorem 112 is completed. Since, from the first part of Theorem 112, all the Everett extensions of R by,A are essentially given by the rings 5' + R, in the following we mainly

take the Everett extensions in the form + R. This has the advantage that for given R, all the JP + R consist of the same elements, thus they differ from one another only in the addition and multiplication of the elements, i.e., in the function quadruples (52.5). In order to give the Everett extension R + R, it is sufficient, by Theorem 112, to determine function quadruples [a, b], ab, a#, which constitute a solution of (52.7) to (52.16).

Two Everett extensions J2 + R of R by A are called equivalent (i.n.s.), if there is an isomorphism between them, which maps every class mod R onto itself and leaves the elements of R fixed. The function quadruples belonging to equivalent extensions are called associated.

STRUCTURES

192

THEOREM 114. If [a, b], , ab, a# is an arbitrary function quadruple. which satisfies the conditions (52.6) to (52.16), then by the formulae

[a, b]+a'+b'-(a+b)', + a'b + ab' + a'b' - (ab)', ab + ab', afi + a'j

(52.26)

are given all the function quadruples (possibly each one several times) associated with it, where a -* a' denotes any mapping of ,2 into R with o' = 0. Between the

two equivalent Everett extensions ll, J'1' belonging to the function quadruple

[a, b],
St ,: lit' ((a, a) -* (a, a - a)).

(52.27)

COROLLARY. For all the choice functions (52.19), (52.20) runs through a

full system of associated function quadruples (not necessarily all distinct).

In order to prove the theorem, we take the Everett extension

91,

belonging to the given function quadruple [a, b], ab, a/, in the form J2 + R. There the compositions (52.4) hold. Further 91' shall denote

an Everett extension of R by JP belonging to a further function quadruple [a, b]1, 1, (ocb)j, (a/I)1. We may assume that lit' consists of the same elements as l and that the compositions

(a, a) -I- (b, /B) _ (a + b, [a, b]1 + a + 9) , (a, a)

(b, /1) _ (a b,
(52.28)

are valid in it. lit, lit' are equivalent if, and only if, an isomorphism lit

91' ((a, a) -(a, a)°)

(52.29)

holds, where (a, a)° is of the form (a, a*) (a* E R) and (o, a)° _ (o, a). If

this is the case, then we put (a, 0)° = (a, -a) with a' E R, o' = 0. Then we have (a, a)° _ ((a, 0) + (o, a))° _ (a, 0)° + (o, a)° _ (a, - a) -T- (o, a)

=(a,a-a'), by which (59.29) passes over into (52.27). Conversely, if a' E R and o'=0 and (52.27) holds, then 91 is evidently equivalent to 51'. Now, because of Theorem 63, (52.27) is equivalent to (a, a) -l- (b, /i) = 11(11-1 (a, a) + II -1(b, /3)) , (a, a)

(b, /1) = II (II-1(a, a) 11-1(b, fl)),

where II denotes the permutation (a, a) --> (a, a - a'). After the computation of the right-hand side these equations become

193

EVERETT RING EXTENSIONS

(a,a)+(b,j)=11((a,a'+a)+(b,b'+9))_

=(a+b,[a,b]+a'+b'+a+fi-(a+b)'), (a, a) is (b, 9) =

_ (ab, + a'b + ab + ab' + afi + a'b' + ab' + a'fl + aq - (ab)'). Hence we can determine the function quadruple [a, b]I, 1, ((xb)1i (a#)1 belonging to R'. Comparison with (52.28) shows that

[a,b]1=[a,b]+a'+b'-(a+b)', 1+((xb)1+(a#)1= +a'b+ab+ab'+a#+a'b'+ + ab' + a'# - (ab)'. If, in the last equation, we first substitute it = fl = 0, then a = o, fi = 0 and then b = o, it = 0 [with the aid of (52.6)] we obtain

1 = + a'b + ab' + a'b' - (ab)', (ab)1 = ab + ab', (a#)1 = a# + a'#. These equations together with the above prove Theorem 114.

In order to prove the corollary, we see that all the choice functions (52.19) may be given by one of them in the form f(a) + a', where a' means the same as in (52.26). If we replace f(a) by f(a) + a', then after easy computation (52.20) passes over just into (52.26). Hence the corollary follows from Theorems 113, 114.

In the trivial case [a, b] = = ab = a# = 0 of (52.5) the Everett sum t' + R becomes the direct sum . 51 ® R. Hence at least one Everett extension ring of Rby,59 always exists, namely the direct sum of the given rings. From (52.26) it follows that the Everett extensions + R is equivalent to R ® R if, and only if, a function quadruple

a' + b' - (a + b)', a'b' - (ab)', ab', a'#

(52.30)

- as a special case of (52.26) - belongs to it. The function quadruples of the form 0, 0, ab, a# and also the Everett extensions belonging to them are called factor free. From Theorem 112 we have immediately the following THEOREM 115. In a factor free Everett extension ring 'R + R of R by'2 ,we have the composition rules

Ia, a) + (b, j9) = (a + b, a +

(a, a) (b, fl) = (ab, ab + a# + ai4) , (52.31)

STRUCTURES

194

where the endomorphism systems ab, aj need satisfy only the conditions a(a + 9) = as + aj9

,

a(a(i) = (ax)f3 ,

(52.32)

(a4B)a = a(fla)

(52.33)

,

(ax)b = a(ab)

,

(52.34)

(2a)j9 = x(af)

,

(52.35)

(a+b)x=as+boc, a(a+b)=as+ab,

(52.36)

(ab)a = a(ba) , a(ab) = (aa)b .

(52.37)

In each of the rings ring

(a + fl )a = as + #a,

with

+ R considered, the elements (a, 0) constitute a sub-

JI

2';:t J9 .

(52.38)

It follows from (52.38) that the residue classes mod R may now be represented by the elements of a subring, i.e., + R "splits" into a schlicht sum of the ideal R and another subring. (Conversely, every ring "splitting"

in this way may be written as a factor-free Everett extension.) The equivalent Everett extensions also have this property, therefore we call them and the corresponding function quadruples splitting. As a special case of Theorem 1 14 the

a' + b' - (a + b)', a'b + ab' + a'b' - (ab)', ab + ab', a(I + a'i3

(52.39)

are all the splitting function quadruples, where ab, a# and a' have the same meaning as in Theorem 115 or Theorem 114. NOTE. The Everett extension theory may be generalized without any essential change for the case where R, J* and R are semirings with zero elements (cf. REDEI (1954b). EXAMPLE. The extension of an arbitrary ring R to a ring with unity element, carried

out in the proof of Theorem 64, was essentially a factor-free Everett extension of

Rby,.

EXERCISE. If R is a ring with characteristic m (> 0), then R admits a factor-free Everett extension by 9/m, which has a unity element.

§ 53. Double Homothetisms

Although the Everett extensions make up the exact ring theoretical analogue of the Schreier extensions, yet Theorem 108 has no analogue in ring theory, simply because in rings generally no inner automorphisms whatever may be defined. An adequate analogue to Theorem 108 is made

DOUBLE HOMOTHETISMS

195

possible by defining certain important mappings, or more precisely pairs of

mappings, of a ring into itself. The combination of two mappings of a set S is, in general, called a pair

of mappings of this set, where we have to pay attention to the order of succession of the mappings, which we distinguish as the "first" and "second" mapping, respectively. A pair of mappings will usually be denoted by only

one letter a orb, etc. If, further, a denotes a pair of mappings of the set S, then we denote the two images of an element a of S, under a, by ac and as where a -> ac and a. -* as mean the first and second mapping, respectively (of the pair of mappings considered). Correspondingly, we shall also give a pair of mappings in the form a --> ax, aa, by which we always mean the pair of mappings a -> aa, a -> aa. By a double homothetism of a ring R we understand a pair a of mappings of R into itself with the properties

a(a+3)=am +af, (a+ #)a=as+j9a, a(a/l) _ (am)#,

(xfl)a = a(/3a)

(xa)(3 = a(a(3)

,

(53.1) (53.2)

(53.3)

.

(ax)a = a(ma)

(53.4)

for all a, i (E R). This means that this double homothetism a consists of the two mappings a -* aa, a -> ca of R into itself. The identical double homothetism of R, which consists of two identical mappings, always exists. Since (53.1) to (53.4) are satisfied, if a is replaced by an element of R, the following definition is meaningful : by the inner double homothetism of R,

induced bye (E R), is meant that double homothetism a of R, for which ax = Oa ,

xa = ao

(x E R)

(53.5)

and we denote it by [e]. According to this we have the rules [O]x = e0 c,

a[e] = xo .

(53.6)

By a double endomorphism of a module M we understand a pair of map-

pings of M into itself, which consists of two endomorphisms of M. We define the sum a + b and the product ab of two double endomorphisms a, b of M by the rules

(a+b)a=as+bx, (ab)a = a(ba)

,

a(a+b)=ca+ab; a(ab) = (ca)b .

(53.7)

In the same way as we proved in § 37 that all the endomorphisms of M constitute a ring, the full endomorphism ring of M, we can prove that the

196

STRUCTURES

double endomorphisms of M, with respect to the addition and multiplication defined in (53.7), similarly constitute a ring, which is called the full double endomorphism ring of M and is denoted by ro 2(M). From (53.71,2) we obtain the direct-sum representation P'2(M) = X(M) HD

X°(M)

(53.8)

where 3'(M) is the full endomorphism ring of M and '°(M) the ring opposed to this. Henceforth R will be a given ring.

From (53.1) it follows that every double homothetism of R is at the same time a double endomorphism of R+, therefore we shall consider the double homothetisms of R as elements of the full double endomorphism ring '2(R+). Of course this means that the sum and the product of two double homothetisms of R are now to be regarded as defined and these are at least double endomorphisms of R+. Yet, in general, they need not be double homothetisms of R. Every subring of ro 2(R+), which consists entirely

of double homothetisms of R, is called a double homothetism ring of R. The previous remark means that, in general, the subring of '2(R+), gener-

ated by a system of double homothetisms of R, need not be a double homothetism ring. Two double homothetisms a, b of R are called amicable if the supplementary conditions

(aa)b = a (ab) , (boc)a = b(aa)

(a E R)

(53.9)

are satisfied by them. This property is symmetric and, because of (53.4), reflexive. Furthermore, we understand by a set or a ring of amicable double homothetisms of R a set or a ring of pairwise amicable double homothetisms of R, respectively. Since "to be a set of amicable double homothetisms of R" is a property of finite character, it follows from the theorem of TEtCHMUCLER-TUKEY (Theorem 17) that every set of amicable double homothetisms of R is contained in a maximal set of the same kind. THEOREM 116. The subring of '2(R+), generated by an arbitrary nonempty set of amicable double homothetisms of R, is always a ring of amicable double homothetisms of R. If a, b are two amicable double homothetisms of R, we may easily compute from (53.1), (53.2), (53.3), (53.4), (53.7), (53.9) that a - b, ab are also

double homothetisms of R and if furthermore c is a double homothetism

of R, which is amicable with a and b, then c is amicable with a - b and a b. Hence Theorem 116 follows from Theorem 44 by induction. According to Theorem 116 every maximal set of amicable double homothetisms of R must itself be a ring. Every such ring is called a maximal ring

DOUBLE IIOMOTHETISMS

197

of amicable double homothetisms of R. If we also take into consideration the previous result, then we have the following COROLLARY. Every ring, and even every set, of amicable double homothetisms

of R is contained in at least one maximal ring of the same kind. The inner double homothetisms of R constitute by virtue of (53.6), (53.7) and (53.9) a ring of amicable double homothetisms of R, which we call the inner double homothetism ring of R. On the strength of (53.2) an arbitrary and an inner double homothetism of R are always amicable. Therefore we have the following THEOREM 117. The inner double homothetism ring of R is contained in every maximal ring of amicable double homothetisms of R. We prove the following THEOREM 118. A ring has a unity element if, and only if, it has only inner double homothetisms, and then it is isomorphic with its inner double homothetism ring. Assume, first of all, that the ring R contains the unity element E. Then if a

is an arbitrary double homothetism of R, we obtain from (53.2) as = a(Ea) = (as) a , as = (ar)a = a(ea) (a E R) . But from (53.3), (sa)E = E(a8), i.e., E a = as. Both together mean that a is

the inner double homothetism of R induced by ae(= ea). Furthermore, P --- [e] (e E R) is obviously a homomorphism of R onto the inner double homothetism ring of R. Because of Theorem 66, this homomorphism must now be an isomorphism (even in the more general case when R has no annihilator distinct from 0). Conversely, we assume that R has only inner double homothetisms. In that case the identical double endomorphism of R must be one of these,

whence the existence of an element e of R with Qa = me = a (a E R) follows, therefore Q is the unity element of R. Thus we have proved Theorem 118.

We consider a ring SO of amicable double homothetisms of R. In this, for arbitrary a, b (E .) and a, j9 (E R), the above equations (53.1), (53.2)1 (53.3), (53.7), (53.9) hold. [The equation (53.4) also holds, although no particular mention need be made of it as it is a special case of (53.9).] We note that the above equations are the same as those given in Theorem 115 from (53.32) to (53.37). Hence we may use the application of the double homothetisms b and a on a and fl, respectively, as endomorphism systems ab, a(3 of a factor-free Everett extension ring . + R, which exists on the

basis of Theorem 115 and in which therefore the compositions are defined by

(a, a) + (b,

(a + b, a + fl) ,

(a, a) (b, j9) = (ab, ab + a/i + ocj9) (a, b E -D; m, 9 E R) . (53.10)

STRUCTURES

198

After these preliminaries we are now able to prove the following analogue of Theorem 108: THEOREM 119. The inner double homothetisms of all the Everett extensions of a ring R induce all the double homothetisms of R.

First, let SR be an arbitrary Everett extension of R. Since R is an ideal of

81, so 91R, R 9i r R. Thus if a is an arbitrary element of R, then R is mapped by both mappings a -- aa, a -+ as (a E R) into itself. Furthermore, since (53.1) to (53.4) are satisfied for a and for all the a, f of R (even

for the a, /3 of 91) it follows that the above two mappings constitute a double homothetism of R. Conversely, we now consider an arbitrary double homothetism a of R. This is contained in a ring _D of amicable double homothetisms of R. (This follows from the above corollary but it is sufficient if we take for -D the subring of ro 2(R+) generated by a.) We form the extension ring +R by _D defined in (53.10). In this (a, 0) (o, a) _ (o, a a) ,

(o, a) (a, 0) = (o, aa)

(a E R)

.

This means that the considered double homothetism a of R is induced by an inner double homothetism of + R. This completes the proof of Theorem 119.

NOTE. The group and ring theoretical analogues manifested in Theorems 108, 109, will be followed up in the subsequent paragraphs. Our double homothetisms are special cases of the "multiplications" in HOCHSCHILD (1947) which were later called by MncLANE (1958) "bimultiplications". The definition of the latter arises from that of the double homothetisms by suppression of the condition (53.4).

§ 54. The Holomorphs of a Ring

A subring of a ring R is called characteristic, if it is mapped by each double homothetism of R into itself. From Theorem 119 we obtain the following "alternative definition" : a subring of a ring R is characteristic if, and only if, it is an ideal in all the Everett extensions of R. A characteristic subring of R must be, a fortiori, an ideal of R. If, moreover, S is a characteristic subring of R, then each double homothetism of R induces a double homothetism of S. By a holomorph of a ring R we mean a factor-free Everett extension ring

_D + R of R by an arbitrary maximal ring -0 of amicable double homothetisms of R, where the compositions in _D + R are defined by

(a, a) + (b,

(a + b, a + /3)

,

(a, a.) (b, /3) = (ab, ab + a(3 + a/3) (a, b E -D; a, /3 E R) . (54.1)

HOLOMORPHS OF A RING

199

The holomorphs of R are consequently special cases of the similarly denoted

rings defined in § 53 (cf. (53.10)). By the corollary of Theorem 116 we have the following refinement of Theorem 119: THEOREM 120. The inner double homothetisms of the holomorphs of a ring R

induce all the double homothetisms of R.

Hence and from the definition of the characteristic subring is obtained the following THEOREM 121. A subring of a ring R is characteristic if, and only if, it is an ideal in all the holomorphs of R.

It is surprising that rings with unity elements are to be regarded as analogues of complete groups as the following theorem, similar to Theorem 111, demonstrates.

THEOREM 122. A ring has a unity element if, and only if, it is a direct summand in all its Everett extensions. Hence every ring with a unity element has only one holomolph. COROLLARY. In a ring with a unity elementy all ideals are characteristic. In order to prove this, let R be a ring with the unity element a and SR an

Everett extension of it. Consider an arbitrary element a of R. By virtue of Theorems 118, 119 there is an element a' of R, for which

ao=a'o, oa=oa'

(oER).

(54.2)

This a' is uniquely determined by x (namely a' = ae = ea). If (54.2) is written in the form

o(a-x')=(a-a')o=0, we see that

x = a' + v ,

(54.3)

where v (E R) has the property that vR = Rv = 0. These v obviously constitute an ideal n of R. Since, on the other hand, the representation of a (E 91) in the form (54.3) is unique, it follows from Theorem 89 that SR is the direct sum of its ideals R, n. So the "only if" part of the theorem is proved.

Instead of the statement "if" we even prove that a ring R must have a unity element if it is a direct summand in any holomorph . + R of R; here Y denotes a maximal ring of amicable double homothetisms of R. From the assumption and (52.30) we infer the existence of a mapping a--> a' of .

into R with the property that

a'+b'- (a+b)'=0, a'b'- (ab)'=0, ab' = ab , a',B = a,B ,

(54.4)

STRUCTURES

200

for all the a, b (E -0) and a, # (E R). From (54.43,4) it follows that consists entirely of inner double homothetisms of R. According to Theorem

117, R consequently has only inner double homothetisms. This means, according to Theorem 118, that R has a unity element. This proves Theorem 122 (together with the above refinement). The corollary is trivial. Regarding Theorem 122 see SZENDREI (1954). For completion cf. REDEI (1954c), SZENDREI (1955-56), LEEUWEN (1958,1965), MAC LANE (1958), EILHAUER-WEINERT (1963) POLLACK (1964).

EXAMPLE. Let R denote the free ring generated by an element a and p a prime number. The elements of the form

p2Ya,a'-hpY b,m' and p2 1

i

I

c,a'+pYdra2'-' (as,bi,c;,d;E 51),

2

each constitute a subring S and T, respectively, of R. Also S is an ideal of R and T an ideal of S, but T is not an ideal of R, therefore T is not characteristic in S. EXERCISE 1. If M is a module with non-commutative (full) endomorphism ring [according to SZELE-SZENDREI (1951) most modules have this property], then the zero ring R with R+ = M has more than one maximal ring of amicable double homothetisms, consequently also more than one holomorph. EXERCISE 2. A zero-divisorless ring, and also a ring R with R2 = R, has only one

holomorph. (Cf. LEEUWEN (1958).)

PROBLEM. Determine all rings having only one holomorph (this problem occurs in all the above-mentioned papers, most of which are devoted to this topic).

§ 55. The Two Isomorphy Theorems By a "substructure" we here always mean a substructure of the same kind.

THEOREM 123 (first general isomorphy theorem). Let a structure S, a substructure S' and a classification C of S be given, in which every class contains at least one element of S', and let C' denote that classification of S' in which every class is the intersection of S' and a class of 9. If C is compatible,

then & is also compatible; furthermore the isomorphy

S'/&

(55.1)

holds and a corresponding isomorphy arises if we assign to each class of C the class of & contained in it.

THEOREM 124 (second general isomorphy theorem). Let a structure S' a homomorphic image S' of S and a classification t ' of S' be given, and let C denote that classification of S, which consists of the inverse images of the

classes of C'. Under this assumption i is compatible if, and only if, C' is compatible; further the isomorphy (55.1) again holds and a corresponding isomorphy arises, if we assign to each class of P its image in p'.

THE TWO ISOMORPHY THEOREMS

201

For the proof of Theorem 123, we first show that the classification C2' of S' is compatible. For that purpose we consider an equivalence relation a = 9 (mod C')

(a, fi E S') . `C))

This holds a fortiori with p in place of C', since the classes in C are oversets of the classes in ?'. Since, further, 9 is compatible, it is true that, e.g., for multiplication

ea=Pj9(mod

(eES')

Since both sides belong to S', this relation holds even with C' instead of

We infer dually that ae - Re (mod p'). Thus we have proved that & is compatible, whence it follows moreover that the factor structure on the right-hand side of (55.1) exists.

Furthermore it is evident that, by the mapping defined at the end of Theorem 123, the left-hand side of (55.1) is mapped one-to-one onto the right-hand side. Since finally the classes assigned to each other always contain a common representative, it follows that this mapping is isomorphic. For the proof of Theorem 124 we denote the given homomorphism of S

onto S' by - '. Then we have, e.g., for multiplication (KA)' = K'/.'

(K,

E S) .

(55.2)

Next we assume that the classification E' is compatible. We consider an equivalence relation

a=fl(mod 0-)

(a,flES).

(55.3)

Hence follows the congruence

a' - 9' (mod

(55.4)

and for each a (E S) also, e.g.,

Bo'a'= O'l'(mod p').

(55.5)

For this, because of (55.2), we may write (9a)' __ (2#)' (mod

(55.6)

This is equivalent to ga = Pfi (mod (() ,

(55.7)

whence we see that the classification C is compatible. Conversely, we assume the latter. We consider an equivalence relation

of the form (55.4). From this follows the congruence (55.3), thus after multiplication also the congruence (55.7). This is equivalent to (55.6) and, because of (55.2), to (55.5). It follows that the classification 9' is compatible. Consequently the first statement of Theorem 124 is proved.

STRUCTURES

202

We now assume that the classifications C, t' are compatible. This implies the existence of the factor structures in (55.1). By the mapping, defined at the end of Theorem 124, the left-hand side of (55.1) is mapped one-to-one onto the right-hand side. We consider two elements of the left-hand side of (55.1), i.e., two classes

a, i (mod e)

(a, 9 E S) .

(55.8)

The class a# (mod p) is the product of these classes. The image of this class is the class (a.9)' (mod C'). Since this is the product of the classes a', #' (mod &) and these are exactly the images of the classes (55.8), we have proved the last remaining statement of Theorem 124. The general homomorphy theorem and the two isomorphy theorems above are the most important theorems in algebra. It is evident that, together with the proofs, they maintain their validity for quite arbitrary structures. These theorems might have been taken as starting point in our investigations, and then for this purpose we should have introduced only the notions of structures and homomorphism (especially isomorphism). All three of these theorems bear the stamp of "fundamental theorems" because of their simplicity and generality, although they are not usually so called.

We shall now apply Theorem 123 to the special case where S is a group and S' a subgroup of S. We may assume the compatible classification in the form G = ocN, jN,... (a, P.... E S) ,

where N is a normal divisor of S. To enable us to apply Theorem 123 we have to assume that each class of C contains at least one element of S'.

This means that we can take for the above representatives elements a, j9,

.

. .

of S'. The classification C' then has the form

Cq'=aD,1D,...

(D = S'f1N).

Consequently (55.1) now transforms to S/N

S'/(S' fl N)

.

From what has been said it follows that S = S'N. Hence our isomorphy may also be written as S' N/N

S'/(S' f1 N)

.

Finally it should be noted that S' and N may now be a subgroup and a normal divisor, respectively, of an arbitrary group. Since, by Theorem 68, it follows that S'N is a group, this may therefore be assumed for S from the first. Hence from Theorem 123 we obtain the following

THE TWO ISOMORPHY THEOREMS

203

THEOREM 125 (first isomorphy theorem for groups). If H is a subgroup and N a normal d i v i s o r o f a group, then H fl N is a normal divisor of H, and we have the following isomorphism

Hf(Hf1N)(eN-HloN)

(oEH).

Theorem 124 may obviously be formulated for groups as follows:

THEOREM 126 (second isomorphy theorem for groups). If between .two groups G, G' the homomorphism

G-G'(e-ye') holds and H, H' denote two corresponding subgroups of G and G', then H is normal in G if, and only if, H' is normal in G', and in this case we have the isomorphism G /H

G'/H'

(eH - e'H').

HN

G

HnN

N

Fia. 2

Fia. 3

We illustrate Theorems 125, 126 by Figures 2, 3. In these the obliques - - - - (downwards) denote subgroups, normal divisors and homomorphic images, respectively; the parallelism and equality of the obliques indicate the isomorphism of the corresponding factor structures. In Theorem 126, because of the homomorphy theorem, we can take the group G' in the form G/N, where N means an arbitrary normal divisor of G; furthermore the group H/N then replaces H'. Hence we obtain THEOREM 126' (second form of Theorem 126). If H is a subgroup of a group G and N is a normal divisor of G contained in H, then H is normal in G if, and only if, H/N is normal in G/N, and in this case the isomorphy G/H ^- (G/N)/(H/N)

holds; a corresponding isomorphism is obtained in which we assign to each coset eH (e E G) the coset oN(H/N) mod H/N as image element. In order that we may establish the corresponding theorems for rings, we .observe by way of introduction the following simple note. If S is a subring

STRUCTURES

204

and a an ideal of a ring, then S + a is a ring. Since S + a is a module, the statement follows from

(S+a)2=S2+Sa+aS+a2 S+a+a+a=S+a. We now obtain the following three theorems similar to the three above. THEOREM 127 (first isomorphy theorem for rings). If S is a subring and a an ideal of a ring, then s n a is an ideal of S, and the isomorphism

(S +a)/ate s/ (S f1 a) (a+a - S fl (a + a))

(aES)

holds.

THEOREM 128 (second isomorphy theorem for rings). If between two rings R, R' the homomorphism

R - R' holds and S, S' denote two corresponding subrings of R and R', then S is an ideal in R if, and only if, S' is an ideal in R', and in this case the isomorphism

R/S

R'/S'(e+S-* o'+S')

holds.

THEOREM 128' (second form of Theorem 128). If m is a subring of a ring R and a is an ideal of R contained in m, then m is an ideal in R if, and only if, m/a is an ideal in R/a, and in this case the isomorphy

R/m : (R/a)/(m/a) holds; a corresponding isomorphism arises if we assign to each residue class a) + (m/ a)mod m/a as image element. + m (Q E R) the residue class SUPPLEMENT. The first statement of the last theorem may be improved as follows: m is a left ideal of R if, and only if, m/a is a left ideal of R/a.

In order to prove the supplement we consider two arbitrary elements (E R), p (E m). m/a is a left ideal of R/a if, and only if, the product of the residue classes 0 + a, p + a is always an element of m/a. This product is Lop + a, therefore the above condition is equivalent to ep E M. This last form of the condition means, in fact, that m is a left ideal of R.

By a simple application of the isomorphy theorems we obtain the following

THEOREM 129. Let S denote a group or a module or a ring and T a substructure (of the same kind) of S, for which the factor structure S/T exists. Let us assume further that S and T are directly decomposed into substructures S1, S2, ... and T1, T.,, ..., respectively, for which Ti S Si, so that T1 = T n Si

(i = 1, 2, ...) .

(55.9)

THE TWO ISOMORPHY THEOREMS

205

Then a direct decomposition of S/T exists, the components of which, in order of

succession, are isomorphic with the factor structures S;/T;

(i = 1, 2, ...) .

(55.10)

It suffices to carry out the proof for the case where S is a ring. From the assumptions we have the direct sum decompositions

.... T=TI®T.,6.... S = S1ED S.-ED

(55.11) (55.12)

We denote by U. (i = 1, 2, ...) that subring of S which arises when we replace all the Sk (k i) on the right-hand side of (55.11) by Tk. From (55.11), (55.12) the direct decomposition

SIT =U;IT ®U,/T®... evidently follows. But

(i = 1, 2, ...)

(Si + T) IT : S;/ (S; (1 T)

(55.13)

follows from Theorem 127. Since Si + T = U;, (55.13) transforms, by (55.9), into U;/T & Si/T

.

Hence we have proved Theorem 129.

§ 56. Simple Factor Structures By a maximal normal subgroup of a group G we understand any maximal element of the set of all the normal divisors of G other than G. A maximal submodule is interpreted as an additive analogue of the former. By a

maximal ideal or maximal left ideal of a ring R we mean any maximal element of the set of ideals or the set of left ideals, respectively, of R other than R itself. (Of course a maximal ideal need not be a maximal left ideal. If a maximal left ideal is an ideal, then it is a maximal ideal.) From Theorem 126' it follows directly that a factor group G/N (N # G) of a group G is simple if, and only if, N is a maximal normal subgroup of .G. The corresponding result holds for modules. Similarly it follows from Theorem 128' that a factor ring R/a (a 0 R) of a ring R is simple if, and only if, a is a maximal ideal of R. . As regards a factor ring R/a there is also the question as to when it is a skew field. By Theorem 79 and Theorem 128' and its supplement it follows.

206

STRUCTURES

immediately as a necessary condition that the ideal a must be a maximal left ideal. This is, by the following theorem, "almost" sufficient. THEOREM 130. If a is an ideal and, at the same time, a maximal left ideal of a ring R, then the factor ring R/a is either a skew field or a zero ring of prime order. COROLLARY. If a maximal ideal of a ring is not a maximal left ideal, then

it is not a maximal right ideal either. It follows from Theorem 128' and its supplement that Theorem 130 and its corollary may also be expressed as follows: THEOREM 130' (second form of Theorem 130). A ring distinct from 0 without proper left ideals is either a skew field or a zero ring of prime order. COROLLARY. A ring without proper left ideals has no proper right ideals.

In order to prove Theorem 130' consider a ring R (s 0) without proper left ideals. For any a (E R, 0) the left ideal Roc is either 0 or R. If Ra = 0, then the ring {a} (t 0) is a zero ring and because R{a} = 0 it is also a left ideal of R, consequently {a} = R. Since according to this R is a zero ring, its submodules are just ideals, so that R contains no proper submodule. Thus it is of prime order. Now suppose Ra = R for each a (E R, 0). If we take another element (E R, 0 0), then Rap = R(3 = R, thus R is zero divisor-free. Because

Rae = R there is an s (E R) with ea = a. For any o (E R), oea = oa, consequently oe = o. Accordingly a is a right unity element and, by the corollary to Theorem 36, a unity element. Since t:a = e (; E R) is now solvable, R is a skew field. Thus we have proved Theorem 130'.

Since this theorem is dualizable, the corollary follows from it. The above proof is from F. A. SzAsz (1953-54). Cf. SZELE (1949) and the Example at the end of § 106. F. A. SzAsz (1960, 1961, 1963) has constructed rings with infinitely many left ideals but only finitely many right ideals.

THEOREM 131 (KRULL's theorem). In a ring with at least one right unity element any proper left ideal is contained in a maximal left ideal. COROLLARY. Among the factor rings of a commutative ring with unity element there occurs at least one field.

For the proof of this theorem we consider a ring R with the right unity element B. Let a be a proper left ideal of R. Consider all the left ideals b of R with the property that

acb

R.

None of these left ideals can contain e, therefore their union set is t R. Since further Theorem 47 is evidently applicable to left ideals of a ring, it follows from the KURATOWSK1-ZORN lemma (Theorem 16) that among the b there is a maximal left ideal. The corollary follows from Theorems 130, 131.

SIMPLE FACTOR STRUCTURES

207

EXERCISE 1. The module of rational numbers contains no maximal submodule. EXERCISE 2. The four-ring has three proper left ideals and only one proper right ideal.

§ 57. Commutative Factor Structures

It is easy to state when a factor group G/N is commutative. This is so

if, and only if, aN 1N = 1N aN, i.e., a(3 N = (3aN for all a, fl (E G). This condition is equivalent to = N, i.e., to afla-1j_1 N

afla_1/3-1 E N

(a, /9 E G)

.

Hence we have the following THEOREM 132. The factor group G/N of a group G with respect to the normal divisor N is commutative if, and only if, N contains the commutator group of G.

We similarly obtain the result that a factor ring R/a is commutative if, and only if,

a# - #aEa

(a,,9ER).

To express this result in a form similar to the above, we define the commutator ideal of a ring R as the ideal of R generated by all the commutators afl - 9a (a, j9 E R). THEOREM 133. The factor ring R/a of a ring R with respect to the ideal a is commutative if, and only if, a contains the commutator ideal of R.

The factor semigroup H/C2 of a semigroup H with respect to the compatible classification C2 is evidently commutative if, and only if, a(3 =_ (3a (mod e)

(for all a, j E H).

EXERCISE. The factor group of a dihedral group with respect to its commutator group is a four-group or a group of order two. The first case occurs if, and only if, the given group is infinite, or finite and of an order divisible by 4.

§ 58. Zassenhaus's Lemma

We consider four subgroups A, A', B, B'

(58.1)

of a group G such that A', B' are each a normal divisor of A and B, respec-

8 R-A

STRUCTURES

208

tively. If D is a subgroup of G such that

A = DA', B = DB',

(58.2)

DfA'=DfB',

(58.3)

then we say that D is a Zassenhaus subgroup for the four subgroups (58.1). We prove the isomorphism A/A'

B/B' (bA' -+ 6B')

(6 E D)

(58.4)

where we have to take the representatives 6, as indicated, from the subgroup D. (It is not necessary that two corresponding cosets should have only one common representative 6 in D.) In fact, a mapping was given in (58.4), which according to (58.2) maps

the left-hand side of (58.4) onto the right-hand side. Since from (58.3) 61A' = 62A' a 61B' = 62B'

(61, 62 E D) ,

the above mapping is one-to-one. It is evidently homomorphic, whereby the statement is proved. Further we consider four subgroups A, A', B, B'

(58.5)

of a ring R such that A', B' are each an ideal of A and B, respectively. If D is a subring of R with the properties

A=D+A', B=D+B', DfA'=DfB',

(58.6)

(58.7)

then we say that D is a Zassenhaus subring for the four subrings (58.5). This again implies the isomorphism A/A'

B/B' (6+A'-+6+B')

(bED).

(58.8)

[The notes on (58.4) similarly apply here.] The following theorem is noteworthy in itself and will be of importance in § 59. THEOREM 134 (ZnssENHAus's lemma). If A, B are subgroups of a group and

A', B' are each a normal d i v i s o r o f A and B, respectively, then A fl B is a Zassenhaus subgroup for the four subgroups

A'(A fl B), A'(A fl B'), B'(A fl B), B'(A' fl B) ,

(58.9)

ZASSENHAUS'S LEMMA

209

whence it follows that

A'(A fl B) /A'(A fl B') x B'(A fl B) /B'(A' fl B) where

6A'(Af B')--SB'(A'flB)

(58.10)

(tEAf B)

(58.11)

is a suitable isomorphism.

If A, B are subrings of a ring and A', B' are each an ideal of A and B, respectively, then A fl B is a Zassenhaus subring for the four subrings

A'+(Af1B), A'+(AnB'), B'+(Af1B), B'+(A'f1B),

(58.12)

whence it follows that

A'+(Af1B))/(A'+(Af B')),. (B' + (AnB))/(B' + (A, nB))

(58.13)

where

b+(A'+(AnB'))--> b+(B'+(A,nB))

0 EAf1B)

(58.14)

is a suitable isomorphism. We prove the first part of the theorem. Since A' is normal in A,

A'(A fl B) = (A fl B)A' , A'(A fl B') = (A n B')A'

.

(58.15)

Both products, by Theorem 68, are groups. By symmetry this is true when A, A' are replaced by B, B'. Accordingly four groups were given in (58.9). We show that the second of these groups is normal in the first. It is sufficient to prove :

a'A'(A fl B') = A'(A fl B')a'

(a' E A),

(58.16)

bA'(A n B') = A'(A n B')b

(b E An B).

(58.17)

By double application of (58.152) we obtain

kA'(AnB')=A'(AnB')=(A f1B')A'=(AnB')A'B'a'=A'(AnB')a' whence (58.16) is true. Furthermore we easily obtain

bA'(Af1B')=A'b(Af1B')=A'(AnB')b, whence (58.17) is also true.

From these results, it follows by symmetry that the fourth group in (58.9) is normal in the third. We have still to prove that the properties (58.2), (58.3) hold f o r (58.9) and A n B instead of (58.1) and D.

210

STRUCTURES

The first of these assertions is trivial. For the second it is sufficient, by symmetry, to show that

Af1BflA'(Af1B') 9Af1Bf1B'(A'f1B). Instead of this assertion, we even prove that

BfA'(Af1B') 9 B'(A'f1B), for which, by (58.152), we may write

B n (A fl B')A' 9 B'(A' fl B) .

(58.18)

Each element a of the left-hand side is of the form

(13E B, 6'EAf1B',a'EA').

e=19= S'a'

Hence a E B and so a' E A' fl B. Since, moreover, S' E B', (58.18) follows. This completes the proof of the first statement of the theorem. The assertions in (58.10) and (58.11) then follow immediately from (58.4) by applying it to the above case "(58.9) and A fl B" instead of "(58.1) and D". The proof of the second part of Theorem 134 is very similar to that of the first part, therefore we leave it to the reader. The second part of Theorem 134 is new. If we want to prove from the theorem only, the isomorphy propositions (58.10), (58.13), without really giving the isomorphic mappings (58.11) and (58.14), respectively, then we can proceed more directly

by using the first and second isomorphy thaorams (for groups and rings). (For groups cf. ZASSENHAUS (1958), KUROSH (1953a).)

EXERCISE 1. Carry out the proof of the second part of Theorem 134. EXERCISE 2. Theorem 134 contains as a special case the first isomorphy theorem for groups and rings. EXERCISE 3. For four subgroups A, A', B, B' (A' c A, B' C- B) of a group,

(AfB')(A'(1B)=AfB'A'flB,

(58.19)

AB' f1 A'B = A'(A fl B)B',

(58.20)

(A fl B)A' = A fl BA'.

(58.21)

(Concerning (58.19) and (58.21) cf. ZASSENHAUS (1958) p. 21.) Both (58.19) and (58.20)

contain (58.21) as a special case. EXERCISE 4 (GRATZER). Retain the foregoing notations. If

is then a matrix, of

type m X n, each element all of which is one of the four subgroups A, B, A', B', then

fl (n a,,) = it (l au-) =1 ,=1 =1

ZASSENHAUS'S LEMMA

211

where v runs through all the mappings of the set <1, ..., m) into the set <1, ..., n). In particular for

AA'

A' B' B B'

(A Bl A, B /JI

we obtain again (58.19) and (58.20), respectively.

§ 59. Schreier's Main Theorem and the Jordan-Holder Theorem

Let S denote a group with the unity element s, or a ring. By a normal series (of length m) of S we understand a finite sequence A0, . . ., Am of subgroups (subrings) of S where

S=Ao

A1?... DAm=s(or0)

(59.1)

and every A,+1 is a normal divisor (ideal) of A. (i = 0, ..., m - 1). The A,/A;+1 are then called the factors of (59.1). Two normal series are called isomorphic, if they are of the same length and their factors taken in a suitable order are pairwise isomorphic. A refinement of the normal series (59.1) of S means a normal series of S, in which all the members of (59.1) occur. A normal series (59.1) is called proper, if everywhere D holds instead of i. By a composition series of S is meant a proper normal series of S which cannot

be refined, i.e., the series has no refinement different from it which is also

a proper normal series of S. By the second isomorphy theorem (Theorems 126, 128) this is the case if, and only if, all the factors of the normal series are simple and consist of more than one element. THEOREM 135. (SCHREIER'S main theorem). Any two normal series of a group or of a ring have isomorphic refinements. In the proof we shall give two such refinements explicitly. We can restrict ourselves to the case of groups, since that of rings can be treated similarly. Therefore, let S denote a group. For the two given normal series of S let one be (59.1) and the other be (59.2)

We apply the lemma of ZASSENHAUS (Theorem 134) with

A=A A'=A,+1, B=Bi, B'=Bi+1

(i=0,...,m- 1; j=0,...,n- 1).

Then

Ai+t (A, n Bi) /At+l (At n Bi+1)

Bi+1(A, n Bi) /Bi+1(A,+1 n Bi) (59.3)

STRUCTURES

212

We write for the sake of brevity AU=A,+1(Ain B;)

(i=0,._ m- 1; j=0,...,n),

(59.4)

B;1= Bi+1(AtnBB)

(i = 0, ..., m; j = 0, ..., n - 1).

(59.5)

(i = 0, ..., m - 1; j = 0...., n - 1).

(59.6)

Then (59.3) becomes A+i /Ai.i+1

Bj;l Bi+1,1

Further by (59.4) and (59.1), (59.2)

A;o=Ai;? AilQ...QA;,,=Ai+1

(i=0,..., m- 1).

Similarly by (59.5) and (59.1), (59.2) Bm1=B1+1

(j = 0,

.., n -

1)

Thus if we insert in (59.1) between Ai, A,+1(i = 0, . . ., m - 1) the members Ail, ..., A,.,,_ 1 and in (59.2) between B1, Bl+l (j = 0, ..., n - 1) the members B1j, ... Bm_ 1, j, then we obtain refinements of (59.1) and (59.2), (of length mn), which are isomorphic by (59.6). Thus Theorem 135 is proved. THEOREM 136 (JORDAN- HOLDER theorem). Let S denote a group or a ring.

If S has a composition series, then every proper normal series of S may be refined to a composition series (so that, in particular, every normal divisor or every ideal of S belongs to at least one composition series). Further all the composition series of S are isomorphic. It suffices to carry out the proof for the case1where S is a group. We con-

sider two proper normal series I, II of S, of which at least one, e.g. I, is a composition series. We then consider according to Theorem 135 two iso-

morphic refinements I', II' of I and II, respectively. Moreover we may assume that the normal series I', II' are proper, since any members which occur more than once in them may be cancelled. Since I is a composition series, then I = I'. Hence it follows that II' is also a composition series, which proves the first part of the theorem. If II itself is already a composition series, then it necessarily follows that 11 = II', which proves the second part of the theorem. From this theorem we can briefly say that every group or ring "essentially" has at most only one composition series. However, it would be erroneous to think that from one composition series we might deduce the others according to a general rule. Of course, not all groups and rings have a composition series, e.g., the infinite cyclic group obviously has none. On the other hand, finite groups and rings always have a composition series. Simple groups and rings are distinguished by the fact that they have a composition series of length 1 (thus they have only one composition series).

THEOREMS OF SCHREIER AND JORDAN-HOLDER

213

An application of Theorem 136 is contained in the following THEOREM 137. The simple direct components of a completely reducible

group (or ring) S are, apart from the order of succession and isomorphy, uniquely determined. S itself has a composition series and all the normal divisors (ideals) of S are direct components of it. The proof is similar for both groups and rings, so we consider only the case where S is a completely reducible group. Let this be decomposed into the direct product

S=S,®... ®S,,

of simple subgroups Si, ..., S. (0 E). Put

(i=0,..., n; To=E,

Ti=S1...Si

These T, are normal divisors of S. Since further Tj/Ti-1

Si

(i = 1,

. .

., n)

,

(59.7)

they constitute a composition series of S. Hence and from (59.7), by virtue of the second part of Theorem 136, the first two assertions of Theorem 137 follow.

In order to prove the last assertion of the theorem, we take a normal divisor T of S. Because S = Sl ... S,,, a fortiori,

S = S1... S T. Put Di

= Sif1Si+i ... S T

(59.8)

(i = 1, ..., n).

Since Di is a normal divisor of S and therefore, a fortiori, of Si, and since Si is simple, it follows that either Di = Si or Di = e (i = 1, ..., n). Hence Si is either a subgroup of Si+1 ... S,, T or s is the only common element of these two normal divisors of S. Hence and from the corollary to Theorem 87 it follows that for every i (= 1, ..., n) Si

... S T = Si+I ... S T or

Si ® S,+1... S T.

If these equations are successively substituted in (59.8), then an equation of the form

S=Si,®...®S,,®T is obtained.

(15i,<...
214

STRUCTURES

Accordingly T is a direct component of S. This completes the proof of Theorem 137. A finite Abelian group is obviously simple if, and only if, its order is 1 or a prime number. Hence it follows that in every composition series of a finite abelian group all the factors are of prime order.

A group is called solvable, if it has a nornial series whose factors are all abelian. (The term "solvable group" is derived from the fact that finite solvable groups in particular are closely connected with "solvable equations". Cf. § 174.) From the above note follows THEOREM 138. A finite group is solvable if, and only if, it has a normal series

with the property that each member of it (from the second on) has a prime number index with respect to the preceding one.

In this connection we wish to point to the recent theorem of Feit and Thomson (1963), which is one of the most important mathematical results of our age; it affirmes that every finite group of odd order is solvable. For a finite group it follows from Theorem 136 that the orders of the factors of its composition series, apart from the order of succession, are invariants of the group. Only this part of Theorem 136 (for groups) is due to JORDAN. With regard to Theorem 135 cf. SCHREIER (1928). For rings Theorem 136 was, as far as we know, unknown

till now. (For further details about this see § 62, Exercise 4.) EXAMPLE. From Theorems 84, 85 it follows that the full permutation group JS

when n = 3 or n Z 5 has only the one composition series ,,,, cA,,, 1. When n = 4 it has only the three composition series ffl4, al4, G4, H, 1, where G4 is the four-group

mentioned in Theorem 84 and H is any one of the three proper subgroups of G4-

§ 60. Lattices Let S denote a set and (S) the set of all subsets of S. In this set (S) are defined the two compositions (1, U , the intersection and the union of sets, often used already. In this sense (S) constitutes a structure which we hen-

ceforth call the lattice of all subsets of S, or a full set lattice. In this both compositions are associative and commutative and have the properties

A()(AUB) = A, AU(A(1B) = A

(A, B S S).

By a lattice we understand in general a set V, in which two associative and commutative compositions A, v are defined such that the rules

aA(av#)=a, av(MAP)=a

(a,flEV)

(60.1)

hold, which we call the two absorption rules. The two compositions A, V in a general lattice are also called the intersection (meet) and union (join), respectively, or just the first and second composition.

LATTICES

215

Lattices constitute an important class of structures: they occurred first in DEDEKIND who gave the name "Verband" to them. However, they have only been studied generally since about 1930, so that lattice theory may be regarded as one of the younger branches of algebra. But it is acquiring more

and more importance with respect to the other structures (group, rings, etc.). We shall deal with lattices only cursorily and we refer to BIRKHOFF (1948), HERMES (1955), DUBREIL-JACOTIN-LESIEUR-CROISOT (1953), SzAsz (1963). A lattice constitutes, with respect to each of the two compositions, a com-

mutative semigroup, which we call the first and second semigroup, respectively, of the lattice; however, these semigroups are not arbitrary, but connected with each other by the absorption rules (60.1). (See also the corollary

of Theorem 142 and the following note.) Because of the commutativity of lattices the duality defined on p. 39 is trivial and therefore should be entirely disregarded. On the other hand, we can speak of duality with respect to lattices, which consists of the interchanging of the compositions. For, from one concept connected with a lattice, there arises another, which we call the dual concept, by interchanging the two composition signs A, V. So e.g., from a A (j9 v y) by dualization

we obtain a v (1 A y). The invariant notions with respect to dualization are called self-dual. Since the lattice axioms obviously constitute a self-dual system, the important duality principle of lattices follows: In lattices, together with any theorem deducible from the lattice axioms, the theorem dual to it also holds.

As a first application, it follows that the dual of a lattice is again a lattice (with the compositions v, A), which we call the dual lattice. THEOREM 139. In a lattice we always have

xnx=x, acvCC =a,

(60.2)

i.e., all elements are idempolent for both compositions. Since (60.2) consists of two dual propositions, it suffices to prove (60.21). According to (60.12) and (60.11)

x = xv(ana), x = aA(av(cAa)). From this (60.21) follows, which proves the theorem. THEOREM 140. In a lattice we always have aA/9 = I GS av(3 = j9 .

(60.3)

The statement consists of the two partial statements "r*", "=", which are evidently dual to each other, therefore it suffices to prove the first of them. From the left-hand side of (60.3), by (60.12), it follows that 8/a

R - A

STRUCTURES

216

#=i9v(i9na)=i9va=xvfi which proves Theorem 140. The absorption rules (60.1) themselves are rather complicated, but (60.2)

and (60.3) express much simpler properties of lattices. Now it is important that, according to the following theorem, in the definition of lattices, (60.1) may be replaced by (60.2) and (60.3). THEOREM 141. A set V, in which two compositions n, v are defined, is a lattice if, and only if, V constitutes a semigroup of idempotent elements with respect to both compositions, and rule (60.3) holds. Because of Theorems 139, 140 we have only to prove the statement "if". Thus we have to show that from the conditions in Theorem 141, (60.1) may be deduced. It is sufficient to deduce (60.11). From the associativity of v and from (60.22)

av (av9)=avfl.

By (60.3) this is equivalent to (60.11). Consequently we have proved Theorem 141.

Before continuing, we consider an arbitrary semiordered set V with the

semiordering relation <. For two elements a, fi (E V) we define their greatest lower bound inf (a, fi) and their least upper bound sup (a, f9) in both cases by the two corresponding properties

inf (a, 9) 5 a, f ; sup (a, 9) >_ a, fl, 1 S a,

l <_

inf (a, fl);

(60.4)

> a, L >_ sup (a, /1) ( E V). (60.5)

For each pair a, fi there is at most one inf (x, fl) and one sup (0c, P), respectively. If, e.g., inf (a, j9) had the two values e, a, then it would follow from (60.41) and (60.51) that e < a and a <_ 0, consequently P = a. On the other hand, inf (a, fl) and sup (a, fl) need not always exist. For instance, Fig. 4

illustrates a semiordered set with five elements, 19

71

Fia. 4

in which inf (a, 9) = 71, but sup (a, 9) does not exist. Such a so-called Hasse diagram (for finite semiordered sets) is to be interpreted in such a way that two

elements K, :. are joined by a straight line from K down to

if, and only if, I is a lower neighbour of K.

It is very important that it is possible to define a semiordering relation in every lattice by which the lattice is uniquely determined. In fact, especially in a full set lattice V

A9BcsAfB=A

(A,BEV)

and in the general case we have the more precise

LATTICES

217

THEOREM 142. In every lattice V a semiordering relation <_ is defined by

CC 5(iaaA#=a

(a,/9EV),

(60.6)

for which inf (a, /9), sup (a, (3) always exist and inf (a, (3) = MAP,

sup (a, /9) = a v (1.

(60.7)

Conversely, if in a semiordered set V inf (a, ,B) and sup (a, /9) exist for all a, /9, then we may define in V two compositions A, V in one and only one way so that V becomes a lattice (60.6) (therefore (60.7) also holds). COROLLARY. In every lattice each of the two compositions is uniquely determined by the other. In order to prove the first part of the theorem we consider a lattice V. First we prove that a semiordering relation < is defined by (60.6). It suffices to show that

a
a5(i, GC :!5;

yy, GC

0C

fl

By (60.6) we write these as

aAa=a, MAP=a, PAY =/aAY=a, anP=cc,9na=/9 cc=/9. The first relation is self-evident. The second is true, since from its lefthand side we have MAY = (an/9) AY = an (j9Ay) = aA# = QC.

The third follows from the commutativity of A. Because of (60.6) and (60.3) we have

a _ /a/_ are dual notions. The same is also true by (60.4) and (60.5), for inf (a, /9) and sup (a, /9). For this reason it suffices to prove e.g., (60.71). We have to show according to (60.41), (60.51) that

ani4a,P, _

a,/9=E<_aA#.

218

STRUCTURES

According to (60.6), these may be written as : (ani4)na = %n/9 ,

(an/9)n f3 = anfl

,

$n(Z = Enf5 = $ =)- 4':n((Xn/9) _ $.

Since the latter are evidently satisfied, the first part of Theorem 142 is proved.

In order to prove the second part we proceed from a semiordered set V, in which all the inf (a, 9) and sup (cc, fl) exist. First we assume that we have

defined in V two compositions n, V such that thereby V has been turned into a lattice and (60.6) holds. By virtue of the previously proved part of the theorem, (60.7) must also hold. Thus if we show conversely that a lattice is defined by (60.7) with the property (60.6), we shall have proved the theorem. Since we evidently have inf (a, i) = inf (9, a), inf (inf (a, 9), y) = inf (a, inf (f, y)), inf ((Z, a) = a and the corresponding results with sup instead of inf, so the compositions n, v, defined by (60.7), are commutative and associative, and (60.2) also holds. From inf (a, 9) = x os sup ((z, P) = and from (60.7)we obtain (60.3). From Oat has been said previously, we see that a lattice is defined by (60.7). Since finally we have

a<jc>inf(a,fl)=a, it follows from (60.7) that (60.6) is also satisfied. Hence we have proved Theorem 142. The corollary is also true, since according to (60.6) only the composition n

enters into the definition of the semiordering relation <, which in turn, by the theorem, determines the composition v, and also the dual of this is valid.

Nom. Briefly, Theorem 142 means that lattices, and those semiordered sets in which all the inf (a, P) and sup (a, P) exist, are essentially identical

concepts, which by (60.6) determine each other. For a lattice we shall always regard the semiordering relation as being given. Because of the corollary we can also consider the lattice as a special semigroup. (For further details in this respect see ELLIS (1949).)

With respect to the compositions n, v in a lattice V, there may be at most one neutral element for each (see p. 29). We denote these by I and 0

219

LATTICES

and call them the unity element and zero element, respectively, of the lattice. Thus one may define them to be such that a n 1 = a or a v 0 = a holds for all a respectively. Of course 1 and 0 need not always exist. In the full set lattice

(S), considered at the beginning, S is the unity element and 0 the zero element. A finite lattice with the elements al, . . ., a has obviously al

v ... v a as unity element and al A ... A a as zero element. We shall now briefly discuss some particularly important kinds of lattices. First of all we note that if a < 9 then inf (a, y) < inf (is, y). This is also true with sup instead of inf. From this and (60.7) it follows that in every lattice we have so-called monotonicity, i.e. QC <_<_ 9 = (any) <_ (f n y) ,

(60.8)

a < P = (avY)

(60.9)

(# V

.

Furthermore we prove semidistributivity, i.e. that an (jvy) _> (MAP) v (MAY),

(60 10)

av (Nny) < (av(3) A (avy).

(60.11)

Since (60.10) and (60.11) are dual, it suffices to prove (60.10). The righthand side of (60.10) is by (60.72), equal to sup (a n j9, a A y). Hence (60.10) is equivalent to the assertion:

an ((ivy) >_ MAP, any . But this follows from (60.8), since 9 v y From (60.11) the rule

y, which concludes the proof.

a<_y=QC v(fny)29 (avP)y

(60.12)

follows immediately (an evidently self-dual property). Later we shall see that every module gives rise to the formation of a certain lattice, where, in addition to (60.12),

a9y

av (9Ay) = (avP) Ay.

(60.13)

For this reason, with DEDEKIND we call every lattice with this self-dual property modular.

If (over and above the semidistributivity (60.10), (60.11)) distributivity holds, i.e.

an (fiv)I) _ (01n(3) v (any),

(60.14)

MV (P n y) = (a v j) A (avy)

(60.15)

STRUCTURES

220

then the lattice is called distributive. But we show immediately that (60.14) and (60.15) are equivalent properties, therefore in distributive lattices only one of these additional properties is required; since further (60.13) is a consequence of (60.15), we see that every distributive lattice is modular. By applying (60.14) and (60.1) several times we obtain

(avfl) A (avy) _((avt3) na)V(((XVJ) ny)_ = av ((acv fl) Ay) = av (any) v (PAY) = acv(flny) . Hence (60.15) follows from (60.14) in every lattice. Because of the duality principle its dual also holds. So the assertion is proved. From what has been said above it follows that the duality principle holds both in modular and distributive lattices.

Figures 5, 6 show lattices, one of which is non-modular and the other modular, though both are non-distributive. FIG. 5

FIG. 6

By a sublattice of a lattice V we mean any

subset of V, which constitutes a lattice under the compositions defined in V. By a set lattice we mean any sublattice (and the

lattices isomorphic with respect to these) of a full set lattice. It is easy to show that every full set lattice, and so every set lattice, is distributive. The converse also holds, i.e., that every distributive lattice is a set lattice; consequently "distributive lattice" and "set lattice" are identical concepts. However we omit the proof. (See, e.g., HERMES (1955).) Every structure S gives rise to the formation of important lattices if we take certain substructures of S as elements of the lattice. First we consider the case where S is a group, and we denote by [S] the

set of all subgroups of S. By A, B, ... we denote elements of [S] (i.e., subgroups of S). Since it is not necessary for A U B to be a group, [S] does not form, in general, a set lattice, with respect to the compositions fl, U. On the other hand A (1 B, {A, B} are always elements of [S] and we easily see that [S] forms a lattice under these two compositions (1, { }, which we call the lattice of subgroups of S. This is in general not modular, as e.g.,

for S = "f4It is quite different for the set [S]o (9; [S]) of normal subgroups of S, which we regard as a sublattice of [S]. This is called the lattice of normal subgroups of S and we prove that it is modular. First, we see that in [S]o the second composition is multiplication, since {A, B} = AB if at least one of A, B is normal. We have thus to prove by (60.13) that

A(B(1C)=ABfC,

(60.16)

LATTICES

221

where A, B, C are normal divisors of S and A S C. But since (60.12) holds in all lattices, it suffices to prove (60.16) with ? instead of We may

take any element of the right-hand side in the form ab = c (a E A, b E B, c E C). Because A c C, b E C and so ab also belongs to the lefthand side of (60.16). Thus we have proved the assertion. Note that § 58, Exercise 3 is a generalization of (60.16). What we have proved relates also to the lattice of submodules of a module. This is also modular. (Hence the name "modular lattice".) Further, in this lattice the second composition is just addition of complexes. For a ring S, the lattice of subrings and the lattice of the ideals of S may

be defined correspondingly. In the latter addition is again the second composition. Further it is modular, the proof of which we leave to the reader. It is noteworthy that the lattice of sub-skew-fields of a skew field is not

modular. (An example of this, which is even commutative, is given in PICKERT (1951), p. 225.)

The set of all subsemigroups of a semigroup S does not constitute, in general, a lattice with respect to the compositions fl, { }, since the intersection of two subsemigroups can be empty. If, however, to the above set is added the empty set 0 as a further element and for this we put {A, 0) = A, then a lattice is formed, which is known as the lattice of the empty set and the subsemigroups of S. The lattice of the empty set and of the sublattices of a lattice is to be interpreted correspondingly. In a lattice V with unity element I and zero element 0, by a complement of an element a we mean an element a' such that

ava' = 1, aAa' = 0

.

(60.17)

If every element a has at least one complement cc', then we call the lattice V complementary (or complemented).

We show that in a distributive lattice every element has at most one complement. If besides (60.17)

ava" = l ,

uAa" = 0,

then from the distributivity it follows that

a.' = a' v 0 = a' v (a A a") = (a' v a) A (a' v a") = 1 A (a' v a") = a'v a" consequently a" <_ a'. By symmetry ac' < a", whence the assertion follows. In particular, the full set lattice (S) of a set S is complemented and in this the complement of A (9 S) is equal to the difference set S - A.

A distributive and complemented lattice is called a Boole lattice or Boole algebra. Thus every full set lattice is a Boole lattice.

222

STRUCTURES

Let V be a semiordered set. If we assume the existence of inf (a, 9) for all a, j3, then V is called a semilattice. Then by (60.71) a composition A is defined in V. With respect to this, V becomes a commutative semigroup VI of idempotent elements. From the first part of the proof of Theorem 142 it follows that, conversely, every commutative semigroup of idempotent elements is a semilattice. If sup (a, f1) exists in V for all at, 9, then the above

may be repeated with (60.72) instead of (60.71), yielding a semigroup V2 with the composition v. Of course V2 is also a semilattice and VI and V2 are interchanged if we introduce in V the semiordering relation > instead of <. We call VI and V2 the first and second semilattice, respectively, connected with V. If, especially, a lattice is determined by V, i.e., if inf (a, $) and sup (a, j9) exist for all a, j9, then V11 V2 are dual and each is essentially, in the sense of the corollary to Theorem 142, just the above lattice. (They are then identical with the first and second semigroups of V previously defined.) As an interesting example, we consider the set [S] of subgroups of a semigroup S. Since the intersection of two elements of [S] can be empty, [S] in

general does not form a semilattice with respect to the composition fl, but because of Theorem 46, if we add the empty set 0 to the set [S] as a further element we obtain the semilattice of the empty set and the subgroups of S. Of course, the general homomorphy theorem (Theorem 60) and the two general isomorphy theorems (Theorems 123124) hold for lattices and semilattices. Also, ZASSENHAUS'S lemma, SCHREIER'S principal theorem and the JORDAN-

HoLDER theorem may, under certain restrictions, be extended to lattices. On the other

hand, the first isomorphy theorem for groups and rings (Theorems 125, 127) and even the three theorems mentioned above contain propositions relating to the lattice of subgroups of a group or subrings of a ring. To all appearances lattice theory is merging into other parts of algebra and in all likelihood lattices will take a leading role. However, it seems that the theory of lattices is not yet sufficiently simplified at present for this to happen in the immediate future. We do not intend to develop lattice theory further, but we shall often make use of it to elucidate various facts. In addition to the literature mentioned previously cf. KORINEK (1942), BENADO (1954, 1955).

EXERCISE 1. A one-to-one mapping a - a' of a semilattice V onto a semilattice V' is an isomorphism (V and V' are regarded as semigroups) if, and only if, a implies

EXERCISE 2. A ring with unity element consisting entirely of idempotent elements

is called a Boole ring. Prove that every Boole ring is commutative and of characteristic 2. A Boole ring R constitutes with respect to the compositions a. A / = = ac, a v fl = cc + ,B + a,8 (a, ,B E R) a Boole lattice R°. A Boole lattice V constitutes with respect to the compositions a + ,B = (a n fl') v (a' A fl), a(3 = = an/ (a, fl E V) a Boole ring V°, where p' denotes the complement of e. We have R = V° if, and only if, V = R°. According to this, Boole rings and lattices may be completely reduced to each other.

CHAPTER III

OPERATOR STRUCTURES § 61. Operator Structures In § 18 we already introduced the notion of operators and at the end of § 19 we mentioned briefly what we mean in general by an operator structure, yet in our considerations so far operator structures have been almost entirely disregarded. We shall now study these. (We advise the reader to revise the contents of § 18.) Let an operator structure S O, i.e., a O -structure S be given; we denote the elements of S and O by a, fi, ... and a, b, . . ., respectively. This means that S is a structure and O an operator domain for S. Applying O as a left operator domain means that, besides the compositions in S, an operator product as (E S)

(a E S, a E 0)

(61.1)

is defined for all a, a. (Other cases, to be examined below, e.g., the case of several operator domains, will not be essentially different.) Let us note that the notion of operator structures includes the structures without operators as a special case, for if O consists of only one operator e and this is the identical operator (for which ea = a) for all a, then S 1 O may be regarded as equal to S. For S 10, those substructures of S will be of particular importance, which themselves constitute operator structures with respect to O. i.e., 0-structures. Accordingly by an admissible substructure or (9-substructure of S 1 (9 we mean any substructure T of S, for which as E T

(a E T, a E (9)

(61.2)

is satisfied. This condition is, in fact, equivalent to saying that O is an operator domain for T. (61.2) is frequently expressed by saying that T is closed

for the operator domain O. By the notation introduced in § 18 we may write (61.2) in a simpler form, as:

aT 9 T

(a E 9) . 223

(61.2)

OPERATOR STRUCTURES

224

If for "S(9, M 9 S we define the "product" , eM to be the set of all my (m E 4, p E M), then (61.2') becomes

OT c T .

(61,2")

In connection with S 10 we shall call special attention to certain homo-

morphic mappings of S, too, which are of great importance. For that purpose for the moment we consider an arbitrary homomorphic mapping a

a' of S into a structure T, where the images are then the elements

of a substructure S' of T. Obviously, we can turn S' into an operator structure S' 0 (with the same operator domain (9), the simplest way being to define the operator product for the elements of S' by aa' = (aa)'. Bearing in mind the result of this consideration we agree to the following definition :

By an operator-homomorphic or 0-homomorphic mapping of an operator

structure S 1 c7 into an operator structure T 1O (with common operator domain (9) we mean a homomorphic mapping a -* a' of S into T such that (aoc)' = aoc'

(a E S, a E 0)

.

(61.3)

If we then denote the substructure of T, consisting of the images a', by S', it follows from (61.3) that all the am' belong to S', i.e., also S' is an 0-structure, which was to be expected from,the above. Note that (61.3) means the interchangeability of operators and mappings. It is therefore quite obvious that by an operator homomorphism or 0-homo-

morphism of an operator structure S 1 0 onto an operator structure S' 10 we understand an operator-homomorphic mapping of S onto S'. If there is at least one such mapping, then we write S 0 - S' 1 O; S' 10 is also called operator-homomorphic with S 10 or an operator-homomorphic image of S 1 c0. Briefly : S' is O-homomorphic with S or is a c9-homomorphic

image of S. If we write

SI0.S'IO

(a-> a'),

(61.4)

this means that a -* a' is an 0-homomorphism of S onto S'. Operator isomorphism, operator automorphism, operator endomorphism, or more concisely (9-isomorphism, etc., means an isomorphism, automorph-

ism or endomorphism, respectively, which is at the same time 0-homomorphic. In conformity with (61.4) S (0

S' I 0

denotes an operator isomorphism which, in the special case S = S', becomes an operator automorphism. Of course, from among the factor structures S J O with respect to the compatible classification C of S, we shall be con-

OPERATOR STRUCTURES

225

cerned primarily with those for which the natural homomorphism of S onto S/C is an operator homomorphism. The condition for this can be formulated as

S 10 - (S/(q) 10

(a - a) ,

(61.5)

where a denotes the class mod t represented by a. For (61.5) to hold it is necessary that we first of all provide S/C with the operator domain 0, i.e., define an operation as (a E S, a E 0). Then (61.5) will hold if, and only if, the condition (61.3) is satisfied, which in this case becomes to

as = as

(x E S, a E O).

(61.6)

We can also apply (61.6) as a definition of the operator product aa, provided that as is uniquely determined by (61.6). This is the case if, and only if, as

depends only upon a and a and so is independent of the choice of the representative a of this class a, i.e., if

a= 8 (mod 6)

as=- a# (mod 2)

(a,PE S; aE (9).

(61.7)

Hence, by an operator-compatible or 0-compatible classification of the operator structure S 10 we mean any such compatible classification ? of S, for which the additional condition (61.7) is satisfied. In this case we call S/C an admissible factor structure or 0 factor structure, which is, by (61.6), an operator structure (S/@) 10. Then from (61.6) follows the validity of (61.5).

It is of the greatest importance that the general homomorphy theorem (Theorem 60) and the general isomorphy theorems (Theorems 123, 124) remain valid with respect to operator structures (with common operator domain 0) when we consider as substructures only the admissible substructures and when we take into consideration only operator homomorphisms, operator isomorphisms and operator-compatible classifications. The. same is true Of ZASSENHAUS'S lemma (Theorem 134), SCHREIER'S main theorem (Theorem 135), JORDAN-HOLDER theorem (Theorem 136) and the theorems about direct sums and direct products.

It will suffice if we prove only the assertion on Theorem 60. The proof of the other assertions would be easier. We have to consider an operator homomorphism (61.4). According to (61.4), S - S' (a -* a'). If we denote the accompanying compatible classification of S by CO, then by Theorem 60 (61.8) S' S/2 (a' -> a), where & is the class mod C which consists of the inverse images of a'. We show that t is operator-compatible, i.e., that (61.7) holds.

226

OPERATOR STRUCTURES

Because of (61.4), (61.3) is satisfied. Since, by the definition of C>, we can write (61.7) in the form

a' = /1' a (aa)' = (aj9)' , (61.7) follows from (61.3). This means that the factor structure S/C is admis-

sible. We have only to prove that the isomorphism (61.8) is an operator isomorphism. The condition for this may now be expressed by (61.3) in the

form (61.6). But since (61.6) is valid, on account of the admissibility of S/C, we have proved the assertion. From what we have discussed above, the reader will see how other concepts are to be adapted to operator structures. For instance, one should call an operator structure S I ( simple, more precisely operator-simple or (-simple,

when every 0-homomorphism of S is either an isomorphism or such that S is mapped onto a structure consisting of one element. Since Theorem 46 obviously remains true for the admissible substructures of S I N7, it is clear what is meant, in general, by an admissible substructure of S I (9 generated by the complex S (S S); for this substructure we shall also later use the notation {S}, or more precisely {S}(9 if necessary. Henceforth we agree that, when dealing with operator structures, - unless otherwise indicated - we consider from among all the substructures and factor structures only the admissible ones and from among all the homomorphic mappings (homomorphisms, isomorphisms, etc-4-only the operator-homomorphic ones. Thus in the terms for these notions the attributives "admissible", "operator-" will be superfluous, so that we can usually omit them.

However, when necessary, they will be used or the attributive "O-" will be used, as has been discussed above. We simplify our notation by writing, e.g., instead of (61.4) simply S - S' (a -> a'), if we have no doubt whatever

that an (9-homomorphism is being considered. In short: after having assumed an operator domain, everything will be related to it. We proceed with the consideration of the above 0-structure S. Hitherto we have not subjected the operator product as (i.e., the "kind" of operation) to any limitation, but on this excessive liberty we now impose some restrictions.

Since for any operator a the mapping a -->- as of S into itself always exists, we call this the mapping connected with the operator a. Further, since

from among all the mappings of a structure only the homomorphic ones are usually considered, we always assume, unless otherwise indicated, that the above mapping a -k am is a homomorphism for at least one of the compositions defined in S, and, consequently, an endomorphism. This condition is expressed with respect to addition or multiplication as a(a. + fi) = as + aj9

or a(a,B) = as aj9 (a, ,8 E S; a E (9)

,

(61.9)

OPERATOR STRUCTURES

227

respectively. We note already here (and turn back to this point later) that for rings we require only the first of (61.91) and (61.92).) We see, e.g., that for a semigroup S, because of (61.92), every subsemigroup of S is mapped by a --> as onto a subsemigroup of S, and especially onto itself if, and only if, it is admissible. The same thing is true of other structures. Further we shall always assume, if not otherwise stated, that the operator domain O is itself a semigroup (later a group, a ring or a skew field) and that (ab)a = a(ba) (a E S; a, b EO) . (61.10) This means that a homomorphic mapping of the semigroup O arises, if to

every operator a the connected mapping a -- as is assigned as image. THEOREM 143. If O is a group with unity element e and S is an 0-structure,

then all the aS (a E 0) are equal to eS; moreover, eS is mapped one-to-one onto itself by each a. For, it follows from (61.10) that aS = (ea)S = e(aS), consequently aS c eS.

Similarly eS = (aa-)S = a(a-'S), consequently eS c aS. Hence aS = eS. Because a(eS) = (ae)S = aS = eS, eS is mapped onto itself by a. We have now only to show that for two arbitrary elements ea, ej9 of eS a(ea) = a(efl)

ex = e fl

(a, fl E S).

It is in fact so, since from the left-hand side, by (61.10), first (ae)a = (ae)j9,

then as = afi, a-' (aoc) = a-' (a#) and finally ea = e,8. This proves the theorem.

If O is a right operator domain, then definition (61.10) is replaced by a(ab) _ (aa)b ,

(61.11)

which is the only change. If further we apply to operators the exponential notation, we use, if nothing else is stated, the definition aab = (aa)b

(61.12)

Comparison with (61.11) shows that this is essentially a version of right operators.

It will often happen that we provide a structure S simultaneously with two operator domains O = . This is not an essentially different case, since we may then consider S (when 0, O' are disjoint) as a O U O'-structure. In principle we can always restrict ourselves to the case where O, O' are disjoint, since any possible common elements may be replaced by other elements such that the kind of operation does not change. However, it is often advantageous to proceed with two "simultaneous" operator domains 0, 0'. Then we speak of a O, 0'-structure,

etc. Usually we then apply 0 as a left operator domain and O' as a right-

228

OPERATOR STRUCTURES

operator domain. Unless otherwise indicated we suppose that 0, 0' are interchangeable (with each other), by which we mean that (aa)A = a(ocA)

(a E S, a E 0, A E 0') .

(61.13)

We will also apply the elements of a set 0 = simultaneously as left and right operators for S, i.e., define both operator products aoc, oca, which may be different from each other (a E S, a E 0). In this case we say that 0 is a double operator domain. S itself is then called a double operator

structure or 0-double structure. We apply the following abbreviations. The elements on both sides of (61.13) are denoted by aocA. Moreover we put

a1 ... a, al ... a,. A,. . . A, _ (a1 ... a.) (al ... a.,) (A 1... At)

(a,ES,a1E(9,A,E0'; i= 1,2,...) (If no multiplication is defined in S, then s = 1.) We apply the correspond-

ing abbreviation in the case where there is only one operator domain, where the At or the at are then absent. Sometimes we also have to deal with operators not having the above-required properties. So, e.g., in a Schreier extension group 4 o G of G by 4 the group G may be regarded as a 4-structure where the operators aredenoted by exponents, but the condition (61.12) is, in general, not satisfied [cf. §§ (50.8), (50.32)]. EXERCISE. Carry out the proof for the two isomorphy theorems (Theorems 123, 124) for operator structures.

§ 62. Operator Groups, Operator Modules and Operator Rings Usually only groups, modules, rings (and skew fields) are provided with

operator domains. From among these three cases that of the groups is essentially known already from § 61 while in the two other cases we make some further conventions. Here we shall consider all three cases successively and, at the same time, illustrate them by several important examples.

Let S = , which will differ from case to case. In any case a multiplication will be defined in 0 (as in § 61), according to which (9 constitutes a semigroup. Unless otherwise stated, we shall apply (9 as a left operator domain (which is, of course, of no importance). All that we shall say may be related to the case of several operator domains; there is no need for us to discuss on this subject, because what has been said at the end of § 61, is sufficent for this case, yet it will occur among the examples. FIRST CASE. Let (S =) G be a group. If

aba = a(ba) , ax/I = as a13

(a, fl E G; a, b E (9)

(62.1)

OPERATOR GROUPS, MODULES AND RINGS

229

then we call G an operator group with the operator domain 0 (or relative to (9). Alternatively we also say that G is an 0-group. We denote this by G 10. EXAMPLE 1. Let 0 = ' be the semigroup of all endomorphisms of G. Since (62.1) is fulfilled, F is in fact an operator domain. The admissible subgroups of G are called full-invariant; these are consequently those subgroups which are mapped into themselves by all the endomorphisms of G.

Accordingly for G I F we call every composition series a full-invariant series. This means thus a chain of subgroups

G=G0 DG1 J... in which every member is full-invariant in G and no further subgroup may be inserted such that this property is maintained. EXAMPLE 2. Let O=ve be the full automorphism group of G. As admissible subgroups of G we have the characteristic subgroups (cf. § 51). The composition series are now called characteristic series. The group G is called characteristically simple, if it has no proper characteristic subgroup. Imo... EXAMPLE 3. Let (9 =t` be the inner automorphism group of G. The admissible subgroups of G are now the normal divisors. The corresponding composition series are called principal series. Note that if we put (9 = G, i.e., G is assigned to itself as operator domain (where the exponential notation is applied to the operators) and the operation is defined by

ae=e-lag

(a,eEG),

then the mappings connected with the operators are just the inner automorphisms of G (each several times, if G has a centre). Hence this case Notice also (9 = G is only unessentially different from the case 0 = that the conditions corresponding to (62. 1), ae° = W)",

(al)e = a°9n

(a, fl, e, a E G),

are satisfied.

EXAMPLE 4. Let 0 = 1, where 1 is the identical operator. All the subgroups of G are now admissible. The composition series are the usual ones. EXAMPLE 5. Let 0 = 7", i.e., the semigroup of integers, where the power a° is defined as operator. (Then according to the terminology introduced in § 20, a natural operator domain is involved.) The conditions (62.1) are aab

= (0C°)b, (a#)Q = aapa

(a, /9 E G; a, b E

")

.

Of these the first is always satisfied, but the second is satisfied only if G is Abelian, since the equation ag3aq = cc2/32, i.e., #a = aq, follows from it, for a = 2. On the other hand, then again all the subgroups are admissible.

OPERATOR STRUCTURES

230

SEcorn CASE. Let (S =) M be a module. First replace (62.1) by the conditions

aba = a(ba)

,

a(a + fl) = as + a#.

We further assume, unless otherwise indicated, that the semigroup taken as

operator domain is (not only a semigroup but at the same time) a ring (0 =) and that for the operation even (a + b)a = as + ba holds. Together with the previous conditions we now have

aba = a(ba) , a(a + f) = as + a,6 ,

(a + b) a = as + ba (a, j E M; a, b E *) .

(62.2)

If these are satisfied, then M is an operator module with the operator domain JP, or relative to A. Instead of this we may call M an. h-module. We denote this by M I /T. More precisely we call M an R-left module. a-right mod-

ules are to be understood in the correspondi1L$ way. From (62.23), Oa = (0 + 0)a = Oa + 0a, thus Oa = 0. Hence the zero element of JP is a zero operator. Therefore from (62.22,3) for _ -a and

b = -a it follows that

a(-a) =(-a)%= -am.

(62.3)

Moreover, again from (62.22,3)

(a+b)(a+/3)=as+afJ+ba+bfl.

(62.4)

(62.2) is satisfied, in particular, if all the operator products as are equal to 0, i.e., J2 consists only of zero operators. Hence it follows that every ring can be turned into an operator domain in at least one way for every module. We see from this example also that in general the existence of an identical operator is not always guaranteed. If V has its unity element e and this is an

identical operator for M (i.e., em = a for all the a E M), then M is called a unitary module or more precisely a unitary 2-module. EXAMPLE 6. Let 39 = 9 be the full endomorphism ring of M. Then the conditions (62.2) [corresponding to (37.1), (37.2), (37.3)] are satisfied. This example resembles Example 1 above. The admissible submodules and the composition series are again the full-invariant submodules or a full-invariant series of M. EXAMPLE 7. Let

= of be that subring of the ring 'referred to in

Example 6, which is generated by the automorphisms of M. This example resembles Example 2, for one can easily show that the characteristic submodules of M are now the admissible submodules. The corresponding is valid for the composition series.

EXAMPLE 8. Let R = 7, where the elements of 7 applied as natural operators (i.e., as now means the additive analogue of the power a°).

OPERATOR GROUPS, MODULES AND RINGS

231

Of course, the conditions (62.2) are satisfied. This example corresponds to Example 5, but likewise also to Example 4 and even also to Example 3, since M has no inner automorphism (s 1). All submodules of M are now admissible. Since Abelian groups are modules in additive notation, they may be provided with a ring as operator domain. For an Abelian fk-group G the conditions (62.2) assume the form:

aba = a(ba), act = as . a'l, (a + b)a = as . ba (a, 8 E G ; a, b E /Q).

(62.5)

THIRD CASE. Let (S =) R be a ring. If R+ is an R-module, where the operator domain JP is a ring, then R is called in full generality an operator ring, or more precisely an eJl-ring. We use for this the notation RIJI. Notice that according to this definition the operation can be entirely independent of the multiplication valid in R. Those JP-rings will be of particular importance, in which the operation and the multiplication defined in R are in some way interconnected. The most simple possibility attainable is that we take for A a subring of R and identify the operation with the multiplication valid in R. More precisely there are three possibilities for this, which consist in applying the operators from the left or from the right or on both sides. These constructions, although apparently trivial, lead to important concepts. These and other cases will be studied in the following examples. But first, we notice the conditions [corresponding to (62.2)] which should be satisfied, according to the above definition, by the operator product in the .59-ring R:

aba = a(ba) , a(a + f3) = as + a#, (a + b)a = as + ba (62.6) (a, ,B E R; a, b E /) .

From this we shall obtain the special cases to be considered under the assumption of various additional conditions. ExAMPLE 9. We see from (62.6) that in an'z-ring the mappings connected with the operators are endomorphisms of R+, but, in general, are not endo-

morphisms of R. If that is later required, it necessitates the assumption of the additional condition

aaq = as aj9 .

(62.7)

Then by (62.6), (62.7),

(a + b)aq = (a + b)a (a + b)19 = (am + ba) (a# + b1) , and

(a + b)afl = aaf3 + bafl = as a# + ba . b p .

For the special case b = a, fi = a we have 2(aa)2 = 0. Thus if there are neither elements of order 2 in R nor nilpotent elements different from

232

OPERATOR STRUCTURES

0, then all the as trust vanish. Since this case is of minor interest we do not, in general, assume that (62.7) is satisfied. [Notice that (62.7) is, in general, also false for natural operations.] EXAMPLE 10. Let 5P be a subring of R, and the operation as (a E,,V, a E R)

be identical with the multiplication defined in R. This operation is called left multiplication (by a). Always when a subring of R is given to R as left operator domain, we understand that the operation is left multiplication. Evidently (62.6) is then satisfied. Further we even assume R = R. The admissible subrings of R are then exactly the left ideals. Because of the anal-

ogy with Example 3, we call the corresponding composition series the left principal series of R. The right principal series are to be similarly inter-

preted. If, furthermore, R is assigned to itself as left and right operator domain, i.e., as a double operator domain, thei the admissible subrings coincide with the ideals. The composition series) are now called principal series. These are the exact analogues of the principal series of groups. EXAMPLE 11. Let Ji = _D be a maximal ring of amicable double homo-

thetisms of R, which we assign to the ring R as double operator domain [which is possible according to (53.1), (53.7), (53.9)]. In this case (according

to § 54) all the characteristic subrings of R are admissible, but there may also be other admissible subrings, since _D does not, in general, contain all the double homothetisms of R. Simplification results if we assign to R all the as operator domains simultaneously, whereupon the admissible subrings are identical with the characteristic subrings of R. The corresponding composition series are called the characteristic series of R. EXAMPLE 12. If for the operation the additional condition

a2 fl = as # = c. a#

(a, P E R, a E -V)

(62.8)

is satisfied, then we call R a hypercomplex ring over * (or relative to ,'$, or with the operator domain *), or we speak of a hypercomplexJ'-ring R. For this we also say that R 1,V is a hypercomplex ring. This means that for the operation the conditions (62.6), (62.8) are satisfied. Since in particular

(62.8) is satisfied for the natural operation as (a E 7), every ring is a hypercomplex ring over 7. EXAMPLE 13. An important special case of that considered in Example

10 arises, if we take for R a field F and as operator domain a subfield .7 of F. The operator structure F 1" is then called a relative field, where the operation is just the multiplication valid in F. This will be referred to again in Chapter VIII. As can be seen from Examples 1 to 12, the JORDAN - HOLDER theorem may have very various applications if the structure considered (group or ring) is provided with a suitable operator domain.

OPERATOR GROUPS, MODULES AND RINGS

THEOREM 144. Let S be an-module, where e. Put ao = cc - ea , al = ea

233

is a ring with unity element

(a : S)

(62.9)

and denote by S, (i = 0, 1) the set of all a,. Then So, S1 are admissible submodules of S with the direct decomposition

S=So®S1.

(62.10)

Further 3 So = 0 and S1 is unitary. If S is also a hypercomplex ring (over Jp), then (62.10) also holds in the ring-theoretical sense.

This special direct decomposition (62.10) is called the Peirce decomposition of S. For the proof we denote by a, fi and a elements of S and JP, respectively. From (62.9) and (62.2) we have ao - fio = (cc - j9)o ,

al - (l1 = (a - j9)t ,

ao o = 0 ,

acct = (aa)1,

eat = at .

Accordingly So, S1 are admissible submodules of S; furtherJz So = 0 and S1 is unitary. Since, because of (62.9), a = ao + al, then S = So + S1. In order to prove (62.10), it suffices to show that So, S1 have no element in common

except 0. This follows, however, from the fact that e is a zero operator for So and an identical operator for S1. This suffices to prove the first part of the theorem. If S is a hypercomplex ring, then it follows from (62.8), (62.9) that

fiao = fia - fi ea = floc - elia _ (j9a)o , 13a1 = (fla)I , otop = ccfi - ea N =

afi - eaf = (aN)o ,

a1N = (aN)1 ,

consequently S Si, Si SE- S, (i = 0, 1). Since, according to this, So, S1 are ideals of S, we have completed the proof of the theorem. EXERCISE 1. If fk is a ring, then for every (non-Abelian) (k -group G with the properties

(62.5) tk G is contained in an Abelian subgroup of G. EXERCISE 2. The quaternion group Q8 and the full permutation group .P,, (n > 2) have only one full-invariant series each. Which? EXERCISE 3. The direct sum decompositions of a ring R are identical with those of the R-double-module R+. Accordingly the notion of the direct product for operator groups includes that of the direct sum of rings. (This and example 10 above give the impression that all the ring theoretical questions are covered by operator groups. However, this is false, since, e.g., the composition series for operator groups do not

include the "customary" composition series for rings. Therefore that part of the JORDAN-HOLDER theorem belonging to rings without operators, is properly concerned with ring theory.)

EXERCISE 4. It is important that every R-module M may be easily turned into a

unitary module by suitable extension of the operator domain R, e.g. the Dorroh

OPERATOR STRUCTURES

234

extension ring R, (see Addition to Theorem 46), with unity element E. The elements of R, are uniquely given by

e=ae+a

(aE.7, aE R)

and, by defining

ea=au+«,u

(TEM),

M becomes a unitary R,-module. Prove this. Cf. KERTESZ (1959).

§ 63. Remak-Krull-Schmidt Theorem This section deals entirely with groups with operators, even when this is not stated explicitly. Let G denote a group. If ue('l is the inner automorphism group of G, then we call the 'e' -automorphisms, i.e., the automorphisms of G, which are interchangeable with all inner automorphisms, the normal automorphisms of G. These evidently form a group.

On the other hand, we call an automorphism a -> a' of G central, if a' = aeea always holds, where the e. (a E G) are central elements of G. In short the automorphism a -* a' is called central if a-la' always belongs to the centre. But we shall prove immediately that the concepts of "normal" and "central" automorphisms are identical. For, an automorphism a -c- a' of G is normal if, and only if, (TaT

i.e., T'a'T _ i = Ta'r-', i.e., T_ 1T'a' = a'T- IT'

(a, z E G) .

Since a' runs through all the elements of G, so the condition obtained means that all the T-'T' belong to the centre, consequently the assertion is proved. Two subgroups, or two direct product decompositions of G are called centrally isomorphic, if one of them may be carried over by a central (i.e.,

normal) automorphism into the other. (The order of succession of the factors is disregarded. Their number must be the same.) THEOREM 145. If a group G has the direct decompositions

G=A®B=A'®B

(63.1)

with the common factor B, then the subgroups A, A' are centrally isomorphic; ,further, the A-component of A' is equal to A (the A'-component of A is equal

to A'). By (63.1) every a E A has a unique representation

a=a&Oc EA',fl E B).

REMAK-KRULL-SCHMIDT THEOREM

235

It follows that a B = a' B, therefore a -* a' is by (63.1) a one-to-one mapping of A onto A'. Hence we have the isomorphism A

A' (a --- a')

.

(63.2)

Further, according to (63.1), B is elemenfwise interchangeable both with A

and A' and so also with f1 = a'-ia. Accordingly #a belongs to the centre of G. It follows that a fl -3 a'j9 (a E A, j3 E B) is obviously a normal automorphism of G and a continuation of (63.2), therefore A, A' are centrally isomorphic.

If Ao denotes the A-component of A', then it follows from (63.1) that first G = Ao B and then Ao = A. Consequently Theorem 145 is proved. In the following the direct decompositions (D

(63.3)

of G into finitely many direct indecomposable factors will be considered. (63.3) is called a Remak decomposition of G. We note the following: For every direct decomposition (63.3) the subgroups AI, ... , Ak (k = 0, . . ., m) form a series of normal divisors of G. If G has a principal series, this may be refined into a principal series of length >_ m. On the other hand, it is easy to show that every group with a principal series has at least one Remak decomposition. (The converse of the last statement is false, since the infinite cyclic group is directly indecomposable, i.e., it is its own Remak decomposition, but has no principal series.) For the following theorem we propose a simple example for illustration. A, B, C denote the three proper subgroups of the Klein four-group G4. Then the Remak decompositions G4 = A ® C = B ® A hold. If the two first factors A, B are interchanged, we obtain one true and one false equation.

THEOREM 146 (REMAK-KRULL-SCHMIDT theorem). In any two Remak

decompositions of a group G with principal series, arbitrary factors of the first decomposition may be replaced simultaneously by suitable factors of the second decomposition; further, the two decompositions are centrally isomorphic.

(Of course this theorem also relates to rings considered as operator modules. In this case we are concerned with operator isomorphism in the customary sense.)

We have only to prove that every factor of the first decomposition can be replaced by a suitable factor of the second, since then by repeated application of Theorem 145 the truth of Theorem 146 follows.

In order to prove this we add all the inner automorphisms of G to the given operator domain of G (as further operators). This has no influence on the Remak decompositions of G, since the direct factors of G are normal divisors, i.e., admissible with respect to the inner automorphisms; the

OPERATOR STRUCTURES

236

principal series, too, remain unchanged. But the above extension of the operator domain results in composition series being identical with the principal series. Therefore every proper normal series may be refined to a principal series. If, consequently, H denotes an admissible proper subgroup of G, then it follows from SCHREIER's principal theorem and the JORDAN-

HOLDER theorem (Theorems 135, 136), applied to the normal series G H D s, that H also has principal series and - if we denote the length of the principal series of a group by 1(...) - then we have the inequality 1(H) < 1(G). Therefore for the proof we make the induction assumption that our assertion is true for smaller /(G). We now prove in advance that if two admissible subgroups A, B of G generate G itself, then 1(G) = 1(A) + 1(B) - 1(A fl B)

(G = A B),

(63.4)

and, in particular, 1(G) =1(A) + 1(B)

(G = A (& B).

(63.5)

From the second isomorphy theorem (Theorem 126') it follows that G/A also has principal series, and two principal series

G/ADG1/A D... LA/A, ADA1D...Ds of G/A and A make up a principal series G D G1 D ... D A D Al ... z) s of G. Consequently (63.6) 1(G/A) = 1(G) - /(A). By the first isomorphy theorem (Theorem 125) we also have 1(G/A) = 1(AB/A) = 1(B/ (A fl B)) = 1(B) - 1(A fl B) .

This substituted in (63.6) results in (63.4). We now consider two Remak decompositions

G=AI®...®Am , G=B1®...®B

(63.7)

of G. (m = n will only follow from the theorem to be proved.) We have to prove that Al may be replaced by some B. In accordance with the two direct decompositions (63.7), we denote by A,(H) and B,(H) the A,- and B; components, respectively, of a subgroup H of G. First of all we consider the case where B1(A1) = B. does not hold for all i (= 1, . . ., n). Then, by (63.72)

G* = B1(A1) ® ... ® B (A1) c G

(63.8)

REMAK-KRULL-SCHMIDT THEOREM

237

is a subgroup of G different from G. Since, again by (63.72), AI 9 G*, it follows from (63.71) that G has the direct decomposition (cf. § 44, Example 7)

G* = Al ® D (D = G* fl(A2 (9 ... (D Am)) .

(63.9)

The factors of G* in (63.8) are admissible subgroups of G, furthermore G*, D are also admissible. Therefore, according to the above observations, the factors in (63.8) and also D admit Remak decompositions. If these are put into (63.8) and (63.9), then we have two Remak decompositions of G*. Owing to l(G*) < 1(G), the induction assumption leads to the equation

G* = X ®D ,

(63.10)

where X is a direct factor of one of the factors in (63.8). We may assume that

X c BI(Al) (c B1)

(63.11)

On account of (63.9), (63.10) it follows from Theorem 145 that X

A1.

(63.12)

From (63.10) X n D = s. We may write for this, because of X 9 G* and (63.9),

Accordingly

Xfl(A2(9 ...®Am)=E.

G**=X®A2®...®Am

(63.13)

is the direct decomposition of a subgroup G** of G. Since, by (63.12), 1(X) = 1(A1), it follows from (63.5), (63.71), (63.13) that 1(G) = l(G**), thus

G** = G. Consequently (63.13) can be written

G=X®A2®...®A,,,.

(63.14)

Since, according to (63.11), X S B1 c G, it follows from (63.14) (again according to § 44, Example 7), that X is a direct factor of B1. But since B1 is indecomposable, it follows that X = B1. This proves the assertion for this case, on the basis of (63.14).

We now only have the case B. (A1) = B;

(i= 1, ..., n) .

(63.15)

AI (B) c Al

(i = 1, ..., 17)

(63.16)

First we show that is impossible. For if (63.16) holds, then according to the case already verified every B; in (63.7.,) may be substituted by a suitable factor of (63.71).

OPERATOR STRUCTURES

238

If we carry out these substitutions successively, all the factors A1, . . ., A. must necessarily be used up. Let B; be the factor substituted for A1. Then AI(B;) = Al follows from Theorem 145. Since this is in contradiction to (63.16), (63.16) is in fact impossible. So we may assume that

A, (B,) = AI

(63.17)

.

Hence and from (63.71) it follows that G = B1(A2 ® ... (D Am) .

(63.18)

Similarly from (63.15) and (63.72) it follows that

G = AI(B2 ®

... ®

(63.19)

(63.71), (63.72), (63.18), (63.19) imply according to (63.5) and (63.4) the equations 1(G) =1(A,) + 1(A2 ® ... (9 Am) , I(G) = I(B1) + I(B2 ® ... ®

1(G) = 1(BI) + 1(A2 ® ... (9 A) - I(Bl f(A2 ® ... (D Am)), 1(G) =1(A1) + 1(B2 ® ... ® Bn) - I(A1 f (B2 ® ... (9 B,..)). From these it follows that in the last two equations the last members must vanish, therefore the products (63.18), (63.19) are direct. Consequently. Theorem 146 is proved. REMAK (1911) was the first to prove the theorem for finite groups. The above proof for the general case originates essentially from SCHMIDT (1928). In this respect cf. also KUROSH (1953a). For an essentially different proof due to FITTING Cf. ZASSENHAUS (1958). For a generalization cf. KUROSH (1953b). EXERCISE 1. Deduce the first part of Theorem 137 from Theorem 146. EXERCISE 2. In two Remak decompositions of a group with principal series, there

is to each factor of the first decomposition a factor of the second decomposition such that these may be interchanged with one another. (Proof by completion of the first

part of above proof: from (63.11), (63.12) and X = B, it follows that B,(A,) = B,, A, B,, whence it is easy to conclude that G = A, a B2 ® ... ® B,,,.) PROBLEM. Is it possible to generalize Theorem 164 in accordance with the previous Exercise 2?

§ 64. Vector Spaces. Double Vector Spaces. Algebras. Double Algebras Throughout this paragraph let,2 denote a ring with unity element e. Let V be the direct sum of n modules J P+ which we consider - and V also as a left R-module (n > 1). In full detail this means that V consists of the elements

a = (a,, ...,

R = (bI.... , bn) 9 ...

(a;, b;, ... E tA)

,

(64.1)

ALGEBRAS AND VECTOR SPACES

239

where two elements a, 9 are to be regarded equal if, and only if, a, = b. (i = 1, ..., n); moreover, addition and operator product are defined by

a+(1=(al+b1,...,an+bn),

(64.2)

ca = (cal, ..., can) .

(64.3)

This left R-module V (and every R-isomorphic s-module) is called an n-dimensional J'-vector space. Instead of a vector space we sometimes call it a left vector space. Right vector spaces are to be interpreted similarly. n itself is called the dimension or the rank of V. (Cf. the note at the end of

this paragraph.) In order to distinguish it from the generalization to be considered later, we also call V a finite dimensional vector space.

From (64.2), (64.3) it follows directly that for arbitrary a, fl (E V), a, b (E

9z')

abcc = a(ba),

a(a + j9) = as + ai4, (a + b)a = as + ba

.

(64.4)

Accordingly the vector space V constitutes in fact a left ,2-module in the usual sense. We see from (64.3) that V is also a unitary R-module. THEOREM 147. For a ring,* with unity element, a left R-module V consti-

tutes an n-dimensional vector space if, and only if, there are elements wl, ..., co. in V such that a = a1cw1 + ... + a(An

(a; E J")

(64.5)

is a unique representation for all the elements of V.

In order to prove the assertion "if", we assume that (64.5) is the unique representation of the elements of V. It suffices to show that by alwwl + ... + ancon - (al, ..., an)

V is JI-isomorphically mapped onto the above vector space. Obviously this mapping is one-to-one. The homomorphy property (a1 co, +

... + an (on) + (b1 c)1 + . .. + bn w.,) - (a1, ... an) + (b1, . .., bn)

is also satisfied, since the left and right-hand sides are equal to (a1 + b)w1 + + ... + (an + bn)w,,, and (al + b1, . . ., a,, + bn), respectively. Finally

c(a1c)1 + ... + a.,wn)

c(al,

. .

., an)

is satisfied, since the left and the right-hand sides are equal to can), respectively. Accordingly the calcol + . . . + cancan and (cat (cal, mapping is in fact an 5-isomorphism. 9 R.-A.

OPERATOR STRUCTURES

240

For the proof of the assertion "only if" we denote the above vector space by V. We consider in this the special elements i

(i=1,...,n).

c0i=(...,0,e,0,...)

(64.6)

For the arbitrary element a of V in (64.1) we have according to (64.2), (64.3), (64.6)

a=

n

i=t

i=1

ai(...,O,e,0,...

=alwi+...+anw . Hence a is representable in the form (64.5). It is evident, too, that this representation is unique. Consequently we have proved Theorem 147. Of course we can apply the (characteristic) property proved in Theorem 147 as a definition of vector spaces. In the following, unless otherwise indicated, we shall use for the vector spaces this second, more explicit definition. The elements wi, ..., cun, with the property described in Theorem 147, form a basis (more precisely an ,A-basis) 91

of the vector space V.

(The reader can see that this basis concept is in keeping with the basis defined in § 45.)

It should be emphasized that for given JP, n there is (to within,-isomorphism) only one n-dimensionals-vector space, as is immediately evident from the first definition of vector spaces. (This is why we have given prefence to this "first" definition.) Thus if we have two R-vector spaces V, Vi with bases wi, . . ., w,, and el, ..., Lon, respectively, then they must be isomorphic. As a matter of fact we see immediately that V I JP N Vi I

(ai wl + ... + an con -* al 1?t + ... + an e.). (64.7)

Note also that the basis of a vector space is not defined uniquely. So, e.g., .., oi,, and Cvl + cw2i CU.a, ..., o j,,, are bases of the same vector space,

(01,

where c is an arbitrary element of 2. It is obvious that an n-dimensional R-vector space is the direct sum of n one-dimensional,5 -vector spaces (which are thus in their turn isomorphic with respect to one another). Algebra has borrowed the term vector space from geometry. As already mentioned,

the vectors of the three-dimensional Euclidean space may be uniquely written by the help of three suitable vectors p, a, a in the form ago + ba + CT,

where a, b, c denote real numbers. The conditions (64.4) are satisfied, therefore in this example we are dealing with a three-dimensional vector space over the field of the real numbers. If we admit as operator domain only the ring f, then the basis to, a, T

will produce a so-called point lattice. In the sense of algebra this is also a vector space. It is evident that all the vectors of the three-dimensional Euclidean space

ALGEBRAS AND VECTOR SPACES

241

do not constitute for any n (= 1, 2, ...) an n-dimensional s-vector space. From this example we see that for a structure it is an important question whether it has an operator domain, and if so, which.

Vector spaces will recur in various topics, and it will become clear step by step that they are of general importance for the whole of algebra. We shall now deal with an obvious and very important application of them in ring theory. We consider the .9Z-vector space V with the basis w1, . . ., w,,. We can turn V into a ring by defining in it a suitable multiplication a.fl (a, fi E V). In short we deal with rings A with the property A+ = V. At least one such ring always exists, for if the multiplication is defined by aq = 0, then this gives the zero ring with A+ = V. In every ring A with A+ = V the product of the basis elements appears in the form (0i wj = , cukl wk k=1

(1, j = 1, ..., n) ,

(64.8)

where the c j) (there are ns of them) are elements of 9Z determined according to (64.5). These are called the structure constants of A (relative to the basis (01...., CO.).

However, the multiplication in A is in general not yet completely determined by the structure constants. Although, it follows from the distributivity that for the product of two arbitrary elements of A

(a1c)1 + ... + anwn) (b1w1 + ...

i,j=1

(aiwi) (bjw) ,

(64.9)

however, we cannot substitute (64.8) on the right-hand side. Yet it is possible, if we subject the operation to a suitable supplementary condition. For a ring , with unity element let an )2-algebra (or algebra over'* or a hypercomplex system over JI) signify a hypercomplex JI-ring whose mod-

ule is an J9-vector space. In other words an R-algebra means a ring A, for which A+ = V is an 2-vector space, moreover for the operation the supplementary condition (see (62.8))

aaq = (ax)9 = a(a9)

(a, 14 E A; a E 92)

(64.10)

is satisfied. By the rank and a basis of the algebra A we mean the rank (the dimension) and a basis, respectively, of the vector space A+ = V. We see at once that from (64.4) and (64.10) it follows that

(aa) (b f) = a(a(b19)) = a(baq) = abaq .

(64.11)

According to this and (64.8), we have, in addition to (64.9), (a1w1 +

... + anwn) (blwl + ... + bnwn) _ i,j,k=1

aibjc,k)wk ,

(64.12)

OPERATOR STRUCTURES

242

whence we see that in algebras the multiplication of the elements is uniquely determined by the structure constants We note further that with respect to the algebra the operator domain is called the fundamental domain or fundamental ring (in particular fundamental skew field or fundamental field). As a consequence of (64.41) and (64.10) we prove the rule abaf3 = baafi

(a, 9 E A; a, b E R )

.

(64.13)

The left-hand side of (64.11) may also be computed in the following manner:

(aa) (b/I) = b ((aoc)fl) = b(aa/I) = baa/I Hence and from (64.11) follows (64.13).

.

The determination of all algebras over a given ring V is an important question which is, however, not answered in full generality. In this respect we content ourselves here with a few simple statements. In the course

of our further considerations we shall deal with certain special cases of great importance. THEOREM 148. Over a fundamental ringR (with unity element) an algebra A with unity elementy can exist only if J is commutative. Then * may be embed-

ded in the centre of this algebra in such a way that after the embedding the operation is identical with left multiplication in A. Conversely, if a ring A with unity element contains a central subring 9z' with the same unity element, which is of such a kind that all the elements of A may be uniquely written in terms of suitable w1, ..., to,, (E A) in the form (64.5), then A is an R-algebra with left multiplication as operation and the unity element ofJI is also the unity element of A.

In order to prove the first part of the theorem, we denote by s the unity

element of A. For fl = s it follows from (64.13) that aba = baa. In particular, this equation holds for every basis element a = to; of A, whence, because of the uniqueness of (64.5), it follows immediately that ab = ba,

i.e. JI is commutative. We prove further that

(a E JI) (64.14) a>as is an T-isomorphic mapping of T into A. This mapping is one-to-one since for a, b (E /9) it follows from as = be, according to (64.10), firstly that

as = asa = (as)oc = (bs)a = bra = ba

(a E A) ,

and then from this, as above, that a = b. According to this 92-isomorphic mapping (64.14), V may now in fact be embedded in A. Since, because of (64.10), (as)at = a(ae), ' 53' will be embedded in the centre of A. Since, further, according to (64.10), as = = (as)a, the operation becomes, after embedding, left multiplication. Hence the first part of the theorem is proved.

ALGEBRAS AND VECTOR SPACES

243

For the proof of the second part of the theorem, we see that (64.4) is satisfied even with a, b, a, # E A. Hence and from the assumption it follows

that A is anil-vector space. Sinceil lies in the centre of A, so for a, b E il (64.10) is also fulfilled. Hence the validity of the second part of Theorem 148 is proved. THEOREM 149. The structure constants ct ? (1, j, k = 1, . . ., n) of an il-algebra necessarily satisfy n

n

E c )cik =

r=1

(ab - ba)c;;1 = 0

1, ..., n),

(i,j, k, I

r=1

(a, b E R; i, j, k = 1, ..., n).

(64.15)

(64.16)

For if co,, ..., con is a suitable basis, then of course

(i, j, k = 1, ..., n)

(wri) wk = w1(W, k)

(64.17)

must hold. Hence by (64.8) and (64.10) n

n

(t

r C&)co:Wk =

t=1

Cor(c)jwr)

.

(64.18)

r=1

By repeated application of (64.8) and (64.10) n

n

t,1=1

t,1=1

c,. c11) uo1.

(64.19)

Since co, 1..., Con is a basis, (64.15) follows. From (64.13), (ab - ba)co;uo1 = 0, i.e., because of (64.8), 11

(ab - ba) E C(Ukl(ok = 0 k=1

(i, j = 1, ..., n) .

Here, because of (64.42), (64.41), we can interchange ab - ba with From the equation obtained (64.16) follows. Consequently we have proved Theorem 149.

We. have seen in the proof that (64.15) follows (among others) from the associativity conditions (64.17) relative to the basis. On that account we call (64.15) the associativity conditions of the structure constants c ,k). We

apply this terminology not only in connection with algebras, but whenever the structure constants CV are defined. It should be mentioned that conversely (64.17) follows from (64.15), but not the full associativity in A (see below). Therefore the term just introduced is not quite correct. Note

moreover that the conditions (64.16) for a commutative ring il are empty. For a non-commutative ring /Q the conditions (64.15), (64.16) do not in general suffice for there to be an algebra belonging to the structure constants considered. (Cf. PLCKERT (1947-49).) On the other hand, for the commutative case the following important theorem holds:

OPERATOR STRUCTURES

244

THEOREM 150. If for some elements c;k) (i, j, k = 1,.. ., n) of a comwith unity element, the associativity conditions (64.15) are mutative ring

satisfied, then there exists (to within isomorphism) only one 39-algebra with the structure constants cO. This will arise from an .-vector space with a basis col, ..., a ,, if one defines the multiplication of the elements by (64.12).

As we have already seen above, (to within /T-isomorphism) at most one

.-algebra can exist with the structure constants el, and in this algebra a suitable basis cot, ... co,, having been established, the multiplication of the elements, is given by (64.12). So it suffices to prove that, conversely, the structure A defined in the second part of the theorem is an *-algebra. First of all, distributivity in A follows at once from (64.4) and (64.12). Since, in particular, according to (64.12) (bcoj)

(aw,)

n

_ t=1 abc()co,

(a, b E V ; 1, j = 1, . . ., n),

it follows from this and from (64.15) because of the commutativity of 39 [by similar computations as in (64.17) to (64.19)] that (a, b, c E JP; i, j, k = 1, . . ., n) . ((ao)l) (bwj))ccok = awi ((bco) (cc)k)) Hence and from (64.12), on account of the distributivity, the associativity

of the multiplication in A follows. Accordingly A is a ring. (64.10) follows from (64.12) and the commutativity of J9. Therefore A is anRalgebra. So the theorem is proved. NoTE. In our definition of algebras, the fundamental ring J2 was subject

only to the condition that it has a unity element. However, this utmost generality is of little use, since according to Theorem 148 an R-algebra with unity element can exist only if R is commutative. Also in Theorem 150 the non-commutative . had to be excluded. This is why one is accustomed to taking into consideration only algebras defined on commutative rings. Looking back, we can see that even condition (64.10) is poorly com-

patible with the non-commutative J9. We shall see that (64.10) may be replaced by other, very simple conditions, which are more applicable to a non-commutative *.

First, we give the following general definition: for an arbitrary ring we call a module M an *-double module, when J2 is simultaneously a left

and a right operator domain for M, i.e., both operator products aa, as (E M) are defined (a E M, a EJP), and for a, j3 E M, a, b E J9 one has the properties

aba = a(boc), a(x + fi) = as + a#, (a + b)a = as + hoc, nab = (oca)b, (a + t9)a = as + floc, a(a + b) = as + ab , (aa)b = a(ab).

(64.20) (64.21)

(64.22)

ALGEBRAS AND VECTOR SPACES

245

(The common value of the left and the right-hand side of the last equation is denoted briefly by aab.) Then we assume in the ring R the existence of a unity element e. As above we denote by V the set of the elements (64.1). Then we define addition of the elements by (64.2) and both the operator products by

ca = (cal, ..., can), ac = (a1c, .

.

., anc)

(c E JP)

.

(64.23)

This operator module V (and every module operator-isomorphic with it) we call the n-dimensional double vector space over (or relative to) JP or the n-dimensional s-double vector space. We call n the dimension or the rank of V. We see at once that conditions (64.20) to (64.22) are satisfied, so that

V in the above sense is an -*-double module. Moreover V is unitary, by which we now mean, of course, that ea = ae = a. We also see that V simultaneously constitutes a left and right vector space over JP. THEOREM 151. For a ring J` with unity element an JI-double module V constitutes an n-dimensional double vector space if, and only if, in V there are elements col, . . ., co, such that every element of V has a unique representation

a = alwl + ... + ancon

(a1, ..., a,, E -4P)

(64.24)

and

ao,=w,a

(aEJP; i = 1,...,n).

(64.25)

In order to prove the assertion "if", we assume that V is an J2-double module with the properties referred to in the theorem. We have to show that V is a double vector space. Let us begin by noting that, because of (64.25), (64.24) is representable in the form

a = w1a1 + ... + wnan

(64.26)

and obviously this representation of the elements of V is also unique. Now it is sufficient to prove that by alwl + ... + anon (= w1a1 + ... + wean) -- - (al, ..., an)

V is mapped operator-isomorphically onto the above-defined double vecto space. This is evident for the left operation from the proof of Theorem 147; the same then follows for the right operation. The assertion "only if" may be proved as in Theorem 147, there being no

need to explain it in a more detailed manner. Hence Theorem 151 is valid.

In the following we shall keep to the definition of double vector spaces in Theorem 151. The elements wl, . . ., co,, are called a basis of the double

vector space. We should note, however, that (as distinct from ordinary vector spaces) in a double vector space with basis w1, . . ., w,, the elements need not, in general, constitute a basis. Though wl + cwt, w2, ..., co,, (c

OPERATOR STRUCTURES

246

the unique representation (64.24) is maintained, property (64.25), on the other hand, may be lost. There is an exception when c lies in the centre of. We should also note that [in spite of (64.25)] the equation as = aa(a E V, a E') is, in general, false. Of course, it is true when 39 is commutative. Hence we see that for a commutative R the double vector spaces differ only insignificantly from vector spaces, in that the operators in them may be arbitrarily written on the left or right. For a ring,-'$ with unity element, a ring A is called -an '92-double algebra (or double algebra over R), when the module A+ is an s-double vector space and for the operators [in addition to (64.20), (64.21), (64.22), (64.25)] ((xa) f3 = a (a(3)

(a, fl E A ; a E JP),

awiw j = (aw;)ao j = w; (aa) j)

(i, j =-/-

.

.

., n)

(64.27) (64.28)

where wl, ..., co,, denote a suitable basis of A+. By the rank and the basis of the double algebra A we understand respectively the rank and the basis of the double vector space A+. Since - as noted above - in the double vector spaces over a commutative

the left and right operations are not essentially different, this is true also of double algebras. Furthermore, we see that the conditions (64.27), (64.28) are then equivalent to the single condition (64.10). Accordingly .

J-double algebras for commutative J' are essentially the same as s-algebras. Thus in this case we may say "algebra" instead of "double algebra". THEOREM 152. The associativity conditions (64.15) necessarily hold for the

structure constants of a double algebra. The proof follows the lines of that of the first assertion of Theorem 149. The only difference is that in the inference from (64.17) to (64.19) we use (64.28) [instead of (64.10)]. THEOREM 153. If for some central elements c(k) (i, j, k, = 1 , ... , n) of a ring,2 with unity element the associativity conditions (64.15) are satisfied, then there exists (to within operator isomorphism) only one.-double al-

gebra with these structure constants c7). This arises from an *-double vector space with a basis co, .

.

., co,,

if we define the multiplication of the

elements by (64.12).

We first prove that, in an JP-double algebra with the given structure constants c(,,,) and a basis co, .

.

., w of it, rule (64.12) necessarily holds for

the multiplication of the elements, whence the uniqueness proposition of the theorem immediately follows. From (64.27), (64.22), (64.25), (64.201). (64.28) it follows that for a, b E A (awi) (bco)

((a o,)b)wj = (a((aib)) wj

(a(bwi))ro; = (abw)w j = abwiw; .

ALGEBRAS AND VECTOR SPACES

247

Hence, and from (64.8), (64.9), (64.20), rule (64.12) follows, in fact. It still remains to prove that the structure defined in the second part of the

theorem, which we denote by A, is an P-double algebra. Distributivity in A follows immediately from (64.12) and (64.20). The associativity of multiplication in A follows then as in the proof of Theorem 150, since (though now R need not be commutative) the c(') lie in the centre of J`z'. Accordingly A is a ring. We then have only to prove the properties (64.27), (64.28).

For a, b, c E A, because of (64.22), (64.25), (64.201) (acol) c = a(co,c) = a(cw,) = acco, , c(bcoj) = cbw j .

From this and from (64.12) it follows that ((aw,)c) (bwj) = (awe) (c(bco))

(i,j = 1, ..., n)

since the elements c;') lie in the centre of 59. This implies, because of distributivity, the validity of (64.27). Since a = ex (aE A), (64.28) follows immediately from (64.12). Thus the proof of Theorem 153 is now completed. Another solution of the problem of extending the notion of an algebra for not necessarily commutative fundamental rings is to be found in PICKERT (1947-49). We note that these "algebras in a general sense" referred to there and our double algebras correspond to each other (cf. WEINERT (1965)).

All that we have said above may be extended to the case of vector spaces, algebras, double vector spaces, double algebras of infinite rank (or infinite dimension). For these structures we take, instead of finite series (al, . . ., in (64.1), infinite series (al, a2, ...) (a, E 9z), where 1, 2, . . . are the elements of an infinite index set and only those series are admitted in which all the terms, with finitely many exceptions, are equal to 0. From this it follows

that instead of the basis w1, .... co, used so far we have now an infinite basis wl, w2. .... The unique basis representation of the elements is then a formal infinite sum

a,w,=alw1+a.2w2+...

(a, E 5)

with only finitely many non-vanishing terms. All that has been said above remains valid with the corresponding modifications. Infinitely many structure constants c,; ) (i, j, k = 1, 2, ...) occur but in such a way that for fixed i, j there are only finitely many c(9 0.

In what follows, for the sake of simplicity, we shall consider, unless otherwise stated, only the above finite case, although our propositions should, if possible, be related to the infinite case. In accordance with our conventions for operator structures we shall say in general of two vector spaces, double vector spaces, algebras or double 9/a R.-A.

OPERATOR STRUCTURES

248

algebras, respectively, that one of these structures is a substructure or a homomorphic image of the other if, and only if, they have the same operator domain (with the same operations) and one is an admissible substructure or an operator-homomorphic image, of the other. In connection with this we want to consider some quite simple cases here, which will soon be applied. Let V be an --vector space with the basis col, ..., co,,. We write this more explicitly in the form V = V(col, .., c. ).

We can arbitrarily arrange the basis elements as follows:

(r + s = n).

01, ..., o al, ..., a,

(64.29)

Evidently then

V(o1, ...,

o,)

is a subvector space and at the same time a homomorphic image of V and we even have the direct sum decomposition V = V (ol, ..., o,) ® V(011' ..., a,).

We consider an R-algebra A with A+ = V, where the former basis col, .

.

., co is to be retained. We again apply the more explicit notation

A=A(co1,...,cu.). Evidently the above ol, ..., o, as basis elements determine a subalgebra of A, which is then representable by A(ol, ..., o,), if, and only if, oioi E V(o1, ..., o,)

(i, j = 1, ..., r) .

Under certain circumstances it is also easy to give a homomorphic image of A as follows: with the above notation (64.29) we assume that

a = V(a1,. , as) is an (admissible) ideal of A, which is - according to what has been said above - obviously equivalent to the condition w;01;, a; c)r E V(al, ..., a,)

(i = 1, . . ., n; j =

1, . . .,

s).

It will then appear that the homomorphic image A/a is an .lp-algebra, which

we now determine more exactly. By (64.8) and (64.29) (with modified notations) oiof

u,Pk +

k 1d+ik)ak (i,j = 1, . . ., r; c ) ms'

, dUk) E

P)

.

(64.30)

ALGEBRAS AND VECTOR SPACES

249

As the al, ... a, belong to a, the following may be seen immediately: if the products Lo&; are defined by

Pre; _

C(k)

k=1

ek

(i, j = 1, ..., r)

(64.31)

[instead of (64.30)], then we obtain an .5s-algebra (64.32)

with the basis Lo1,

...,

and the structure constants c(uk), for which A/a

A'.

(64.33)

Hence A/a is in fact an .fir'-algebra, and it may be identified with A.

Similar simple statements, needing no further explanation, also relate to double vector spaces and double algebras. The important .5s-structures introduced above are all subject to the condition that the ring .J`z' has a unity element. However, we can apply these

constructions with a slight modification to the case of an arbitrary ring ,2, as follows: on account of Theorem 64 we extend . to the Dorroh ring Y with unity element. Then let V be the .?-vector space with the basis

coi, ..., co,, and A an .9-algebra with A+ = V. The basis representation of the elements of V (and also of A) is then a = aico, + ... + a,pJ

(a, E 9')

.

If we subject the a; to the restriction a, E 39, then the rest of the elements a obviously constitute an R-module V'. If, moreover, the structure constants cU' of the algebra A, or at least all the products abckt

(a, b E 9; i, j, k = 1, ..., n)

(64.34)

lie in 5i', then the above a constitute a subring A' of A (with A'+ = V7, which is an e. -ring. Of course V' and A' are, in general, not a vector space

or an algebra, respectively. NoTm. According to Theorem 147 the rank (or the dimension) of an Rvector space V with the basis O) , . . ., co is equal to the number n of the basis elements. It is, however, possible that V has other bases with different

numbers of elements. This means that the rank of V is not, in general, uniquely determined by V, in other words it is not an invariant of V. (See Ex-

ercise below.) Thus if we speak of a vector space of rank n, then it should only mean that it has (among other things) a basis of n terms. However, we shall later prove for certain important rings (especially for all skew fields) .5P the invariance of the rank of the .2-vector spaces, but in the gen-

OPERATOR STRUCTURES

250

eral case only the "set of the ranks" of a vector space has an invariant meaning. The same may be said of infinite dimensional vector spaces and of double vector spaces, algebras and double algebras. EXAMPLE 1. The direct sum of arbitrarily many infinite cyclic modules is an 9-vector

space. Every ring with such a module is an 7-algebra. EXAMPLE 2. The field of complex numbers is an algebra of rank two over the field of real numbers with the basis I, i and the multiplication table of the basis elements is I

1

1

i

i

i

-1

The associativity conditions of the structure constants are now obviously satisfied. This example accounts for the terms "hypercomplex systems" (= algebras) and "hypercomplex rings", which are to be regarded as generalizations of the field of the complex numbers. EXAMPLE 3. The quaternion field considered previously (see § 24, Exercise 1) is a (noncommutative) algebra of rank four over the field of the real numbers. EXERCISE 1. Show that over a ring /Q there exists a left vector space if and only if /,Q has a right unity element (cf. PICKERT (1947-49)). EXERCISE 2. We wish to find the rings /Q with unity element, which are constructed

so that the /Q-vector spaces of rank one are at the same time of rank two. For this purpose we consider an /Q-vector space with the basis element w (which particular

w is irrelevant.) The ring /Q has the required property if, and only if, there are two elements a a. in it, for which a,w, aEw constitute a basis of the considered vector space, i.e., if its elements may be uniquely written in the form x(a,w) + y(a,w) _ (xa, + ya2)w

(x, y E ti).

For this, it is obviously necessary and sufficient that there should be two further elements a a4 in GQ with a3a, + a4a2 = e

(e is the unity element of t)

(64.35)

and that the equation

xa, + ya2 = 0

(64.36)

(x, y E /Q)

can only be solved trivially, i.e., with x = y = 0. If we multiply (64.35) on the left by a, and a2, then we obtain (a,a3 - e)a, + a,a4as = 0, a$a3a, -+- (a$a, - e)a2 = 0.

Hence we obtain the further necessary conditions

a,a, = e,

a,a4 = 0,

a2a3 = 0,

a,a4 = e.

(64.37)

Show that, conversely, the ring 12, defined by (64.35), (64.37) has the required property,

0 and (64.36) is only trivially solvable in it. [More precisely there is the question of the ring rk, defined by the equations (64.35), (64.37) and ea; = a, e = = a, (i = 1, 2, 3,4). Cf. REDEI-WELNERT (1966).] Of course all finite dimensional vector spaces over this ring are isomorphic with one another. For a construction i.e., rQo

of another ring with this property cf. EVERETT (1942b).

CROSS PRODUCTS

251

§ 65. Cross Products We shall here consider certain special Schreier group extensions, which are suitable for assisting in the construction of a ring. Let a skew field .F and a group G be given. First, we consider an arbitrary Schreier extension

0=Go.7*

(65.1)

of .9'* by G. According to Theorem 102 the multiplication of the elements in CAS is defined by

; a, b

(a, a) (fl, b) _ (afl, mpd b) (a, # E G ; a, b E

0)

,

(65.2)

where

a'3 ao (E T*)

(65.3)

is the function pair belonging to this extension. Hence

a--> a'

(65.4)

is, for every a, an automorphism of the group ."*. If necessary, (65.4) should be extended, by the stipulation that 0" = 0, to a mapping of .7" onto itself.

Now we want to admit only such extensions (65.1), for which (65.4) is an automorphism even of the skew field .7 for every a. Under this assumption we define a ring R as follows: take all the distinct elements al, a21 ... of the group G as the basis of an 7-right-vector space and take this to be the module W. Accordingly

A = yajai

(a, E .7)

(65.5)

is the unique basis representation of the elements of i, where in (65.5) there are only finitely many a; different from 0. The operation is now given by

Ac = Z a;ac

(c E 7).

(65.6)

The next stop is to define the product of the "one-term" elements of 1l [as in (65.2) ] by as ,Bb = a fla,'a''b

(a, # E G ; a, b E 7),

(65.7)

where " . " denotes the multiplication in T. Finally we define the multiplication in R with respect to (65.5) and (65.7) by alai

ajbj = j, j

(ajaj a jb j) _

0

a;a j c 'al'b; .

(65.8)

252

OPERATOR STRUCTURES

[It should be noted that (65.7) is in fact a special case of (65.8).] We denote the structure 91 so defined, by

N = G x.7,

(65.9)

which we call a Noether cross product of the skew field .7 with the group G,

and prove that this is always a ring. We have only to prove that multiplication is associative in 91 and that distributivity also holds. We have

a(a + b) = as + ab, (a + b)x = ax + bx;

(65.10)

the latter follows from the fact that (65.4) is an automorphism of Jr, conse-

quently, a fortiori, of .9r+. From (65.8) and (65.10) distributivity in 91 follows. For the associativity of the multiplication, it is sufficient to prove that it holds for the one-term elements of J1. This assertion follows immediately from the fact that (65.7) is of the form (65.2).

NoTE. In applications we may denote the basis elements of G by x,,, xx,, ..., instead of al, m2.... Then (65.5) and (65.7) assume the form

A=

xx,a;

(65.11)

and

xxa xsb = xxp ocsasb ,

(65.12)

respectively; furthermore (65.8) has to be transformed accordingly. For the completion of the definition of 91 let us note that according to Theorem 102 the function pair (65.3), in addition to what has been said, is still subject to the conditions

ex=a,=ex=e, a' = a,

(65.13)

axsas = aB(ax)s ,

(65.14)

a9yjr

= (am''(as)'' ,

(65.15)

where s, e denote the unity elements of G and .92 respectively (a, j9, y E G ; a E .5z').

We see that in the product formula (65.7) the two functions aa, a° appear (a E G, a E .9"). In both cases we are concerned with an operation, where either a or a is the operator, i.e., either X or G is the operator domain. This change of roles explains the term "cross product". Cross products were introduced by NOETHER (1934). The cross products introduced

above are more general than the original ones of Noether.

253

CROSS PRODUCTS

It should be noted that (c E 7)

xaac xxb = x.#oc'(ac)sb

follows from (65.12). According to this, the conditions (64.10) are, in general, not satisfied in R for the operation, not even when. is commutative. Therefore the cross product is not generally an algebra over Y. (In the

Noether case, cross products occur which constitute an algebra over a subfield of 9'.) As a very simple case we may have a" = a, i.e., we have an automorphismfree Schreier extension (65.1). If moreover 3 is commutative, then the cross product (65.1) is evidently an .3-algebra (with .7 as right operator domain). Cross products are a special case of "monomial rings" (cf. § 66). EXAMPLE. Let . " denote the field of complex numbers and a' the conjugate of a (E .F); further, denote by G the group of order two which consists of the unity element

e and one further element w. We now define the function pair (65.3) by means of a

fixed real element r of F' by

e`=a =e'°=1, w"=r; a`=a, a°'=a'. Then a -- a`, a -- a' are automorphisms of X, moreover conditions (65.13), (65.14), (65.15) are satisfied. Accordingly the Schreier extension (65.1) now exists. In the connected cross product Ill, rule (65.12) may be given by the multiplication table x,,b

x°,b

xE a

x, ab

x°,a

x°,ab

x°,a'b xsra'b

Consequently the product rule in 91 is

(x,a + x°,b) (xac + xmd) = xe(ac + rb'd) + x°, (a'd + be)

(a, b, c, d E .

").

It is easy to see that 5111, in the special case r = - 1, is isomorphic with the quaternion field previously defined (§ 24, Example 3). That is, J1

Q

x.a+xmb

a-b (b,

a)1-

§ 66. Monomial Rings

Let R be a ring whose module R+ is an /-vector space or 9-double vector space with finite or infinite basis WA, coB, ... where A, B, ... run through the elements of an index set H. The operator domain JP itself shall be an arbitrary ring with unity element. A very simple special case is that in which for the product of the basis elements we have WAOB = CA, B wAB

(CA, BE JI ; A, B E H)

(66.1)

OPERATOR STRUCTURES

254

where wAB again denotes a basis element of R (depending on A, B). In this case we call cod, (A)B, ... and the CA, B a monomial basis of the ring R or the connected factor system, respectively. In this case, i.e., if it has a monomial

basis, the ring R is called a monomial ring (over or relative to,2). If R constitutes either an s-algebra or an 2-double algebra, then the terms monomial algebra or monomial double algebra are used, respectively. As yet AB in (66.1) does not mean a product, but we subsequently define in the index set H a (not necessarily associative) multiplication, even with the condition that (66.1) remains true also if AB now denotes the product of A and B. Henceforth this will be assumed. It should be noted that AB is uniquely defined, because of (66.1), if WAwB 0; on the other hand, AB may be equated to any element of H if WWACOB = 0 (i.e., CA, = 0). Later below we shall consider only those monomial rings which are monomial +Jp-algebras or J9-double-algebras (for a more general investigation cf. WEINERT (1965)).

We want to determine the associativity conditions of the structure constants CA, B. For that purpose, we write (66.1) in the form

WAwB = A BWC C

where the summation is over all C E H and cA, B

_ JCA,B for C = AB, 0

for C

AB.

(66.2)

The required associativity conditions according to (64.15) then run as follows : c CA, B(

CA, B

(A, B, C, D E H) .

(66.3)

We distinguish the four cases

D = (AB)C = A(BC); D (AB)C, A(BC); D = (AB)C A(BC); D = A(BC) (AB)C. In the first case D may be cancelled in (66.3), because of (66.2). In the second

case, both sides of (66.3) vanish, therefore (66.3) is now identically satisfied. In the third and fourth cases one side of (66.3) vanishes, so that the other side must also vanish. The result is expressed in the following theorem. THEOREM 154. If wA, COB

... is a (monomial) basis of a monomial ring

R over the ring. with unity element, where A,B, . . . are the elements of an index set H, then the product of the basis elements may be taken in the form (66.1), where AB denotes a (not necessarily associative) multiplication in H; the associativity conditions of the factor system CA, B are then CA. B CAB, C = CB, C CA. BC

(A, B, C E H)

(66.4)

MONOMIAL RINGS

255

with the additional condition that if (AB)C A(BC), both sides of (66.4) are equal to 0. In the following we consider the special case of this theorem, where (AB)C = A(BC), i.e., the index set H constitutes a semigroup. In this case (66.4) itself (without the above additional condition) expresses the associativity conditions of the factor system CA,B. Hence, if we have a semigroup H and a set of elements CA, B, for which (66.4) holds and which are in the centre of V, then there exists, by Theorem 153, a monomial .m`1'-double algebra R with this factor system. If, furthermore, V is commutative, then R is, by

Theorem 150, even a monomial R-algebra. These facts afford comparatively simple possibilities for the construction of special double algebras and algebras. For that purpose, for a given semigroup H, we need only solve the equation system (66.4) in the centre of 9P. We first consider the trivial but important case, where all the CA, B are equal to the unity element of 2. Then (66.1) assumes the form wAw, = wAB

Hence it follows that the basis elements wA, &)S.... themselves constitute a semigroup isomorphic with H, so that one can take for it the elements of H. If we now denote these by col, w., ..., we obtain the following definition. Let a ring Jz with unity element and a semigroup H be given, consisting of the elements co,, 0 ) 2 ,-- . . Assume furthermore that and H contain no common element. An .5P-double algebra R is obtained if we form the s-double vector space with the basis wl, co., ... and in it define the product of the basis elements exactly as in H. Hence E a;coi = a1w1 + a.p2 +

...

(a, E dl)

(66.5)

is the unique basis representation of the elements of R, where only finitely

many a; different from 0 are admitted. The sum and the product of the elements of R, and both the operator products, are given by the following formulae :

aiwi + > bpi = bi o;, _

atwi

cY a,wi = > cacoi

,

(ai + bi)wi ,

(66.6)

aibwiw;

(66.7)

(> aiwi)c = > a,cwi

.

(66.8)

The monomial R-double algebra R so defined is called the semigroup ring of H over R or the A-semigroup ring of H and for this we often employ

the notation (R =) 2 (H). If in particular H is a group, we use the term group ring. IfHH is a group or a semigroup with the unity element e, then dl can be embedded in R (H) by substituting a for as (a E dl). Later we often

OPERATOR STRUCTURES

256

assume this embedding. Then (66.5) becomes

al + a2w2 + a3w3 + ..., where w2, w3 , ... are the elements of H different from E, and (66.6), (66.7),

(66.8) are also modified accordingly. In particular we call 7(H) the natural semigroup ring or natural group ring of H ; we may also call it "the semigroup ring" or "the group ring" of H. NoTE. If the semigroup H has a zero element, this is never the zero element of the semigroup ring _ (H). This is evident, for if co denotes the zero ele-

ment of H, which is thus a basis element of 2(H), then all the aw (a E A, # 0), including to, are different from the zero element 0 of A (H).

In this case we note the following: the vector space of rank 1 formed by the aw (a E R) is an admissible ideal of (H), which we denote by (co). Consequently we can form [cf. (64.32), (64.33)] from R = t-*(H) an

s-double algebra R', for which R/(w) : R' and R' arises from R by cancelling in (66.5) to (66.8) the terms with wi = co and w;w; = w. We express this concisely, but rather inaccurately, by saying that R' arises from R by cancelling from the basis elements of R the zero element co of H. (But if the zero element co is cancelled from H, H may cease to be a semigroup. For the same reason R' is in general not a subring of R.) We now turn to the more general case, where in Theorem 154 the index set H constitutes, as before, a semigroup, but the CA, B subjected to the conditions (66.4) are arbitrary central elements of 2. As we have said, a (monomial) double algebra then also arises, which we call, in order to distinguish it from the special case considered in (66.5) to (66.8), a semigroup ring with factor system. More precisely, it is the ' 5P-semigroup ring of H with the factor system ca, B. In this ring R, we can take the elements A, B.... of H as basis elements, too. The basis representation of the elements of R then assumes the form

aAA + aBB + ...

(aA, aB, ... E -V),

(66.9)

and the rule (66.1) for the product of the basis elements in R becomes A B = cA, B AB ,

(66.10)

where A . B and AB denote the product of A, B, in the ring R and the semigroup H, respectively. For the sum of two elements of R and for operator products rules similar to (66.6) and (66.8) hold, but for the product of two elements of R we have

Z axX x

x

bxX =

axbycx, yXY, x, Y

(66.11)

MONOMIAL RINGS

257

where we have to sum over all X, Y E H. On account of its importance we repeat a part of this result in the following THEOREM 155. If.91 is a ring with unity element and H asemigroup, then the

J -semigroup ring of H with the factor system cA B is defined if, and only if, the CA, B are central elements of .5P and CA, B CAB, C = CA, BC CB, C

(A, B, C E H).

(66.12)

We now turn to the general case of Theorem 154, where the multiplication

in H need not be associative. Let the unity element of V be denoted by 1. However.53' may be constituted, 0, 1, -1 certainly lie in its centre. Accordingly a monomial'-ring R is always a double algebra if the factor system consists of the elements 0, 1, -1, i.e., in R, for the product co, co j of two basis elements cot, c'j (E H), only the three possibilities ul, co j = 0,

cu,roj, - co,rv j

are admitted. (Of course, the associativity conditions of the factor system must also be satisfied.) In this respect we go on with the following THEOREM 156. Let a ring .9' with unity element be given and one of the four semigroups Hl = < c1, a..,

...

(66.13)

H2 = <w,01,229... ,

(66.14)

H3 = <«I, a2, ..., Yat, Ya2, ... i.

(66.15)

H I = <w, al, a2...., Y0E1, Yx2, ... >,

(66.16)

where each time the zero element is denoted by co and a central element by y, for which y2 is the unity element. (In (66.15) and (66.16) each a; appears

twice, alone and in the product ya;). In all four cases the products a;a j (i, j = 1, 2, ...) are uniquely given by one of the three possibilities a,at = w, ak, yak

(66.17)

for some k (= 1, 2, ...) determined by i, j. If a multiplication, distributive with respect to addition, is then defined in the V-double vector space having the basis al, M2.... so that for the product of the basis elements similarly as in (66.17)

M! aj = 0 , ak, - ak then an R-double algebra results in every case.

(66.18)

OPERATOR STRUCTURES

258

This is trivial, since from the associativity of the multiplication (66.17) that of the multiplication (66.18) follows. NOTE. Theorem 156 is of fundamental importance, since it includes as

particular special cases - as will be seen in the following paragraphs among others polynomial rings, full matrix rings, alternating rings, quatemion fields and complex number fields, all double algebras which are of considerable importance and having various properties, which will be considered in the last part of this chapter. Moreover, Theorem 156 contains essentially all monomial rings with factor systems consisting of the elements 0,1 and - 1 only (cf. WELNERT (1966) (to appear)). In other respects we see that the first two cases of the theorem are not new, since in these we

either have the semigroup ring of Hl or the double algebra which arises from the semigroup ring of H2 by cancelling from the basis elements the zero element w of H2. EXAMPLE 1. We apply Theorem 156 to the case where the given semigroup is the cyclic group {a; of order six. By putting y = a", the elements of {a} are representable in the form e, a, a2, ye, yx, 7,a2 (where e is the unity element),

so that {a} is an H3 in (66.15). For e, a, a2 we have the multiplication table e

e

e

a

a

a2

a2

x

Thus if we form the (k-double vector space with basis g, a, z and define in it the product of the basis elements by the multiplication table O

a

T

e

O

6

a

Q

a

r

T

-p T -n -a

then a double algebra is obtained. In this we have for the product of the elements the rule

(ae + ba + cT) (a'B + b'a + c'l) = (aa' - bc' - cb') e + + (ab' + ba' - cc') or + (ac' -1- bb' + ca') a

(a, ..., c' E /Q).

EXAMPLE 2. If a group G contains an element a (#e) of finite order, then every group ring of G contains zero divisors, since

(e-0)(e+a+...+a"-`)=0

(o(a)=n)

be a ring with the unity element e, and 4 a subgroup of (,Qx with the same unity element e, together with an arbitrary group G which has no common element with Q. Take an automorphism-free Schreier extension CS = G o (4 of 4 by G. We now denote the associated factor system by EXERCISE. Let i

cx,0 (E 4)

(a, ,B E G)

MONOMIAL RINGS

259

(instead of (0). Then, according to Theorem 107, the ca,0 lie in the centre of cob. Assume that they even lie in the centre of /Q, and prove that the 2-group ring of G with this factor system c,,,p exists. (This bears a certain similarity to the construction of cross products.) We note here (cf. § 50, note 3) that both this theorem and

the cross product [(65.9)] may be generalized to the case where we take a semigroup instead of the group G.

§ 67. Polynomial Rings We consider the semigroup H for fixed n (>_ 1), defined by the equations

ex1 = x,s = x, (i, k: = 1 , ..., n)

x ; x k = xkxi ,

and call this a free commutative semigroup with unity element (generated by finitely many elements). The unity is s. If is a ring with the unity element e, which contains no element of H, then we call the. -semigroup ring of The xl, ..., xn H a polynomial ring over 92 and denote it by *[x1, ...,

themselves are called indeterminates and we say that 9p[xl...... ] is obtained from R by the adjunction of the indeterminates x1..... x,;. We perform the usual embedding of,9Pin R [xl,...,

by replacing aE (a E

by a. Henceforth we always assume this embedding. As a result, since s = es is replaced by e in the embedding,,R and H have the common unity element e and no other common elements, a circumstance which (by adapting the original assumption) will always be assumed later. The elements of ,..9P[x1,..., are called the polynomials in the indeterminates xl..... xn over ,A. These may be uniquely written in the form a,i...rn x ... xn

.f = .! (XI, .... xn) _

(aj,...in E JP),

(67.1)

ill ..., in ;_> o

... = x,,°, = e. The a,,.,,,, are called the coefficients of the polynomial f, for which, of course, only finitely many where we are to understand that x° _

different from 0 are admitted. The fundamental ring ,59 is accordingly called the coefficient ring of ,J`2[x1, . . ., We call the elements of .5' constants (or constant polynomials). Since the power products

x`... x;°

(il, ..., iry ? 0), (67.2) i.e., the elements of H, constitute a basis of ,I [xI, ..., Y ], we emphasize that the polynomial and the system of its coefficients determine each other uniquely, i.e., (67.1) and a further polynomial

g=

b.,... in x1..l ... -r,°

(67.3)

are equal if, and only if, at,...in _ b,,...,

(il, ..., in = 0, 1, .. .)

(This fact is also called the principle of coefficient-comparison for polynomials.)

OPERATOR STRUCTURES

260

Further, from the definition, the rules for addition and multiplication are as follows:

f+g= fg =

(67.4) a,,...1. bl,...1. xi

+.i,

... x;; +J..

(67.5)

The operator products are computed according to the rules cf=

caj,...te x 1,. .. x;; ,

fc =

aj....;,, c4 ... x;," .

(67.6)

(These are, in fact, special cases of (67.5) on account of the embedding.)

It is important to note that

,R [x,, ...,

[yi, ...,

is (to within isomorphism) uniquely defined by , . . ., and n. Thus in this sense the indeterminates are of subordinate significance, only their number being important. It should be noted that the above polynomial rings are tR-double algebras (in the commutative case 2-algebras) of infinite rank. It is often useful to extend the notion of a polynomial ring .W [x1, . . ., to the case where the ring V has no unity element. This is done according to the method discussed in § 64 so that we extend to a ring Y with unity element and retain from Y [x1, . . ., only the polynomials with coefficients therefore .51 [x1,

from J. These polynomials constitute [according to (67.4) and (67.5)] a subring of Y [xi, . . ., depending only upon LR and xl , . . . , x,,, which we likewise denote by V [x1, . . ., and call a polynomial ring. Of course this is, in general, not a double algebra nor an algebra and need not con-

tain the power products (67.2). In the following we continue with the above case, though the reader will see that some of our statements can also be related to the more general case.

We first want to make a more detailed study of the case n = 1. If we refer briefly to a polynomial ring, we mean this case. Write x instead of x1. The elements of R[x] appear in the form k

f = f(x) = E a;xi = ao + alx + ... + akxk (a, E ) .

(67.7)

1=0

This is to be interpreted to mean that the coefficients of xk+', xk+z, . are equal to 0. If ak 0, then k is uniquely defined by f (x) and called the degree of this polynomial. This definition does not apply to the zero polynomial f(x) = 0, since in this all coefficients are equal to 0. It is often

POLYNOMIAL RINGS

261

useful to define the degree of this polynomial by k = - co. Accordingly we also say that the zero polynomial is of lower degree than all the other polynomials. The constants different from 0 are all the polynomials of degree zero. If in (67.7) we also admit ak = 0, then k is called a formal degree of J(x). But this is not uniquely defined by f(x), since by taking some terms 0xk+l, 0xk+2, ... we can (arbitrarily) enlarge the formal degree. The formal degree is always Z 0. The totality of polynomials with the formal degree k is just that of the polynomials of degree <_ k. In order to distinguish the degree from the formal degree, it is usually called the proper degree. But

we shall mostly use for both cases the term "degree", an inaccuracy without danger, as the reader will see from the context which degree is intended. We repeat the rules (67.4), (67.5) for the polynomials of x in the following form :

(ao + alx + ...) + (bo + blx + ...) _ (ao + bo) + (a, + bl) x + ... , (67.8)

(ao+alx+...)(bo+blx+...)=aobo+(aobl+albo)x+ + (ao62 + albs + a2bo) x2 + ....

(67.9)

For most purposes one takes f(x) [instead of (67.7)] in the form "ordered according to decreasing exponents"

f(x) = ao xk + ... + ak .

(67.10)

aoxk and ao are called the leading term and leading coefficient, respectively, of f (x). If the latter is the unity element e, we call f(x) a principal polynomial. The non-constant principal polynomials are therefore of the form

f(x) = xk +

... +

ak

(k > 0).

The only constant principal polynomial is f(x) = e. All the polynomials different from 0 may be (uniquely) written in the form cf(x), if, and only if,

is a skew field, where c (E *, # 0) is a constant and f(x) a principal polynomial. The multiplication rule (67.9) becomes, with respect to (67.10),

... + ak)(boxr + ... + b,) = = 40 xk+' + (41 + albo)

(aoxk +

xk+!-1 +

... +

akb,

.

Accordingly, the degree of a product of polynomials is at most equal to the sum of the degrees of the factors and exactly equal if the coefficient ring is zero-divisor-free. Furthermore, in this case, the leading coefficient of the product is equal to the product of the leading coefficients of the factors. It follows also that a polynomial ring over a zero-divisor free ring is likewise zerodivisor-free.

262

OPERATOR STRUCTURES

By the number of terms of a polynomial f(x) we understand the number of the non-vanishing terms of the form axJ of this polynomial. In particular the product rule for one-term polynomials is as follows:

ax'. bx' = abx' +' .

(67.11)

From this and from the distributivity the general rule (67.9) follows. We see that the product of polynomials over a zero-divisor; free ring is one-term if, and only if, all the factors are likewise one-term. In the course of our considerations polynomials will be applied in various

ways. Here we want to study one of their most important applications. For this purpose we consider a polynomial f(x) E R[x], which we take in the form (67.10); further we consider an extension ring Y of and an element a of Y. The right-hand side of (67.10) has a meaning even if the indeterminate x is replaced by the element a, and becomes an element of Y, which is denoted simply by f(a), i.e.,

f(a) = aoak + ... + ak

(a E .9')

.

(67.12)

We emphasize that here we have to consider a term of the form x' in (67.10) always as ex'; this is necessary, since ea' and a' are not necessarily equal. We call f(a) (by analogy with function theory) the substitution value of f(x) at the place x = a. We also refer to the substitution principle for polynomials according to which a function f(a) (a E 50) is defined by any polynomial f(x) (E J2 [,YD in any extension ring Y of 2. In this function f(a), a then plays the role of a variable. We can call f(a), in order to distinguish it from f (x), a "polynomial function"; however, we seldom use this

ambiguous and unnecessary expression. Instead of this we call f(a) the .function assigned to the polynomial f(x). The substitution value f(a) is occasionally incorrectly called a polynomial in a. For a a (E 50) such that f(a) = 0, we call a a zero of the polynomial f(x) or a root of the equation f(x) = 0 (in Y). We can substitute a g(x) (E t[x]) for x in f(x) (E J2 [x]). (This is in accordance with the case Y= J 2[x] of (67.12).) From this a polynomial f(g(x)) (E fi[x]) is obtained, which we call an iterated polynomial. We also say that f(g(x)) results from f(x) and g(x) by iteration. Of course we may conceive the iteration as a (third) composition in P[x]. (Cf. Exercise 1.) In order to make the notions clear we observe that in general the functions f(a), g (a) (aE Y) assigned to the two different polynomials f(x), g (x) (E fi[x]) may be equal (cf. Example 1). This may be interpreted as saying that polynomials constitute a more "differentiated" notion than polynomial functions. (In other words: the algebraic polynomial notion is "superior"

to the function theoretical one.)

POLYNOMIAL RINGS

263

In connection with (67.12), for a fixed a (E _9') the mapping

f(x) --f(a)

(67.13)

of V [x] into 9', i.e. the substitution x --> a, plays an important role. It is by the help of this that the module.52[x]*, by (67.8), is mapped homomorphically into Y. On the other hand this is in general not valid for the semigroup

,59[x]' because on the one hand we have (67.11), but on the other hand aa$a' aba'+j may occur, unless a and b are permutable. This remarkable circumstance is explained by the fact that, on account of (67.9), x is a central element of .R[x], while a may be an arbitrary element of Y. It is different if a is permutable with the elements of JI, since then, as in (67.9),

(a0 + ala + ...) (bo + bla + ...) = aobo + (aobl + albo) or + ... , which means that the mapping considered is homomorphic with respect to multiplication. Hence we have the following important THEOREM 157 (substitution theorem for polynomials). If. is a ring with unity element, Y an extension ring of .59 and a an element of Y permutable with all the elements of S°, then 2[x] is mapped homomorphically into .9' by (67.13).

The equations valid between the elements of a polynomial ring, as for instance (x + a)2 = x2 + 2ax + a2 are usually called identities (or more precisely polynomial identities). It is a consequence of Theorem 157 that an identity in J' [x] by the substitution x -> a becomes a true equation, if a is an element in an extension ring of. ' elementwise interchangeable with Jp.

We call 7 [x] the ring of polynomials with integer coefficients or the natural polynomial ring (of x). The elements of 7 [x] with a vanishing constant term constitute a subring 7[x]o. This ring has the following important property. If f(x) = aoxk ... +ak-ix (ao, ..., ak-I E 7) is an arbitrary element of .7[x]o and .9' an arbitrary ring, then the substitution x-i-a may be carried out with any element a of So, so that the element

f(a) = aoak + ... + ak-la of .9' is obtained. It is likewise evident here that .7 [x], is mapped homomorphically into Y. for arbitrary n. We now consider the polynomial ring V[xl, ..., We immediately extend the above definitions to this more general case, in so far as they remain meaningful. Also our previous statements may be suitably generalized but this generalization does not need to be discussed in detail; moreover, some new questions arise. If necessary n > 1 is tacitly assumed.

OPERATOR STRUCTURES

264

First of all it should be noted that, according to (67.5), for the product of one-term polynomials

ax ... xin - bx ... x;,n = abx't+i' ... x`n +ia . This together with (67.4) and distributivity results in rule (67.5). We can have (67.1) rearranged in the form

f= ao + alxn + ... (a, = ai(x1i

... , Xn-1) E J2` [X1, ..., xn-1])

,

(67.14)

where the "coefficients" a, are uniquely defined polynomials. If we take a further polynomial

g = bo + blx,, + ... in 5i [x1,

. .

(bi = bi(x1, ..., xn-1) E * [x1, ..., xn-1])

., xn], then from the definitions (67.4), (67.5) the equations

f + g = (ao + bo) + (a1 + b1)xn + ..., fg = aabo + (aobl + albo) xn +

...

follow. Comparison with (67.7), (67.8), (67.9) shows that JP[x1,.. .... , xn] xn_l] by the adjunction of the further indetermiarises from nate xn, i.e.,

,

[x1, ..., Xn] _ (,5p[Xl, ..., xn-1])[xn] .

(67.15)

Repeated application implies that we may obtain R [x1, . . ., xn] from by successive adjunctions of each indeterminate. For this reason we can make use of the above statements, in particular of Theorem 157, concern-

ing the case n = 1, for an arbitrary n. There is no need to carry it out more exactly. By the degree of the polynomials (67.1) we understand the maximum of the sums i1 + . . . + in of the exponents in the terms actually occurring (i.e., terms provided with a coefficient different from 0). It is obvious that we understand by the degree of (67.1) in or relative to x, the maximum of the exponents i where again we have to consider only the actually occurring terms. If f(x1, . . ., xn) = 0 we again have to take - co for both degrees. The polynomials of degree 1, 2, 3, 4 are said to be linear, quadratic, cubic or

quartic, respectively. Now and then f° and f,, denote the degree or the degree relative to x of a polynomial f. A polynomial is said to be homogeneous if its terms (different from 0) are of the same degree. Of course this degree is then the degree of the polynomial. The homogeneous polynomials are also called forms. Evidently the product of homogeneous polynomials is again homogeneous.

The linear forms of the indeterminates x1, ..., xn overt are as follows:

alx1 + ... + anxn

(a1, ..., aE R) .

(67.16)

POLYNOMIAL RINGS

265

These evidently constitute an n-dimensional J-vector space (even an Jpdouble vector space). This is why vector spaces are also occasionally called linear form modules. If f(x1, . . ., is a form of degree k, and t a (new) indeterminate, then evidently

.f(tx1, ..., txr) = tkf(x1, ..., x,,) .

(67.17)

This may also be inverted : The polynomial f(x1, . . ., is homogeneous of degree k if the equation (67.17) holds for an indeterminate t. For, every polynomial is uniquely representable as

f=f0 +f1 + ...

+fr,

where fk denotes a form of degree k, or 0. The f, different from 0 are said to

be the homogeneous components of f. Through the substitution xi - txI (i = 1, . . ., n) the right-hand side becomes

fo + tf1 + ... + t'f, .

(67.18)

If now f is not homogeneous, then among the fo, . . ., f, at least two of them are different from 0. Then (67.18) cannot, for any k, equal the right-hand side of (67.17). So the statement is proved. It is necessary for more than one reason to arrange the terms of a poly-

nomial (of several indeterminates) in a fixed order. We first of all point out the following: Let S denote an ordered set and C5 the set of n-term sequences with terms from S. If for two elements of S

a=a1,...,a,,; b=b1,...,b the relations a1 = b1, . . ., ai-1 = bi-1, a, < bI

hold for any i (1 <_ i <_ n), then we put a < b. Evidently c then becomes an ordered set. We say for this that the lexicographical ordering in S has been introduced. (The words of a lexicon are arranged on this principle.) If the terms of the polynomial (67.1) are now lexicographically ordered according to the exponent series i1, ..., then we call it a lexicographically ordered polynomial. For instance, the lexicographical ordering of a polynomial in three indeterminates and of degree two runs as follows :

f(x,y,z)=a+bz+cz2+dy+gyz+hy2+kx+lxz+ + mxy + nx2.

OPERATOR STRUCTURES

266

The notion of the polynomial ring is easily generalized to the case of infinitely many indeterminates. For this purpose we have to proceed from an infinite set <x1, x2, ...> (instead of <x1, . . ., x";). The ring [xl, x2, .. arising in this way is called a polynomial ring of infinitely many indeterminates. Its elements may be uniquely written in the form

as,,, ... xl x' [cf. (67.1)], where in every term only finitely many exponents and only finitely many coefficients are different from 0. Accordingly the elements of [x1, x2, ... ] may be written in the form (67.1), where we always have to take a suitable finite subset <x1...., x,;) of <x1, x2, ...;; . Of course (67.4), (67.5) keep their validity. It is important that finitely many given polynomials from a polynomial ring with infinitely many indeterminates should always belong to a suitable A [x1.. . ., x j. EXAMPLE 1. If rk. = is a finite commutative ring with unity element, then f(x) = (x - a,) ... (x - a") V_ 0 is a polynomial in rk [x], for which the function f(a) (a E rk.) vanishes everywhere.

EXAMPLE 2. The polynomial theorem (23.6) may be expressed as an identity in the

natural polynomial ring .1 [.Y,, .. ., x,], namely as the "expansion formula" for (x, + ... + x,)". The original form of the theorem arises by substitution. EXAMPLE 3. The maximal number of terms of a form in n indeterminates of degree

k is

n + it - 1

I

,

namely the number of the combinations of k elements with repe-

titions from n ellements. EXAMPLE 4. The above ring J [x]o is identical with the free ring generated by x. According to DYCK's theorem (corollary to Theorem 101) the homomorphic images of 1[x]o are the rings generated by one element. EXAMPLE 5. The polynomial ring Q [x] can most easily be obtained as a skew

product by defining in the set of sequences (a,,, a,, ...) (a, E /k) with finitely many non-vanishing terms, the compositions

(a,,, a,....) + (bo, bl....) _ (a,, Jr b,,, a, + b ...), (a., aI, ...) (b,,, bt.... ) _ (aobo, aobi + albo, ...).

By the notation x = (0, e, 0, ...) we obtain the above form of (P-[x] (prove it!), which is more useful for applications. EXAMPLE 6. The homogeneous components of an element of an ideal generated by homogeneous elements of a polynomial ring likewise lie in this ideal. EXAMPLE 7. The notions defined in polynomial rings may be extended to the free ring 91, ... generated by the free generators x, y, . . . The remarks in the preceding example are retained. Otherwise ... is equal to the natural semigroup ring of the (free) semigroup generated by x, y.... (1n particular "`X = .9[Y]o, cf. Example 4.) EXERCISE 1. In t [x] the iteration of the polynomials is an associative, (in general) non-commutative composition; further it is I eft -distributive with respect to addition. (Cf. § 146.) EXERCISE 2. Give polynomials f(x) with real coefficients, for which the number of terms in f(x)2 is less than that in f(x). (Cf. RENYI (1947), VERDENIUS (1949), SEIBT (1965).)

LINEAR MAPPINGS

267

§ 68. Linear Mappings

We consider two left *-modules M, N over a ring with unity element. (These will later be specialized to two vector spaces.) The operator homomorphisms of M into N are called linear mappings (of M into N). It should be noted that in the special case M = N these are the operator endomorphisms of M. (This case will be studied in detail in

the next paragraph.) For a linear mapping A of M into N, we denote by .

eA (E N)

(O E M)

(68.1)

the image of a under A. (Note that A is written on the right-hand side of n. which is usual for mappings.) From the definition it follows that (e + a)A = OA + aA (ae)A = a(eA)

(o, a E M)

(e E M, a E

,

(68.2)

),

(68.3)

i.e. the mapping A is operator-homomorphic. Accordingly all the linear mappings A of M into N are characterized by (68.1), (68.2), (68.3). The common value of both sides of (68.3) can be denoted by a0A which, on account of (68.3), is unambiguous. Linear mappings may also be added and multiplied. We define the sum A + B of two linear mappings A, B of M into N by e(A + B) =- OA + eB

(o E M) .

(68.4)

Since from (68.2), (68.3), (68.4) it follows that

(e+a)(A+B) =(e+a)A+ (O+a)B= OA +aA+OB+aB =O(A+B)+a(A+B), (ae)(A + B) _ (ae)A + (ae)B = a(eA) + a(OB) _ = a(oA + LOB) = a(o(A + B)),

we see that A + B is also a linear mapping of M into N. We define the product AB of two linear mappings A, B where M, N, P are three left J'modules and A, B are linear mappings of M into N and N into P, respectively, by (O(AB) = ) OAB _ (OA)B

(o E M) .

(68.5)

OPERATOR STRUCTURES

268

Since it follows that

eABEP, (Q + a)AB = ((e + a)A)B = (LOA + aA)B = Q0A)B + (aA)B = = QAB + aAB, (ae)AB = ((aQ)A)B = (a(eA))B = a((eA)B) = -a(oAB)

then AB is a linear mapping of M into P. It should be noted that according to (68.5) in the product AB we have to

carry out first A, then B, that is in the reverse order to the usual for the multiplication of mappings. [This variation is connected with the notation (68.1) of the images.]

Hereafter M, N will denote two left s-vector spaces of rank m and it, respectively. For this important special case we want to consider linear mappings more closely. For this purpose we assume in M and N bases µl, ..., .um and v1, ..., vn, respectively, to which we shall keep referring in the following. Further, we denote by A a linear mapping of M into N. Since the elements of M are uniquely representable in the form m

e = r1µ1 + ... + rmlum= Y- ri iul i=1

(ri E JP)

(68.6)

and, because of (68.2), (68.3), m

BOA =

rr,,,,

ri (UiA) ,

(68.7)

it is sufficient for the determination of LOA, if we give the images µ,A (i = 1, ..., m) of the basis elements µl, ..., p,n. By the help of the basis of N we can take these in the form

µ1A = all v1 + ... + al vn , n

(or µ;A = Y aijvj; i = 1, ..., m) j=1

(68.8)

PmA = aml vl + ... + amn vn ,

where the aij are uniquely defined elements of R. By (68.7) we then have m mm m oA = I (r1 E a; jv! ra jv,. (68.9) i=1 j=t = i=1j=1

Conversely, if the aij are arbitrary elements of R (i = 1, ..., m; j = 1, .,

n), then (68.6) and (68.9) define a mapping A of M into N, which evi-

LINEAR MAPPINGS

269

dently fulfils the conditions (68.2), (68.3) and so is linear. In short the linear mappings A and the coefficients aI j define each other uniquely. This implies the following definition: a system

(68.10)

of elements a;j of the ring tV is called a matrix of type m x n (over R). A matrix of type m x n is an mn-term sequence of elements from a ring, which is divided into a sequence of m partial sequences each consisting of n terms. Its rectangular arrangement shown in (68.10) is perhaps immaterial

however, since it is usual, matrices are also called rectangular matrices.

Accordingly, matrices of type ml (i.e., of type m x m) are said to be square.

The matrix (68.10) is also denoted by (a; j)m,,, or, if the type is known or to be disregarded, by (a; j). The a; j are said to be the elements of the matrix

and the element au itself the Y-element (of the matrix). The ij-elements with fixed i or fixed j form the i`h row or thejth column of the matrix, respectively.

The rows and columns are also matrices, each of type 1 x n or m x 1. These matrices are called row vectors and column vectors, respectively.

The elements a,; (1 < i<_ min (m, n)) constitute the diagonal of the matrix. If the elements vanish outside the diagonal, then we speak of a diagonal matrix. This has for the cases m = n, m < n, m > n the following forms :

all

0

0

all

amm

0

0

0

all

am.. .. 0

1

0

0

a,,,

0

0

1

If all the elements of a matrix are equal to 0, then it is called a zero matrix. Zero matrices are often denoted by 0. From the matrix (68.10) we consider the rows with the indices

il,...,i

(1
(68.11)

and the columns with the indices

f1, ...,.i.

(1 <jl < ... <j":5 11).

(68.12)

270

OPERATOR STRUCTURES

If we retain only the elements of the matrix which lie simultaneously in these rows and columns, then we have a matrix ai,i, . .. ai,i,

ai.;,

ai,,;,

of type u x v, which we call the submatrix of (68.10) belonging to the row and column indices (68.11), (68. 1-7). Accordingly, the rows and columns

of a matrix are also submatrices of the matrix. We assign the matrix A in (68.10) to the linear mapping A described in the formula (68.9). From what has been said above this correspondence is one-to-one. But as linear mappings may be added and multiplied, it is convenient to define the sum and the product of matrices in such a manner that the sum and the product of two linear mappings should be in accordance with the sum and the product of the assigned matrices. For that purpose we consider two matrices

A = (a,), B = (b1).

(68.13)

First they should be of the same type m x n. We take as starting-points the above vector spaces M, N with the bases Ft1,. .., µm and vl, ..., vn, respectively, and denote by A, B the linear mappings of M into N assigned to the matrices A. B. According to formulae (68.4), (68.9) then [for the p in (68.6)] m n

e(A + B) = E E ri(a,, + bi,) i; i=Ij=1

Since, according to this, the matrix (ai; + bi.) (also of type m x n) is assigned to the linear mapping A + B, we define the sum of two matrices of the same type (over the same ring JI) to be (aij)mn + (bij)mn = (ail + bij)mn .

(68.14)

In order to define the product of the matrices (68.13), we now suppose that these are of type m x n and n x p, respectively. We then take three Pi, ..., v,1; 471, ..., ap, vector spaces M, N, P with the bases ,ul, ..., the linear mapping A of M into N and the linear mapping B of N into P assigned to the matrices A, B respectively. Then A is given by (68.8) and B similarly by P

via = Y bi;,T1

(i = 1, ..., n).

LINEAR MAPPINGS

271

furthermore by (68.5), (68.7), (68.9) m

n

QAB = (OA)B = m

p

n

riair(1rB) _ nr

E Y_ riau EJ=1 b1Jm1 J 1=11=1

n

P

L 11',a,,btJ'rr . i=1 t=Ij=1

If we compare this with the formula (68.9), we see that to the linear mapping

AB of M into P is assigned that matrix of type m x p whose (j)-element n

is equal to E ai,b,J. Therefore we define the product of two matrices of r=11

type m x n and n x p (over the same ring 2) by n

(aij)mn (b,J)np = (CO., (cii

=rZ

ai,b,1l

.

(68.15)

By the way in which the definitions (68.14), (68.15) were derived we see that the required connection between sum or product of linear mappings and the assigned matrices now exists. We repeat that we have defined the sum and the product of two matrices

only for the case where they are of type m -x n, m x n and m x n, n x p respectively.

The product rule (68.15) for matrices may be given in another, more, comprehensible form. For this purpose we consider two sequences

a=a1,...,a,,;

b=b1,...,b,,

(a1,b,E9; t = 1, . . , n)

of the same number of terms. By their inner product we understand the sum a1b1 + a2b2 + ... + a,,b,, = Y, a,b, . t=1

We also apply this definition to the case where a and b are row vectors, or column vectors, or one of them is a row vector and the other a column vector, where we consider them as sequences. (Of course a, b must consist of the same number of elements.) Now we can formulate rule (68.15) as follows: the product AB of two matrices A, B may be formed by taking the inner product of the ith row of A and of the jih column of B for the -ele-

ment of AB. (In this rule it is implicitly understood that the types of A, B and AB must be of the form m x n, n x p, m x p, respectively.)

It is evident from (68.14) that matrices of type m x n (over JI) constitute a module. If, moreover, we define both the operator products c(aiu) = (cab) ,

10 R.-A.

(ali)t = (a,Jc)

(c, aJ E JP)

(68.16)

OPERATOR STRUCTURES

272

for the matrices, this module becomes a double-vector space. A basis is formed by those matrices whose elements vanish with the exception of one element and for which the non-vanishing element is the unity element of 'R. The operator c in (68.16) is called a scalar, more precisely a left or righthand side scalar factor, respectively. Therefore the operations in (68.16) may be called left and right multiplication, respectively, by a scalar. For three matrices A, B, C overR the associative property of the multiplication (AB)C = A(BC), (68.17)

holds provided that at least one side of (68.17) is meaningful (then the other side, too, has a meaning). The truth of this assertion follows from the fact that the multiplication of matrices agrees with that of linear mappings. From (68.14) and (68.15) both the distributive properties

A(B + C) = AB + AC, (B + C)A = BA + CA ,

(68.18)

follow at once, again with a similar restriction for each of these equations as in (68.17). By the transpose of a matrix we understand the matrix resulting from interchanging the rows and columns of the given matrix. If we denote the

transpose of an arbitrary matrix A by A', then we can formulate the definition of transposes as follows:

all ... aln

all ... am,

am,

a]

amn

amn

If one of the two matrices A, A' is of type m x n, then the other is of type n x m. We likewise see that the ij-element of A' is equal to the ji-element of A. Hence the transpose of (ai;) may be denoted briefly as (ai). We mention the two trivial rules

(A')=A, (A+B)'=A'+B'. For matrices over a commutative ring the rule

(AB)' = B'A' is obviously valid. The transpose of a row or column vector is respectively a column or row

vector. Therefore a column vector may always be taken in the form a', where a is a suitable row vector.

LINEAR MAPPINGS

273

We often identify a matrix (a) of type 1 x 1 with the element a. This implies that we may take the inner product as a special matrix product, i. e.

alb, +ab 2 g + ... + anbn = al,

a

Is

., b.)'.

It is important that the equations (68.8), which determine a linear mapping,

can be compressed into just one equation, in which only matrix products occur.

For this purpose we generalize matrices (especially row and column vectors) so that their elements may be entirely arbitrary (elements of a module, mappings, operators, etc.). For these generalized matrices we define the notions "sum, product, multiplication with a scalar factor, inner product, transposes, etc." formally just as above, provided that something meaningful results. Then applying rule (68.15) we can write for equations (68.8) the single equation p1A

all ... aln

vl

IAmA

aml .

amn

vn

If we simultaneously introduce the notation (68.10), then this may subsequently be written as:

(µl, ..., µJA = A(vl, ..., V.), .

(68.19)

If, moreover, we introduce the row vectors

(µ) = (Is ..., µm) , (v) = (v1, ..., vn) ,

(68.20)

which we call basis vectors of the vector spaces M and N, respectively, then (68.19) is reduced to the elegant form (cc)'A = A(v)'.

(68.21)

Note that in this equation the image of (µ)' produced by the linear mapping A is on the left, where we consider A as a scalar factor, while on the right is a matrix product where A denotes the matrix belonging to the mapping A. If w e write (68.6) as o = (rl, . . ., rm)(a)', so that B O A = ( r 1 . . . . . then because of (68.21) we obtain QA = (r1, ..., r.)-4(v)'

(P =r1µ1 + ... rmµm rI, ..., rm E -*), (68.22)

by which the image oA of an arbitrary element of M is given in a concise form. (68.22) is just another form of (68.9).

274

OPERATOR STRUCTURES

EXAMPLE 1. We consider the two polynomial rings

(kx=(1X1,...,X,,,], As stated above, the linear forms in these constitute an /Q-vector space of rank in or n, respectively. All the linear mappings of the first into the second are given by xi

any, + . . . + alnyn

(i = 1, . . ., m)

(68.23)

where the au are arbitrary elements of T. Now we can consider (68.23) as a substitution in the ring /Q,,. By this every polynomial x,,,) (E /Q.,) is changed into a polynomial + alayn, , amlyl + + amnyn) (E t ,) , Yn) = f(allyl + Therefore the linear mapping (68.23) is called a linear transformation, and one says that f(x1, . . ., x,,) is transformed by it into g(y1, , yn). g(yl,

EXAMPLE 2. Over the ring .1 we have 1

2

(l

-2

3

3)

6

0

2 20

3 50

13

10

(17

-10

-2' 6)'

(_,

123

(1,2,3)(l,2,3)'=14, (1,2,3)'(123)= 1246 369 EXAMPLE 3. We consider two J-vector spaces with the bases Jul, ,u, and v1f v2, vj

respectively and the linear mapping given by I -3

1-1) of the first into the second. 2

3

The image of 2µ1 - 3µ2 will be, according to (68.22),

(2,-3) (-1

2

1

)(v1, v2, v3)' _ (5, -8, 11) (v1, v2, vol = 5v1 - 8v2 + I I v2.

§ 69. Full Matrix Rings We consider all the square matrices of type n2 over a ring with unity element e for fixed n (>_ 1). These constitute a ring according to the addition and multiplication defined in § 68, which is called the full matrix ring of rank n2 (over *) and denoted by R,,. We shall see that J9, constitutes a

monomial R-double algebra of rank n2. From the preceding results it follows that all the linear mappings of a vector space into itself can be given by the matrices of this will be dealt with in more detail below. The elements of ,i are the matrices

all ... ai A = (au) _

(au E J2).

an

(69.1)

FULL MATRIX RINGS

275

The diagonal matrix

(69.2)

occurring among them obviously has the property

E is the unity element of 5,, and may be called the unity matrix (of A). (If we want to emphasize that E is of type n', then we write it more precisely as E,.) The zero element of s,, is the zero matrix 0. Let El; (E lfin) denote that matrix, in which the ij-element is equal to e and all the other elements are equal to 0 (i, j = 1, . . ., n). Evidently n

(au) =Eau E,!

(69.3)

1'j- I

It is likewise obvious that the scalar factors of on the right-hand side are uniquely determined by the left-hand side. Hence we conclude that e2, is an ,*-double vector space of rank n2 and has the E;j as a basis, where, of course, left and right multiplication by a scalar hold as operations. We have only to prove for the operations that the conditions (aA)b = a(Ab) , aE,1= EEja

(a, b E

$; A E 2,,; i, j = 1, ..., n)

are fulfilled. As, in fact, this is so, the statement is true. Furthermore we show that .5?', is a double algebra. Because of Theorem 153 it suffices to prove that the structure constants of e39n relative to the basis Et; belong to the centre of JP. Since now, as is obvious, EjiEk

_ Ellforj=k, to for j & k

(I,j, k, l = 1, ..., n) ,

(69.4)

all the structure constants are equal to 0 or 1. From this the statement follows. Moreover it has been proved that 51,, is monomial. In order to express (69.4) more conveniently, we define the Kronecker

symbol (which is also useful for other purposes) bij (= Sj,) by bil

I fori=j, I 0 for irsj

(i, j = 1,

..., n).

(69.5)

By the help of this (69.4) may be condensed to EUEkr = bikEjI

(i, j, k, l = 1, . . ., n) .

(69.6)

OPERATOR STRUCTURES

276

In particular E12

= E t t, E E = 0

(i s j; i, j = 1, ..., n) ,

E = Eu + E22 + ... En,.

(69.7) (69.8)

The Eli are called matrix units. The special diagonal matrices

= aE = Ea

(a E JP)

of ,.,*n are said to be scalar matrices. Because

aE + bE _ (a + b) E , aE bE = abE

(a, b E JP)

these constitute a subring of Rn isomorphic with J. A suitable isomorphism is aE -->a. Since

b(aE) = baE -* ba , (aE)b = abE --> ab , this is an operator isomorphism. Consequently the fundamental ring 'R can be embedded in 'Pn by substituting a (a E JP) for the scalar matrix aE, by which ,2, becomes an extension ring of V. However, this embedding in matrix rings is not always performed. We have seen in § 68 that the linear mappings (i.e., operator homomorphisms) of an m-dimensional vector space into an n-dimensional vector space are given by the matrices of type m x n. The consequence of this, as mentioned above, is that the matrices of *n give the linear mappings of an n-

dimensional *-vector space V into itself, i.e., the 2-endomorphisms of V. This will now be considered more closely. By way of introduction we note the following: although the linear mappings of V into themselves and the 92-endomorphisms of V are the same, the fact they are oppositely multiplied (viz., the first are multiplied from left to right, the latter from right to left) results in their being rings anti-isomorphic to each other. More precisely they are opposed rings. It will be sufficient to examine the first of them. For this purpose we give in V a basis vector (w) _ (wl, ..., Co.) .

(69.9)

If we denote by A all the linear mappings of V into itself and by A the matrices in e/dn, it follows from the preceding results [see (68.21)] as a special

277

FULL MATRIX RINGS

case that the A and A may be assigned to each other, one-to-one, in such a manner that (co)'A = A(w))'

.

(69.10)

(On the left and right, as for inverse images and images, the same basis vector (69.9) will do. One could have taken different bases, but this would only involve useless complications.) We consider a second assigned pair B, B: (co)'B = B(co)'

Then [because of (68.2) to (68.5) ] (co)'(A + B) = (CO)'A + (co)'B = A(cv)' + B(co)' = (A + B)(c))' , (co)'AB = ((w)' A)B = (A(c))')B= A((co)'B) = A(B(co)') = AB(co)'.

These signify that the one-to-one assignment A -* A obtained through (69.10) is an isomorphic mapping of the ring of all the linear mappings of V We refer to some of the results in the following into itself onto the ring TiEoREM 158. The full matrix ring'Rp is isomorphic with the ring of linear mappings of an n-dimensional. -vector space V into itself, i.e., it is antiisomorphic with the-endomorphism ring of V. EXAMPLE 1. Define the semigroup, consisting of the n2 + 1 elements 0, E,, (1, j =

= 1, ..., n), by the equations (69.6) and OE,, = E#O = 0 (i, j = 1, . . ., n). The application of the second case of Theorem 156 to this semigroup then yields exactly the full matrix ring /k,,. EXAMPLE 2. By the help of the Kronecker symbol d,, the square diagonal matrices may be given in the form (S,, au). In particular E _ (8i,) is the unity matrix. EXAMPLE 3. In 9z the elements a = E12, # = Ell, y = kE22, satisfy

a =0,#2=9,y1=ky,afl=O,fla=a, ay=ka,ya=0,fly=O,yfi=0 and therefore generate for each k (E .) a subring R with the basis a, i, V. Moreover aR = kJa, Ra = 7'a. (Cf. § 42, Exercise 2.) EXERCISE 1. The full matrix ring /k over a commutative ring is isomorphic with the opposed ring V,,. (Cf. § 39, Example.) EXERCISE 2. For the matrix units E,, (i, j = 1, ..., n), E12, EE3...., is a generating system (n > 2). § 70. Linear Groups

Let a ring with unity element be given. We ask how from a given basis vector (co) = (col, . . ., of an J`'-vector space V we can give all its (likewise

n-term) basis vectors (p) = (el, . . ., Q.). We call this the problem of the basis transformation of vector spaces.

OPERATOR STRUCTURES

278

In order to find the required basis vectors, we first of all take, for the 01, ..., en, arbitrary elements of V and denote the matrix (pl, ..., on) of type 1 x n by (O). Since (co) is a basis vector, we can uniquely write every ioi

in the form Ni = ail Wl + ... + ain (on

(i = 1, ..., n; aij E *):

If we unite the aij in a matrix A (E ,h'), then what was said above may be expressed in the single equation (o)' = A(c))'

(70.1)

where (a)', (co)' denote the transposes of (Lo), (co). Since the given (e) and the A determine each other uniquely by (70.1), to set up the basis transformations

we have to define only those matrices A, to which, by virtue of (70.1), belongs a basis vector Qo). If this is the case, then we say that the basis ol, ..., ", arises from the basis W.1, ..., Wn by transformation with the matrix A. For the above purpose we assume first of all that (e) is a basis vector. Then, besides (70.1), (W)' = B(P)' ,

(70.2)

for a suitable B (E .n). If we put (70.1) into (70.2) and (70.2) into (70.1), then we obtain (co)' = BA(W)',

(N)' = AB(N)' .

Because of the above uniqueness, it hence follows that

BA=AB=E, where E is the unity matrix. Therefore A has, according to Theorem 26, a uniquely determined inverse A-' (in the sernigroup R'). We show that, conversely, if A-1 exists, then (70.1) defines a basis vector (o). It has to be proved that every a1W 1 + ... + anco

(al, ..., an E -)

may be uniquely written in the form

b1o1 + ... + bnon

(b1, ..., bn E tR) .

This assertion means that the equation (bl, . . ., bu) (N)' _ (a1, ..., an) 0-1'

LINEAR GROUPS

279

for an arbitrary (a,, . . .,

has exactly one solution (bl, ..., b,,). This equation, because of (70.1), is representable in the form (b1, . . .,

(a,,... ., an) (CO)" .

This is, since (co) is a basis vector, equivalent to the equation

Since this has obviously the unique solution

i our assertion is proved. Those matrices of R. which have inverses are called invertible matrices. Our result may therefore be expressed as THEOREM 159 (theorem of the basis transformation of vector spaces). All the n-term basis vectors (o) = (eel, ..., of an R-vector space are obtained from an n-term basis vector (w) = (wl, . . ., co.) by the formula (70.1), where A has to run through all invertible matrices in the full matrix ring J9.With respect to this theorem we call (70.1) the formula for basis transformations of vector spaces. The invertible matrices of An constitute, by Theorem 34, a multiplicative group with the unity element E. This important group is called the full linear group of degree n over R. The origin of this terminology lies in the fact that matrices yield linear mappings of vector spaces into vector spaces. THEOREM 160. The JI-automorphism group of an JP-vector space V of rank n is isomorphic with the linear group of degree n over R.

The R-automorphisms of V are now uniquely given by the basis transformations (70.1), i.e., by the invertible matrices A (E JI,.). If the matrices A, B belong to the automorphisms A, B, then the matrix BA belongs to the automorphism AB (according to § 69). Hence, and from Theorem 67, Theorem 160 follows. In the linear group the scalar matrices obviously constitute a subgroup. This is normal if the fundamental ring is commutative. For a commutative ring R with unity element the factor group of the linear group (of degree n over JI) with the subgroup of its scalar matrices is called the (full) fractional linear group (of degree n over R).

We wish to complete the results of § 68 on linear mappings by the following theorem:

THEOREM 161. In two 9-vector spaces M, N, let bases and i',, ..., v,,, be given respectively, and a linear mapping A of M into N together

with the associated matrix A of type m x n over 9. If the bases are each 10/a R. - A.

OPERATOR STRUCTURES

280

transformed by matrices M (E gym), N(E t2") respectively then after the transformation the matrix MAN-1 belongs to A. Let

(u) =

(IM

IA.) ,

(v) _ (v1, ...,

1um) ,

(V*) = (vl , ..., vn)

(W1 i t ', .

be the basis vectors of M, N before and after the transformation. Then

(µ)'A = A(v)', (a*)' = M(#, (v*)' = N(v)'. Hence

(µ*)'A = (M(ju)')A = M((ci)'A) = M(A(v)') = MA(v)' _

= MAN` (v*)' . This proves the theorem. As a special case when M = N, m = n, u, = vi (1 = 1...., n) we obtain the following COROLLARY. The matrix A (E R") associated with a linear mapping of an

s-vector space into itself is replaced after a transformation of the basis by means of a matrix S(E ") by the matrix SAS-1. We call SAS-1 a transformed matrix, which arises from A by the transformation with the matrix S. § 71. Alternating Rings

We shall show that to every ring 39 with unity element and to every n (>_ 1) there is a monomial *-double algebra A,, of rank 2" such that with suitable elements Et,

..., E. (E A") having the property

e2i = 0, EIEj= -ejei

(i0j;i, J= 1....,n)

(71.1)

a basis of A,, consisting of the 2" products

ei,...ei,

(1<11<...
(71.2)

may be given. (The empty product (71.2) for the case r = 0 means the unity element of A".) We call A. the alternating ring of rank 2" over tR. Besides

we should note the following: If the factors of the product ei ... e, are permuted, then because of (71.1) only the sign may be changed and this remains invariant if, and only if, the permutation belongs to the alternating group that is why we call A. an "alternating ring". From among the basis elements (71.2)

61....,8 are called alternating units (or alternating numbers).

(71.3)

281

ALTERNATING RINGS

We have to prove the existence of A. With this end in view we define a semigroup H4 with n + 3 generating elements e1, ..., E"

w, e, Y,

(71.4)

by the help of the equations Y = e, e? = w,

(i

ei e! = Ye1 Ei

J; i9 j = 1, ..., n)

(71.5)

and the further conditions that w should be the zero element, a the unity element and y a central element of H4. (Of course, these additional conditions may also be expressed by further equations, so that we have a semigroup

defined by equations.) We note that the el, ..., e" in (71.4) so far have no connection with the above elements. The products of the form (71.2) will be denoted by ai (i = 1, ..., 2"), which, however, are now elements of H4. Among these ai we also have the empty product which is to be identified with s. All the distinct elements of H 4 are (see the exercise below) given by (71.6) (I = 1...., 2'). w, ai, yai

(Accordingly 0 (H4) = 1 + 2"+'.) The fourth case of Theorem 156 is applicable with this semigroup H4. An s-double algebra of rank 2" is obtained with the basis elements (OCj, a2, ...,), i.e. (71.2), and for the ele-

ments (71.3) because of (71.52,3) the equations (71.1) also hold. This proves the existence of the alternating ring A. It is likewise obvious that this is uniquely determined, since one can compute from (71.1) the product of two arbitrary basis elements (71.2). EXAMPLE. The multiplication table of the basis elements of A2 is: E

el

e2

£.1 e2

£

e

£1

E2

£1 £2

£1

el

0

E1 E2

0

e2

£2

£1 ez

-el e2 0

£1 £2 1

0

0

0

0

EXERCISE. In order to show that (71.6) yields all the different elements of H4, it is sufficient, according to Theorem 101, to give a semigroup of order 1 + 2"+1 belonging to the equations (71.5) which define H4. For that purpose we denote by F one of the 2"

sequences il, ..., i, from (71.2). Let (F) be their set. If two sequences F', F" from (F) have no common member, then their members suitably joined again form a sequence from (F), which is denoted by [F', F"]. On the other hand, the juxtaposition of F and F" results in a sequence, which we can obtain by a permutation from [F, F"]. According as this permutation is even or odd, we put sign (F', F") = 1 or -1. We now consider the 1 + 2"+l different symbols Q, Fe

(FE (F); e = 1, -1)

OPERATOR STRUCTURES

282

(in this e is not an exponent), denote their set by M and define in this a multiplication as follows. If F, F" have the above meaning, then we put

F4F"o = (F', F,"psisn(F'. F")

(C, a = 1, -1).

All other products in M are defined to be equal to £ . Prove that M is then a semigroup. If we denote by F0 the empty sequence and by Ft the sequence consisting of

the single member i (i = 1, ..., n), then

Q,FF,Fo',F',...,F. are generators of M. They constitute) in place of (71.4)) a solution of the equations (71.5) to which therefore M belongs.

§ 72. Determinants

In this paragraphR will throughout denote a commutative ring with unity element. The a, b, ... (even if provided with indices) always denote elements of , . We consider a square matrix

(1,1= 1, ..., n)

(ay)

(72.1)

of type n$ from the full matrix ring R,,. We make use of the alternating ring

A. over R with the alternating units er, ..., e,,. Then

ej = 0 ,

-eJel

(i # j; i, j = 1, ..., n) .

(72.2)

Hence it follows immediately that every product

a,,..el.

(Ik=1,...,n; k=1,..., n)

is equal to 0 or el ... en or -el ... e,,. Thus r i

e

(a;lel + ... + ai

(72.3)

where D is an element of 2, which is uniquely determined by the aiJ, i.e., by the matrix (72.1), for er ... e. is one of the basis elements of A.. We call this D the determinant of the matrix (72.1) or a determinant of order n (in .J), and denote it by

an ... ai

=IajI.

D=

(72.4)

a41... a,,,, (Of course, the order of a determinant D differs from o(D) and from o+(D).)

DETERMINANTS

283

If necessary, we use instead of I au I the more precise notation I au In By the determinant of order zero we understand the unity element of J9, which is, however, usually disregarded. Some of the notations used for matrices are also applied to determinants. E.g., au is called the ij-element of the determinant I au I , as for matrices. We likewise adopt,the terms itb row", "row vector", "diagonal", "transpose" of a determinant. For n = 1 the right-hand side of (72.3) is equal to a11E1 Hence the determinant I all I of first order equals all. For n = 2 if we compute the left-hand side of (72.3), because of (72.2) we obtain a11a22E1E2 - a12a21E1E2 -

Hence all a12

= alla22 - al2a21

(72.5)

a21 a22

In the general case the left-hand side of (72.3) is evidently equal to n

a111 ... a,, ej, ... Ef,, II....,41=1

Because of (72.2) those terms in which the il, ..., in are not all different disappear. So only the terms are left in which i1, ..., in is a permutation of 1, ..., n. Hence alnl ... amen

DE1 ... an ,

(72.6)

where we have to sum over all the permutations ai (E ..9,,) and ni denotes the image of i. We show that Enl ... Ean = sign 7L El ... En.

(72.7)

For, the left-hand side is obtained from the product

E1...En, by carrying out the permutation n on the factors. Thus, if n is a transposi-

tion of the form (i, i + 1), then the left-hand side of (72.7), because of (72.2), is equal to -El... E,,. In the general case we decompose n into a product of t transpositions of the form (1, i + 1). By repeated application of the preceding we then obtain for the left-hand side of (72.7) the value

(-0,61 ... S.-

OPERATOR STRUCTURES

284

On the other hand, by Theorem 83, (- 1)' = sign n, whence (72.7) follows. Putting this in (72.6) results in

D= j auj _ Z sign n a>1 a2s1 ... a,,,,,, , 'E6

(72.8)

or, more concisely,

D = I a j I = Y ± a11. a22, ... a,,,;

,

(72.9)

where we have to sum over all the permutations 1', . . ., n' of 1, . . . , n and take the sign + or -, respectively, according as this permutation is even or odd. The right side of (72.8) [or (72.9)] we call the expansion or the

"value" of the determinant. The terms occurring in it are briefly the terms of the determinant. Their number is n!. In particular, the term

.a

is called the leading (principal) term of the determinant. It is . evident that the diagonal determinant is reduced to its leading term. For the present the afj will denote indeterminates, which define the natural polynomial ring .7 [all, a12, ..., a,,,,]. Then the determinant D will be aform of degree n with coefficients 0, 1, - 1. Moreover, D is a homogeneous linear polynomial in the elements of an arbitrary row or column. We call this polynomial the (nth) determinant polynomial. From this single determinant all the possible determinants of order n arise by substitutions. a11a22

THEOREM 162. A determinant is equal to its transpose.

If D' _ I aj, I denotes the transpose of D, then according to (72.8)

D' _

sign n a,1 1 a.2,2 ... an,,,,, .

(72.10)

Since n-' runs through all the different elements of .9s as n does so in (72.10) we may substitute oz-1 for n. Then the pair

n-'i, i

(i = 1, ..., n)

(72.11)

takes the place of the index pair ni, i . On the other hand, ni also runs through the elements 1, ... n as i does so, but, in general, in another order of succession. Therefore the pairs (72.11) agree to within the order of succession with the pairs

i,ni

(i= 1,. .., n).

But since in (72.10) the order of succession of the factors is irrelevant because of the commutativity of ., it follows that D.

Lr sign n

n-

a1n1 aznz ...

DETERMINANTS

285

Finally, since according to the corollary to Theorem 83 sign n-1 = sign n we obtain, on comparison with (72.8), the equation D' = D. Consequently Theorem 162 has been proved. As a consequence of Theorem 162, each proposition, which follows from the definition of determinants, remains true, if the notions "rows" and "columns" (of determinants) are interchanged with each other. This is called the principle of duality for determinants. THEOREM 163 (multiplication theorem for determinants). For the full matrix ring *n over a commutative ring J` (with unity element)

An ti Rx ((aij) - I a,1 I) ;

(72.12)

therefore two determinants of order n (in 59) may be multiplied according to the rule n

aij I bij I = I cij I

(cij = Y a,, bj; i,J = 1, .... n) . t=1

(72.13)

Nom. Because of duality, (72.13) remains unaltered if, in the sum, at, is substituted for a,, or bj, for b1 or both simultaneously. According to this, from (72.13) three further formulae for the multiplication of determinants may be obtained. We can call (72.13) itself the "multiplication formula of the determinants according to rows and columns" and the three other formulae accordingly. The dualization of any further theorems is left to the reader. Since obviously all the elements of ,1I occur among the I a,j I, it is sufficient io prove (72.13). To do this we consider the elements

(i = 1, ..., n)

1t = b,1E1 + . . . + bine.

(72.14)

of An. From (72.2) it follows at once that

ej = 0,

'i,rlj= -ijrli

(i 0.1;

i,j= 1,...,n).

Since these equations are of the form (72.2), (72.3) remains valid under the substitution of %, . . ., % for e , ..., En: n

n

H I airrit = I aij I rh

i=.L t=1

. n,,,

(72.15)

where (72.4) is also inserted simultaneously. On the one hand, according to (72.14) [with the ca defined in (72.13)] [n

[n

n

Y cijEj = c, r1 + ... + cinsn L aitbtjej =j=1 t=1.1=1

[, air211 = Lr t=1

On the other hand, according to (72.14) and (72.3)

th...?In

En.

OPERATOR STRUCTURES

286

These yield after insertion in (72.15) the equation n

r] (ctle + ... + ci e) _ aij I i=1

Ibij I El .

The left-hand side, because of (72.3), is equal to

I

. . En .

cij I El ... E,,. Hence

(72.13) follows, by which the theorem is proved.

THEOREM 164. Interchanging two of the rows (with djerent indices) of a determinant D changes it to - D. So if the rows of D are subject to a per-

mutation n, then sign a D is obtained. It is sufficient to prove the theorem for the case where two neighbouring

rows are interchanged. But for this case the theorem follows easily since, by (72.2), the interchange of two neighbouring factors of the lefthand side of (72.3) changes only the sign of this product. THEOREM 165. A determinant with two equal rows vanishes.

For, on account of (72.2), the square of a factor vanishes on the lefthand side of (72.3), therefore the theorem is true for the case where two

neighbouring rows are equal. Hence, according to Theorem 164, the general truth of theorem 165 follows. THEOREM 166 (Theorem for addition of determinants). If for a fixed i (I < i < n) the determinants Di, D2 (in JT) of order n are identical except for their it' rows, then

D1+D2=D,

where the determinant D is obtained from D1 by the addition of the it' rowof D2 to the i`s row of D1 (term by term) while the other rows remain unaltered.

THEOREM 167. A determinant D (in JP) is changed by the (scalar) multipli-

cation of a row by a factor c (E ) into cD. Both theorems follow immediately from the fact that the determinant polynomial is homogeneously linear in the elements of a row.

THEOREM 168. A determinant D (in J2) is invariant with respect to the addition of c times one row to another row (c E 09) . This follows from the three preceding theorems. We consider a determinant D = I aij I of order n. By a subdeterminant of this determinant D we understand any determinant of the form

d=

:

I

(1
1<Ji<...<j,Sn). (72.16)

DETERMINANTS

287

More precisely d is called the subdeterminant (of order r) of D belonging to the row indices ii, . . ., i, and column indices jl, ..., j,. If for this d and a further subdeterminant of order (n - r) airtilr+l

al,+.j.

d* _

(72.17)

..

a,,,j,+.

aials

1<jr+i<...
the adjoint ofd. (For r = 0, r = n we mean d = e, d* = d = D and d = D, d* = d = e, respectively, where e is the unity element of 'R. We mostly disregard these cases and consider only the cases 0 < r < n.) It should be noted that the subdeterminant (72.16), in the special case i k = j k (k = 1, ..., r), is called a principal minor of order r of the determinant.

There are nl of these. I rllll

THEOREM 169 (LAPLACE's theorem). For a determinant D

D= Y dA,

(72.19)

d

where d runs through those subdeterminants of order r of D, which may be formed from r arbitrarily given rows of D, and d denotes the adjoint of d. (72.19) is called the Laplace expansion (according to the given rows) of D. In order to prove the theorem, let us consider a permutation of the special kind

(l.:.n ll

il.

i

JJ

(1 _< it <

.

. < it < n; 1 <_ it+l < ... < i < n1

(72.20)

and denote by (i) the number of its inversions. It is evident that in (72.20) only those pairs ix, is, form an inversion, for which

i., >iy (1 S x<_r; r+ 1
OPERATOR STRUCTURES

288

For fixed 4 these i,, are exactly the terms of the sequence 1, . . ., iX - 1 different from the il, ..., so that their number is 1, - x. Accordingly

+...+i,-(1+ ...+r).

(72.21)

If now the rows of D are subjected to the permutation (72.20), then D becomes, by Theorem 164, (-1)'°D. Hence and from (72.3) it follows that n

fl (aiiel + ... +

(-1)()Del ... e .

t=1

(72.22)

Since, on the other hand, r arbitrary elements among the alternating units el, . . ., en of A,, assume the role of a full system of alternating units of the alternating ring Ar, it follows from (72.3) that

fi (ai,lel + ... + airnen) _

dej, ... ejr Y lsli<...<jr5n where d denotes the determinant (72.16). Likewise n / d *ej,+ ... E II (a,,lel + .. . + ai,nen) _ Z t=1

15j'+, <... <jnan

t = r +1

with the determinant d* in (72.17). If both are inserted in (72.22), then we obtain (-1)(''dd* £j, ... ejn = D el ... en. (72.23) l5h<... <jrran 15j,+,<...

On the left, because of (72.2), it is sufficient to retain only those terms in which all the jl, . . ., jn are distinct. This is equivalent to

(1...??)

jl

(72.24)

jn

being a permutation. It should be noted that then d* means the complement of d. Similarly to (72.21) the number of inversions in (72.24) is

(1)=jl+...+j,-(1 +...+r).

(72.25)

With this notation, on account of (72.2.9),

ejl ... ejn = (-1)a' el ... en .

(72.26)

We can write (72.18), by (72.21), (72.25), in the form

A = (_ 1)cMJ) d* If we insert this and (72.26) into (72.23), then a f t e r omitting 8 l . .. en we obtain equation (72.19), i.e., Theorem 109 is true. .

DETERMINANTS

289

The special case r = 1 of this theorem is particularly important. In this respect we notice that the subdeterminants of order I are the elements ay themselves. We denote the adjoint of a,j by Arj. On account of (72.18)

Arj - (- l)r+j Dij

(i, j = 1, ..., n)

,

(72.27)

where Drj denotes the complement of a,j, i.e., that subdeterminant of D which remains after cancelling the ith row and the jth column. The special case considered in (72.19) then becomes : n

D=

at3 A,j

(72.28)

(i

This formula is called the (Laplace) expansion according to the ith row of the determinant D. We see also from (72.28) that D is homogeneously linear in the elements of the ith row. Moreover (72.28) means that the adjoint Arj is the "cofactor" of the element a,j.

We repeat and complete formula (72.28) in the following THEOREM 170. For the elements arj of a determinant D of order n and for their adjoints Arj 11

E arj Akj = b;k D

j=I

(i, k = 1, ..., n),

(72.29)

where ark is the Kronecker symbol. In matrix calculus this is denoted by (arj) (Aj,) - D(bj) .

(72.30)

Because of (72.28) we have to prove (72.29) only for i 0 k. With this end in view let Dl denote that determinant, which arises from D by taking the ith row in the latter for the kth row. The left-hand side of (72.29) is then the expansion of Dl with respect to its kth row. Since, on the other hand, Dl has two equal rows, on account of Theorem 165 Dl = 0. Consequently we have proved Theorem 170. The matrix ((A,1)' =) (Ajr) occurring in (72.30) is called the adjoint matrix of the square matrix (a,). These are always interchangeable since from the dual of (72.29) it follows that (Aj,) (au) = D(b,j).

We denote, in general, the adjoint matrix of a (square) matrix M by adj M. Thus we have the important formula

M= M I denotes the determinant of M, and E the unity matrix (of the same type as M).

OPERATOR STRUCTURES

290

Similarly, by the adjoint determinant of the determinant D = ai; I is meant the determinant I Ai11, which we denote by adj D. THEOREM 171. For the adjoint determinant of a determinant D of order n

adj D = D".

(72.32)

It is sufficient to prove the theorem for the special case D = I ai; I with indeterminates aid. Now by applying (72.31) to M = (ac) and from Theorem 163 [equation (72.13)] it follows at once that

Since the polynomial ring 7 [all, a12, ..., a"] is zero-divisor-free and D 96 0, we can cancel D. Consequently we have proved Theorem 171. Determinants were introduced by LEIBNIZ (1693), but have come into general use only since CRAMER (1750) and were denoted by CAYLEY in the present form. For an extension of the notion of determinants to non-commutative rings cf. DIEUDONN$ (1943).

§ 73. Cramer's Rule The problem of algebraic equations and equation systems in rings (in particular, in skew fields and fields) may be formulated as follows: let a system

(i = 1,. .., m)

f = A XI, ..., x")

(73.1)

of m polynomials be given in a polynomial ringV [x1, ..., x,] of n indeterminates x1, . . ., x". We have then to determine those systems at, ..., a"

j q),

(73.2)

for which all the equations f (al, .

.

.,

0

(i = 1, ..., m)

(73.3)

are satisfied. In connection with this problem, we call the equations

f(x1,...,x") =0

(i= 1,...,m),

(73.4)

formed from the polynomials (73.1), an algebraic equation system or, if m = 1, an algebraic equation (in or over *) with the unknowns xI, . . ., x,,. Since (73.4) are not true equations, for the f need not be zero polynomials, we should properly use in (73.4) a sign other than "_", but this inexactitude

will cause no misunderstanding. Any system (73.2) with the property (73.3) is said to be a solution or root (in the case n > 1, more precisely

CRAMER'S RULE

291

a solution system or root system) in R of the algebraic equation system (73.4). We frequently require solutions of (73.4) in an extension ring 9' of,R. In principle this is no generalization, as this case may be reduced to the former, by replacing R by So at the outset. Essential generalizations arise if we admit infinitely many equations or unknowns. It should be noted that in (73.4) the unknowns x1, . . ., xn are as a matter of fact indeterminates, which, however, we often consider as variables. Furthermore we often denote the solutions by the same x1, ..., xn. If in particular all the polynomials f are linear, then we call (73.4) a linear equation system. After taking the constants to the right-hand side, (73.4) appears in the form

(i = 1, ..., m)

.f (xv ..., x,,) = cr

(73.5)

where the f, now denote linear forms. Here we shall consider only the special case of (73.5) a11x1 + ... +

cl (73.6)

an1x1 + ... + annxn = cn,

where the number of equations and unknowns is equal and the underlying ring T, containing the coefficients ari, c,, is commutative. We call (73.7)

D = I art I

the determinant of the linear equation system (73.6). Further we denote by Di that determinant which arises from D when the elements of the j'h column are replaced by the cl, ..., c,,. This means, according to LAPLACE's theorem, n

Di = c1A11 + ... + c,Ani = E c,Au

(j = 1, ..., n),

(73.8)

where Ari denotes the adjoint of ari.- We call D1 the jth codeterminant of (73.6).

THEOREM 172 (CRAMER'S rule). In the case of a commutative ring JP, for

any solution x1, ..., xn of the linear equation system (73.6) with the determinant D and the codeterminants Di (j = 1, ..., n) we must have

Dxi=Di

(j= 1,...,n).

(73.9)

If D is a regular element ofd, then (73.6) has, in the quotient ring of,*, the unique solution

xi=D-1Di

(j= 1,...,n).

(73.10)

OPERATOR STRUCTURES

292

These x.j...., xn lie in._ if, and only if, the divisibility conditions

(j= 1,...,n)

DID

(73.11)

are also fuflled. We write (73.6) in the form =1

a;x, = c,

n).

(73.12)

If we multiply by A, , then after addition we obtain n

n

Q= 1, ..., n),

E E a11A;jxt = Dp

1=1t=1

(73.13)

where we have also taken (73.8) into consideration. This agrees with (73.9), on account of Theorem 170, by which we have proved the first assertion of Theorem 172. Likewise, it follows from (73.9) that (73.6) for a regular

D can have at most the one solution (73.10) in the quotient ring of tV. We suppose now that D is regular and want to show that (73.10) is then a solution of (73.12). After substituting (73.10) into the left-hand side of (73.12) we obtain n

D-1 E a;D, t=1

This, because of (73.8), is equal to n

n

D-1 I=lk=1 E E aitckAk, , so, on account of Theorem 170, it is equal to c; by which the assertion is proved. Consequently Theorem 172 is also proved, since the last assertion is trivial. NOTE. Only that part of Theorem 172 is due to CRAMER in which .R is

a field. For this case it follows from the theorem that, if D # 0, (73.6) has the unique solution (73.10) (in

).

EXERCISE. If we introduce the notations (x) = (x,, . . ., xn), (c) _ (c,...., cn), which are to be considered as one-rowed matrices, then the linear equation system (73.6) may be expressed by the unique equation (au) (x)' = (c)',

(73.14)

where (x)', (c)' denote the transposes. Prove Theorem 172 proceeding from (73.14) by the help of matrix calculus.

CHARACTERISTIC POLYNOMIALS

293

§ 74. Characteristic Polynomials Let J' be a given ring with unity element. (Later we also require commutativity.)

By way of introduction we give the following definition: an element a of an extension ring of ' is called algebraic over 2, if there is a nonconstant principal polynomial f(x) in R[x] with

f(a) = 0,

(74.1)

i.e., if a is a root of the algebraic equation f(x) = 0. The minimum degree of these polynomials f(x) is then called the degree of a (over R). Obviously all the elements of J' are algebraic of degree 1 over R. In the following we shall employ the full matrix ring R[x],, of rank n2 over the polynomial ring Ji [x]. The elements of J [x]n are the matrices (fjj(x))

(fii(x) E R [x])

of type n2. Accordingly they may be written as polynomials in x with coefficients from the full matrix i.e., as elements of the polynomial ring -*n[x]. It is then immaterial whether we make calculations with these

as elements of R[x],, or of Rn[x], because these rings are identical. For a matrix

A=(a,1)(EL')

(74.2)

we call the following matrix formed in 2[x]. (=

x - all

- al;, ... - a21 x - a22...

- uln - a2n

= Ex - A

-anl

(74.3)

-a,_...x-am

the characteristic matrix of the matrix A, where E is the unity matrix. From now on let 2 be commutative (further on with a unity element). We then call the determinant

f (x) = I Ex - A I (E JP[x])

(74.4)

the characteristic polynomial of the matrix A. Here f(x) is a principal polynomial of degree n. Whenever necessary, we assume that f(x) is replaced

by the scalar matrix Ef(x) (= f(x)E), i.e., we consider f(x) as an element of.A[x],,. Indeed, it is compatiblewith the fact that the customary embedding

of JP[x] in R[x],, consists in the replacement of every scalar matrix E9(x) by 9(x) (9(x) E -V[x]).

OPERATOR STRUCTURES

294

TimoREm 173 (CAYLEY-HAMILTON theorem). Every matrix A in the full

matrix ring *n over a commutative ring OP with unity element is algebraic over R of degree at most n, and is a zero of its characteristic polynomial

f(x)

Because of (74.4) [and (72.31)]

adj (Ex - A) (Ex - A) = Ef(x).

(74.5)

n-1

The first factor is a polynomial

i=o

A;xi in On[x] with A0, ..., A,_1 E A.

Then according to (74.5) n-l

t=0

(A,x`+i - A,Ax)= Ef(x)

The left-hand side vanishes for x = A, whence Theorem 173 follows. The transforms of A are likewise zeros of f(x), because of the following Tm;oit 1 174. Under transformations of a matrix its characteristic polynomial remains invariant.

Let SAS-1 be a transform of A, where S is an arbitrary invertible matrix in ' n. Then

S(Ex - A)S-' = Ex - SAS-'.

(74.6)

From this, after passing to the determinants, and from the multiplication theorem for determinants (Theorem 163) we obtain

ISIIEx-AIIS-'I=IEx-SAS-'I But since I S-1 I = I S I` holds, so because of (74.4) Theorem 174 follows. EXAMPLE. The characteristic polynomial of

b d)

over a commutative ring with

unity element 1 is:

x-a -b = x2 - (a + d) x + (ad - bc). - cx-d

Therefore

1ab)x

cd

-(a + d)

ab c

d'

+ (ad - be)

0 01

This can also be easily obtained by direct computation.

=0.

NORMS AND TRACES

295

§ 75. Norms and Traces

Let an algebra A be given of rank n over a ring,59 with unity element. (Later J9 will be assumed to be commutative.) First, we give a fixed basis w1, . . ., w,, in A. For arbitrary a (E A) It

wia = E aJwJ

(i = 1, ..., n; au E 2)

(75.1)

J=1

with uniquely determined a,J. Accordingly

a -+ (al)

(75.2)

yields a mapping of A into the full matrix ring,*,,. We show that this mapping is an *-homomorphism of A, which we call the regular representation of the algebra A (in,*j (for the basis co, ..., (0j. Since a linear mapping of A+ into itself is given by the matrix (a,1), and

w,(a + fi) = wia + 00, w,afi = (w,a)f (4 E A), the mapping (75.2) is, first of all, an "ordinary" homomorphism. Since A is an algebra, we obtain from (75.1) It

(a E 2),

w;(ca) = cwia = E ca,JwJ j=1

therefore the image of ca is the element c(a,J). Consequently (75.2) is in fact an %R-homomorphism.

Now this representation (75.2) is evidently an isomorphism if, and only if, for a 0 one has (a,J) ; 0. Because of (75.1) and the linear independence of wt, . . ., co, this means that not all of the w,a (i = 1, . . ., n) vanish. But since w1, . . ., w are generators, this means that a is not a right annihilator of A. This proves the following THEOREM 175. The regular representations of an algebra (of finite rank) are true if, and only if, it has no right annihilator except 0.

Retaining the previous suppositions, the fundamental ring R will be commutative from now on. Then we call the characteristic polynomial

f(x) = I lf;;x - a,J 1

(75.3)

of the matrix (a,J) in (75.2) the characteristic polynomial of the algebra element at. From the corollary of Theorem 161 and from Theorem 174 it follows that the polynomialf(x) (for fixed n) does not depend on the particular choice of the basis co, . . ., co, So in this sense the respective coefficients of f(x) are also invariant. Now, because of (75.3),

f(x)=x"-(a11+a22+...+a,,,,)x"-1+..+ + (- 1)" I

at. I

.

(75.4)

OPERATOR STRUCTURES

296

Above all the two explicit coefficients of f(x) are important (viz., the constant term and the coefficient of x`). With some alteration of sign we denote them by

N(a)=Ialll, T(a)=all+a12+...+a"m

(75.5)

and call these respectively the regular norm and trace of the algebra element a. These are likewise independent of the particular choice of the basis (01, ..., w (for fixed n). (In the following we shall tacitly relate N(a), T(a) to a fixed n.) Ttioa t 176. For the elements of an algebra A over a commutative ring with unity element

N((x#) = N(a)N(A

T((x + f3) = T(a) + T(J) (a, P E A) .

(75.6)

For the proof we assume P -* (ba;), just as in (75.2). Because of the homomorphy property of (75.2)

a + fl -* (a,) + (bj), aft -+ (aj) (bat). Hence on account of (75.5) and the multiplication theorem for determinants Theorem 176 follows. EXAMPLE. We define the regular representation of Q. relative to the basis [cf. (69.6)]

E11, ..., Elm.

The image of a matrix A =

E21, ..., E2n; .

,

En1, ..., E,,,,.

(E 12.) must lie in 'Q,., consequently it must be a

matrix Al of type n2 X nz. Since n

Eu k,1=1

n

n

ak1Ek1 = Eakak,Ei, = F, k.1=1

(i,J = 1, ..., n)

1=1

we see that Al =

C where n matrices A are joined diagonally and the other elements of Al vanish. For a commutative ring /R we obtain from this, for the characteristic polynomial, the norm and trace of (a;,) in this regular representation respectively.

IadI", n(all+asa+...+amm),

N0(A)=Iall I, T0(A)=all+a21+...+a,, (75.7) are called, respectively, the reduced norm and trace of the matrix A = (a ). Then for the regular norm and trace N(A) = N0(A)',

hold.

T(A) = nT0(A)

(75.8)

COMPLEX RINGS

297

§ 76. Complex Rings

It is simple to define a double algebra R over a ring R with unity element, if we take two basis elements 1, i, of which 1 is the unity element

of R and i is such that i2 = - 1.

(76.1)

That this definition is reasonable can either be seen directly or by application of the third case of Theorem 156 with the cyclic group of order four.

After the usual embedding of JP in R the basis representation of the elements of R becomes:

a = a + bi

(a E R; a, b E ,)

(76.2)

Furthermore, because of (76.1), the formulae for the sum and product of two elements of R are :

(a+bi)+(c+di)=(a+c)+(b+d)i,

(76.3)

(a + bi) (c + di) = (ac - bd) + (ad + bc)i.

(76.4)

We call R the complex ring over 2. Since, according to (76.2), is = -b + ai, so

a-*

b a)

(76.5)

is the regular representation of R. Now let,3P, and so also R, be commutative. Then because of (76.5)

x2-2ax+a2+b2 is the characteristic polynomial of a, consequently N(a) = a2 + b2, T(a) = 2a.

(76.6)

We define the conjugate of a by

a' = a - bi.

(76.7)

With this notation (76.6) is also representable as N((x) = aa',

T(a) = a + a'.

Evidently a -9. a' is an automorphism of R. If the complex ring is a field, then we call it a complex field.

(76.8)

OPERATOR STRUCTURES

298

THEOREM 177. The complex ring R over a commutative ring J` with unity

element is zero-divisor-free if, and only if, V is zero-divisor-free and the equation

a2 + b2 = 0

(a, b E

)

(76.9)

has only the trivial solution a = b = 0. Further this ring R is a field if, and only if, for (76.9) the above condition is fulfilled and J is a field. That 2 should be zero-divisor-free is evidently a necessary condition, so that we may assume it from the very first. If R is zero-divisor-free, then

aa' = 0 (a E R) is possible only for a = 0, consequently (76.9) is then, because of (76.61), (76.8), only trivially solvable. Conversely, for every a (E R, 0) the product aa' is an element of V, different from 0. So if, for a /9 E R, the equation aq = 0 holds, then a a fi =

= 0, so that /9 = 0, i.e., R is zero-divisor-free. This proves the first part of the theorem.

In the proof of the second part we may assume that (76.9) is only tri-

vially solvable. If R is a field, then every equation a = b (a, b E '2; a 0) has a solution l: in R. However, this must lie in 2, therefore 9 is also a field. Conversely, every equation z (a, J3 E R; a 0) has, since a'a 0,

a'a E,,*, the solution $ = (a'a)-1 a'j9 in R, therefore R is a field. Thus we have proved the theorem. EXAMPLE. In the field of real numbers (76.9) is only trivially solvable. Consequently over every real number field rk the associated complex field exists; this is isomorphic with a complex number field. But conversely, not every complex number field is a complex field in the above sense. In particular, the field of complex numbers a + b E .moo) is not a complex field, as is obvious. EXERCISE. If rk is a skew field, then the complex ring R over ' is zero-divisorfree if, and only if, the equation x2 + I = 0 is insolvable in' and R is then a skew field.

§ 77. The Quaternion Group The group defined by the equations a2 = /92 = y, y2 = a, flap-1 = ay

(a the unity element)

(77.1)

is called the quaternion group. We prove that it is of order 8 and denote it by Q8.

It is evident that any element of Q8 may be written in the form a`j'yk

(i,.l, k = 0, 1).

(77.2)

Hence O(Q8) <- 8. In order to prove the statement, it is then sufficient, because of Theorem 101, to give a group of order at least eight, which

THE QUATERNION GROUP

299

belongs to the equations (77.1). Such a group is the subgroup of the full permutation group 958 generated by the elements a' = (1 2 3 4) (5 6 7 8). (1 5 3 7) (2 8 4 6), y' _ (1 3) (2 4) (5 7) (6 8),

(77.3)

as can be proved by easy computation. (As a secondary result we see that the permutations (77.3) and even a' and i' generate the quaternion group.) From what has been proved it follows also that in (77.2) all the different elements of Q8 were given. We see that y is a central element of order two of Q8. EXERCISE. In Q8 the rule at glyk

at tt'Rt+!'yktk}llholds

for the product of two elements. The subgroup (y) of Q8 is at the same time the centre, the commutator group and the Frattini subgroup of Q8. All the subgroups of Q8 are normal. § 78. Quaternion Rings

We give a ring JP with unity element, which will later be assumed to be commutative. To this and to the quaternion group Q8 we wish to apply the third case of Theorem 156. The s-double algebra, to be obtained in such a manner, is called the quaternion ring over tR and we denote it by R.

If,*is commutative and R a skew field, then we call this the quaternion field over,2. This case will occur, as we shall see, in particular for.5p=Y0; the quaternion field over 9'0 is called the ordinary quaternion field and we denote it by Q. If later on we speak of the quaternion field, without indicating the fundamental domain, we mean Q. First of all we have to prove the existence of R in general. At the same time we shall give R explicitly. With this end in view we write down fully the elements of Q8 (according to § 77): E, a, fl, afl, Y, ocy, fly, 7-fly.

(78.1)

For the first four elements [according to (77.1)] we have the multiplication table e E

i

e

a

f

a

a

fi

afB

afl

fly

Y aY

a

Y afy p

y

(78.2)

OPERATOR STRUCTURES

300

Since y is a central element of order two of Q8, and on account of (78.1), the third case of Theorem 156 may in fact be applied, by which the existence of R is confirmed. We could denote the basis elements of R (according to Theorem 156) by e, a, i, afl; however, we wish to write them as 1, i, j, k.

For these basis elements we obtain from (78.1), if we replace y by -1, the multiplication table i

i

-1

j j k

-j

j

j

-k

-1

i

k

k

i

1

1

1

j -i

k k (78.3)

-1

Thus the unity element of R is 1. (Later on we shall dispense with the above notations e, a, 9, y.) Having carried out the usual embedding of in R, we obtain for the elements of R the unique basis representation in the form

a=a+bi+cj+dk

(aER; a,b,c,dE59).

(78.4)

Thus and by (78.3) the quaternion ring R over R is completely defined. The elements (78.4) are usually called quaternions. The basis elements 1, i, j, k are also called the qualernion units. We now restrict ourselves to a commutative R. Then R forms an Ralgebra. We call [cf. (76.7) ]

a'=a - bi - cj - dk

(78.5)

the conjugate of the quaternion a. From (78.4) and (78.5) it follows that

a + a' = 2a.

(78.6)

Further we obtain, from (78.3),

as'=a'a=a2-(bi+cj+dk)2=a2+b2+c2+ d2.

(78.7)

If (78.6) is multiplied by a and then (78.7) inserted, we obtain

a2-2aa+(a2+b2+c2+d2)=0 (a = a + bi + cj + dk).

(78.8)

According to this all the elements of R are algebraically of degree at most

two over A.

301

QUATERNION RINGS

Considering (78.6), (78.7), (78.8) we put

t(a)=a+a'=2a

n((x)=aa'=a2+b2+G +d2,

(a = a + bi + cj + dk)

(78.9)

and call these respectively the reduced norm and reduced trace of the quaternion a. With this notation (78.8) assumes the form

a2 - t(a) a + n(a) = 0.

(78.10)

n(a') = n((x), t(a') = t(a).

(78.11)

Furthermore We show that a -> cc' is an anti-automorphism of R. Since R is obviously mapped one-to-one onto itself by a -> a', we have only to prove that

(a#)' = #'a', (a + fl)' = a' + ft'

(a, fl E R).

(78.12)

The second of these is trivial. The first need only be proved for the cases where a, P each mean one of the elements i, j, k. Since, according to (78.5),

i' = -i, j' _ -j, k' = -k, we immediately see that (78.121) holds for the rest of the cases of (78.3).

Consequently the assertion is proved. The formulae (cf. Theorem 176) n(afl) = n(a)n(j9),

t(a + 9) = t(a) + t(9)

(a, fl E R)

(78.13)

also hold. From (78.91) and (78.121) it follows that

n(afl) = afl(aj' = agfl'a' = an(f)a' = aa'n(8) = n(a)n(fl), i.e., (78.131); (78.132) follows at once from (78.92),(78.122). THEOREM 178. The quaternion ring R over a commutative ring ,- with unity element is zero-divisor-free if, and only if, * is also zero-divisor-free and the equation

a2+b2+c2+d2=0

(a, b, c,dE 2)

(78.14)

has only the trivial solution a = b = c = d = 0. Further this ring R is a skew field if, and only if, for (78.14) the former condition is fulfilled and is a field.

OPERATOR STRUCTURES

302

The wording of this theorem is similar to that of Theorem 177, and so is the proof, so that it is unnecessary to repeat it. (Though R is now noncommutative, this causes no change in the proof.) Since (78.14), in the field .5"0, is only trivially solvable, the existence of the ordinary quaternion field Q follows.

Since i2 = -1, it is evident that the special quaternions a + bi constitute a subring of R, which is a complex ring over JP. This subring according to Theorem 177 is a field, if R is a quaternion field. THEOREM 179. Let F be the quaternion field with the basis 1, i, j, k over a field Y and G the subfield of F, which consists of the special quaternions a + bi (a, b E . "). Then the special matrices (78.15)

E G)

in the full matrix ring G. of rank four over G form a quaternion field Fl isomorphic with F, and

-a+ bi c+dil I

c + di a - biJ

(a, b, c,dEY) (78.16)

is a suitable isomorphism. Put f 0011'

i*=f 0

For these four matrices a

0I,

.l*=

k*=(ii

0)

multiplication

table similar to (78.3) holds,

as can be easily computed. They may thus be applied as a basis of a quater-

nion ring over Y. Since, furthermore, by their help the right-hand side

of (78.16) may be expressed in the form al* + bi* + cj* + dk*, so the similarity to the left-hand side of (78.16) renders the theorem evident. NOTE 1. The quaternion ring R over a ring5l is evidently commutative if, and only if, 59 is commutative and the i. j, k are interchangeable with one another. A glance at (78.3) shows that this last condition is equivalent

to i= -i, j= -j, k= -k, i.e., to 2i=2j=2k= 0. This is again equivalent to the fact that the characteristic of is equal to 2. It should be noted, too, that then, by (78.3), the quaternion units 1, i, j, k constitute

a four-group. Consequently we have obtained the statement that R is commutative if, and only if, JI is a commutative ring of characteristic 2, and then R is the JP-group ring of the four-group. NOTE 2. As regards Theorem 156 we have already noted, immediately after its proof, that we may derive important structures from it by application to certain special semigroups. This we have since done and have used

303

QUATERNION RINGS

the following semigroups: the free commutative semigroup with unity element,

the semigroup generated by the matrix units or alternating units respec-

tively, the cyclic group of order four and the quaternion group. It is noteworthy that the rings so obtained from a common source are otherwise connected with one another. This is why it is desirable to find further interesting applications of Theorem 156. EXAMPLE 1. From (78.4) it follows that

ia=-b+ai-dj+ck, ja=-c+di+aj-bk, ka=-d-ci+bj+ak. Accordingly, the matrix, which consists of the 16 coefficients on the right-hand side of (78.4) and these last three equations, is the image of a in the regular representation of the quaternion ring R over rk. For a commutative Q the regular norm and trace of a are as follows: N(a) =

d

a

b

c

-b -c

a

-d

c

d

a

-b

b

a

-d -c

,

T((x)=4a.

Hence we obtain without great trouble N(a) = (a2 + b2 + c2 + d2)2. Because of (78.9), N(a) and T((x) are expressed by the reduced norm and trace of a as follows: N(a) = n(a)2,

T((x) = 2t(a).

EXAMPLE 2. In the group Q* of the natural quaternion field Q we determine the ele-

ments a = a + bi + cj + dk of order 2 and 4. If o((x) = 2 then a 96 1, e = 1. Hence and from (78.8) it follows that

2aa=1+a2+b2+c2+d2. Since then as E . 'o, we must have a ; 0, b = c = d = 0, a = a = -1. Thus -1

is the only element of order two. Consequently o(a) = 4 is equivalent to ag = -1. By (78.8) we may then write:

2aa=-I+a2 +b2+c2+d2. If a 96 0, then b = c = d = 0, and hence a2 _ -1. Since this is impossible, we have a = 0, consequently

a=bi+cj+dk

(b2+c2+d2=1; b,c,dE.` p).

Conversely a2 _ - I follows then from (78.8), hence by these conditions all the a with o(a) = 4 are characterized. Of course there are many of these. Give them explicitly.

CHAPTER IV

DIVISIBILITY IN RINGS After considering various structures in the two preceding chapters in general, we now proceed to particular studies. As the ring Y, and contained in the latter, play primary roles among the subsemiring structures and, as operator domains, may affect every structure, first of all it is necessary to make a more detailed survey of their properties. The examination of the multiplicative semigroupdiscussions 7x will not be made in isolation but in conjunction with general on semigroups of certain zero-divisor-free rings with unity element. It is possible to relate some fundamental notions and simple statements to arbitrary rings and even to semigroups, but, for the sake of brevity, this will be omitted. Moreover, some problems on rings without unity element will be discussed.

§ 79. Factor Decompositions and Divisibility In this paragraph R denotes a ring with unity element. Small Greek letters (also provided with indices) always denote elements of R, while the unity element is denoted by 1. We also require that R be zero-divisor-free (and later commutative). We set ourselves the task of examining the semigroup Rx of R, but we shall acquire sufficient knowledge of Rx only in certain special cases to

be stated later in detail. In other words, our task is to make a survey of all the factor decompositions fl = Nl...

k

(79.1)

of the elements of R. We often disregard the trivial case k = 1. The factor decompositions (in the case k = 2)

#=ao

(79.2)

into two factors are of fundamental importance. For, if in (79.2) a or 0, are again decomposed into a product of two factors and this process is repeated, then all the factor decompositions (79.1) occur in this way. The connection between factor decompositions and divisibility is evident. 304

FACTOR DECOMPOSITIONS AND DIVISIBILITY

305

For given a, 9 a factor decomposition (79.2) exists if, and only if, a I , i.e., a is a left divisor of 9. Therefore the study of divisibility relations a I is practically the first step in the exploration of factor decompositions. [By the way we note that if, in (79.2), a # 0, because R is zero-divisor-free e is uniquely determined by a and P.]

We leave to the reader the proof of the following almost trivial, but

at the same time, fundamental divisibility properties : a 10,

(79.3)

0 I a ra a = 0,

(79.4)

I a,

(79.5)

a I a,

(79.6)

ally, f1IYY,

(79.7)

I l fi

(79.8)

1

a I fly,

aIfYaIYfl

(700),

alfl,Y=IIfl+Y,

(79.9) (79.10)

aal,...,In alale]L+.+(79.11) For a commutative R we may add :

aI t, Yl

(79.12)

(n>_ 1),

(79.13)

which we wish to note already here. We shall apply these simple rules later without comment. In particular, (79.6) and (79.7) express the reflexivity and transitivity of the divisibility relation. Because of the latter, from a chain of divisibility

a1 1 a21 ...Ia", which means namely the collection of the relations ai I a;+1 (i = 1, ..., n - 1), all the relations a; I a1 (1 < i < j < n) follow. We note with respect to (79.11) that the expression alLOj + ... + M"e", occurring there, runs through all the elements of the right ideal (al, ..., a"). Therefore in questions of divisibility the notion of an ideal also plays a role (see § 80).

Since by (79.3), (79.4) the divisibility properties of the element 0 are entirely explained, in what follows we shall often tacitly disregard the element 0.

DIVISIBILITY IN RINGS

306

We have already stated that the divisibility relation is reflexive and transitive. Now it is useful to derive an equivalence relation, which we call associatedness (more precisely left associatedness), denoted by "ass" and defined by (79.14)

ass

If a ass fl, then we call the elements a, 9 left-associated. It is evident that left associatednes is in fact an equivalence relation. (Accordingly, the rightassociated elements are to be interpreted as a dual notion.) The definition (79.14) may be formulated as follows : a, j9 are left-associated if, and only if, each of them is a left divisor of the other. We prove the following important rule: GE

I P a a' I fi'

(a ass a', fl ass j9').

(79.15)

I.e.: if a, a' and 9, fl' are left associated, then one of the relations a a' I (f' implies the other.

It is sufficient to prove from (79.15) the assertion "=". This is expressed as follows : a ass a',

9assi',

cI fl=a'

ft'.

From the left-hand side it follows that a' I a, a I f3, ,B I P'. Hence the right-

hand side follows because of the transitivity. Consequently (79.15) is proved.

The equivalence classes of R connected with the left associatedness are called the classes of left-associated elements (of R). Here this classification is denoted by C and the class containing a by C(a). The class

C(0) consists only of the element 0, therefore we also denote this class by 0. It is often disregarded. The classes of right-associated elements are to be considered as a dual notion. In particular the class 0 is self-dual. The class C (1) represented by the unity element 1 is of great importance. Its elements are said to be the units of R. This term is justified by the fact that according to the following theorem the units constitute a self-dual notion. (If this were not so they would have to be called something like left units.) THEOREM 180. In a zero-divisor-free ring R with unity element 1 the units

are identical with the left divisors of 1 and they form a subgroup with the same unity element 1, consisting of the invertible elements of R; further, a product of elements of R is a unit only if all the factors are units. Since P is a unit if, and only if, e ass 1, i.e., if a 1, 11 o, and since the latter of these is always satisfied, the first assertion of the theorem is true.

FACTOR DECOMPOSITIONS AND DIVISIBILITY

307

Now we consider a unit 9. Then o 11, i.e., there exists a further element a with Qa = 1. Hence it follows on account of Theorem 27 that 9 is invertible. If this is so, then conversely oo-I = 1, thus P 11, therefore then a is a unit. Since, accordingly, the units are identical with the invertible elements

and these on account of Theorem 34 constitute a group, we have proved the second assertion of Theorem 180. The last assertion needs to be proved only for the case of two factors.

We decompose a unit P into a product: P = ajI. Then a I LO, 9 11, thus a 11. Accordingly a is a unit. Since a, P are units, it follows from the group property of the units that j9 is a unit, too. Consequently we have proved the whole of Theorem 180.

The group formed by the units is called the group of units of R and we denote it by U. Its elements, the units, are thus the invertible elements of R, i.e., the left divisors of 1 and, at the same time, the right divisors of 1. It follows also that the units and only they are left divisors (thus also right divisors) of all elements of R. THEOREM 181. The classes of left-associated elements of a zero-divisor-free

ring R with unity element are the left cosets with respect to the group of units U.

In the proof we need only consider those elements different from 0. If a ass jI, then a I f, ;3 I a. Hence follows the existence of elements Lo, a

with ae = f, 9a = a. We obtain from these

cc

cr = a, thus Loa = 1.

Because of Theorem 180 it follows that o is a unit, consequently a, fi belong to a left class with respect to U. Conversely, we assume that a, f lie in a left class with respect to U, i.e.,

there is a unit P with oae = P. Then a = fl e', thus a ,B, ,9 I a, a ass Consequently Theorem 181 has been proved. Now we can express rule (79.15) by saying that in a left-hand divisibility relation a I fl all depends only upon the classes C (a), C (f). This involves

the important principle that, with regard to left-hand divisibility of the element, the left associated elements may be regarded as equal. We shall apply this principle, usually without noting it explicitly, from time to time.

The above principle may be applied so that we separate in each class of left-associated elements a certain element which we call the normed elements, and after this norming we restrict ourselves, in questions relative to left-hand side divisibility, to the normed elements. The set itself of the normed elements is by virtue of Theorem 181 identical with an (arbitrary) system of left representatives with respect to the group of units U. In different rings norming is carried out in very different ways. We shall give examples later, but we want to quote here two general observations. First, we shall often carry out the norming only in certain classes, which we then call the normed classes; this will always be done so that the omission of the other classes will mean no essential loss. Second, we shall always

308

DIVISIBILITY IN RINGS

arrange the norming so that the product of two normed elements will again be a normed element, i.e., the normed elements constitute a subsemigroup of R; only the fulfilment of this requirement can guarantee the usefulness of norming.

It should be emphasized that according to Theorem 181 all the leftassociated elements of a are given by the products oco (e E U).

Among the left divisors of an element a occur all the units and all the left-associated elements of a. The other left divisors of a are called its proper left divisors.

If 61 al, ..., an ,

(79.16)

we call S a common left divisor of the elements al, . . ., a,,. If, apart from the

units, there is no such 6, then we say that the elements al, ..., an have no common left divisor.

If in a set of elements of R there are elements such that they are left divisible by all the elements of the set, then these aforesaid elements are pairwise left divisible with respect to each other, thus they are left-associated.

This observation is to be taken into consideration in the following definition :

If, among all the S which satisfy the condition (79.16), one 60 occurs such that (79.17)

bl60

for all the existing b, then we call this uniquely (apart from left-associated elements) determined b0 the greatest common left-divisor of the elements al, ..., an. We denote it by (al, ..., an)1 . (See Example 1.) The following two definitions are similar to the previous two. If 11,...,7-nI

Iu

(79.18)

we call p a common left multiple of the elements aI,..., an. p = 0 is always such an element. (See Example 7.) If, among all the p which satisfy the condition (79.18), one µo occurs such that µo I g

(79.19)

for all the above µ, then we call this uniquely (apart from associated elements)

determined po the least common left multiple of the elements We denote it by [al, . . ., an]l. If, in R, finitely many elements always have a greatest common left

divisor or a least common left multiple then we call R a ring with greatest common left divisor or a ring with least common left multiple. respectively.

309

FACTOR DECOMPOSITIONS AND DIVISIBILITY

We prove the two rules

e[aI, ...,

[PaI, ...,

(e # 0)

(79.20)

in the case where

S = (al, ..., an)1, M = loci, ..., an]i,

S1 = (Pal, ..., Pan) i or /A1 = [Pal, ..., el"I

exist; we prove furthermore that from the existence of bl that of 6 follows, also that the existence of one of ,u, µI implies the existence of the other. (Strangely enough, on the other hand, S may exist, according to Example 1, even in the commutative case without the existence of 61, cf. WEINERT (1961b).)

In the proof, rule (79.9) is continually applied. First of all we assume the existence of 61. Evidently e I bl, consequently there is a

(E R) with

(]=o.

Since, moreover, bl 1 Pal, ..., 9a,,, it follows that

Eial,.. ,an. On the other hand, we consider an arbitrary ' (E R) with ' I at, ..., an.

It follows that e!' I ell, ..., ean, then eE' 1 bl. Consequently Pse' then

'

.

According to this b exists and

$= b. Accordingly it also follows that bl = eb, i.e., (79.201). Then we assume the existence of It. Because al, ..., an I µ

eal, ...,

ea

Conversely, consider an 17 (E R) with 921, ..., Pan I r1

Then e 1 17, consequently 17 = Prl' for some rj (E R), then al, ..., cc,, follows, thus 1A I

q,.

Hence eµ j eal', i.e., eµ I i. Consequently pI exists and iuI = PPThis implies the validity of (79.20.,).

310

DIVISIBILITY IN RINGS

Finally we assume the existence of µl. We have only to prove that from this the existence of ,u follows. Because gal, ..., ga. I ,ul, e I µl, consequently there is a C (E R) with

It follows that gal, ..., ca,,

thus

Conversely, consider a S' (E R) with al,

. .

., a C'. Then gal, ..., ga, I gc', I

consequently µl I gg'. It follows that gg

i.e.,

CC'. Hence p exists (and is equal to 0. Consequently all the assertions concerning (79.20) are proved. A factor decomposition

a=al...ak

of a ( 0) is called proper if k >_ 2 and no factor is a unit. An element of R different from 0 and the units is called irreducible, if it has no proper factor decompositions. Otherwise we call it reducible. It is

evident that the left-associated elements of an irreducible or reducible element are again irreducible or reducible, respectively. We call

a=gwl...ajk

(k>0)

(79.21)

an irreducible factor decomposition (of a), if g is a unit and cot, ..., cok are irreducible elements. [In particular, for k = 0, (79.21) becomes a = g.] If in R all the elements different from 0 have an irreducible factor decomposition, then R is said to be a ring with irreducible factor decomposition. It means, in other words, that the regular semigroup R* (which, because R is zero-divisor-free, consists of the elements different from 0) is generated by the units and the irreducible elements. (See Example 8.) In the following, R will also be considered to be commutative, and by consequence an integral domain. All the above concepts are then self-dual, therefore the distinctions "left", "right" are omitted. The meaning of e.g., the associated elements, the common divisor or multiple, the greatest common divisor and the least common multiple is clear. The last two are denoted, of

course, by ()* and [ ]*. The rule (al, ..., an)* 1#11..., N,,]* = Y (a1N1 = a9 j9.. _ ... = anfln = Y 0 0), (79.22)

311

FACTOR DECOMPOSITIONS AND DIVISIBILITY

always holds, if the second factor of the left-hand side exists, whence the existence of the first factor follows. (On the other hand the first factor, according to Example 1, may exist without the existence of the second.) Let

We have to prove that

µ=

*

79 23

.

6 = (a1, ..., a)*

(79 . 24)

6Y=y.

(79.25)

exists and From

u and y = MA it follows that

(i= 1,...,n).

VI mitt

(79.26)

On the other hand, consider a y' of R with Y, I mitt

).

Q

It follows that y' j9; I a;9;µ, i.e.,

Y'fIIYp

(i= 1,...,n).

But since, according to (79.23) and what was proved in (79.202), [y'31,

...,

also exists and this is equal to y'µ, it follows that y'p I yµ, i.e., y' I y. Because of (79.26) this means that

exists and

(alt,

..., amµ)* = )'

-

According to what was proved in (79.201), (79.24) also exists and (79.25) then follows. Consequently the assertion is proved. We now note that the existence of the second factor in (79.22) follows

by a similar proof if (not only the first factor but more generally) (pal, .

. . ,

exists for all O E R.

THEOREM 182. An integral domain with greatest common divisor is one with

least common multiple and conversely.

For the proof, we consider an integral domain R and we use in it, for the sake of brevity, the notations (79.23), (79.24). At first we suppose the existence of every u and prove the existence of each 6. In the proof

we may assume that al .. ak # 0 and ak+I = ... = a = 0 (0 <_ k <_ n). In the case k = n the assertion follows from the statement in (79.22) applied to y = al ... a,,. In the cases k = 0 and 0 < k < n we have simply 6 = 0 or 3 = (at, ..., ak)*, respectively. For the second time we 11/a R.-A.

312

DIVISIBILITY IN RINGS

suppose the existence of every S and prove the existence of each µ. Accord-

ing to the foregoing note, this can be inferred in a similar way if flI ... fl,, 0, while in the opposite case y = 0 holds trivially. Thus the theorem is proved. An element n, different from 0 and the units, of an integral domain R is called prime (or a prime element) if

nIa9=nIa or aII9,

(79.27)

i.e., if in every product aj3 divisible by n at least one factor is divisible by n.

Of course then similar results are valid for the products with arbitrarily many factors. Evidently, the associated elements of a prime element are again prime. We prove that every prime element is necessarily irreducible. (However, see Example 2.) If the prime element n were reducible, then it could be decomposed into a product

n=all,

where neither a nor 9 is a unit. Hence obviously n X a, P. But since on the other hand n I afl, we have a contradiction, which proves the assertion. If in an irreducible factor decomposition (79.21) the cot, . . ., co, are prime, then we call it a prime (factor) decomposition. We now consider an integral domain R with irreducible factor decomposition. Let (k >_ 0) (79.28) a= eW1...COk

be the irreducible factor decomposition of an arbitrary element a (# 0), where cot, ..., OIk are irreducible and io is a unit. On account of commutativ-

ity the Cot, ..., OIk may now be replaced by arbitrary associated elements,

where we also replace o by another suitable unit, the order of the factors being unimportant. If in R for every a ( 0) the irreducible factors on the right-hand side of (79.28) are, apart from order and associated elements, uniquely determined, then we say in short that in R the irreducible factor decomposition is unique. (See Example 3.) THEOREM 183. In an integral domain R with irreducible factor decomposition

this is unique if, and only if, all the irreducible elements in R are prime. First, assume the required uniqueness. We have to show that every irreducible element n of R is prime. For this purpose we consider two elements

a, 9 with

nIa9.

This means that for some further element y

ny=aj9.

FACTOR DECOMPOS*TIONS AND DIVISIBILITY

313

Now we replace each of a, j3, y by an irreducible factor decomposition. Because of the supposed uniqueness n must then occur among the irreducible factors of the right-hand side, whence it follows that n 1 a or n f. This means that n is prime. Conversely, now let all the irreducible elements of R be prime. We consider an equation

oni...nk=ow1...wl,

(79.29)

where e, or are units, ni, ..., nk and wi, ..., wI prime elements. We have to prove that these, apart from order and associated elements, are identical, whence of course it follows that k = 1. If k = 0 obviously 1= 0, therefore the assertion is true for this case.

If k > 0 we assume that the assertion is already proved for k - 1 instead of k. From (79.29) it follows that the right-hand side is divisible by n1. Since ni is prime, this means that for some w, we have ni I wI. As the order of the factors is unimportant, we may assume that ni I wl. Since, on

the other hand, wi is prime and consequently irreducible, it follows that ni , co, are associated. Therefore we may assume that nI = (01.

(79.30)

From (79.29) on account of (79.30) we obtain by cancellation

9n2 ... nk = owl ... wI .

Hence because of the induction assumption n2, ..., nk and w2, ..., (Oil apart from order and associated elements, are the same. Because of (79.30) the same then holds for ni, . . ., nk and wi, . . ., wl. Consequently the proof

of Theorem 183 is complete. According to this, an integral domain with unique irreducible factor decomposition is simply called a ring with prime decomposition. This termi-

nology indicates satisfactorily that an integral domain is involved, since we have defined prime elements only in such rings. There is no need to call particular attention to the uniqueness of the prime decompositions, since it follows from Theorem 183. In a ring with prime decomposition "irreducible element" and "prime element" are equivalent terms. For historical reasons, the first term is more common in many such rings, while in others the second is more common. The concept should, more correctly, be referred to by the second term in these rings.

We now consider a ring R with prime decomposition. The prime decomposition of an element a (A 0) is then of the form

a = t)nl ... nk

(k >_ 0).

(79.31)

DIVISIBILITY IN RINGS

314

where a is a unit and all ..., nk are, apart from order and associated elements, uniquely determined prime elements. We call these the prime factors of a. The uniqueness is more precise if we single out in every class of associated elements a representative and admit only these prime elements as prime factors. Henceforth we assume this. Then the prime factors of a are, apart from the order, uniquely determined. Simultaneously Q is then uniquely determined in (79.31), therefore we call this o the unit factor of a. The a with the unit factor 1 and the zero element 0 obviously constitute a full system of normed elements and at the same time a subsemigroup of R. Only normings of this kind are customary in rings with prime decomposition. After

norming, "prime element" should mean a "normed prime element", "different prime elements" are then also non-associated. In general we call the powers n' (i >_ 1) of prime elements n prime powers.

(Sometimes we also admit i = 0.) We often assume the prime decomposition (79.31) in the form

a = e4' ... ii

(e1, ..., er ? 1; r ? 0)

,

(79.32)

where the prime elements vI.... , n, are pairwise distinct. These are called the distinct prime factors of a. [To make the distinction more obvious the ni, . , nk in (79.31) may be called "all the prime factors" of a.] We briefly call (79.32) the prime power decomposition of a. The n;' occurring in it are called the prime power factors of a. Often (79.32) is given in the form

a = eni' n2' ...

(el, e2, ... > 0) ,

(79.33)

where all n2, ... mean all the distinct prime elements of R and among the ell e2, ... only finitely many may differ from 0. In passing to another a, only e and the exponents el, e2, ... change. If the associated elements are irrelevant, then we omit the unit factor in (79.31), (79.32), (79.33), as in the case when one is interested only in the divisors of an element or in the greatest common divisor of elements. All the divisors of the a given in (79.33) are evidently the following:

ni, nz' ...

(0 <_ xi _ e;; i = 1, 2, ...) .

(79.34)

As application we prove the following THEOREM 184. Every ring with prime decomposition is one with a greatest common divisor.

In the proof we shall give an operation by the help of which an arbitrary may be computed in the given ring. greatest common divisor (a,, ...,

FACTOR DECOMPOSITIONS AND DIVISIBILITY

315

We may assume that al, ..., an differ from 0, their prime decompositions being as follows :

x, _ 4h

...

2

(i = 1, ..., n).

(79.35)

Any common divisor of a,, ..., an may then be given in the form (79.34), where the Xk have to satisfy the conditions

OSxk;5 elk

(i= 1,...,n; k= 1,2,...).

Consequently (79.34) gives, with the exponents

(k = 1, 2, ...) , (79.36) the greatest common divisor (al, ..., an)*. Thus the theorem is proved. xk = min (elk, ..., enk)

One similarly proves that for xk = max (elk, ..., enk)

(k = 1, 2, ...) ,

(79.37)

the least common multiple [al, . . ., an]* of the elements (79.35) is given by (79.34). A converse of Theorem 184 states that an integral domain with greatest common divisor, in which no infinite chains of proper divisors exist, is a ring with prime decomposition (cf. WEINERT (1961b)). EXAMPLE 1. There are many integral domains "without greatest common divisor" (i.e., integral domains in which not all greatest common divisors exist). A very simple example of this is the ring of complex numbers a + b, J- 5 (a, b E 9), which we denote by d [J--5] and call the Dedekind ring, as Dedekind was the first to use this ring to illustrate the "irregular divisibility relations" in integral domains; these are other "Dedekind rings" which we will not investigate. (We have chosen the

notation .7 [ J-5] because the elements of this ring are the substitution values f(J - 5) of the polynomials f(x) E a [x], as is obvious.) First, we wish to examine the conditions under which, in .7 [.J-5], the divisibility

m+ n

1 a+ b J:: 5

(79.38)

holds; (79.38) holds if, and only if, there are integers x, y with

(m+n 1-5)(x+ya±b,J-5, i.e., if the linear equation system

mx - 5ny = a,

nx + my = b

is solvable for the unknowns x, y. The condition for this, according to Cramer's rule (see the end of Theorem 172), is that

mr + 5n2 I ant + 5bn, bin - an,

(79.39)

which is to be understood, of course, in the ring J. Thus (79.39) is the necessary and sufficient condition for (79.38). On the basis of this criterion we obtain firstly the

proposition that in :7 [,J-5] the group of units U is
DIVISIBILITY IN RINGS

316

table can be easily computed, where in every column all the divisors of the above elements of ,7 [,J-5] are enumerated:

2 + J-5

3

1

2+J-5

3

3(2 -l- ,J-5)

9

l

1

3

3

2+J-5

9

2± J-5

3(2+ J-5)

Accordingly the elements

9,32+ of 0 [.J-5] have only the common divisors

1,3,2+5 Since among these elements (again according to the above table) none is divisible by the others, we have in fact proved that the ring 3 [,,1-5] is without greatest common divisor.

Since (9, 3 (2 + J-5))* does not exist, although (2 -

3)* (= 1) exists,

it follows that for

e=2+ J-S, a,=2- J-5,

a$ = 3

only the left-hand side of (79.20,) exists. For a,=2+J-5,

we have

a2 =3, N,=2-.J-5, #s=3 001 =a2#2 =9

and

0,

(al, a2)* = 1.

On the other hand [#,, /3g]* does not exist. For, if [,8,, #,]* existed, because of (79.22) we should have [YIP /21* = 9;

but this is false, since /, and Y8 also have the common multiple fll.42, which is not divisible by 9.

EXAMPLE 2. In the ring 7 [,1--5] the element 3 is irreducible but not prime, since

3I(2

(2- J15), 3X2+,J-5, 3,Y2-V!-5.

All this can be seen from the above table. EXAMPLE 3. :; [ /-fl is a ring with irreducible factor decomposition (prove it!), yet without prime decomposition. The latter follows immediately from Theorem 183

FACTOR DECOMPOSITIONS AND DIVISIBILITY

317

and Example 2, but may be directly seen from the above table, where, from the equation

32 =(2+.15)(2-.J-5 two different irreducible factor decompositions of the element 9 of 0 [,,,/ _-5] are given.

EXAMPLE 4. Let %, ..., a (#0) be elements of an integral domain with greatest common divisor, thus (because of Theorem 182) with least common multiple. If we put Of IX, = al ... a!_L at+] .

(I = 1,

. .

, n),

then from (79.22) the following special rules are obtained

(aI, ...,

a

[al, ..

(79.40)

EXAMPLE 5. The ring of real numbers a + b , J3 (a, b c .9) contains infinitely many units

- (2 + 3)k

(k = 0, ±1, ...),

which form the group of units. EXAMPLE 6. The complex numbers 2 (a + b

'_-3)(a, b E 0; 2 1 a + b) form a ring

(prove it!) in which the group of units consists of the elements

(k = 0, ..., 5), and so is cyclic of order six. EXAMPLE 7. Let R denote the ring defined by the equations

Ix=.r1 =x,

ly=y1 =y.

(We may perhaps call R the free ring with unity element generated by x, y.) This ring

is non-commutative and zero-divisor-free. Prove the latter! It is obvious that the equation

xX = yY

(X, Y E R)

has only the trivial solution X = Y = 0. This means that the elements x, y have no common left multiple except 0. Consequently [x, y]; = 0. Note that in non-commutative rings the notion of least common left multiple is scarcely ever used; but the greatest

common left divisor is used (see Theorem 188 and § 89), yet there is no analogy with Theorem 182. EXAMPLE 8. The elements other than 0 of a skew field F are units. Therefore the group of units U = F*. (See also § 155, Example 10.)

EXAMPLE 9. In the full matrix ring 9 of rank n1 over ,7, all the units are those matrices (a#) whose determinant l anI is equal to ±1. In this respect it should be noted that the matrix units E., of an arbitrary matrix ring [see (69.7), (69.8)] have nothing to do with the units of the matrix ring. Likewise, the alternating units (71.3) are not

units in the sense of divisibility. On the other hand, the quaternion units of a zerodivisor-free quaternion ring [(78.4)] are always units of the ring in the usual sense of divisibility.

318

DIVISIBILITY IN RINGS

EXAMPLE 10. The regular semigroup R* of a ring R with prime decomposition is

the direct product of an Abelian group E and a free commutative semigroup H with a unity element: R* = E a H. This is true if E is the group of units and H the subsemigroup generated by I and the normed prime elements of R*. EXAMPLE H. The fundamental concepts relative to divisibility in an integral domain R can be expressed very clearly in terms of lattice theory, as will now be shown briefly. Let a denote the class of elements associated with x. With respect to the multiplication

aft =aft these form a semigroup V. (This statement is false if R is not commutative!) Define in V a divisibility relation by I/t. i i i fl This is a semiordering relation in V according to which V forms a lattice (see Theorem 142 and the note after it) if, and only if, R is a ring with greatest common divisor, conse-

quently, according to Theorem 182, with least common multiple. In the given case define

in V the greatest common divisor and the least common multiple by (a1, ..., an)* = (a1, ..., «n)*,

[a1, ...,

[a1, ...,

Then in particular [cf. (60.4)] inf (a, ft) = (a, ft)*, sup (a, ft) _ [a, /t]*, therefore the first and second compositions in V are defined by

aAfi = (a, f)*, av(t = [a. /t]*. Here 1 is the zero element and 0 = 0 the unity element of the lattice V. One would think that only the first or only the second semilattice V1 or V. exists (cf. the end of § 60), yet on account of Theorem 182 this is impossible. (A lattice-theoretical proof of this theorem seems impracticable.) EXAMPLE 12. According to ISBELL'S theorem (see the Problem at the end of § 26),

in a ring with prime decomposition, which is not a field, the number of the nonassociated prime elements cannot be finite if the group of units is finitely generated.

In this respect what is the behaviour (1) of the ring .7, (2) of the ring of rational numbers with a prime denominator p? EXERCISE. From part of the proof of Theorem 183, we take the THEOREM: In an integral domain the irreducible factor decomposition of an element, apart from order and associated elements of the factors, are always uniquely determined if among these factor decomposition a prime decomposition also occurs.

§ 80. Ideals and Divisibility We will see that the divisibility relations just discussed in zero-divisorfree rings with unity element may also be described by the help of the ideals of the ring. This will reduce the studies of divisibility to the study of ideals. It will be proved moreover that the ideals are in many respects more suitable

for research than divisibility so that ideal theory may be regarded as the "natural continuation" of divisibility theory, even for arbitrary rings. This does not detract from the importance of the previous paragraph.

IDEALS AND DIVISIBILITY

319

For the present R will be taken to mean a zero-divisor-free ring with unity element. The notation for the elements will be that at the beginning of the previous paragraph. (Later we allow R to be an arbitrary ring.) We prove that

. (a), R (N)r ,

aIP

(80.1)

according to which a is a left divisor of t4 if, and only if, the principal right

ideal generated by a contains the principal right ideal generated by i4. Since R has a unity element, a right principal ideal (O), of R consists of all the left multiples e6 of o. Now the left side of (80.1) means the solvability of the equation a$ = P, i.e., that (a), 3 8 is satisfied. This is equivalent to the right side of (80.1), which proves (80.1).

Hence all the concepts derivable from left divisibility can in fact be described in the language of ideals. So we prove first of all for left association that

aass Pa(a),= (f),.

The left side means that a

and

(80.2)

a, whence, because of (80.1),

(80.2) follows. Since (1), = R, the special case

= 1 in (80.2) means that a is a unit if, and only if, (a), = R, which is equivalent both to (a)1= R and (a) = R. For a common left divisor it follows immediately from (80.1), that b I a1,

..., a 'C* (Or ? (a1, ...,

(80.3)

exists It also follows that the greatest common left divisor (a1, . . ., if, and only if, in the set of principal right ideals containing the right ideal (a1, ..., a,,), there is a minimal one, contained in all the others. If this is a S. If, moreover, (a1, ..., equal to (b) then (al, . . ., principal right ideal, then for this case

(al, ...,

Mr

(a1, ..., ar,)i = S .

(80.4)

After all this we formulate the principle that ideal theory in arbitrary rings may be regarded as a generalization of divisibility theory. In practice this means that in cases where the concepts of divisibility are insufficient,

we introduce ideals (or left and right ideals). This can be useful even if not absolutely necessary, since the concept of ideals is in some respects simpler and often more appropriate than notions of divisibility. It must be noted that rule (80.4), especially for an integral domain with greatest common divisor, simply reads as follows : (011,

..., an) = (b) = (a,, ..., an)* = 6 .

(80.5)

DIVISIBILITY IN RINGS

320

We now wish to extend the notion of prime elements to ideals of an arbitrary ring. In an integral domain an element was called prime if from n 1 aj9 it always followed that n I a or n 9. This can also be expressed as follows :

a#E(n)=aE (n) or PE(n). In accordance with this rule an ideal p of an arbitrary ring R is called a prime ideal if for any ideals a, b of R

ab9; p=acp or b 9 p ;

(80.6)

further, we call it a complete prime ideal if for any elements a, f3 of R

of E4= ocE4 or /9Ep.

(80.7)

THEOREM 185. A complete prime ideal is always a prime ideal, and in a commutative ring these concepts are identical. For the proof we consider a complete prime ideal p in a ring R. Let a, b

be two ideals of R with

ab9p,aSE p.

(80.8)

Hence there is an element a with

aEa, aEp.

(80.9)

From (80.91) and (80.81) ab S p. Then a1 E p for every f9 E b. Hence and from (80.7), (80.92) it follows that f1 E p, i.e., b 9 p. Since we have inferred this from (80.8), we have established that p is a prime ideal. We have now only to prove that in a commutative ring R every prime ideal p is completely prime. We assume two elements a, fl with

a f1 E p, a( p.

(80.10)

On account of the commutativity of R, (a) (is) = (aq). From (80.10) then follows (a)(f) S p. The ideal (a), by (80.102), is not contained in p. Hence and from (80.6) it follows that (f) c p, then i4 E p. This means, by (80.10), that p is completely prime. Consequently we have proved the theorem. NOTE. On account of Theorem 185 we use only the first of the terms "prime ideal" and "complete prime ideal" in commutative rings. In this case we generally keep the second of the two equivalent definitions (80.6), (80.7) of prime ideals. THEOREM 186. In a ring R a maximal ideal to is prime if, and only if, R25p in.

(80.11)

IDEALS AND DIVISIBILITY

321

Of this the "only if" part is trivial. In order to prove the "if" part we assume that a, b are two ideals of R with ab S m

Hence

.

(a+m)(b+m)9m.

At least one factor must be different from R because of (80.11). Because of the maximality of m this means that either a or b is contained in in. Consequently

Theorem 186 is proved. EXAMPLE 1. In an integral domain the principal ideals generated by the prime elements are prime. EXAMPLE 2. An ideal p of a ring R is completely prime if, and only if, the factor ring R/p is zero-divisor-free or, equivalently, if the difference set R - .p is a subsemigroup of R x . The first statement follows if we write (80.7) in the form

afl - 0 (mod p) _- a - 0 (mod p) or # = 0 (mod p), the second is obtained from (80.7) by logical inversion. EXAMPLE 3. An ideal p of a ring R is prime if, and only if, the multiplicative semi-

group of the ideals of the factor ring Rip is zero-divisor-free. This follows from the definition and the first part of Theorem 128'. EXAMPLE 4. Every ring R is a complete prime ideal of itself. Further, 0 is a prime ideal or complete prime ideal of R if, and only if, the ideals of R form a zero-divisorfree multiplicative semigroup or if R itself is zero-divisor-free, respectively. EXAMPLE 5. In a ring with unity element every maximal ideal is prime. This follows trivially from Theorem 186. EXAMPLE 6. In the polynomial ring :7 [x1, x2,. .l (x1) C

CS) C (x1. x2, x3) C ...

is an infinite chain of prime ideals.

§ 81. Principal Ideal Rings We retain what was said at the beginning of § 79, but we now include rings R without unity element. We call a ring a principal left ideal ring, if all its left ideals are principal left ideals.

In any ring R if (al, a2, ...)I is even an ideal, then (7-1, 12,

...)I =

(a1, a2, ...)

(in words: if the left ideal generated by the elements ocl, a.,, then this is equal to the ideal generated by aI, a2, ...).

(81.1)

... is an ideal,

On the one hand (81.1) evidently holds with S instead of =. Since, on the other hand, (MI, a2, ...)1) al, a2, ..., it follows that (81.1) then holds with ? instead of =. Consequently the statement has been proved.

DIVISIBILITY IN RINGS

322

If R is now a principal left ideal ring, then, in particular, the ideals of R are just principal left ideals. Therefore (81.1) gives, as a special case, the following

THEOREM 187. All ideals of a principal left ideal ring are principal ideals.

If R is simultaneously a principal left and right ideal ring (whence from Theorem 187 - which holds under even weaker assumptions - it follows that all the ideals of R are principal ideals), then we call R a principal ideal ring. (In the commutative case "principal left ideal ring" means, of course, the same as "principal ideal ring".) THEOREM 188. Every zero-divisor-free principal right ideal ring R with unity element is a ring with greatest common left divisor, and in it

(a1, ..., aa)r = (b)r

(at, ..., an)1 = S .

(81.2)

We already know (cf. (80.4)) that (81.2) is true with instead of Hence the first statement of the theorem follows. We have still to prove that (81.2) holds with

instead of a. For that purpose we assume that (MI, ...,

= (b1)r .

(81.3)

Hence according to the preceding it follows that

This, and the right-hand side of (81.2), imply that b, bl are left associated, i.e., (b)r = (b1)r. Consequently, because of (81.3) the proof of the theorem is complete. THEOREM 189. Every integral domain R, which is a principal ideal ring, is a ring with prime decomposition, and in it (81.4)

furthermore a factor ring R /(a) is for the elements a of R, other than 0 and the units, a field if, and only if, a is a prime. COROLLARY. In the above ring R the equation

a1s1 + ... + anEn = b

($t, ... $n E R)

b. is solvable for every greatest common divisor (al, . . ., First of all we prove that R is a ring with irreducible factor decomposition.

To do this we assume that in R there is an element al, other than 0 or the units, which admits no irreducible factor decomposition. Since then al is reducible, aI = a2N2

PRINCIPAL IDEAL RINGS

323

for elements a2,fl2 (# 0) of R, which are not units. By hypothesis, at least one

of a2, P2 does not admit an irreducible factor decomposition. We may assume that this is the case for a2. On the other hand, (a1) a (02). The repetition of this inference results in the existence of an infinite sequence a1, a.:, ... with lai) C (a2) C ... . The join of these ideals (01), (02) is again an ideal, thus a principal ideal (a). Moreover, because a E (a), a is contained in an (a;), whence it follows that (a) c. (a;). This contradiction proves that R is a ring with irreducible factor decomposition.

In order to prove the first statement of the theorem, we have only to show that every irreducible element n of R is prime. We take two further elements a, fl with 7V I aft , n ,f' a .

(81.5)

From this we have to deduce that n I P. According to Theorem 188 we put (n, (X)*= e-

(81.6)

eIn, ela.

(81.7)

This gives

Because of (81.71) a is a unit or an associate of n. Since the latter is impossible because of (81.72) and (81.52), a must be a unit. Consequently we may

put e = 1. Because of (81.6) and (81.2) it follows that (n, a) = (1), hence I E (n, a). Accordingly the equation

with suitable $, 71 is satisfied. After multiplying by # we obtain

nf + aft = 0 . Hence and from (81.51) n I fi, which is the first statement of the theorem. Statement (81.4) follows from Theorem 188.

With regard to the last statement, according to Theorem 130 (and the preceding remark) R /(a) is a field if, and only if, (a) is a maximal ideal. Since the condition (a) c (+3) is equivalent to fi I a, it follows that (a) is maximal if, and only if, a is irreducible. Since the irreducible elements are now the primes, we have proved Theorem 189. The corollary follows from (81.4).

According to Theorem 189 we may call an integral domain, which is a principal ideal ring, a principal ideal ring with prime decomposition.

DIVISIBILITY IN RINGS

324

NOTE. (81.4) may be expressed as follows: in integral domains which are principal ideal rings, the ideal (i.e., principal ideal) generated by the greatest common divisor (a1, . . ., is equal to the ideal (a1, . . ., generated by the al, ..., a,,. Accordingly we may say that in these rings and (al, ..., Mn) are "essentially" equal, therefore in these 61, . . ., rings, furthermore, instead of (al, . . ., we write (a1, . .., If necessary, we state in what sense this symbol is used. In particular, if we write

b or (al,

(al, . . .,

.

.

., cc,,) = (b) ,

then we mean by the left sides the greatest common divisor or the ideal, respectively. It must be noted here that although we have only defined the

common and greatest common divisor of finitely many elements, the generalization to infinitely many elements is trivial and all the previous statements retain their validity. This also applies to common and greatest common left divisors in zero-divisor-free principal right ideal rings (and to some extent in arbitrary zero-divisor-free rings) with unity elements. However, (81.2) shows that for ( )i we must write ( ), and not ( )1.

Since in a ring R with unity element 1 the ideal (1) generated by I is equal to R, in an arbitrary ring R the ideal R is called the unity ideal (of R) and it is sometimes denoted by 1, even if R has no unity element. (Note that the unity ideal is the unity element of the lattice of all ideals of R). Since for two distinct prime elements e, a of an integral domain (e, a)*

= 1, we call, more generally, any (provided there are at least two) nonempty subsets (elements, ideals, etc.) S1, S2, . . . of an arbitrary ring relatively prime, if (Si, S,, . . .) = 1, i.e., if the ideal generated by them is the given ring. In particular, if (SI, S2) = I we also say that S, is prime to S2 (or S2 to S1). Relatively prime elements of a ring have, of course, no common divisor. The converse of this also holds according to Theorem 189 for integral domains which are also principal ideal rings. THEOREM 190. If the elements al, . . . a of a ring with prime decomposition have pairwise no common divisor (or they are pairwise relatively prime in a principal ideal ring with prime decomposition), then [aI,

...,

xl ... a .

(81.8)

This follows at once from the procedure used for the determination of the least common multiple [cf. (79.37)]. EXAMPLE. In arbitrary integral domains (al, . . ., greatly divergent behaviour. The Dedekind ring .9

and (a,, ..., a,)* may be of 5

is a simple example. Accord-

ing to § 79, Example 1, (3, 2 + J-5)* = 1. But (3, 2 + -5) 0 1, since

(3,2+5)2=(9,3(2+ J-5), (2+,J-5)2)= (2 + J-5)(2 -

_,/mmmw

3, 2 + -5) = (2 + J -5) # (1).

EUCLIDEAN RINGS

325

§ 82. Euclidean Rings

The process of division in the ring 7 due to Euclid is well known. It can be greatly generalized, as we shall see here. The special case just mentioned will be dealt with in § 84. Let R = be a zero-divisor-free ring. Neither unity element nor commutativity is required. If there is a mapping a -- a°

(a

0)

of the set R - 0 into the semiring <JY, 0> of non-negative integers such that (GO)° >_ 0,fl° (82.1) (a, f # 0) and for each pair of elements a, P (f1

0) there are further elements co, o

with

a = o)# + N

(o = 0 or o° < fl'),

(82.2)

then we call R a left Euclidean ring. Equation (82.2) itself is said to be a left

Euclidean division carried out on a, j9. Furthermore co and 0 are called the quotient and residue, respectively. The function a° is itself called a (left Euclidean) absolute value. We say also that R is a left Euclidean ring relative to this absolute value a.°. On account of the duality principle the meaning"of a right Euclidean ring is evident. If furthermore a ring is simultaneously left and right Euclidean (each relative to a suitable absolute value), then we call it a Euclidean ring. (In the commutative case this terminology is always sufficient.) If R is a skew field, then co = aft-', o = 0 is a solution of (82.2). Hence it follows that every skew field is left and also simply Euclidean. To clarify the concepts, it should be noted that co and e in (82.2) are not necessarily uniquely determined.

THEOREM 191. Every left Euclidean ring R is a principal left ideal ring with unity element. If the ring R is, moreover, Euclidean (thus a principal ideal ring), then it is a ring with irreducible factor decomposition and its ideals are the principal ideals generated by the elements of the normalises of R. In the commutative case R is a ring with prime decomposition. First we assume that R (relative to the above absolute value a°) is left Euclidean. We show that R has a unity element. For that purpose let f1( 0) denote an element with minimal 9°. Then in (82.2) we must have e = 0, thus R = Rfl. Consequently there is an element E with

N=efi.

DIVISIBILITY IN RINGS

326

Since, on the other hand, R is zero-divisor-free, R" - 0 is, by Theorem 36, its regular semigroup. Hence, if we apply the last proposition of Theorem 27 to the previous equation, it follows that s is the unit of R.

Further we show that an arbitrary r (A 0) is a unit if, and only if, r° is minimal.

For, if r is a unit, then it has an inverse n-'. It follows from (82.1) that E° =

(Err_1)°

(Er)° > s°

But since sr = r, so no = s°. Thus r° has, for all units r, a constant value. Consequently it is sufficient to show that j9° is not minimal if 4 ( 0) is not a unit. To do this we apply (82.2) with a = 6. Then we have 0 and therefore o° < fl', which proves the statement. We now prove that every left ideal a (0 0) of R is a principal left ideal. To do this we take an element P (0 0) from a, for which fl' is minimal (within a). We show that the equation a = top is solvable for this fi and any arbitrary a (E a). In the other case in (82.2) we had a o, other than 0, for which o° < fi° On the other hand, it follows from (82.2) that o = a - wfl, so that o E a. This contradicts the minimal property of 19°, so the first part of the theorem is proved.

In addition, we show the following refinement of part of (82.1): (at4)° > j9°

(a 0 0, P # 0, a a non-unit).

(82.3)

For, because of (82.2),

= waft + o

(o = 0 or o° < (a#)°)

(82.4)

is solvable. Since a is not a unit, it follows that o 0 0, o° < (afi)°. On the other hand, (82.4) is expressible in the form o = (s - wa) fi, whence, according to (82.1), o° >- fl°. This, with (a#)° > o°, proves (82.3). We now assume that R is two-sided Euclidean. We first show that every element A of R, which is neither 0 nor a unit, has an irreducible factor decomposition. In addition to the retention of a° we denote by a° a suitable right Euclidean

absolute value in R. We assume that we have already proved our assertion for smaller values of the sum A° + A.

.

If A is irreducible, then the assertion is trivial. If not, we can put

A = PI''

(82.5)

EUCLIDEAN RINGS

327

where the factors are not units. Thus (82.3) and the dual of it are applicable to (82.5). So we have

A°>v°,%°>It°. On the other hand, from (82.1) it follows that 2° z µ°, )'o Z P. . Consequently

A°+h°>Fi°+ p., v°+ vo. Accordingly, because of the induction assumption, U, v have each an irreducible factor decomposition, so that, by (82.5), the same follows for 7., as was asserted.

Further, we consider an ideal a (0 0) of R. By the above a is both a principal left and a principal right ideal (whence, according to Theorem 187, it follows that R is a principal ring). Hence

a=aR=Rr

(82.6)

for two suitable elements a, T. Accordingly

a=ar, r=a#

(82.7)

for two further elements a, fl. It follows that

r=arfl. If at were not a unit, then on account of (82.3), we would have r° > (r#)°. But since this is impossible by (82.1), it follows that a is a unit. By (82.71)

Rr = Ror = Ra , so that (82.6) assumes the form

a=aR=Ra. Accordingly a lies in the normalizer of R, moreover

a = aRR = RaR = (a). Consequently we have proved the second part of Theorem 191. Finally, because of Theorem 189, the last part of Theorem 191 is true. For a particularly simple proof of the uniqueness of the prime decomposition in the ring 3 cf. KRBEK (1954), p. 121. For a generalization of Euclidean rings, cf. MOTZKIN (1949).

328

DIVISIBILITY IN RINGS

§ 83. Euclid's Algorithm

If we start with two elements ao, al (a1 # 0) of a left Euclidean ring R relative to the absolute value o c', then, in R, we may obtain equations of the form afl = w1a1 + a2 a1 = w2a2 + F a3 (83.1)

an-2 = wn-lan-1 + an

an-1 = wnar. such that

i>a'>...>a

(83.2)

(n > 1). These equations (83.1) originate from the following: the first equation is the result of a left Euclidean division carried out on ao, a1, where the quotient and the remainder are denoted by w1 and a2, respectively;

whenever for an i (>_ 2) the (i-1)uh equation in (83.1) has already been reached and moreover a; 0 0, then the i`h equation in (83.1) is the result of a left Euclidean division carried out on ai_1, al, where the quotient and the residue are denoted by w; and a:+1, respectively. Since from this ai > az > > ... follows and the ai, az, . . . are non-negative integers, the equations (83.1) necessarily constitute a (non-empty) finite sequence. We have denoted the number of these equations by n. That means that an+1 = 0, as has already

been taken into consideration in the last of the equations (83.1). We call the equations (83.1) a left Euclidean algorithm carried out on «o, a1, or in the commutative case, a Euclidean Algorithm.

In general neither the equations (83.1) nor their number n are uniquely determined by the pair of elements ao, al. However, according to the following theorem the element a,, has an important invariant meaning.

THEOREM 192. For the greatest common right divisor of two elements 0) of a left Euclidean ring R

aq, at (al

(a0, (xl)r = an

(83.3)

where a,, denotes the last term other than 0 in the sequence al, a2, ... occurring in the left Euclidean algorithm (83.1). So in the commutative case in particular (MO, a1) = an .

From the first equation in (83.1) it follows that (ao, at)t = (wlal + a2, (X1)1 = 0/2, al)t = (a11 a2), -

(83.4)

EUCLID'S ALGORITHM

329

By repeated application we obtain from (83.1) (a0, a1)/ = (a1, (X2)1 = ... = (xn-1, 001 = (wean, an]l = (an)[

Hence and from Theorem 188 follows Theorem 192.

§ 84. The Ring of the Integers

We show that the ring 3' of integers is Euclidean and that the usual absolute value I a I of a (E 7, 0) is a suitable Euclidean absolute value. To that end, we choose two elements a, b (b 0 0) of 7. Evidently there are elements q of 3' with a - bq 0. Henceforth let q be chosen such that

a-qb=r(>_0)

is minimal. Certainly we then have since otherwise

r-Ib{=a-(q±l)b>-0,

contradicting the minimal property of r. Consequently we obtain as a result that for the given elements a, b there are two further elements q, r of 3"

such that

a=qb+r

(0
(84.1)

Since, on the other hand, jabj?Iaj, Jb! (a 0, b # 0), it follows [see (82.1), (82.2)] that 3' is Euclidean. It should be further noted that the group of units of 3' obviously con-

sists of the two elements 1, - 1. Accordingly, any two elements a, -a

(a E 7,

0) form a class of associated elements. Consequently in 3' every such class contains exactly one non-negative element. With regard to this we introduce in .7 that norm in which the non-negative elements are normed. Since, according to Theorem 191, 3' is a principal ideal ring, with prime decomposition, the irreducible elements are prime, so that we may reason-

ably call the (normed, i.e.,) positive irreducible elements of 3" briefly prime numbers. (This is not a new definition, since it agrees with § 20.) This accounts for the validity of THEOREM 193 (fundamental theorem of elementary number theory). Every integer a (0 0) is representable in the form

a=(-1)p1,...pn where p1,

. .

(i=0,1),

(84.2)

., p apart from the order are uniquely determined prime numbers.

DIVISIBILITY IN RINGS

330

If we talk of the prime factors of the integers a (# 0), we mean the pl,... p in (84.2). An important extension of Theorem 193 is THEOREM 194 (EucLID's theorem). There are infinitely many prime numbers.

It will be sufficient to prove that for arbitrary prime numbers Pi, ..., Pk (k >_ 0) there is at least one prime number different from them. For this purpose we consider the number 1 + P1... Pk (> 2).

According to the former theorem, this is divisible by at least one prime number. Since this is certainly not one of pl, ..., pk, Theorem 194 is proved. The powers p` (i > 1) of a prime number are called prime number powers, or prime powers, as in the general case considered previously (§ 79). (Some-

times i = 0 is also admitted.) We now wish to apply the results obtained about .7 to group theory. This will enable us to extend the study of .7. THEOREM 195. If a is an element of finite order of a group G and

o((X) = m = P1 ... Pk

(84.3)

is the prime power decomposition of its order, then a has only one factor decomposition of the form

a=a1...Mk (a;E G;o(a,)I PI; a;x,=aia;;i,j= 1,...,k).

(84.4)

All the a; belong to {a} and

o (a;) = P,

(i = 1, ..., k)

(84.5)

holds, which is stronger than (84.4). Let

m;=P1...P;-1P;+1...Pk Then m11

(i= 1, . . ., k).

(84.6)

..., Mk are relatively prime. Consequently the equation m1c1 + ... + mkck = 1

(Cl, ..., ckE .7),

(84.7)

because of the corollary to Theorem 189, is solvable. We put

a1=am"

(i = 1,...,k).

(84.8)

From (84.3), (84.6), (84.7), (84.8) it follows that (84.4) is satisfied. From this, and (84.3), (84.5) also follows.

THE RING OF THE INTEGERS

331

It only remains to be proved that the aI, . . ., ak are uniquely determined by (84.4). From (84.4) and (84.6) it follows that a" = a;" and

amjci=a"""

(i= 1,...,k).

But by (84.7), m,c; - 1 (mod P;), so [by (84.4)] a fortiori m;c1 = I (mod o(a,)). Hence, it follows that (84.8) must necessarily hold. By this the uniqueness,

and consequently Theorem 195 is proved. A group is called periodic (or a torsion group) if all its elements are of finite. order. A group is called torsion free if its elements, other than the unity element, are of infinite order. Finally a group is said to be mixed if it is neither periodic nor torsion free. Of course, this terminology will also be applied to modules. A ring is called periodic, torsion free or mixed, in conformity with the type of its module. In an abelian group the elements of finite order obviously constitute a subgroup; this is called the periodic kernel of the group. This definition will also be applied to modules, where we understand by the periodic kernel of a ring that of its module, which is obviously an ideal of the ring. Special cases of periodic groups are the primary groups ; a group is so called in which the orders of the elements are powers of a fixed prime number. If this prime number is p, then we call the primary group more precisely a p-group. In an Abelian group G the elements of orders 1, p, p2, .. .

constitute a subgroup, which we call the p-component of G. In a similar way we speak of the p-component of a module,, and we understand by the p-component of a ring that of its module. The p-components belonging to the various prime numbers p are called the primary components.

THEOREM 196. A periodic Abelian group G is the direct product of its primary components. COROLLARY. If this group G is finite and O(G) = PI ... Pk is the prime power decomposition of its order, then for each Pt = p;' (the P; being prime numbers) the pi component of G has the order p;'. The theorem follows from Theorems 87 and 195. The corollary follows from the counting of elements. THEOREM 197. A periodic ring is the direct sum of its primary components.

From the previous theorem and Theorem 89 it will be sufficient to prove that aj3 = 0, if a, j9 are elements of a ring, of relatively prime orders. We put

o+ (a) = a,o+(8) = b

((a, b) = 1)

and solve the equation

ax + by= 1

(x, y E ,7)

.

332

Then

DIVISIBILITY IN RINGS

afl=axaq+byafi=aa x3+ya b4=0.

Consequently Theorem 197 is proved. We now revert to the study of .Y, considering first the factor rings of . Y. Since .7 is a principal ideal ring, the principal ideals (m) (m >_ 0) are all its ideals. All the factor rings of .7 are then the .7l(m), where we may restrict ourselves to the cases m >_ 0. (Cf. § 42, Example 1.) THEOREM 198. If

m=ml...mk

(84.9)

is a decomposition of a natural number m into a product of natural numbers relatively prime in pairs, then .71(m) has the direct sum representation

,7/(M) ; 7/(Ml) ® ... ® 71(mk) .

(84.10)

COROLLARY (Chinese remainder theorem). For natural numbers H11'...' Mk

relatively prime in pairs and arbitrary integers a1, . . ., ak the system of congruences

x=_ a, (mod m)

(i= 1,...,k)

(84.11)

always has solutions x in .7; further, these constitute a residue class mod m (m = mt ... Mk)For the proof we may assume that (84.9) is the prime power decomposition of m. We apply Theorem 197 to .7/(m). We obtain a direct decomposition . 1(m) = RI ® ... ® Rk ,

(84.12)

where the R, are subrings (ideals) of .7/(m) and the orders of the elements of R; are divisors of mi (i = 1, . . . k). Let a denote the unity element of .71(m) (i.e., the residue class 1 (mod m)). Then

O(.9'/(m)) = o+(s) = m According to (84.12)

.

e=el+...+ek,

(84.13) (84.14)

where e, is the unity element of Ri, whence o+(e,) I m,

(i = 1, .

.

.,

k).

(84.15)

Further, since a is a basis of . 1(m), e; is a basis of R;, so that O(R,) = o+ (e,)

(i = 1, . . ., k).

(84.16)

THE RING OF THE INTEGERS

333

Now on account of (84.12) 0(7/(m)) = O(R1) ... O(Rk) . Hence and from (84.9), (84.13), (84.15), (84.16) it follows that

O(R) = m1

(i= 1,. .., k) .

(84.17)

Since, finally, the elements of R, are added and multiplied according to the rules

ae, + bsi = (a + b)s;, ae,

(a, b E 7),

be; = abet

it follows from (84.17) that R;

.9'/(m,)

(i = 1, ..., k).

(a--t - a (mod m))

This, with (84.12), proves Theorem 198. The corollary is most simply proved as follows: if (84.14) is multiplied by x (E 7), then we see that (84.11), because of (84.17), is equivalent to Xs = a1e1 -l- ... + akek .

(84.18)

Now it follows from (84.13) that the x satisfying the condition (84.18) constitute one residue class mod m. This proves the corollary.

Since the factor ring J'/(m) for m > 0 is finite, its regular elements, by Theorem 33, constitute a group, which we denote by 71(m)* and term the group of regular (or prime) residue classes mod m, its elements the regular (or prime) residue classes mod m. The number of these residue classes, i.e., the order of the above group we denote by ga(m) = 007/ (m)*) .

(84.19)

This qo is called Euler's function (of m). Now from Theorem 198 and the corollary to Theorem 90 we immediately obtain THEOREM 199. If I/(m)* has the direct product representation S"/(m)*

J/(ml)* ®

... ®.7/(mk)*

(m = m1 ... mk),

(84.20)

then ml, . . ., mk are natural numbers relatively prime in pairs. We want to compute 9'(m). First, it follows from (84.19) and (84.20) that 9'(m) = 99(Po

...

(m = Pl ...

(84.21)

DIVISIBILITY IN RINGS

334

where m = P1... P is the prime power decomposition of m. On the other hand it is clear that the residue classes

((a, m) = 1; 0<-a5m- 1)

a(mod m)

constitute all the different elements of the group .7/(m)* (hence the term prime residue class). Accordingly WW(m)* is equal to the number of terms prime to m in the sequence 0, . . ., m - 1. In particular, for m = p' (p a prime number, i > 1) in the sequence 0, . . ., p' - 1 only the

(k = 0, ..., p'-1 - 1)

pk

are not prime to m, therefore for this case

(Pt)=Pi-p'-1=p (iI

1

1

P

Hence and from (84.21) we obtain

f 1 -p

gy(m)=m p

1

) ,

(84.22)

where p runs through all the different prime factors of m. From the corollary to Theorem 69 the formula ao(G) =

E

(a E G)

(84.23)

follows for every finite group G, where a is the unity element of G. Hence and from (84.19) we have THEOREM 200 (FERMAT-EULER theorem). We have

a'"(') _- 1 (mod m) (a, m E,7; m > 0; (a, m) = 1) ;

(84.24)

in particular a"-1

= 1 (mod p)

(a E -7; p,{' a; p a prime number).

(84.25)

EXAMPLE 1. From Theorem 193 it follows that the elements of the group J%'0* may

be uniquely written in the form

(l = 0, 1; 'I,'2' ... E J), (- l Ypip'i .. where pI, p2 , ... are all the different prime numbers. Accordingly . "o is the direct product of a free Abelian group generated by a countably infinite number of elements and a group of order two. EXAMPLE 2. 91(m) (m >_ 1) is (zero-divisor-free, i.e.) a field if, and only if, m is a prime number. EXAMPLE 3. In a zero-divisor-free ring R with the group of units U the norming can often be carried out as follows (although thereby only some of the classes of left-

THE RING OF THE INTEGERS

335

associated elements are normed): take, if possible, an ideal a of R, such that the regular semigroup R/a* (= (R/a)*) of the factor ring R/a is a group; moreover, suppose that the residue classes, which constitute this group, are represented by the elements of a subgroup of R, which is the direct product of U and a further subgroup G of R. These residue classes are then y,7 (mod a)

(y E G, tic U).

(84.26)

Of these the residue classes

y (mod a)

(y E G)

(84.27)

constitute a subgroup of R/a* (isomorphic with G), and the elements contained in the residue classes (84.27) obviously constitute a system of representatives of those classes of left-associated elements, which are contained in the union set of the residue classes

(84.26). Consequently they may be chosen for normed elements. These elements (with fixed a) are usually called the primary elements of R. Evidently the primary elements constitute a semigroup (that this is a self-dual notion is obvious). Note that, for a, only those ideals are considered for which the residue classes 77 (mod a)

(77 c U)

(84.28)

are distinct. Mostly we choose the group G finite and of lowest possible order. (This norming process will be applied in rather more generalized form in § 89; cf. § 89, Example 5.)

In particular, if R = $ (U = (1, -1)) then a = (4) is an appropriate ideal

to carry out the required norming in 0, because 7/(4)* (i.e., the group of prime residue classes mod 4) consists of the two residue classes

1, -1 (mod 4)

(84.29)

[cf. (84.26)] and these are represented exactly by the elements of U. Since now G = 1, i.e., (84.27) is reduced to the one residue class I (mod 4), we call the elements of this residue class the primary numbers (of .7). The primary prime numbers are then the

following: -3, 5, -7, -11, 13, 17. .... These with 2 constitute a full system of prime numbers which are not associated with one another (as with the usual, i.e., the positive, prime numbers). EXERCISE 1. The ring of complex numbers of the form x = a + bi (a, b E 7) is usually called the Gauss ring. Show that this is Euclidean. (As Euclidean absolute

value we can take the norm N(a) = a2 + b2.) According to the procedure in the earlier Example 3, norming may be carried out with the help of the ideal (2 + 2i), with the result that the elements of the residue class 1 (mod 2 + 2i) are the primary numbers of the Gauss ring. EXERCISE 2. Define all the subgroups, submodules and subrings of .7-0 and select

the non-isomorphic ones. (See REDEI-SZELE (1950b).) Find the derived groups, modules and rings of J V., and determine the non-isomorphic ones.

§ 85. Szendrei's Theorem

Theorem 64 has left undecided whether or not every zero-divisor-free ring has an extension ring of the same kind with a unity element. THEOREM 201 (SZENDREI'S theorem). Every zero-divisor-free ring R (# 0) has zero-divisor-free extension rings with unity element. Among them 12 R.-A.

336

DIVISIBILITY IN RINGS

there is one, which we denote by S, such that the others contain a subring isomorphic with S. Moreover R is an ideal of S and S = {e, R}, where a is the unity element of S.

Because of this we may call S the minimal extension ring (of R) with unity element. For the proof, let RI denote the Dorroh extension ring of R with the unity element a whose elements are uniquely given by

(aEJ,aER)

ae+a

(85.1)

(cf. the Supplement to Theorem 64). The as + x with (ae + a) R = 0 obviously constitute an ideal of R1, which is denoted by r. Because R is zerodivisor-free

rnR=0.

(85.2)

We show that this ideal r is completely prime.

With this end in view we consider the product of two elements of RI such that (ke + K) (le + A) E t

(k,1,

E

.

; K, A

E R) .

(85.3)

We have to show that at least one factor of the left-hand side lies in r. Because a(ks + K) E R and (ke + K) a E R hold for every a(ER, # 0), a(ke + K) (is + A) a = 0

(85.4)

follows from (85.2) and (85.3). If (Is + A)a = 0 for all a, then Is + 7, E r. Therefore we have only to consider the case for which there is a a with

(Is + A)a 0 0. But then it follows from (85.4) that a(ke + K) = 0. For every r (E R) it follows that a(ke + K)r = 0, then (ke + K)r = 0, so that ke + K E r, which proves that r is completely prime. This means that the factor ring RI/r is zero-divisor-free. Since RI is a ring with unity element, so also is the factor ring. Finally this contains, by (85.2), (cf. Theorem 294), a subring isomorphic with R. Consequently, the first assertion of Theorem 201 is proved. In order to prove the other assertions we consider an arbitrary zero-

divisor-free extension ring of R with unity element e. The subring T = {e, R}

(85.5)

is then a ring with the same properties. Its elements are representable (not necessarily uniquely) in the form (85.1). First of all, from this, it immediately (follows that R is an ideal of T. If we now show that T, to within isomorlhism, is uniquely determined by R, then the theorem is proved.

SZENDREI'S THEOREM

337

To do this, we examine under what conditions the equation

ae+a=0

(85.6)

is valid for an element of T of the form (85.1). If y is a non-vanishing element

of R, then, because T is zero-divisor-free, (85.6) implies that (as + a)y= = 0, i.e., ay + ay = 0. Now this condition is entirely independent of the specials, i.e. of the special ring T just considered. Since for the elements (85.1) of RI we have the rules

(as+a)+ (be+P)=(a+b)e+(a+ fi), (as+a)(be+P)=abs+(ba+a#+a#), the asserted isomorphism is evident. By this the proof of Theorem 201 is completed. This proof originates from a written communication from 0. STEINFELD. The original proof may be found in SZENDREI (1949- 50). For other studies on ring extensions with unity elements cf. BROWN-McCoy (1946), WEINERT(1961a). EXERCISE. If R is a zero-divisor-free ring and for two elements r (E 5), a (E R)

one of the equations

e=r

holds for some ; (E R, 96 0), then both equations hold for all (E R). For every r there is at most one such e. The suitable r constitute an ideal of J. Henceforth r will denote the non-negative generator of this ideal and a the element of R belonging to r. Then r I p, where p denotes the characteristic of R (cf. Theorem 38). Lo = 0 if, and only if, r = p. The ideal mentioned in the above proof consists of the set of all k(re - e) (k E 3). The ring S in Theorem 201 is an Everett extension of R by .71(r) and in it e = re. Cf. SZENDREI (1949-50).

§ 86. Polynomial Rings over Skew Fields THEOREM 202. Every polynomial ring F[x] over a skew field F is Euclidean.

The proof will go somewhat beyond what is stated in this theorem. The proper degree of a polynomial f(x) ( 0) is denoted by f ° . We prove that for any two polynomials

f(x), g(x) (E F[x]; g(x)

0)

(86.1)

there are two further polynomials q(x), r(x) uniquely determined in F [x], for which

f(x) = q(x)g(x) + r(x) , (r(x) = 0 or r° < g°) .

(86.2)

DIVISIBILITY IN RINGS

338

Since, according to this, left Euclidean division is possible in F [x] and so also its dual, the theorem follows. The polynomial r(x) in (86.2) is briefly called the left residue polynomial, in the commutative case simply the residue polynomial. We note that the assertion in (86.2) remains valid if we replace F by an arbitrary ring with unity element and g(x) by a principal polynomial. First, we prove the possibility of satisfying (86.2). It is trivial in the cases where f(x) = 0 or f° < g°. If f° >_ g° we assume the assertion for all f(x) of lesser degree. We denote the leading coefficients of f(x) and g(x), respectively by a and b. For the polynomial f i ( x ) = f (x) - ab -' x 1 ° ' g(x)

.

either fl(x) = 0 or f i < f. By the induction assumption there are then two polynomials q, (x), r (x) in F[x] with

fl(x) = g1(x)g(x) + r(x)

(r(x) = 0 or r° < g°) .

Then this r(x) and q(x) = 91(x) + ab-lxt°-g° satisfy (86.2).

In order to prove the uniqueness, we assume that besides (86.2)

f (x) = q(x)g(x) + r(x)

(r(x) = 0 or To < g°)

.

By subtraction we obtain

((x) - q(x)) g(x) = r(x) - r(x) . The left-hand side is either 0 or of degree >- g°. On the other hand, the right-hand side is either 0 or of degree < g°. Consequently both sides must vanish, thus q(x) = y(x), r(x) = i(x) . This implies the asserted uniqueness, so completing the proof of the theorem.

It is evident that the group of units of F[x] is F*. Accordingly all the left-associated elements of an f(x) (E F[x], # 0) are the f(x)c-1 (c E F, # 0). Among them there is only one principal polynomial, which is obtained when we take the leading coefficient of f(x) for c. Therefore we introduce in F [x] the norming, which consists in taking for the normed polynomials (other than 0) the principal polynomials. This norming will suffice for general requirements, since the product of principal polynomials is again a principal polynomial.

We should note that F[x], on account of Theorems 202 and 191, is a principal ideal ring with irreducible factor decompositions. In particular, all the linear polynomials in F[x] are irreducible.

POLYNOMIAL RINGS OVER SKEW FIELDS

339

The most important case of F[x], where F is a field, will be discussed in greater detail. By Theorems 183 and 189 all the irreducible polynomials are prime and F [x] is a ring with prime decomposition. A prime polynomial f(x) (over a field) therefore means the same as an irreducible polynomial.

(However, of these two equivalent notations we shall mostly use the second.) F[x], by Theorem 184, is also a ring (more precisely an integral domain) with a greatest common divisor. In F[x] we shall tacitly assume the irreducible polynomials, the common divisors and the greatest common divisors (if these are other than 0) and, more generally, all polynomials other than 0, to be principal polynomials if not otherwise stated. Especially we may take the unique prime-power decomposition of the elements of F[x], other than 0, in the form

f(x) = c(fi(x))' (f2(x))" ...

(c E F, 0 0; i1, i2, ... >_ 0) ,

where fl(x), f2(x), ... denote all the different irreducible polynomials. (This may be regarded as an analogue of Theorem 193.) Here we have to discuss an important property of the greatest common divisor in the polynomial ring F[x] over a field F. First of all we consider the greatest common divisor d(x) = (f(x), g(x))

of two polynomials which are assumed to be different from 0. We observe by way of introduction : if we pass from F to an extension field, then the

irreducible factors of f(x), g(x) may be further decomposed. Therefore we might also think that d(x) changes, too. But this never happens, since, according to Theorem 192, the determination of d(x) may be obtained by Euclid's algorithm. Hence the coefficients of d(x) are certain expressions in the coefficients off(x) and g(x), from which it follows that they are completely invariant under an extension of the fundamental field. Because of the trivial rule ((fl(x), ..., fn(x)) _ ((f1(X).... fn-1(x)), f. W)

for the greatest common divisor, this also applies to the general case. Consequently we have proved the following theorem: be polynomials in the polynomial ring THEOREM 203. Let fi(x), ..., F[x] over afield F, not all of which are zero polynomials, and their greatest common divisor d(x) be taken in the form of a principal polynomial. Then d(x) lies in F[x], where F (c F) denotes the field generated by the coefficients

of f1(x), ...,f (x), and does not depend on the special choice of F (? F'). The property we introduced in this theorem is called the invariance of the greatest common divisor of polynomials (under fundamental field extension).

DIVISIBILITY IN RINGS

340

EXAMPLE 1. The circumstances in factor decompositions of the elements of a polynomial ring R [x] over a ring R with unity element and zero divisors may be very complicated, even in the commutative case. Then not even the linear principal polynomials

need be irreducible. In the special case R = 9/(6), because

x + I =(3x+ 1)(-2x+ 1) (mod 6), the polynomial x + 1 decomposes into the product of two linear polynomialsIn the polynomial ring (.7/(p2)) [x] (p a prime number), because

(px + 1)° = 1 (mod p$),

the linear polynomial px + I is a unit. EXAMPLE 2. Let F be a skew field with centre Z. We show that the ideals, other than 0, of F[x] are the principal ideals (1(x)) generated by the principal polynomials Ax) (E Z [x]). Because of Theorem 191, it will be sufficient to prove that the normalizer of F[x] consists of the elements of the form aog(x)

(ao E F, g(x) E Z[x])

(86.3)

of F[x]. For, the elements other than 0 of the normalizer are the polynomials

h(x)=a0e+...+a"

(a000; a0,. ,a.E F)

having the property h(x) F[x] = F[x] h(x).

Since then also h(x)F = Fh(x), for every a (E F) there must be an a' (E F) with h(x)a = a'h(x). Written in full, this condition becomes

as = a'a,

(i = 0, ..., n).

Then a0a = a'a0, i.e., a' = a0aao I, so that the previous equations may be written as as = a0aao Ia,, i.e., a o Iaa = aao Ia,

(i = 0, ..., n).

That means that all the c, = ao Ia,

(i = 1, ..., n)

He in Z. Since from this it follows that

h(x) = a0(x' + clx"-I + ... + c"), we have shown that the elements of the normalizer of F [x] are necessarily of the form (86.3). The converse is trivial, whereby the proof is concluded.

§ 87. The Residue Theorem for Polynomials THEOREM 204 (residue theorem for polynomials). In a polynomial ring R[x] over a ring R with unity element let an arbitrary polynomial

f(z) = aox" + ... + a

(87.1)

RESIDUE THEOREM FOR POLYNOMIALS

341

and a linear principal polynomial x - c be given. The results of the left and right Euclidean division by this linear polynomial are respectively f(x) = fi(x)(x - c) + rI

and

(87.2)

f(x) = (x - c)f2(x) + r2 ,

where

rl=aoc"+...+a", r2=c"ap+. ..+a".

(87.3)

COROLLARY. J(c) = 0 if, and only if, x - c is a right divisor off (x).

NOTE. The dual of the corollary is likewise true, if we define f(x) by f(x) = x"ao + ... + a,, (instead of (87.1)). In order to prove the theorem we note that the polynomials

x - c, xt__l+cx'_2+...+c'_I

(i>1)

are permutable and their product is x' - cr. Hence and from (87.1), (87.3) it follows that of the polynomials

AX) - rv AX) - r2 the first is divisible by x - c on the right and the second on the left. This proves the theorem. The corollary then becomes obvious. THEOREM 205. A non-constant polynomial f(x) (E R [x]) over an integral domain R vanishes at the distinct places c1, ..., ck (E R) if and only if

f(x) = (x - ci) ... (x - ck)9(x)

(9(x) E R [x]) .

(87.4)

Consequently the number of the (distinct) zeros off(.r) in R is at most as great

as its degree. The assertion "if" is a consequence of Theorem 157. In order to prove the "only if" part we make use of the quotient field F of R. Let

f(x) = cfi(x) ... f, (x)

(c E R ; fi(x), ..., AX) E F [x])

(87.5)

be the prime decomposition of f(x) in F[x]. Since for the zeros cl, ..., Ck of f(x), according to the above corollary, we have x - cI, . . ., x - ck I f (x) in R[x], this is so F[x], too. Hence and from the uniqueness of (87.5) it follows that the factors x - cl, ..., x - ck must appear on the right-hand side of (87.5). This means the existence of an equation of the form (87.4) with a g(x) from F[x], which, however, on account of the assumption must necessarily lie in R [x]. Consequently the proof of Theorem 205 is complete.

DIVISIBILITY IN RINGS

342

THEOREM 206. If R is an infinite integral domain and f(xj, ....

(A 0)

is a polynomial over R, then in R one may find elements cl, . . ., cn with

f(c1, ..., co 0. For n = I it follows from the last theorem. If n >- 2 we assume that the assertion is true for n - 1. Since is also an (infinite) integral domain, it follows, by hypothesis, that there exist polynomials g.-,(x.) (E R[x ]) with

f(91(xn), ..., gr.-1(xn),

00

and so there exists an element c (E R) with .f(91(c), ..., g,,-1(c), c) 0 0 . Consequently Theorem 206 is proved. EXAMPLE. Theorem 205 is not true either for the polynomials over a skew field, or for those over a commutative ring with zero divisors. E.g., the polynomial .x2 + 1 over a quaternion field with the quaternion units 1, i, j, k has the six zeros ±1, ±j, ±k.

(It has in fact, infinitely many zeros, cf. § 78, Example 2.) The polynomial x1 - I over 31(15) has the four zeros 1, 4, 11, 14 (mod 15). EXERCISE. Prove Theorem 173 by applying the corollary to Theorem 204.

§ 88. Gauss's Theorem For a ring R with greatest common divisor we call, as did KRONECKER, the greatest common divisor of the coefficients of a polynomial f (x1, . . ., ]) the content of this polynomial. If this is the unity element, then we call f(xl,..., a primitive polynomial.

x

THEOREM 207 (Gauss's theorem). If R is a ring with prime decomposition, then so is the polynomial ring R[x]. Both rings have the same group of units and the prime elements of R [x] are those of R together with those primitive polynomials of R[x], which are prime over the quotient field of R. COROLLARY. The product of primitive polynomials in R[x] is itselfprimitive.

NOTE. The theorem and the corollary are also true for R[x1, ..., as well as R[x], as can be seen by induction immediately. The second assertion of the theorem is trivial. We denote the quotient field of R by F and show that every polynomial f(x) (E F[x]) other than 0 can be represented in the form f(x) = 09(x)

(e E F, 9(x) E R [x])

(88.1)

so that g(x) is, apart from associated elements, a uniquely determined primitive polynomial in R [x].

GAUSS'S THEOREM

343

For the proof take an a (E R) other than 0, for which of (x) lies in R [x]. Then MAX) = #g(x)

where i denotes the content of of (x). Thus g(x) is a primitive polynomial. This means that (88.1) is satisfied by P = a-1 fl. If, in addition to (88.1),

f(x) = ah(x) for some primitive polynomial h(x) (E R[x]), then @g(x) = ah(x)

.

We now take an w (E R) other than 0, for which we, ma lie in R. Because w Og(x) = wah(x) .

we, as well as wa, is the content of the same polynomial in R[x], therefore they must be associated. Hence it follows that g(x), h(x) are also associated, whereby assertion (88.1) is proved.

Now we prove the statement that every prime element n of R is also prime in R[x]. To do this we consider two polynomials f(x), g(x) (E R[x]) not divisible by n. We have to show that n is not a divisor, of f(x)g (x). By hypothesis, certain congruences hold, i.e.

f (x) - ax' + .... g(x)

flxk +

... (mod n)

(n ,}' a,

where the degrees of the unwritten terms are less than i and k, respectively.

It follows that

f(x)g(x) = a(ix'+k + ... (mod n) where the degree of the unwritten terms on the right-hand side is less than

i + k. Since, by hypothesis, a X a9, it follows in fact that n f' f (x)g(x). Since a non-primitive polynomial in R[x] is divisible by at least one prime element of R, the corollary follows immediately from the preceding proof. Now, to conclude the proof of the theorem, we consider a polynomial f(x) other than 0 in R[x]. For this the prime decomposition f(X) = h f1(X) ... fk(x)

(K E F)

should hold in F[x]. Because of what was proved for (88.1) we may assume

that the fl(x)...., fk(x) are primitive polynomials in R[x]. According to 12/a R.-A.

DIVISIBILITY IN RINGS

344

the corollary, already proved, their product is also primitive, so that K must lie in R. Thus

.f(x) = nI ... nr fi(x) ... fk(x) ,

(88.2)

where nI, ... n, denote prime elements of R and fj(x),... fk(x) (nonconstant) primitive polynomials in R[x], which are prime in F[x]. If we show that by these properties the factors on the right-hand side of (88.2) are,

apart from order and associated elements, uniquely determined, then we shall have proved the theorem. Since in (88.2) the product fl(x) ... fk(x), by the corollary, is primitive, it follows from (88.2) that the product nI ... n1 is the content of f(x). By the assumption of the theorem the factors nI, . . ., nl are, apart from order and associated elements, uniquely determined. Since, further, the

fl(x), ..., fk(x) are the prime factors of f(x) in F[x], they are, apart from order and constant factors in F, uniquely defined. But if we apply what has been proved for (88.1) to f1(x), . . .,fk(x) (instead of f(x)), it follows that these constants are necessarily units in R. This completes the proof of the theorem. EXAMPLE 1. The polynomial ring R[x] in Theorem 207 is Euclidean only if R is a field and (somewhat stronger) Rix] is a principal ideal ring only if R is a field. If R is

an arbitrary integral domain, but not a field, then it contains an element a, which is neither the 0 nor unity, and then (x, a) is not a principal ideal. It is to be carefully noted that as special cases the polynomial rings , [x], .F"0 [x, y] are not Euclidean

and are not principal ideal rings, although they are, according to Theorem 207, rings with prime decomposition. EXAMPLE 2. The last part of Theorem 207 and the corollary do not apply to the

Dedekind ring J[,/ 5]. The polynomial 3x2 + 4x + 3 is irreducible over this, but, because

3(3x2+4x+3)=(3x+2-]-.J-5)(3x+2-.J-5), it is reducible over the quotient field of 7 [,,,/ 5]. EXERCISE. In the ring 9[x] the polynomials

f(x) = ax + b, g(x) = cx + d

(a, b, c, d E .7; a, c not both 0)

are relatively prime if, and only if, (in 3) (b, d) = I and there is an e (E 9, > 0) with

ab c dl

(a, c)e.

If e is then chosen as minimal, then e is at the same time the minimum of the degree of the polynomials u(x), v(x) (E 9 [x]) with

f(x)u(x) + g(x)v(x) = 1. How can we obtain such u(x), v(x)? (Cf. RUDEI (1958c).)

THE RING OF INTEGRAL QUATERNIONS

345

§ 89.* The Ring of Integral Quaternions The ring to be studied here is a very interesting example of non-commuta-

tive Euclidean rings. A knowledge of this paragraph is not necessary for the study of subsequent parts of this book. In the ordinary quaternion field Q (over .r}'0) we shall denote the quaternion units, usually denoted by 1, i, j, k, by (89.1)

1 , l1, i2, l3 .

The basis representation of the elements of Q then becomes

a = ao + alit + a2i2 + asi3

(ao, a1, a2, as E Yo) .

(89.2)

The special elements

a = 2 (aa + alit + a2i2 + a3i3)

(a.,a1,a2,asE 97;

ao

(89.3)

as (mod 2))

of Q are called integral quaternions. (For the justification for this term cf. Example 1.) Later we shall show that the integral quaternions constitute a subring of Q, which we call the ring of integral quaternions and which is here denoted by Q1. Its group of units is denoted by U. We wish to study Q1 and, in particular, the semigroup Q?, in detail. Of course our results may be useful for Q itself (cf. Example 2). An integral quaternion is said to be primitive if it is not divisible by any natural number greater than 1. Since the two elements

2 = 1 + i1,

(89.4)

0 +11+i2+ i3)

(89.5)

of Q1 will play an important role, we denote them always as above. We complete the notations introduced. in (89.1) by

it=i,

(r,s,E7; r=s(mod 3)).

(89.6)

Then 1r2

=-I,

1r1r+1 = 4-I ,

it+11r = -lr-1

(r E Y).

(89.7)

From (89.5) and (89.7) we obtain the formulae

lra _ -a + tr + It_ l , ai, = -a + 1r + it+1 ,

(89.8)

a2=a- 1 =-a+i1+i2+i3.

(89.9)

DIVISIBILITY IN RINGS

346

Now we see from (89.3) and (89.5) that the integral quaternions may be

written in the form a = aoa + alit + a2i2 + axis

(ao, at, a2, as E 7) .

(89.10)

More precisely: the integral quaternions constitute an 7-vector space with the basis a, it, i2i is. Hence and from (89.7), (89.8), (89.9) it follows that the integral quaternions constitute a ring. For the conjugate, the (reduced) norm and trace', respectively, of the element a in (89.2) we apply the previous notation

a' = ao - alit - a2i2 - a3is

(89.11)

,

n(a)=aa'=aa=ao+ai+az+a.1, t(a)=a+a'=2ao. (89.12)

Hence and from (89.3) it follows that the norm and trace of an integral quaternion are integers. We see that the conjugate of an integral quaternion

is also an integer. Further we note that, of the rational numbers, only the integers are integral quaternions. THEOREM 208. The ring Q, of integral quaternions is Euclidean.

For the proof we consider two integral quaternions a, fi (8 # 0) and write

aQ_I=xp+... +xsis

(xo,...,xsEYo)

Evidently integers ao, . . ., a3 are determinable with the property

(r=0,1,2,3),

ao=-...=a3(mod 2), I2x,-a,I<1

and we may even postulate that here for at least one r the inequality holds. Then

w=

1

2

(ao + ... + axis)

is an integral quaternion and for the quaternion

= MP-1 - W =

1

2

(2xo - ao) +

... +

1

2

(2x3 - as) i3

we have n(s) < 1. If we put e = aft, then the result becomes

a=cofi+ e, n(e)
THE RING OF INTEGRAL QUATERNIONS

347

Here, together with a, P, co, the element 2 is also an integer. Since, for two arbitrary non-vanishing integral quaternions a, P n(ab) = n((x)n(8) Z n (a), n(f3) ,

it follows that Q, is left Euclidean. The proof is dualizable and this proves Theorem 208. The quaternions (89.2) with integral coefficients ao, . . ., as are called quaternions with integer coefficients. These obviously constitute a subring of Q, of -index 2. THEOREM 209. An integer quaternion a is a unit if, and only if,

n(a) = l

,

(89.13)

and the group of units U of Q; consists of the 24 units

1, ± il, ± i2, ± is, 2 (f 1 ± it + i2 ± is) .

(89.14)

COROLLARY. Every integral quaternion has a left-associated element with integer coefficients. If (89.13) holds, then mot' = 1, so that a is a unit. Conversely, if a is a unit,

then a-I is an integral quaternion. Since, further, as-1= 1, then we have n(a)n(a I) = 1, so that (89.13) follows. In order to complete the proof of the theorem, we have only to de ermine for the a in the form (89.3), all the solutions of (89.13). In tertms of the coefficients of a, (89.13) is as follows:

a05+...+a32 =4. This equation has exactly 24 solutions in 7, which substituted in (89.3) yield the units of (89.14). In order to verify the corollary, we consider an arbitrary integral quaternion a, again assumed to be in the form (89.3). If the ao, ..., as are even numbers, then a already has integer coefficients. If ao, ..., as are odd numbers, then the validity of an equation of the form

a=q+2j9, follows from (89.3) and (89.14), where n is a suitable unit (with non-integer coefficients) and j9 is a quaternion with integer coefficients. Then

an, = I + 2#,q'.

DIVISIBILITY IN RINGS

348

Here ri' is a unit, furthermore the right-hand side is obviously a quaternion with integer coefficients. So the corollary is proved. Because of Theorems 208 and 191, Qj is a principal ideal ring with irreducible factor decomposition. THEOREM 210. In the ring QI of integral quaternions an element n is irreducible if, and only if, n(n) = p (p a prime number). (89.15)

By (89.15) exactly p + I irreducible elements n (apart from left-associated elements) belong to every odd prime number p, but only the one irreducible element A (= 1 + lj) , apart from left-associated elements, belongs to the prime number 2. For the proof we need several propositions. PROPOSITION I (LAGRANGE'S theorem). For every prime number p the congruence

I +a2+b2=0 (mod p)

(a, b E 7)

(89.16)

is solvable.

This can also be expressed by saying that the equation 1+a2+b2=0

is solvable in the field F = 7/(p). In the proof we may assume p # 2. From the second part of Theorem 205 it follows that the polynomials 1 + x2, - x2 (E F[x]) for all x (E F) may assume each value at most twice. Since O(F) = p, it follows that the range of both polynomials consists of at least 2 (p + 1) different elements. Hence these ranges must have at least one common element. This means that there are elements a, b (E F) with 1 + a2 = -b2. Consequently Proposition 1 is proved. PROPOSITION 2. For every odd prime number p the congruence

xo +

... + x3 =- 0 (mod p)

(xo, ... xa E J)

(89.17)

has exactly

(p + 1) (p2 - 1) + 1

(89.18)

different solutions mod p. Instead of this it will suffice to show that (89.18) is the number of solutions

of the equation

x2 +y2+z2+ t2 = 0

in the field F = 7/(p). This number does not change when y, z are replaced by

-ax +y, -bx+z,

THE RING OF INTEGRAL QUATERNIONS

349

where a, b are any integers. We choose them so that the congruence (89.16)

holds. Since F is of characteristic p, our equation then takes the form 2x(ay + bz) = y2 + z2 + t2.

(89.19)

Firstly, we consider only the solutions of (89.19) with y = z = 0. Then,

necessarily, t = 0 and x is arbitrary. Consequently there are exactly p such solutions. Secondly, we consider the solutions of (89.19) with

y = bu, z = -au

(uEF,00).

The numbers of these pairs y, z is p - 1. For each such pair we have

ay+bz=0, y2+z2=-u2. Therefore x is now arbitrary in (89.19) and only the two values ± u are possible for t. Accordingly we have 2p(p - 1) further solutions. Thirdly, we have the solutions of (89.19), for which

ay+bz:0. There are exactly p2 - p such pairs y, z. For each of these pairs we may choose t arbitrarily in (89.19), and then x in one, and only one, way. That means p(p2 - p) further solutions. Accordingly

p+2p(p- 1)+p(p2-p) is the total number of solutions of (89.19). Since this number is that in (89.18), Proposition 2 is true. It should be noted that, because of Theorems 208 and 188, Q, is a ring with greatest common left divisor. Now we can show the following: PROPOSITION 3. If p is a prime number, a an integral quaternion with

p, a, p I n (a) ,

(89.20)

n = (p, (Z)i

(89.21)

and

denotes the greatest common left divisor of the elements p, a, then

n(n) = p

(89.22)

DIVISIBILITY IN RINGS

350

It follows from (89.21) that

nIP,a. Hence, because of (89.201), p f' a. Thus

P=nj9 with an integral quaternion 9, which is not a unit. On the other hand, it follows from (89.21) that

a'n = (a p, a'(X)i

.

Because a'a = n(a) we have p I a'n, therefore, by (89.201), n is not a unit. Since then, according to Theorem 209 n(S) > 1, n(n) > 1 and, on account of the above, P2 = n(n)no ,

(89.22) follows, i.e., Proposition 3. An integral quaternion a is called even or odd according as 2 1 n(a) or

2 r n(a). PROPOSITION 4. The even quaternions have integer coefficients, and they

are all the ao + ... + a3i3

(2 1 ao +

... + a3; ao, ..., a3 E

7) .

(89.23)

Furthermore they are both left and right divisible by 2 (= 1 + il). For the proof we consider an arbitrary integral quaternion a in the form (89.3). If ao, . . ., as are odd, then 2 r n(a), so that a is also odd. If, conversely, a is even, then ao, . . ., a3 must be even, whence follows the first statement of Proposition 4. The second statement is then obvious. For the third statement we note that n(A) = 2, therefore A is a left and right divisor of 2. With regard to (89.23) it will evidently suffice if we show that in Ql all the equations

(1 + il)1 = 1 j it ,

ri(l + i l ) = 1 4- it

(r = 1, 2, 3)

are solvable. The solutions are:

=

2

(1 - i1) (1

4) ,

y1 =

2

(1 ± ir) (l - i1) .

These occur among the units (89.14), so they lie in Q,. Consequently Proposition 4 is proved.

THE RING OF INTEGRAL QUATERNIONS

351

PROPOSITION 5. If p is an odd prime and a is an integral quaternion with p,}'a, p I n (a), then the solutions $ (E Q) of the congruence

a - 0 (mod p)

(89.24)

constitute exactly p2 residue classes mod p.

Since p is odd we can take both a and as quaternions with integer coefficients. We write

a = ao + ... + a3i3 .

(89.25)

Further, for the solutions we may restrict ourselves to the

(xo,...,x3=0,...,p- 1)

(89.26)

since these constitute a system of representatives mod p. We have to show that in (89.26) there are exactly p2 among the for which (89.24) holds. To do this we introduce the notations

N=ao+a171

y=a2+a3i1

(89.27)

µ = xo + x1i1,

v = x2 + x3i1.

(89.28)

[Note in the following that 19, y, µ, v are elements of the complex field .5ro (il) and also that the conjugates v lie in Y0 (il). The commutative law needs to be borne in mind.] From Theorem 179 it follows that the congruence (89.24) means that in the matrix product

-Y v, 'lµ'+/9v

Yl

V' t,1 - I - (Yap + j9 v)' CAµ - Y v)'

all four elements on the right-hand side of this equation are divisible by p. This is identical in meaning to

flu - yv'==-0,

y'µ + fl'v =0(mod p).

(89.29)

We have to show that this system of congruences is satisfied by exactly p2 pairs µ, v. First we consider the case p,j' j319'. From p I n(a), (89.25) and (89.27) it follows that /3,9' + yy' = 0 (mod p). Hence it follows from (89.291), after multiplying by 19'y', that j319'(y'µ + fI'v) _- 0 (mod p) .

352

DIVISIBILITY IN RINGS

But this congruence is identical in meaning with (89.292) so that (89.29) is now reduced to (89.29). This, after multiplication by P', becomes

flp'p - j9'yv'

0 (mod p) .

This congruence has for every v exactly one solution u, thus proving that (89.29) has the required number of solutions in the case under consideration. The other case p I jfl' may be reduced to the former. According to (89.271) it is characterized by

pI ao+ai.

If moreover

p I ao + a',

(r = 1, 2, 3)

(89.30)

(89.31)

2, that all the ao, . . ., a3 are divisible by p. This is impossible because p,f' a. Therefore there is an r for which (89.31) is false. Since the three quaternion units il, i2, is by a cyclic permutation become a system with the same properties and moreover the a1, a2, a3, according to (89.25), undergo the inverse permutation, we have shown that the case (89.30) may be avoided. Consequently Proposithen it would follow, since p I n(a), p

tion 5 has been proved. We now wish to verify Theorem 210. First we show that the irreducibility of n follows from (89.15). Indeed, from an equation

n=a/9

(a,I9EQ)

p = n(a) n(fJ), by (89.15), so that either n(cc) = 1 or n(f) = 1. This

means, by Theorem 209, that either a or fi is a unit, consequently n is irreducible.

We now show that p itself is reducible. We take a solution a, b of congruence (89.16) and write

a = 1 + ail + bi2. Then (89.20) is satisfied, whence, according to Proposition 3, there exists an integral quaternion n with n(n) = p. So p is in fact reducible. In order to complete the proof of the first statement of Theorem 210, we have only to show that n(n), for every irreducible quaternion n, is a prime number. Let p denote a prime factor of n(n). According to what has been said above p is reducible, thus p I' n. Consequently Proposition 3 is applicable for a = n. This implies that for n1 = (p, n)1

we have n(nl) = p. Thus, nl is irreducible. Since nl I IT and IT, nl are both irreducible, they are left-associated. We have n(n) = n(nl) = p, by which the first statement of Theorem 210 is proved.

THE RING OF INTEGRAL QUATERNIONS

353

For the case p = 2 it is easy to complete the proof of Theorem 210. For, if n(A) = 2, it follows from the former that A is irreducible. If, on the other hand, n(n) = 2 and moreover n is an integral quaternion, then n is even. From Proposition 4 we have A I n, therefore n is left-associated with A.

In the remaining part of the proof we need only consider the case p s 2. We make n (E Q,) run through all the solutions of (89.15), so that leftassociated elements are disregarded. We have only to verify that the number

of these n is equal to p + 1. To do this we make $ run through all the Xp+...+X313 (X0,...,X3=0,...,p- 1; 5 0,PIW)) (89.32)

We first show that to each of these tioned n, such that

there is only one of the aforemen-

nI.

(89.33)

On the one hand, by (89.15) and (89.32), the existence of n = (p, E), follows from (89.33), whereby n (apart from left-associated elements) is

uniquely determined. On 'the other hand, the existence of an integral quaternion nI with n(n1) = P , n1 I E follows from (89.32) by Proposition 3. Moreover, in nI a right unit factor is unimportant, so any proper left-associated element of nI is such a n. The statement is thus proved. Conversely, it is evident that for every n there is at least one E satisfying (89.32), (89.33), for we can take that E from (89.32), for which we have _- n (mod p). (Here we have taken p ,r n into consideration.) According

to Proposition 2 the number of E in (89.32) is equal to (p + 1) (p2 - 1). Consequently, if we show that for each of our n there are exactly p2 - 1 elements E from (89.32) with the property (89.33), then we shall have proved Theorem 210. For that purpose we transform (89.33) into n'n I n'E, for which we may also write

n' = 0 (mod p)

(89.34)

by (89.15). It should be noted that the condition p I n(E) is included in this. If we now admit in (89.34), in addition to the E from (89.32), also $ = 0, which gives one more solution, then on account of Proposition 5 (applied to a = n') the number of the solutions is p2. The number of solutions of the required type is p2 - 1. Consequently we have proved Theorem 210. THEOREM 211. All the distinct ideals, different from 0, of the ring Q; of integral quaternions are the principal ideals: (89.35) (mAt) (m = 1, 2, ...; A = I + iI; t = 0, 1).

DIVISIBILITY IN RINGS

354

First we show that the ideals (89.35) are distinct. It follows from Proposition 4 that AQ; as well as Q,A consists of all the even quaternions, thus AQ; = Q;A. More generally mA`QI = Q,mk', therefore the principal ideals (89.35) may be written in the form

(n.') = mA'Q;. It is evident from this that they are distinct. Now we have only to show that each ideal (a) of QI (a E Q,, 0 0) is one of the ideals (89.35). Let m denote the greatest natural number with m I at.

According to Proposition 4, either m-a or m-'A-1 is an odd integral quaternion. So we can write (89.36)

at = mA'co ,

where m, t have the same meaning as in (89.35) and co is a primitive odd quaternion. Since [according to (42.25) ] (rA') (c)) = (mA'co,mA'Q co) _ (m)49, QimA'co) = (m.ttco) ,

so by (89.36) ((X) = (ma`) (CO)

Consequently, if we show that (co) = 1 for all the primitive odd quaternions w, then the theorem is proved. Because of the corollary to Theorem 209 we may restrict ourselves to an w with integer coefficients, therefore we can put co = ao + a)l! + a2!2 + a313 (a0, ..., as E 7; (ao, ..., a3) = l; 2,}' n (co)). Then

i,wi, = to - 2a0 - 2a,.i,

(r = 1, 2, 3),

so that w - ilcoil - i3U)i3 - 13co i3 = 4a0 .

From this it follows that 4ao E (w) .

If we apply this with - i, co instead of w, then because (- i, w) = (co) we have 4a, E (c))

(r = 1, 2, 3) .

This implies that (4ao, ..., 4a3) ` (c)) .

THE RING OF INTEGRAL QUATERNIONS

355

... a3) = 1 it follows that 4 E (w). But since co is odd, we obtain (w) = 1. Consequently Theorem 211 is proved. Because (a0,

THEOREM 212. For .1 = I + it the factor ring Q./(2A) is of order 64, the group (89.37)

Q1/(2A)*

of its regular elements is of order 48, and

(1+2o')'al

(AEU; t=0,1; a=

1

2

(1+iI+i2+i3)) (89.38)

is a system of representatives of the residue classes mod 2A contained in (89.37).

Because A = I + 11, we have il, (89.3) it is obvious that

I (mod 1). Hence and from

i2, i3

0, 1, a, a'

(89.39)

is a system of representatives mod A of Qi. So if a, f, y run through the four elements (89.39) independently of one another, then

2a + A# + y

(89.40)

yields a system of representatives mod 22 of Q;. Accordingly, O(Q;/(2A)) _

= 64. Since the odd elements in (89.40) arise. in such a way that y is subjected to the restriction y 0 0, the order of the group (89.37) is indeed 48.

On the other hand, the elements (89.38) are odd; furthermore their number is likewise 48. Therefore it will suffice to verify that they are distinct

mod 2A. Since then constitute a group, it is sufficient to show that n - 1 (mod 2A) only for 71 = 1, and that 71 - I + 2a (mod 22) for no 71(71 E U). This can be verified directly from (89.14) so that Theorem 212 is proved. We call an odd quaternion a primary, if a - 1 or I + 2a (mod 21)

(a =

1

2

(1 + iI + i2 + i3)) .

(89.41)

Since x - 1 is then even, it follows from Proposition 4 that every primary quaternion has integer coefficients. Since further (1 + 2a)2 = I + 4(a + a2) - I (mod 2A)

,

(89.42)

the primary quaternions constitute a subsemigroup of Q,. It follows, more generally, from every equation a = ,9y (a, fl, y E Q,), in which two of the

three elements x, 9, y are primary, that the third must also be primary.

DIVISIBILITY IN RINGS

356

The importance of primary elements consists in the fact that they assist in norming the odd elements of Q;, where the primary elements are the normed elements. (We always assume this norming in Q;.) For, by Theorem 212, every class of left-associated odd quaternions has exactly one primary element. In particular, 1 is the only primary unit in Q,.

NoTE. Part of Theorem 210 implies that exactly p + 1 (irreducible) primary quaternions n with n(n) = p belong to every odd prime number. Of course, in Q; the irreducible factor decomposition of the elements is not unique. Nevertheless, a certain kind of uniqueness holds in the sense of the following theorem. THEOREM 213. If a is a primitive primary (and so odd) quaternion, then there is, for every prime decomposition

n(a) = PI ... Pk

(89.43)

of its norm, exactly one decomposition

a = nl .

.

. nk

(n (nr) = p,; r = l , ..., k)

of a into (irreducible) primary factors nI,

(89.44)

.. nk

This is trivial for k = 0. If k > 0 we assume the statement for k - 1 instead of k. It will suffice to prove that there is exactly one primary quater-

nion nI with

nl a , n(n) = Pi .

(89.45)

For then it follows, on the one hand, that in (89.44) the first factor nl is uniquely defined. On the other hand, from (89.45), we have the existence of

an equation a = n1j9, where / is again a primitive primary quaternion and for this n(f) = p2 ... Pk. Since PI I n (89.45) is now identical in meaning to nl = (pl, oc)i according to Proposition 3. If, moreover, we postulate that nl also is primary

then nl is uniquely determined by these conditions. Hence Theorem 213 is proved. EXAMPLE 1. Consider all the subrings R of Q with the property that 1, il, is, is belong to R and all the n(a), t(a) (a E R) are integers. Evidently Q, has this property; further it is easy to show that Q, includes all the subrings to be considered. This is why the elements of Q, are called "integral quaternions". EXAMPLE 2. Evidently .% is the centre of Q,, so that Q is the quotient skew field of Q. EXAMPLE 3. A = 1 + it is, to within associated elements, the only element, other than 0 and the units, of Q,, which has the property that for a product aq8 (a, # E Q,) A I a8 if, and only if, A I a or A I P. (Therefore and because of Proposition 4 we could call A the only "prime element" of Q,.) EXAMPLE 4. All the distinct right ideals of Q, are the (a), = aQ,, where a runs through a system of representatives of the classes of left-associated elements of Q,.

357

THE RING OF INTEGRAL QUATERNIONS

The dual holds for the left ideals of Q,. In comparison with Theorem 211 we may thus say that Q, is comparatively rich in one-sided ideals and poor in two-sided ideals. EXAMPLE 5. Regarding primary quaternions we note: it is obvious that

(r = 1, 2, 3).

1, a = a i, (mod A)

Hence, because of (89.10), more generally

qa = aq (mod R)

(q E U),

so that

q(1 + 2a) _ (1 + 2a)q (mod 2R)

(q E U).

Since, moreover,

(I + 2a)2

I (mod 21)

we see that a subgroup of the group Q,/(2A)* is represented by the elements (89.38),

which is the direct product of a group isomorphic to U and a group of order two. Hence we understand that the norming carried out above in Q, is obtained by an easy generalization of the method described in § 84, Example 3. EXAMPLE 6. A kind of converse of Theorem 213 is the following THEOREM. If =I, ..., nk is a sequence of primary irreducible quaternions such that

(n,+I 96 ± n',) (i = 1, ..., k - 1) and in the sequence of prime numbers n(ni), ..., n(nk)

(89.46)

between any two equal terms lie only terms equal to them, then the product

a = al ... nk

(89.47)

is a primitive quaternion.

If k = 0 it is true, since then a = 1. If k > 0 we assume the statement for k - 1. Then

= ns ... nk

(89.48)

is primitive. We have a = nif. We suppose that a is not primitive. Then there is a prime number p with Hence

We must have

P I nifl P I niniN

n(1) = mini = P,

(89.49)

since otherwise we could infer that p I P, although fl is primitive. By the above, 221/9 = PY

for some integral quaternion V. Since p = nlni, we have (89.50)

Since now p I n(,8), p occurs among the prime numbers n(n$), ..., n(nk). Hence and from (89.49) it follows, because of the assumption in (89.46), that n(7r2) = p, so that n(n2) = n(ni).

DIVISIBILITY IN RINGS

358

(89.48) and (89.50) imply, by Theorem 213, that n2i zi are left-associated, i.e., for a suitable unit r/

ni = nsyt.

(89.51)

By (89.49) we then have P = nlnz7l-

Since p or -p and r11 z2 are primary, it follows that , = ± 1. This contradicts, because of (89.51), the suppositions of the theorem. This completes the proof. EXAMPLE 7. As an application we prove the following THEOREM (JACOBI'S theorem). For every natural number a the number of solutions of the equation (89.52) a = x1 + y2 + zE + t2 (X, Y, z, t E %)

for an odd a is 8 times, and for an even a 24 times the sum of the odd divisors of a. Let f(a) denote the number of the solutions of (89.52), i.e., that of the quaternions

a with integer coefficients a such that (89.53)

n((x) = a.

First let a be odd. If g(a) denotes the number of the primitive solutions a of (89.53) with integer coefficients, then evidently

f(a) = E g(d-2a),

(89.54)

a'la

where d runs through the natural numbers with d2 I a. Since every primary quaternion has exactly 8 left-associated elements with integer coefficients, so

g(a) = 8h(a),

(89.55)

where h(a) denotes the number of primary primitive solutions a of (89.53). In order to determine h(a), we use the prime-power decomposition

a=pi'...pk,

(s> 0; k1,...,k,>0)

of a, where the order of succession of the factors is retained. If a is a primary primitive solution of (89.53), then according to Theorem 213, there is a uniquely defined factor decomposition of this,

a = n ... T,

(89.56)

in which the factors n, ..., z are primary irreducible quaternions. and the sequence n(n), . . ., n(r)

is of the form

Pt, ..., p ; k1

PS1 ...I P$; k,

, Pa;

P3. k,

since a is primitive, so in (89.56) no two neighbouring factors, neglecting signs, may be conjugate. Conversely, if these are satisfied, then a, according to the theorem of Example 6, is a primary primitive solution of (89.53). Theorem 210 implies that the number of

primary irreducible quaternions with norm PI is equal to p1 + 1. It follows that the number of possibilities for the choice of the ith factor on the right-hand side of

359

THE RING OF INTEGRAL QUATERNIONS

(89.56) (i = 1, . . ., k,) is equal to p, + 1, pa, . . ., pl. Altogether there are exactly

(Pt + 1)pi'-, = pi '(1 + pi-') possibilities for these factors. The same is true for the other factors, so that

h(a)=afl(l+p-1), pla

where p has to run through the different prime factors of a. Accordingly h(a) is the sum of those m -la, in which m is a square free divisor of a. Because of (89.54), (89.55),

f(a) is then 8 times the sum of all the divisors of a, therefore the theorem is true for this case. Secondly let a be even. Then

a=2'b

(r >t1)

for an odd natural number b. The quaternions a with integer coefficients in (89.53) may obviously be taken in the form (89.57)

where i4 is an integral quaternion, for which

n(9) = b.

(89.58)

Conversely, if i is an integral quaternion and (89.58) holds, then (89.57) yields, according to Proposition 4, a quatemion a with integer coefficients, for which (89.53) holds. Taking into consideration that all the 24 left-associated elements of an even quaternion

(according to Proposition 4, all the even quaternions) have integer coefficients, we finally obtain f(a) = 3f(b). Consequently the theorem is proved. EXERCISE 1. A quaternion given in the form (89.2) is primary if, and only if, its coefficients a o, ... , a3 are integers and

ao - 1 = a, = ap =- a3 (mod 2), au + ... + a3 = I (mod 4). E.g., the primary irreducible factors of (p =) 3, 5, 7 are in order as follows:

-i+j+k, i-j+k, i+j-k, -i - j - k -1 ± 2i, -1 + 2j, - L ± 2k

±2+i-f-k, ±2-i+j-k, ±2-i-f+k, ±2+i+j+k

(p

3),

(p = 5),

(p=7),

where we have written i, j, k instead of i,, i2, i3. EXERCISE 2. Prove Theorem 211 with the help of Theorem 191 so that one shows that the normalizer of Q, consists of the elements a, ad (a E 9) and their associated elements. For further details on quaternions cf. HURWITZ (1919).

CHAPTER V

FINITE ABELIAN GROUPS Finite Abelian groups constitute an important chapter in algebra, which is practically covered by the FROBENIUS-STICKELBERGER main theorem.

However, even today it still provides many interesting problems and this has led to its expansion recently by several new important studies. § 90. Cyclic Groups Cyclic groups have already been mentioned more than once in § 21 but their properties deserve a more detailed study. Let a cyclic group, either infinite or finite, of order m be given, such that Zo = {co}

(o(w) = 0),

(90.1)

Z. = {a}

(o(a) = m > 0).

(90.2)

We know that these (apart from isomorphism) give all distinct cyclic groups. In general we call a generating element of a cyclic group a primitive element of this group. In the above cases, co and a are primitive elements of Zo and Zm, respectively.

First, we shall consider the group Z0. It is evident that wk

(k E -7)

(90.3)

are all its distinct elements, among which only co and co-' are primitive. Because of the isomorphism

4 ., 7+ (co' --*. k)

(90.4)

we can at once give the subgroups of 4. Since the submodules of t7+ are the ideals of 7, and 7 is a principal ideal ring, it follows that all the subgroups of Zo are the cyclic groups {w"'}

(m 360

0).

(90.5)

CYCLIC GROUPS

361

With the exception of the trivial case m = 0 these groups are infinite, and so isomorphic to 4. Since any two proper subgroups of Zo have an intersection other than the unity element (which is again an infinite cyclic group), it follows that Zo is indecomposable.

The homomorphic images of a finite or infinite cyclic group are, of course, likewise cyclic. Hence follows the isomorphism (m Z 0).

Zn/{w"'} ;ze Z.

(90.6)

Consequently the Z. (m z 0) (to within isomorphism) are all the homomorphic images of Z0. We now consider the finite cyclic group Zm (m > 0). We have

ak = a! As a special case

k =_ I (mod m).

(90.7)

ak=EamIk,

(90.8)

m 0((X,) = (k, m)

(90.9)

where a is the unity element. Hence we have

and the special case

o(ad) = d

(d I m).

(90.10)

In order to prove (90.9) we write n = o(ak). Then n is the least natural number with akn = E.

By (90.8) this equation is identical in meaning to m I kn, which may be written as m k (k, m) Since

I (km) ,

(kkm) 1

(k, m)

n.

= 1, this is equivalent to m (k, m)

n,

and (90.9) follows.

From (90.9) it follows that a" is a primitive element of Zm (m > 0) if, and only if, (k, m) = 1. According to this and (90.7) the number of primitive elements of Zm is given by Euler's function pq(m).

FINITE ABELIAN GROUPS

362

We prove that all the distinct subgroups of Zm (m > 0) are the groups {ad}

(d I m);

(90.11)

since the order of (90.11) is d-' m, it follows that exactly one subgroup of Z. of order m' belongs to every divisor m' of m. Every subgroup of Zm is, because of (90.6), derivable from Zo, and so by Theorem 58 is a homomorphic image of a subgroup of Zo. It is.therefore always a cyclic group. Hence any subgroup of Zm may be written in the form {a"}. This group obviously consists of all the a", where x runs through the elements of the ideal (k, m) of ._7. Now since this ideal is a principal ideal (d) containing (m), where necessarily d I m, the above assertion follows.

If, in particular, m = p' (p a prime number), it follows that the groups {a''}

(i=0, ..,t)

are all the subgroups of Zm. Since these constitute a chain of subgroups. the cyclic groups of prime-power order are indecomposable. On the other hand, by Theorem 196, the other cyclic groups of finite order are decomposable.

§ 91. Frobenius-Stickelberger Main Theorem THEOREM 214 (FROBENIUS-STICKELBERGER main theorem). Every finite

Abelian group is a direct product of cyclic groups of prime power order (> 1).

This theorem is also called the first main theorem for finite Abelian groups. For the proof, we denote a finite Abelian group by A and its unity element

by e. Let B be a subgroup of A, for which Theorem 214 holds; such a B is, e.g., B = e (where the number of the factors is 0). It will suffice to verify

that for B 0 A there is an element a of A not contained in B, such that the theorem holds for {a, B). To show this we first take an arbitrary element at of A not in B. Our method will be to interchange a with another element of the same property until we finally reach a satisfactory result. By Theorem 195, a is a product of elements of prime-power order. At least

one of the factors must lie outside B. a is interchangeable with this, so that we may assume that o(a) is a power of a prime number p. Since the element e then appears in the sequence a, a', a'2, ... this sequence has a last term, which does not lie in B. If we take this for a, we now have a' E B.

(91.1)

O({a, B}) = pO(B).

(91.2)

Then evidently

MAIN THEOREM

363

On the other hand, by hypothesis B has a direct product decomposition

B={YI}®...®{,a,},

(91.3)

are of prime-power order. With suitable where the elements order of the factors in (91.3) we may write, by (91.1),

ap = fll ... A, (ik= 1,...,o(tlk)- 1; k= 1,...,t; 05t5s). (91.4) From this it follows that

By (91.3) all the factors of the right-hand side must be equal to E, so that, to-

gether with o(a), all the o(1),.. ., o(,B,) are powers of p. If, in particular, t = 0, then, according to (91.4) ap = s, so that {a, B} _ {a} ® B. Hence and from (91.3) it follows that the theorem is now true for {a, B). Let t > 0. We can ensure that none of the exponents il, ..., i, in (91.4) is divisible by p. For if, e.g., p I i1, then (91.4) is representable in the form (a 9-P-11,)P

= A. ...Nit

Since the element in brackets is not in B, at may be replaced by it. I this way, by finitely many steps, we reach our desired conclusion.

Further, among the #I, ..., t, we may assume that

Yl

is of maximal

order. Then, by (91.4), o((x) = po(fl1), so that by (91.2) and (91.3), O({ a, B}) = o(a)o(f2)

... o(f).

On the other hand, according to (91.4) we have YI E {a, N2, that {a, B} = {a, P2, ..., j9,}. Consequently

so

{a,BI ={a}®{#2}® ...®{j,}. Hence Theorem 214 is proved. We now consider a finite Abelian group A and its decomposition

A=Z1®...®Z,

(91.5)

FINITE ABELIAN GROUPS

364

into the direct product of cyclic subgroups of prime power order (> 1). Since these are indecomposable, their orders

0(Z I), ..., 0(Z,)

(91.6)

are uniquely defined, apart from the order of succession, because of the REMAK-KRULL-SCHMIDT theorem (Theorem 146), so that we call them the invariants of the finite Abelian group. We shall show that these invariants constitute a complete independent system of invariants for finite Abelian groups.

The completeness of the invariants (91.6) means that by (91.6) A is determined to within isomorphism. In order to prove this, we consider a further finite Abelian group B with the same invariants as A. Then we may write

A={(xi) ®...®{ar}, B={i3}®... ®{flr}, o(ar) = o(fli)

Q= 1, ..., r).

Hence the following is clearly an isomorphism A N B (x ... (Xr - lei ...

rl')

The independence of the invariants (91.6) means that no proper subsystem of them determines the group A uniquely. This is clear, since for arbitrarily given prime powers P1, . . ., Pk, there is one finite Abelian group whose invariants are exactly these PI, ..., Pk. Indeed, the direct product of cyclic groups, of order PI, ..., Pk, respectively, is such a group. It should be emphasized that according to (91.5), (91.6) the product of the invariants of a finite Abelian group is equal to the order of that group. We write (91.5) explicitly in the form : A = {al} ®

... ®

{ar} ,

(91.7)

where each of the direct factors is given by a primitive element aI, ..., ar We then call aI, ..., ar a prime power basis of the finite Abelian group A. This means a system of elements acl, . . ., ar of A, which has the property that the orders P1 = o((Xi), ..., Pr = o((Xr)

(91.8)

are prime number powers and that (91.7) holds; this is equivalent to saying that the elements of A may be uniquely written in the form

a=at'...a

(ik=0,...,Pk- 1; k= 1,...,r).

(91.9)

We call (91.9) a basis representation of the element a (corresponding to the given prime-power basis). For the sake of convenience we admit arbitrary

PRINCIPAL THEOREM

365

integers for the exponents, which are (automatically) considered only mod Pk (k = 1, . . ., r). For the sake of completeness we write down in A the multiplication rule for two elements with basis representations, al ... a 1 '

(X... a;' = ai+i.... a;+i, .

(91.10)

To this it may be added that the prime-power bases are obviously a special case of the previously (§ 45) defined bases.

We now wish to draw a series of important consequences from the main theorem. Since the product of the invariants of a finite Abelian group A is equal

to O(A), it follows that every prime factor of O(A) must be a divisor of at least one invariant. Thus if the prime number p is a divisor of O(A), then A certainly contains elements of order p. This result will be generalized to arbitrary finite groups in the following THEOREM 215 (CAucHY's theorem). If the order of a finite group G is divisible by a prime number p, then G contains at least one element of order p.

(Since conversely, according to the corollary to Theorem 69, the orders of the elements of G are divisors of O(G), it follows that the finite p-groups are identical with those finite groups whose orders are powers of p.)

In order to verify the theorem we assume it for those groups of order less than O(G). We denote by C1, ..., Ck the Frobenius classes of G. Let C, consist of n, elements (i = 1, ... , k). Then [cf. (42.32)]

O(G)=nl+...+nk. (91.11) We take from each of the classes CI, ..., Ck, an arbitrary element yl, ..., yk and denote their normalizers by N1, ..., Nk respectively. Then n, is equal to the index of Ni in G :

n, = O(G : NI)

(i = 1, . . ., k).

(91.12)

if there is an N, with Ni G and p I O(N;), then, from the induction assumption, it follows that Ni and therefore G contains an element of order p. Thus we have only the case to consider in which, for every Ni, either Nr = G or p,r O(N,). According to (91.12) this implies that for every n, either np = 1 or p I n,. Consequently, because of (91.11) and p I O(G), the number of cases for which n, = 1, i.e., the order of the centre of G, is divisible by p. But since, by the preliminary note, the theorem is true for the Abelian case, it follows in general. THEOREM 216. In a finite Abelian group A with the unity element a every equation (91.13) (p a prime number) 0(° = E

366

FINITE ABELIAN GROUPS

has exactly p!° solutions, where tp is the number of invariants of A divisible by p. For the proof, we give the elements a of A by their basis representation

(91.9). After substitution in (91.13) we obtain a''n 1

...

This implies that all the equations

ak°=E

(k = 1, . . ., r)

(91.14)

hold. Now by (91.8), (91.14) is equivalent to

p-1PkI ik according as p I Pk or p,f'Pk

.

or Pklik,

For the value of the k`h factor ak in (91.9)

there are exactly p possibilities or only one possibility (k = 1, ..., r), respectively, whence Theorem 216 follows. THEOREM 217. A finite Abelian group A is cyclic if, and only if, its invariants

are pairwise relatively prime, i.e., if every equation a° = e in it has at most p solutions (p a prime number, a the unity element).

Let Z1, Z2 be two factors from a Remak decomposition of A. Then ZI ® Z2 is a subgroup of A. If A is cyclic, then its subgroups are likewise cyclic, whence it follows that the orders of Z1, Z2 must be relatively prime. This means that the invariants of A are pairwise relatively prime. Conversely, if the al, ..., ar constitute a prime-power basis of A, then it follows directly that the order of the element cc ... ar is equal to the product of the invariants, i.e., equal to the order of A. Accordingly A is now cyclic. So, from Theorem 216, the proof of Theorem 217 is complete. In general we call a periodic group elementary, when its elements other than the unity element are of prime order. A finite Abelian group is evidently elementary if, and only if, its invariants are equal to a fixed prime number. We denote by Lk

(0<_k5n)

(91.15)

1

the number of the subgroups of order pk of an elementary abelian group of order p". In order to determine this number, we proceed as follows: Let A denote an elementary Abelian group of order p'. We denote by nk the number of those systems of k elements al, ..., ak of A, for which O({al,

..., (Zk}) = pk (0:!9 k 5 n). It is evident that for the choice of aI

PRINCIPAL THEOREM

367

there are exactly p" - 1 possibilities, then for the choice of a2 exactly p" - p possibilities, etc., so that

nk = (p" - 1) (P" - p) ... (p" - pk-I).

(91.16)

On the other hand, obviously nk

n

=

k

Ikk,

so that from (91.16), after cancellation by a power of p, n

- 1) (P- 1)(P2- 1)...(pk- 1)

(p" - 1)

Lk]

(pn_I

- 1) ...

(p»-k+I

(91.17)

As regards a detailed theory of finite Abelian groups cf. CHATELET (1925). For the original proof of the first main theorem cf. FROBENIUS-STICKELBERGER (1879). From

among the numerous other proofs, that of FucHs (II952a) is particularly concise.

in (91.17) to all integral k

EXAMPLE 1. Extend the definition off the symbol [ k

and non-negative integral n so that I k 1 = 0, if k

J

>L

n or k < 0. Then we obtain the

o rmulae

[Al

= [n

kj(n>_0)'

[iJ

Pk[flk

1]+[k-1](">1) (kE.9).

(91.18)

These and (91.17) remind us of the properties of binomial coefficients. EXAMPLE 2. Because of Theorem 196 it would have been sufficient to prove the first main theorem for finite Abelian groups only for the special case of p-groups, but this would not have shortened the above proof. EXAMPLE 3. The characteristic of a finite Abelian group is the least common multiple

of its invariants. The elementary p-groups are identical with the groups of characteristic p. EXAMPLE 4. The module of a skew field of prime characteristic is an elementary p-module.

EXAMPLE 5. The first main theorem for finite Abelian groups is easy to prove for the special case of elementary groups; further, it is true for all elementary Abelian groups including infinite ones. The proof is as follows: Consider the set of all sets of independent elements of the given group. Since this independence is a property of finite character, the TEICHMULLER-TUKEY lemma (Theorem 17) may be applied. The existence of a maximal set of independent elements follows, which is obviously a basis. EXAMPLE 6. Let A be a finite Abelian p-group with n equal invariants p". If al, ..., a" is a basis of A, then all its bases w, ..., a), are given by

where the exponents au are subject to the restriction that their determinant is not divisible by p. The proof is furnished by CRAMER's rule (Theorem 172).

13 R.-A.

I a., I

FINITE ABELIAN GROUPS

368

EXAMPLE 7. The order of the full automorphism group of an elementary Abelian group of order p" is, according to (91.16), equal to (n" =)

(p"-

1)(P"-p)...(p"-

EXERCISE 1. For an arbitrary group G we denote by G" the set of (distinct) nth powers of the elements of G. (This definition contradicts that of the product of complexes, but will cause no misunderstanding, since G", as a power in the sense of the multiplication of complexes, is trivially equal to G, so that G" is only used in the sense defined here.) We call G" the nth power of the group G. A counterpart to this is the nth radical Je of the unity element e of G, by which we now mean the set of those elements of G whose nth power is equal to e. In particular, for an Abelian group G, since a"fl" = (an)" (a, t3 E G), both G" and Z ° are groups, and we have the homomorphism

G-G"(a-a"), whose kernel is V Fe. Now let G be a finite Abelian p-group with the invariants

Pl>...>P" and let n be a divisor of P1. If t denotes the number of invariants which are greater than n, then

n-1P1,...,n-`Pr or

t

n, ..., n, P1+1, ..., P, are the invariants of G" or

of V ; respectively.

EXERCISE 2. Let A be a finite Abelian group with the invariants P,

P11

,

Q1,

, Q,

and let (91.19)

respectively be divisors of them. The invariants of a subgroup of A always constitute a system (91.19), and conversely, every system (91.19) constitutes the invariants of a suitable subgroup of A. (It is intended that the Q. = 1 are to be disregarded.) A similar

theorem holds for the homomorphic images instead of the subgroups of A. Hence and from Theorem 216 it follows that in a finite Abelian group the pth radical of the unity element (p a prime number) consists of at least as many elements as the pth radical of the unity element in a homomorphic image of the given group. This is not a trivial result and does not hold for infinite A belian groups, since e.g., a torsion free Abelian

group also has mixed and periodic groups (different from the unity element) as homomorphic images. The theorem is not valid for non-commutative finite groups either, as the example of the quaternion groups shows, since it has for a homomorphic image the four-group. EXERCISE 3. Prove the uniqueness of the invariants of the finite Abelian groups by the help of the first part of the theorem in Exercise 1, without applying the REMAKKRULL- SCHMIDT theorem.

HAJOS'S THEOREM

369

§ 92. * Hajos's Main Theorem Let an arbitrary finite Abelian group A with the unity element a be given. A complex of A of the form -1>

(o(a) >_ e >_ 2) (92.1) (a)e = 00'a, ..., ae is called an (e-term) simplex. This is a (cyclic) group if, and only if, it contains ae, i.e., if o(a) = e. If a schlicht product (al)e, ... ((Xn)en

of simplexes is a group (which must then be of order el ...

(92.2)

then

we call it a Haj6s product (or Haj6s decomposition).

For the moment let t, S, g ( ) denote a Hajos decomposition of a given A, a simplex factor occurring in it and the property of being a group, respectively. We consider both logical formulae (cf. REDEI (1965c))

(3 ') (V S)g(S), (V sf') (3 S)g(S), (3 = there is; V = for all) which arise from one another by logical dualisation i.e. by interchange of 3, V. The first formula is identical with the FROBENIUS-STICKELBERGER

main theorem; the second one is also true and is written as THEOREM 218 (HA.r6s's main theorem). In every Haj6s product at least one factor must be a group.

This is called the second main theorem for finite Abelian groups. Its proof will be surprisingly difficult. If in a schlicht product (92.2) of simplexes all the partial products (al},, ... (Mk).,,

(k = 1, ..., n)

(92.3)

are groups, then we call it a normal product of simplexes. It should be noted that akk then obviously belongs to (al)e, ... (ak-1)e,_, (k = 2, . . ., n). By a seminormal product of simplexes we mean a product which may be turned into a normal product by a permutation of the factors. THEOREM 218' (refinement of HAJbs's main theorem). The Haj6s products are identical to the seminormal products. Since in a seminormal product at least one factor is a group, Theorem 218 is a consequence of Theorem 218', therefore we need only prove the latter. The first part of it, namely that every seminormal product is a Hajos product, is trivial. For the proof of the other part we need several preliminary steps. If e is a prime number, then (92.1) is said to be a prime simplex. PROPOSITION 1. If theorem 218 holds for the Hajos products of prime simplexes, then it holds in general. Because (a)or = (a)e(ae)f

(o(a) ? ef; e, f ? 2)

FINITE ABELIAN GROUPS

370

every simplex - and therefore every Hajos product - may be converted into a schlicht product of prime simplexes. Therefore it will suffice to show the following: if a simplex (a), is decomposed into a schlicht product

(a)e = KL

of two complexes, of which K (s s) is a group, then (a)e is a group, too. Let K (A e) denote an element of K. Because KK = K then K(a)e = (a),,

Hence K = ak (1 -< k <_ e - 1), ae = Kae-k, ae E (a)e, therefore (a)e is in fact a group. This proves Proposition 1. We call o(a) ... the height of (92.2). PROPOSITION 2. If a factor of a Hajos product (92.4)

(a)e, ... (an)e,,

is a group and the Haj6s products of smaller height than (92.4) are seminormal, then (92.4) is seminormal, too. For the proof we may assume that (ai)ej is a group. Let Q --> Q' be the natural homomorphism of the group (92.4) with its factor group mod This is then equal to (a2)e. ... (a;)e,,, therefore the latter is a Hajos product

[in the factor group A/(ai)e,], and of smaller height than (92.4). As the order of succession of the factors is immaterial, because of the assumption,

it may be supposed that all the (a)e, .

.

(k = 2, ..., n)

. (ak)e,E

are groups. But since (aI)ej is also a group, it follows that the products

(k= 1, ..., n)

(al)e, ... (ak)ek

are likewise groups, i.e., (92.4) becomes, after the rearrangement of the factors, normal. Hence, Proposition 2 has been proved. PROPOSITION 3. If (92.4) is a Haj6s product of prime simplexes for prime numbers ei = pi then pi I o(ai)

(i = 1, . . ., n),

(92.5)

and

(al)p1 ... (ann)p =

(a), .

. .

(a.),

(p14'ti ;

i = 1, .

erefore the left-hand side of (92.6) is a Haj6s product.

.

., n).

(92.6)

HAJOS'S THEOREM

371

It will suffice to verify that: if a schlicht product (a)PK of a prime simplex (a),, and a complex K () s) is a group, then p I o(a),

(a°)PK = (a)PK

(pf't).

(92.7)

Because a E (a)PK we have a(oc),, K = (aa), K, and therefore , after cancellation of identical elements we obtain aPK = K. Consequently in general

aPkK = K

(k E 3).

(92.8)

Since, according to this, a'K depends only on the residue class i (mod p), (92.72) is true.

If (92.71) were false, then it would follow from (92.8) that MK = K. This contradiction proves Proposition 3. For a prime p we call every normal product of the form

(al)P ...

(p2 I o(a2), ..., o(a,J)

(92.9)

briefly a p -product (n Z 1). Since (92.9) is then a group, all the power products a

-'Y)

(92.10)

belong to this group. We call (92.10) a special power product, when it is prime with respect top and every ik (k z 2) equals 0 or is prime with respect to p. PROPOSITION 4. Every element a ( s) of a p -product (92.9) may be written as a special power product (92.10). We make the induction assumption that the assertion is true for smaller n. We can write at (uniquely) as

a=acM . . . a;

,"

( 1 <Jm

For the case m = 1 the proof is complete. If m > 1 then am is an element other than s of (ai, ... (an, _1)P. Since the latter is a p-product, because of the induction assumption, then

ate,=ai...aM-1 where it is prime top and every ik (k > 2) equals 0 or is prime to p. From

a=ai ... am-, Proposition 4 then follows.

mPtlmam+l...ann

FINITE ABELIAN GROUPS

372

PROPOSITION 5. If in' two p -products the first factors are different and all their factors occur among the factors of one and the same Hajds product, then these p-products have no common factor. For the purpose of the proof we assume that the assertion is false. After

cancelling some of the last factors of the two given p-products we reach a point where only the last factors of both p-products are equal. Let this common factor be (cc)p. Then the p-products obtained are of the form P(a)p and Q(a)p, respectively, where P, Q are some p-products. It follows that ap (A s) is an element of P as well as of Q. But the factors of PQ are different and are factors of one and the same Hajbs product. This contradiction proves Proposition 5. LEMMA. I f al, ..., a" (n >_ 1) are elements of an Abelian p -group with the property that for G. = {ocl, . . ., a,,) and for all the Gk = {a1,, ..., a;k)

(1<_'I<...
O(G") = p" , O(Gk)

pk

(92.11)

(1 < k < n - 1) ,

(92.12)

then there are powers sI, . . ., s" of p for which the product

(al')p ...

(92.13)

is seminormal.

This is true for n = 1, since then Gl = (al) = (ml)p. If n > I we assume

the assertion for smaller values of o(al) ... o(a,J. It should be noted that, by (92.12), o(al), ..., o(an) >--_ p.

First we consider the case, where > always holds in (92.12). Then O(Gk) Z pk+1

(92.14)

for all the Gk with 1 <- k < n - 1. In particular, by (92.11), {a2, ..., a") _ = G", so that even R

Hence, and from (92.14), it follows that the assumption of the lemma remains valid, if oci is replaced by ai. Since o(afj) < o(al), the lemma follows for this case by the induction assumption. We have still to consider the case where equality occurs in (92.12), i.e. where, for a fixed Gm (with 1 < m <_ n - 1) O(G,") = p".

(92.15)

For a suitable order of the al, ..., a" we may assume that Gm =loci, ...,am).

(92.16)

HAJOS'S THEOREM

373

From (92.15), (92.16) and from (92.12) (for 1 5 k < m - 1) it follows, by the induction assumption, that there are powers sl, ..., sm of p such that (92.17)

(ai-')p ... (xm ),,

is seminormal. For a suitable order of the al, ..., am, (92.17) will even be normal. Consider the natural homomorphism e - O' of G. onto Let Gk denote the image of Gk. By (92.11), (92.12), (92.15)

O(G')=L'

(m + 1 Sk<_n - 1).

m,

If we now take into consideration that, by (92.16), Gn = {a;,,+L, ..., cc, then we see that the assumptions of the lemma are satisfied with

a;, (instead of al,

. .

Since o(a,,+1) ... o(a;,) < o(a1) ... s of p such that the product

.,

it

follows that there are powers sm+l, ..., / '8mi-l

lamt L )p ... (a'Sn)p

is seminormal. For a suitable order of succession of the am+L, ..., x it will even be normal. Since (92.17) is also normal, the same follows for (92.13). Consequently the lemma is proved. We are now able to prove Theorem 218'. It is only necessary to show that every Haj6s product is seminormal. By induction it follows from Propositions 1 and 2 that we may then restrict ourselves to the Haj6s products of prime simplexes. As the first case we consider a Haj6s product of the kind

(a)p ...

(p a prime number).

(92.18)

Then, evidently, the lemma is applicable to these al, . . ., a,,. If thereby in (92.13) we have st = ... = s,, = 1 then the proof will be complete. Otherwise a contradiction will arise. We may assume that in (92.13) for some m (>_ 1) P l sl, ..., Sm ;

sm +1 = ... = S. = 1.

Temporarily we make use of the group ring 7(G) of G. Put

[a]p = s + a +

... + ap-1 (E 3"(G))

(a E G).

Since the group (92.18) contains the element al, it follows that (s - a) [aJ]p... lot. 1P = 0.

(92.19)

374

FINITE ABELIAN GROUPS

Because of (92.19), we then have so much the more

(E - ai) ... (E - am) am+1Jp ...

0.

On multiplying out, the term s must be cancelled by another term. This contradicts the fact that the product (92.13) is schlicht. Accordingly Theorem 218' is true for the case considered. As a second case we consider a Hajos product ((X>)' ...

((X"),.,

(92.20)

where pl, . . ., p are not all equal prime numbers. We have to show that

(92.20) is seminormal. We make the induction assumption that the assertion for similar products of smaller height is true and assume that it is false for (92.20). According to Proposition 2, no factor of (92.20) is then a group. From Proposition 3 it follows that pi is a proper divisor of o(a) (i = 1, ..., n). Furthermore, it easily follows from Proposition 3 and the induction assumption that every o(ai) must be either the product of pi and another prime number or a power (> pi) of pi. Accordingly we can write (92.20) with a different notation in the form in

F1 (ail'f)P, n (7j)rl ,

j=I

1=1

(92.21)

where

o(al) = p, s o(fl1) = q1

(i = 1, ..., m),

(92.22)

o(yj) = power (> rj) of rj

(j= 1, ... , n)

(92.23)

and all the pi, gi, rj are prime numbers. Not all the pI, . . ., pm, r1, ..., r are equal.

From Proposition 3 it follows that (92.21) is equal to all those m

n

(cc?'fii) 1=1

1

II (yj)r,

(92.24)

j=1

in which p1,f'al (i = 1, ..., m), rj,f'cj (j = 1, ..., n). The case m = 0 is impossible, since then the product of those factors of (92.21), in which rj = r1, would be a Hajos product. but this is false according to what has been proved previously. Consequently m > 1. We consider a certain i (= 1, . . ., m). By (92.24), after cancellation of (92.21) remains unaltered. But since the height of (92.24) is diminished, a seminormal product is obtained from it by the induction assumption. We arrange the factors so that this product is normal. Then the first factor

HAJ6s'S THEOREM

375

(which is a group) must be exactly (a;),, by (92.22) and (92.23). Therefore this normal product, on account of (92.22) and (92.23), necessarily has the form (92.25) (Yw)p, (xr, fl;)p,.... (P; # p;), (x;)p, (Yu)p,

is guaranteed by the fact where the occurrence of the factor (x;., that the group (92.25) is equal to (92.20), and so is not a p;-group. On the may be lacking. From this it necessarily other hand the . . ., follows that m > 2. We write P; = (x;)p, (Y.), ... (Y.)p,

This is a prproduct, thus a p; group; furthermore, we must have (a; AY" E P; . By (92.22) first [3?'' E P;, then [again by (92.22)] f;- E P5

(92.26)

.

Of course, several P1 and i' may belong to an i, but we choose a fixed P; and i' f o r every i. Then i --- i' will be a mapping of the set <1, . . ., m> into itself without fixed elements. Consequently <1, . . ., m> has a subset fit,

which consists of at least two elements and is permuted by i -Now, according to (92.24), H (x;)p, fT (Yj)r,

1'.

(92.27)

j=1

i=1

is a Hajos product. The factors of Pi appear among those of (92.27). So it follows from Proposition 5 that the P; (i E fl) have pairwise no common factor. On the other hand, Proposition 4, by (92.26), yields a representation of i4;, as a special power product o,

cu

a; Yu

.vcw

iw

Then we have

..., yx ca, ..., change, depending on i). If we multiply and take into consideration that i' also runs these equations for i E through the elements of 9R, we obtain on the right an expansion term, other than e, of one of the Hajos products (92.24). This contradiction proves Theorem 218'. (Of course, y,,,

The above proof is from REDEJ (1955b). For the original proof cf. HAJ6s (1942). This was simplified by REDE1 (1949) and SZELE (1949-50a), cf, also REDEI (1954b)

For the connection between Haj6s's main theorem and others, see HAMS (1942),

13/a R.-A.

FINITE ABELIAN GROUPS

376

R9DEI (1949-50, 1955c). Similar questions were dealt with in HA16s (1950a, 1950b), FARM (1949), BRUUN (1953a, 1953b), R9DEI (1947, 1950b, 1965b) and SANDS (1957,

1959, 1962a, 1962b, 1964). For the extension to infinite Abelian groups cf. FucHs (1958a, 1958b). For a generalization in the finite case, see RfDEI (1965d).

§ 93. The Character Group of Finite Abelian Groups

We first consider two modules 4, 8 (without operators). Furthermore,

let G be the module of all linear mappings of 4 into 8. We denote arbitrary elements of c.Q and G, by a, b and a, f4 respectively. The image of

a under a is denoted by as (E $) and we agree to write as for as (i.e., we put as = aa). Then the linear mapping a consists of all the correspondences

a --ma (= am)

(a E d4).

As defining properties (according to § 68) we have a(a + b) = as + ab, (a + #)a = as + fia.

It is important to ensure that under interchanges of a, b with a, fi these equations are interchanged, i.e., as a whole they remain invariant. This implies, conversely, that the correspondences a

as (= aa)

(a E G)

(for each a) constitute a linear mapping of G into $. For a given 4 with suitable 8, it may happen that in this way all the distinct linear mappings of G into 8 arise, so that cQ and G may be simultaneously considered as modules of all the linear mappings of the other into 8. This implies an important concept which we wish to utilize below in the theory of finite Abelian groups. Jt is only a formal difference that in 4, G, $ we pass to the multiplicative way of writing the compositions and introduce the term "character" instead of "linear mapping". It is essential that ,$ should be suitably chosen. From now on c.P will thus designate an arbitrary finite Abelian group. Its characteristic, i.e. the maximum of the orders of its elements, is denoted by m. Furthermore let 8 be a finite or infinite Abelian group, which we,

however, subject to the condition that those elements in it whose order is a divisor of m constitute a cyclic subgroup of order m. Let G be the set of all the homomorphic mappings of 4 into $. The image of a (E c0) under a (E G) is denoted by aa,or alternatively by am: as = aa.

(93.1)

Then (a(ab) _ )

aab = as ab

(a, b E (p; a E G).

(93.2)

CHARACTER GROUP

377

We define the product a# of two elements a, 8 of G by afl E G and ((a#) a =)

(a E 4; a, fl E G).

afJa = as - #a

(93.3)

According to the above, in this way G is now an Abelian group, which we call the character group of 4t according to 8 and its elements are said to

be the characters of c0. (Cf. the note at the end of this paragraph.) We repeat that the character a means the homomorphism

a-->as(=ax)

(aEd)

(93.4)

of 4 into 3. Because of the homomorphism (93.4), o(oca) I o(a). On account of the assumption made with respect to 8 it follows from this that for a fixed a (E d) there are only finitely many possibilities for the images aa. Because of the finiteness of cot, G is also finite. Since, by (93.2) and (93.3), ama =

= aa' and a' is the unity element of c.0, the unity element of G is a"'. This means that the characteristic of G is a divisor of m. Hence it follows that 8 satisfies the above conditions in connection with G (instead of C4). Therefore, according to our definition the homomorphic mappings of G

into 8 are the characters of G. Now it follows from the symmetry of (93.2) and (93.3) that the mappings a -> aac (= ma)

(a E G)

(93.5)

certainly are characters of G. If in this way all the different characters of G occur, it means that, conversely, c0 may be considered as the char-

icter group of G. In fact, this case always happens, as is implied by THEOREM 219. Every finite Abelian group d is isomorphic to its character

group G, and furthermore by (93.5) cPi may be regarded as the character group of G. In the proof we shall actually construct the character group G of 4t. For this purpose we need a prime-power basis

al, ..., a,

(93.6)

of d1. The invariants of dt are then the orders Pi = o(al), . . ., P, = o(ar);

(93.7)

m = [PI, ..., P,]*.

(93.8)

furthermore, we have

By,8," we denote the cyclic subgroup of ,g of order m, which consists of those elements of 8 whose order is a divisor of m. We select from $.

378

FINITE ABELIAN GROUPS

arbitrary elements j,

., 3r with

o($I) = F1,. .., o0r) = Pr

(93.9)

which shall remain fixed throughout the following. We now write the elements a of 4 in their basis representation

a=a...ar

(ik=0,...,Pk- 1; k=

(93.10)

and prove that all the characters of 4, i.e. the elements a of the character group G, are given by a

as = $i'' ... $:'r

(jk = 0, ..., Pk - 1; k = 1, ..., r).

(93.11)

On the one hand, by the homomorphism (93.4), every character a of C4 satisfies o(aak) I o(ak); hence, by (93.7) and (93.9), the existence of certain equations (jk = O, ..., Pk - 1; k = 1, ..., r) aak = $k follows. This shows, by (93.2) and (93.10), that a is in fact given by (93.11).

On the other hand because of (93.9), it is evident that only distinct characters, i.e., homomorphisms a of 4 into 8, are given by (93.11). This proves the statement relative to (93.11). Now let us take the elements ak (k = 1, . . ., r) of G, which are obtained

from (93.11) by putting jk = 1 and j, = 0 (10 k; 1 < l 5 r). By (93.3) and (93.11) it follows that

a = al' ... aJrr

(jk = 0, ..., PA: - 1;

k = 1, ..., r)

(93.12)

are all the distinct elements of G, and furthermore that

o(ai) = PI, ..., o(ar) = Pr

(93.13)

follows from (93.9). Hence we see that ai, . . ., ar is a prime power basis of the character group G and its invariants agree with those of (4. Consequently the first assertion of Theorem 219 has been proved.

Since according to this O(G) = 0(d4), it will suffice to prove with respect to the second assertion of the theorem that distinct mappings (of G into 8) are provided by (93.5) for different a. Because of (93.2), it is sufficient to show that as can be the unity element of $ for all the a (E G) only if a is the unity element of c.0. As this is a trivial consequence of (93.10) and (93.11), Theorem 219 has been proved.

This theorem means, among other things, that all the characters of At and G are simultaneously given by (93.4) and (93.5) respectively. Since (93.4), (93.5) arise from each other simply by interchanging a and a, we have the important duality principle for finite Abelian groups, which is as

CHARACTER GROUP

379

follows: Every proposition, which follows from the definition of the character

group of a finite Abelian group, retains its validity after interchanging the roles of these two groups. In future, co and G will have the same meaning as in Theorem 219, while rand V will designate the lattice of all the subgroups of CO and G, respectively. For a subgroup V of 4 we denote by the set of those elements

a of G, for which

as = 1

(for all a E X) .

(93.14)

From (93.3) it follows that these a constitute a group, so that A' is a subgroup of G (i.e., A' is the group of those characters, which map onto 1). Accordingly

' ->

'

(,° E ?)

(93.15)

is a mapping of the lattice ?' into the lattice V. Similarly let the dual mapping

(of V into ?) be denoted by H -> H'

(93.16)

(H E V),

in which the group H' (c c0) of those elements a of 4, for which

as = 1

(for all a E H)

(93.17)

is assigned to the subgroup H of G (i.e., H' is the group of those elements of (4, which are mapped onto 1 by the characters in H). Since characters were first

considered by Dirichlet, we call (93.15) and (93.16) the first and second Dirichlet mappings of ? into V and of V into r, respectively. Furthermore "' and H' are said to be the Dirichlet images of and H, respectively. THEOREM 220. The first Dirichlet mapping maps the lattice ?' of a finite Abelian group 4 one-to-one onto the lattice V of the character group G of 4t. The dual proposition holds for the second Dirichlet mapping where the roles of ?, V are interchanged. These two mappings are inverse to each other Finally, the first (second) semilattice of C.4 and the second (first) semilattice of G are mapped isomorphically one onto the other. The following two (dual) results follow immediately from the definitions:

H c K H' K'

,

(93.18)

where,

and H, K denote subgroups of 4t and G, respectively. Similarly from the definitions follow the (dual) properties

l°" Q X, H"

H

(93.19)

380

FINITE ABELIAN GROUPS

for every subgroup X and H of c4 and G, respectively, where we understand that A"' = (a°')' and H" Then we prove the (dual) properties

c4/,Y :

G/H ., H',

(93.20)

for the mappings (93.15), (93.16). It will suffice to prove the first of these.

We consider a character A (with respect to 8) of the factor group cP/'P and define a mapping a --> as of ci (into 8) by

as = A(a')

(a E (P)

.

(93.21)

Then for a, b E c

aab= Therefore we have a E G. In particular, for all a E 2f°, as = AX = 1, so that a even E X'. Conversely, if a E :V', then as = 1 (a E t'). Hence, because of (93.2), it follows that for two elements a, b (E d) with aX = bX we have as = ab. Thus if we put A(a ) = ma, we have defined a (one-to-one) mapping a2P --> A(aX) of cP/X (into 8). Here, because of (93.2), A(a 2P ba") = A(ab

') = aab = as ab = A(a2f°) A(bX )

.

Hence A is a character of c4/.T. Consequently we have shown that (93.21) gives exactly all the characters

a E a', if A runs through the characters of d /X. Since different a evidently correspond to different A the character group of c0/V is mapped one-

to-one onto 7' by A --* a. Finally, if B, 9 is a second pair of characters assigned to each other, then

afla = as #a = A(a.P) B(aT) = AB(aMj . This completes the proof of (93.20). This easily leads to the proof of Theorem 220. From (93.20) it follows that

O(P)O(Yt') = 0(c4),

O(Y')O(Jt") = O(G).

CHARACTER GROUP

381

Because 0(4) = O(G) it follows that O(X") = 0(,7). This means that (93.19) is refined to

X"=', H"=H.

(93.22)

Accordingly (93.18) may be refined to

ar c 7 a 2L°' DH c K a H' D K'.

(93.23)

Since the lattices r, V are finite the first assertion of the theorem follows

from (93.22). The second (dual) assertion is then also true. The third follows from (93.22) and the last from (93.23). Consequently we have proved Theorem 220. NoTE. In our definition of the characters of finite Abelian groups, we can take the group of the field of complex numbers for the group 8. Further valuable properties of characters will arise from this specialization, so that usually only this case is referred to. (Cf. § 155, Example 9.) EXAMPLE 1. The first and second semilattices r1, rY of all subgroups of a finite

Abelian group 4 are isomorphic. Consider two subgroups 7, .7 of 4 with . ° c X. For their images 7, .7' in the character group G of 4 we have 7' z .7' by Theorem 220 or (93.23). On the other hand, take an isomorphism a -- q'a of 4 with G according to Theorem 219. Then q '.5Z'' are subgroups of 4, and evidently 97-'. °' op-',5I'"'. Hence we have the isomorphism

V1 - A (Jr -

(93.24)

EXAMPLE 2. From Example I it follows that in a finite Abelian group of order n the number of subgroups of order d is exactly the same as that of the subgroups of

order d- In (d I n) (91.181 is a special case of this). EXAMPLE 3. For an element a of a finite Abelian group 4 and for an element at of its character group G aa" = a"a

(93.25)

because of (93.2) and (93.3). As an application we prove that the Dirichlet image of 4" is equal le, where a denotes the unity element of G and -Je the n1° radical of e. Firstly 4" consists of all the a" (a E 4). The Dirichlet image of 4" consists of those a (E G), for which the equation ma' = 1,

i.e.,

a"a = 1,

is satisfied by all the a (E 4). This is so if, and only if, a" = e, i.e., a E . IE. This proves the assertion.

§ 94. The MSbius-Delsarte Inversion Formula Let a finite or infinite set C5 of finite Abelian groups be given such that, if a group is an element of C5, all its subgroups belong to S. Take an arbitrary module M. Let f and F be functions with the domain of definition C and the

FINITE ABELIAN GROUPS

382

range M. The problem raised and solved in a special case (see below) by MOBius, and in general by DELSARTE (1948), is as follows : What is the condition for these two functions f, F to satisfy the system of equations E f(H) = F(G) (G E C5) . (94.1) HcG

("H c G" here means that H runs through all the subgroups of G.) Of course, we are concerned with the case where the function F is given and the function f is unknown and so may be called the solution of the problem. It is almost trivial that exactly one solution always exists; the problem is to obtain the formula for solving the equation system (94.1). This can be done using the Delsarte function µ(G), which we define for

any finite Abelian group G as follows: If G is elementary of order p'° (p a prime number), then µ(G)

_

(-1)kp(2)

(k

(94.2)

0) .

If G is a non-elementary p-group, then p(G) = 0. Finally, if G is arbitrary and P,, ..., P, are its primary components, then u(G) = p(PI)

... p(P,)

(94.3)

.

THEOREM 221 (DELSARTE'S theorem). The system of equations (94.1) is equivalent to

f(G) _ Y u(G /H) F(H)

(G E C)

.

(94.4)

HcG

Further for every finite Abelian group G

_ HC-G µ(H

1 for O(G) = 1 0 for O(G) > 1

(94.5) .

Before proving this, it should be noted that (94.4) is the formula for the solution of (94.1). We call (94.4) the Delsarte inversion formula. We see that (94.5) itself constitutes that special case of (94.1), where S denotes the set of all finite Abelian groups, where isomorphic groups are regarded as

equal, and we have M = 7+. Furthermore F(G) denotes that function which equals I for the unity group and vanishes for other G. We begin the proof with the verification of (94.5). The left-hand side of (94.5) will be denoted by Y_ G. If

G=P,®...®P,

383

MSBIUS-DELSARTE INVERSION FORMULA

is the direct product decomposition of G into its primary components PI, ..., Pr, then we obtain all the different subgroups of G in the form

H=Q1®...®Qr, where Q1 has to run through all the subgroups of P1 (i = 1, ..., r). Since then, according to (94.3), we have µ(H) = µ(QI) ... µ(Q,), >G = Y- P,

.. Y- P,

follows.

Accordingly it will suffice to prove (94.5) for the case where G is a pgroup. Evidently, from the definition of the function µ, it follows that I G = Y- G,,,

where Go denotes the subgroup of those elements of G whose order is at mostp. Thus it will suffice to prove (94.5) for the case where G is elementary of order p" (n >-- 0). Of this the case n = 0 is trivial. For n >_ 1 the assertion becomes by (94.2)

(n = 1, 2, ...)

k J (- l )k p(k) = 0 k=O

(94.6)

,

L

I denotes the number of subgroups of order pk of an ele-

where

k mentary Abelian group of order p". Let us generalize 1

1 as in (91.18).

I

k

Then we may write the left-hand side of (94.6) in the form pk I

n- 1 k

k

1)k p(_)

+E

Ink

-1

- 1, (- 1)k

p(:)

,

(94.7)

where( we have to sum over all the k (E J). The first sum in (94.7), because 11,

k + 2I = Ik

is equal to

2

In - 11(-

k-1

I)'-' p(2)

Hence we see that (94.7) vanishes. Hence (94.6) and (94.5) have been proved.

We now see that, by (94.5),

µ(G/H) _ H9G

1

for O(G) = 1,

0 for O(G) > 1

.

(94.8)

FINITE ABELIAN GROUPS

384

For, if we apply (94.5) to the character group of G instead of G and apply the formula in (93.201), then we obtain (94.8).

The first assertion of Theorem 221 still remains to be proved. For this, we substitute, on the one hand (94.1) into (94.4), and, on the other, (94.4) into (94.1). This gives

f (G) = E E µ(G/H) f(K)

(G E (S) ,

E Y µ(H/K) F(K) = F(G)

(G E Cam),

HcG KcH

(94.9)

or

(94.10)

HcG KcH

respectively. It will suffice to prove that (94.9), (94.10) are identical, i.e., they are satisfied for all the functions f and F, respectively. These assertions come from the following: Kc

Gµ(

K SHSG

G

l

(1 for K=G, = to for KcG,

µ(H/K) =

94.11

(

)

1 for K= G, j0 for K c G,

where K is a subgroup of G and H has to run through those subgroups of G,

for which K c H S G. Now (94.112) is true, because this formula is obtained if we apply (94.5) to G/K instead of G. Since, according to

the second isomorphy theorem (Theorem 126') G/H

(G/K) / (H/K)

,

(94.111) follows from (94.8), by applying it to G/K instead of G. Consequently we have proved Theorem 221. We shall now apply this theorem to the special case where C5 is the set of all finite cyclic groups. Since these groups are essentially given by their

orders, so for this case we may introduce the notations u(n), f(n), F(n) instead of µ(G), f(G), F(G), where n is the order of the (cyclic) group G (n = 1, 2, . . .). We call µ(n) the Mobius function, which has, according to the above general definition, the following meaning: µ(n)

_

-

( {

1)k

0

if n is the product of (k >_ 0) different prime numbers, if n has at least one multiple prime factor. (94.12)

The system of equations (94.1) for this case is

E f(d) = F(n)

(n = 1, 2, ...) .

dIn

Again from Theorem 221 we obtain the following

(94.13)

MdBIUS-DELSARTE INVERSION FORMULA

385

THEOREM 222 (MSBIus's theorem). The system of equations (94.13) is equivalent to

f(n) = > u(d-'n) F(d)

(n = 1, 2,...) ;

(94.14)

din

furthermore

µ(d) din

1 for n = 1

= 0forn>1.

(94.15)

This special case (94.14) of (94.4) is called the Mobius inversion formula. EXAMPLE. The formula

v(n) _ E d1(nd-1)

(94.16)

din

for Euler's function follows from (94.15) and (84.22). EXERCISE. Formula (94.8), and consequently Theorem 221, may be proved without using the first main theorem for finite Abelian groups and the character concept.

§ 95. Zeta Functions for Finite Abelian Groups We shall define here certain functions, which allow the lattice of subgroups of a finite Abelian group to be examined more closely.

In this paragraph we denote by [S] the number of the elements of an arbitrary finite set S. (Consequently, if G is a finite group we could write [G] for O(G), nevertheless we retain for this case the notation O(G).) We denote an arbitrary non-negative integer by n and the set <1, ..., n> by 1.

(For n = 0 we take t = O.) Let z (= 1,2 ....) denote a positive integral variable. Let a finite Abelian group G be given with unity element e and arbitrary

(not necessarily distinct) subgroups A1, . . ., A. of it. We denote by Ag

(C c %) the product of the subgroups Ai with i E C5. (If e = 0 or
e(z) = e(z; A1, ...,

Z (-1)l61(O(A,,))-Z .

(95.1)

This sum consists of 2" terms. In particular, the term 1 corresponds to e = O.

If n = 0, e(z) = 1. For certain reasons to be explained later (see Example 8) we call

s(z) = S(z; A1, ...,

e(z; A1, . . ., An)-1

(95.2)

a zeta function (for finite Abelian groups). [If e(z) = 0, then one has to take C(z) = oo ]. Of course, of the functions (95.1), (95.2) it will suffice to examine only the first, which we may, somewhat inexactly, also call a zeta function.

FINITE ABELIAN GROUPS

386

To do this, we define the following function +p(z, G). If G = P is a p-group and t the number of its invariants, then ,-I

+p(z, P) = ri (1 - pl_)

(t >_ 0).

(95.3)

r=0

(For t = 0 we take y,(z, P) = l.) If, in the general case, P1, ..., P, are the primary components of G, then we define 'V(z, G) = r l 'p(z, Pk)

(95.4)

k=1

THEOREM 223 (main theorem for zeta functions for finite Abelian groups). We have

e(z) = o(z; A1, ..., An)

y'(z, G/H)O(H)--- (n >_ 1) ,

(95.5)

HcG

where the sum extends over those subgroups H of G, which contain no AI. In the proof H, K, L will always denote subgroups of G. Further, denote by Nk(K) the numbers of those sets C (9 91), for which

From (95.1),

[C5] =k, AS=K = n N(z)

(k=0,...,n).

(- 1)kNk(K)O(K)

(95.6)

k=0 KcG

For any K we denote by 9 the set of i (E fit) with At 9 K. Thus AC, (S c %) is a subgroup of K if, and only if, C5 c R. Consequently, if we take for a fixed k (= 0, ..., n) all the S with CF S St, [C5] = k, then Ac, produces every subgroup H of K exactly Nk(H) times. Then

H(k = 0,..., n). ()=Nk(H) k 191

According to the Delsarte inversion formula (Theorem 221) we obtain from this

Nk(K) =HE µ(K/H) I[l]l k

(k = 0, ..., n),

-K

where j denotes the set of those i (E 'Tc), for which Ai c H. If we substitute this in (95.6), then we obtain

(1) A(z) = I E E (- I)k µ(K/H) k O(K). k=0 KcG HSK

(95.7)

ZETA FUNCTIONS

Now

387

ik=o

-1)k (E1j k

equals 1, if [fl = 0, i.e., if H contains no A;. Otherwise, this sum vanishes. If we take (95.7) into consideration, then after a rearrangement of the terms we obtain MO(K)_z

P(z) = I' E 1z(KI

,

HcG HcKcG

where we have to interpret the first sum as in (95.5) and in the second sum we have to sum over the K with H c K c G. Since O(K) = O(K/H) O(H)

and if, for the sake of brevity, we put

f(z, G) = Y µ(L)O(L)-z, LcG this is transformed into

e(z) = Y' f(z, G/H)O(H)--.

(95.8)

(95.9)

HcG

Now we have to show that f(z, G) = ip(z, G), for then (95.9) implies the validity of (95.5). From (95.8) and (94.3) we see that the function f(z, G) has the property (95.4). Therefore it will be sufficient to prove the assertion for the special case where G = P is a p-group. We denote by Q the group of those elements ofP whose order is at mostp. From (95.8) and the definition of the Delsarte function p we have

f(z, P) = E µ(L)O(L) -z (= AZ' Q)) .

(95.10)

LcQ

We write O(Q) = p`.

Then t is the number of invariants of Q and also of P. Because of (95.10) and (94.2) becomes

f(z,P)= E (-

1)1

(b[tJp-Iz LJ

I=o

We denote the right-hand side by g(z, t). It will suffice to show that this

function for t = 0, 1.... is equal to the right-hand side of (95.3). If we generalize the symbol [n ], as in § 91, Example 1, then we may put

9(z, t)

1)1p(2) [ t]p-Iz,

(95.11)

L

where we now have to sum over all the integers 1. We shall show that

FINITE ABELIAN GROUPS

388

(95.11) is equal to the right-hand side of (95.3) for all the integers z (not only for the positive ones). If t = 0, both are equal to 1. Therefore it is sufficient if we prove that

9(z, t) = 9(z - 1, 1 - 1) (1 - p-Z)

(t = 1, 2, ...) ,

(95.12)

for then the assertion follows by induction from (95.3). Because of (95.11) and (91.182) we have

p(2)p [ t

9(z, t)

t

1 I P-1-

1

+ E (- 1)p( [ 1 -

lip-rZ

If we write, in the second summand, I + 1 instead of 1, then the validity of (95.12) follows from (95.11) with a little computation. Consequently we have proved Theorem 223. THEOREM 224 (theorem of inertia of finite Abelian groups). 0:!9

I

(n>-1; z = 1,2,...)

(95.13)

is always true. Since, according to (95.3) and (95.4), we always have ip(z,G)?0

(z = 1, 2, ...),

(95.14)

so the left-hand inequality of (95.13) follows at once from Theorem 223. In order to prove the right-hand inequality, we need the formula E 0(Ac) -Z P(z; A;1 Ac, /AC-; , ..., ArAc/ Ac) = 1,

(95.15)

where we sum over the C5 (S 91) and it < ... < ik denote all the different elements of 91 - S. Before giving the proof let us observe that the left-hand side of (95.15) in particular contains the term a (z; A1, . . ., A,.), corresponding to the case S = 0; from this and from the left-hand inequality of (95.13), already proved, the right-hand inequality will follow. Therefore it suffices to prove (95.15).

With this end in view we write the left-hand side of (95.15), according to (95.1), as follows:

E

E

csst sst-c

(-1)[cl] 0(A51A51Ac)-Z0(Ar) Z

389

ZETA FUNCTIONS

We write for this

E (- 1)151 O(AG Ac)

E

After a rearrangement of the terms, making use of the fact that ACtAG = AC,us, we obtain

E E (- 1)121

O(A,,)-z.

919;91 a39%

The inner sum is equal to 1 for S21 = 0, and vanishes for 910 0. So (95.15) and also Theorem 224 have been proved. EXAMPLE 1. For n = 2 the definition (95.1) (using a simplified notation) becomes

p(z; A, B) = 1 - O(A)-` - O(B)-` + O(AB)--', where A, B are arbitrary subgroups of G. The formula O(AB) =

O(A)O(B) O(A fl B)

(95.16)

follows from AB /B u Al (A n B). Accordingly

e(z; A, B) = 1 -

O(A)1

6(B)1

+

O(A) O(BB

whence the truth of Theorem 224 for the case n = 2 becomes obvious. EXAMPLE 2. In an integral domain R with a greatest common divisor we have (95.17)

((X, fi E R, # 0),

[a,

which is completely analogous to (95.16). This formula (95.17) may be generalized to arbitrarily (though finitely) many elements. So, e.g., [a,#, V l* =

a#y((Z, fl, y)*

(aA*(a, y)*(fl, y)*

((X, fl, y E R,

0).

(95.18)

(95.16) cannot be similarly generalized, since for three subgroups A, B, C of a finite Abelian group, in general, O(ABC)

O(A)O(B)O(C)O(A fl B fl c)

O(A fl B)O(A fl noon C)

(95.19)

holds. [If, e.g., A, B, C mean the three proper subgroups of the four-group, then the two sides of (95.19) are equal to 4 and 8, respectively.] No generalization of (95.16) for three or more subgroups of a finite Abelian group is known. Therefore the truth of Theorem 224 in case n > 3 is not immediately evident.

390

FINITE ABELIAN GROUPS

EXAMPLE 3. We are able to generalize (95.1) to the case where G is a (finite) noncommutative group, if we define A(Z as the subgroup of G generated by the A, (i E G). Theorems 223 and 224 are not then valid. (See Example 5.)

EXAMPLE 4. Apply (95.5) to the case where Al, ..., A. are all the subgroups P, Q.... of G of prime number order. In this case, on the right-hand side of (95.5) only H = I is admitted, consequently e(z; P, Q.... ) = i,(z, G).

(95.20)

EXAMPLE 5. By three counter-examples we show in what manner Theorem 224 cannot be improved.

1. Let G = D2p be the dihedral group of order 2p (p an odd prime number). It has p subgroups Al, . . ., A, of order two. We compute e(z; Al, ..., A,) (as defined in the generalization in Example 3). We now have 2° terms in (95.1), half of

which are positive, the other half negative. One of these terms is 1, and p terms are equal to -2-z. As two different At, Ak generate the whole group G, so the other terms are all equal to + (2p)-z. Hence

e(z; Al, ..., Ac) = 1 - P + P(2p) =

1

(95 21) .

2°

We see that the right-hand side has no lower bound for z (= 1, 2,. ..), thus Theorem 224 is false for finite non-commutative groups. 2. Let G be an elementary Abelian p-group of order p` (t Z 2). If A,, ..., A are all the subgroups of order p of G, then, according to (95.20) and (95.3), (95.22) e(z; AI, ..., A,,) = (1 - P-=) (1 - p1 ) ... (I - pr-1-`). This is positive for z > t, but vanishes for z = 1, ..., t - 1. Consequently, the

lower bound given in Theorem 224 for e(z) is exact.

3. Extend the definition (95.1) to all real z (> 1). Then Theorem 223 retains its validity, but - which is rather surprising - Theorem 224 is false and even e(z) is neither bounded below nor above, as is seen from Equation (95.22). [We see likewise that (95.22), in the successive open intervals (1, 2), . . ., (t - 1, t), is of alternating sign. ] EXAMPLE 6. The formula

e(z; A2, ..., An) = e(z; A....., e(z; AlA2/ A,, ..., A1A

/AI)O(AI)-2

(n > 1)

(95.23)

is easy to prove. From this and from the first part of Theorem 224 [from the lefthand side of (95.13)] we obtain the monotonicity of the zeta functions i.e.

e(z; All. . ., AA) < e(z; A2,. .

., An)

(n > 1, z = 1, 2, ...),

(95.24)

which we cannot verify without Theorem 224. EXAMPLE 7. Definition (95.1) may be extended to the case where G is an infinite periodic Abelian group and a countable sequence of subgroups AI, A2, ... takes

the place of AI, ..., A. We define

e(z) = e (z; Al, A2, ...) = I'm e(z; Al, ..., An), n-'co

(95.25)

provided that the limit on the right-hand side exists and is invariant with respect to permutations of Al, A2, .... This is always the case for z = 1, 2, ..., as can be

ZETA FUNCTIONS

391

seen directly from the monotonicity (95.24) and from Theorem 224 [from the righthand side of (95.13)]. Moreover we see that Theorem 224 retains its validity for this case.

EXAMPLE 8. Following Example 7 we compute e(z) = e(z; A1, As, ...) for the special case where O(A,) = p; is the ith prime number in the sequence of numbers

1, 2, ... (i = 1, 2, ...) and G = Al a A2 o ... holds. According to (95.20), (95.3) and (95.4) this gives n

e(z; A1, ..., An) = 1 1 (1 - A), 1=1

so, according to definition (95.25), we have

9(z) =F1 (1 - Pi z). r_i

Since now C(z) = e(z)-1 (for complex z with real part > 1) is equal to the wellknown Riemann zeta function, we see why (95.2) was called--a zeta function. It should

be noted here that we may consider (95.2) with complex z, and that zeta functions may be defined not only for groups but also for certain other structures. (Cf. REDEI (1955a, 1955c).) EXAMPLE 9. We see that the functions (95.1) are adapted to the second semilattice of all the subgroups of G and are determined by them. We can dualize (95.1) by taking

for AS the intersection of all the A, (i E C) and by substituting O(G : H) for O(H). In this case (95.2) is said to be a dual zeta function. Yet these are not essentially different

(because of § 93, Example 1) from the above zeta functions. On the other hand, for infinite Abelian groups the dual zeta functions are essentially different from zeta functions. E.g., in the infinite cyclic group the zeta functions are unimportant, but

not the dual zeta functions. If we take all the subgroups of prime number index in this group, the related dual zeta function is again the Riemann zeta function. EXAMPLE 10. From Theorem 223 it follows that e(z; A,, ..., An) = 0, if the unity group occurs among the A1, . . ., An. Hence and from (95.23) it follows that an arbitrary function e(z; A,, . . ., An) remains unaltered, if in it an A, is cancelled, which contains

another Ak. From Theorem 223 it follows also that e(z; A1, ..., An) is not constant,

if no A, is equal to the unity group. EXAMPLE 11. By Theorem 223 the zeros z (= 1, 2, ...) of a non-constant function o(z; A1, . . ., An) always constitute an unbroken sequence 1, ..., k, which may, of course, be empty.

EXERCISE. Show that Theorems 223 and 224 are independent of the first main theorem for finite Abelian groups.

§ 96. The Group of Prime Residue Classes mod m An important finite Abelian group is the set of prime residue classes mod in,

considered earlier, i.e., the group 7/(m)* of regular elements of the factor ring .7/(m). We now wish to examine this group more extensively, restricting ourselves to the case m >- 2. We know that O(,7/(m)*) = qq(m). If m = PI ... P, is the prime-power decomposition of m, then ,7/(m)* is, according to Theorem 199, the direct product of .7/(P)*, . . ., 7/(P,)*. Therefore the case where m is a prime number power still remains to be considered.

FINITE ABELIAN GROUPS

392

THEOREM 225. The group 7/(pt)* (p a prime number, t >_ 1) is cyclic,

if p # 2 or p = 2 and t < 2, while, for p = 2, t >_ 3 it has the invariants 2'-2, 2 and so is not cyclic. In the proof we denote the common unity element of .7/(p') and .7/(p')* by s. Accordingly s is the residue class 1 (mod p'). This meaning of s is, however, unimportant; we only take into consideration that -'//(P') is

generated by its unity element e and is of order p', so that this ring consists of the elements as (a = 0, . . ., p' - 1). From this, the as with p,f'a form the elements of 71(p')* . Consider first the case t = 1. Since .7/(p) is a field, every equation = s (n > 0) has at most n roots according to Theorem 205. In particular,

this holds in the group .7/(p)*, whence, according to Theorem 217, it follows that it is cyclic.

Secondly let p # 2 and t > 1. Because (p') c (p), we have The left-hand side contains only nilpotent or invertible elements. Such elements are mapped by a homomorphism onto elements of the same kind. Thus

17l(p`)* - 7/(p)*. Since the right-hand side is cyclic of order qq(p) = p - 1. so the left-hand side contains an element of this order. It will suffice to prove that it also

contains an element of order p'-1, for then, because (p - 1, p'-1) = 1, it contains an element of order (p - l) p'-' = q4(p'), which implies that it is cyclic.

We show that

o((1 + p)e) = p'-I.

(96.1)

For this we prove first that for any integers a, u (p,r a, 1 < u <- t - 1) ((l + p"a)s)P = (1 + p"+Ib)e for

(96.2)

The left-hand side, because of the binomial theorem, is equal to e + pu+Iae + E

(P)iui

i=2 i

Since furthermore p is a prime number, so from pI

= i!

(P

- i)! (,p)

(96.3)

PRIME RESIDUE CLASSES MOD /1J

393

we obtain the following important property of divisibility P pl

(1 5 i _< p - 1).

(96.4)

Hence and from u > 1, p >_ 3 it follows that the summand in (96.3) is divisible by putt. Consequently (96.2) is proved. Because o+ (E) = p' the repeated application of (96.2) leads to ((1 + p)e)P'-2 # E,

((1 + p)e)''-2 = 6.

Accordingly (96.1) is, in fact, true.

Thirdly let p = 2, t = 2. Since the group J/(4)* is of order two, it is cyclic.

In the fourth place let p = 2, t Z 3. Since the group L7/(2')* is of order q,(2`) = 2'-1, it will suffice to prove that the maximum of the orders of the elements in it is equal to 2'-'. For this purpose we prove first that for any integers a, if (2 < u <- t - 1) we have ((1 + 2ua)E)2 = (1 + 2"+Ib)e

(96.5)

for some number b, where 2 ,4' a implies 2 f' b, and ((1 + 2a)E)2 = (1 + 23c)E

(96.6)

for some integer c. Since the left-hand sides of (96.5) and (96.6) are equal to

(1 + 2u+(a + 2' 'a2))e and (1 + 22a(a + 1))E, respectively, so the assertions of (96.5) and (96.6) are true.

Beginning with (96.6), by repeated application of (96.5) it follows that

((1 +

2x)E)21_Z

=E

(for all x E,7)

and

(5E)2'-3 = ((1 + 22)E)21-3 # 6.

These imply the truth of the above assertion, by which the proof of Theorem 225 is complete. For a prime residue class a (mod m) we denote its order (in the group L7/(m)*) by (96.7) o(a (mod m)), which is always a divisor of qq(m). We call (96.7) the order of the number

394

FINITE ABELIAN GROUPS

a mod in. It should be noted that this order is the least natural number n, for which

a" = 1 (mod m). In particular, we call a a primitive number mod m, if o(a (mod m)) = q?(m),

(96.8)

i.e., if the group J1(m)* is cyclic and the residue class a (mod m) is a primitive element of this group. In other words it means that 1, a,...,Q

-1

is a system of representatives of the prime residue classes mod m. THEOREM 226. Among all the natural numbers only for

m = p', 2p', 4

(p an odd prime number, t >_ 0)

(96.9)

is there a primitive number mod m.

For the first and third case in (96.9) it follows from Theorem 225 that a primitive number mod m exists. The same holds for the second case by Theorem 199. If, on the other hand, a number m is not of the form (96.9), then it is divisible either by 8, by four times of an odd prime number or by two distinct odd prime numbers, whence, according to Theorem 225 and

the preceding note, it follows that the group ,7/(m)* has at least two invariants divisible by 2. Then by Theorem 217 this group is not cyclic, so that there are no primitive numbers mod m. Consequently Theorem 226 is proved. EXAMPLE. If m is of the form (96.9), i.e., if primitive numbers mod m exist, then

exactly q;(q(m)) of them are distinct mod m. In particular, for a prime number p there are q'(p - 1) distinct primitive numbers mod p. Because cp(q' (13)) _ = q' (12) = 4 there are four (mod 13) distinct primitive numbers mod 13; these are: 2, 6, 7, 11. EXERCISE 1. (ZslGMONnv's theorem). If a (> 2), n are natural numbers there is no prime number p with n = o (a (mod p)) if, and only if, we have

a=2; n=1, 6 or a=2'- l(t=2,3,...), n=2 (cf. SACHS (1956-57), Mom (1958c)). EXERCISE 2. Call a ring cyclic, if its module is cyclic. The finite cyclic rings with unity element are (to within isomorphism) exactly all the 31(m) (m > 0). Prove that if we take all the pairs of non-negative integers m, d with d I m and the ring defined

by the equations ma = 0, aQ = da, then we obtain all the non-isomorphic cyclic rings.

CHAPTER VI

OPERATOR MODULES Operator modules are of great importance for the whole of algebra. The theory of operator modules is also called linear algebra and includes a great part of ring theory. We have already met operator modules. Although

this chapter is exclusively devoted to them, only a small selection of the important problems concerning them can be discussed here. § 97. Operator Modules and Vector Spaces

Let an s-module M be given. This means that M is a module and the is a ring, which we apply - unless otherwise indicated operator domain as a left operator domain; this means that for a (EJ2), a (E M) the operator product as (E M) is defined and for arbitrary a, b (E-I), a, fi (E M)

a(a + P) = as + a#,

(a + b)a = as + ba,

aba = a(ba).

(97.1)

Mainly we shall consider only a unitary $-module M (cf. Theorem 144). This means that JP has a unity element e which is an identical operator, i.e., em = a (a E M). For the present, we take the former more general case into consideration. As submodules of M only the admissible submodules M' will be considered, for which therefore

ME M'

(aEt.,aEM')

is satisfied, i.e., which are s-modules. Likewise, a homomorphism of M will always mean an R-homomorphism a - a', for which therefore (aa)' = am'

(a E J2, a E M)

is satisfied.

Any elements al, ..., a of M are called linearly independent (with respect 0 for each i and an equation such as

to or over tR), if,

alai + ... +

an E JP) 395

(97.2)

OPERATOR MODULES

396

can hold only if

slot, = ... =

(97.3)

0.

(This amounts to an extension of the definition of linear independence in modules without operators, given in § 45.) If, in particular, JP is a skew field .7 and the module M is unitary, then we can replace this definition by a simpler one. For, from an equation such as as = 0

(a E Y, 0 0)

it follows that a-laa = 0, i.e., a = 0. Accordingly, in this special case the linear independence of the elements al, ..., a means that from the equation (97.2) we obtain (97.3')

Here it need not be assumed from the very first that al, from 0, for if, e.g., al = 0, then (97.2) is fulfilled for

..., a are

distinct

a100,

a the definition of linear independence to arbitrary subsets M

of an .'-module M, where we call M linearly independent, if all finite subsets of M are linearly independent. Then linear independence is a property of finite character. The.'-module {a} generated by the element a is called a cyclic J-module or a cyclic module. This consists of all the

is + as

(i E 7, a E .),

(97.4)

which need not all be different. In the unitary case the first term may be omitted in (97.4).

Let A (9 M) be a complex of M. The elements a (ER) with aA = 0 then constitute, because of (97.12,3), a left ideal a of JP, which we call the

annihilator left ideal of A in 2. The elements of a are briefly called the annihilators of A (in JP). If, in addition, A is an admissible submodule of M then a is also an ideal, as is obvious. In this case, as well as for a commutative J2, a may be called the annihilator ideal of A. The definition of the annihilator left ideal can also be applied to the case where A is a complex of an arbitrary ring R, so that we consider A as a complex of the R-module R+. Therefore the annihilator left ideal a of A will then consist of those elements a of R, for which aA = 0. If, in addition, A is an admissible submodule, i.e., a left ideal of R, then a is

OPERATOR MODULES AND VECTOR SPACES

397

an ideal. In the special case A = R, a consists, according to our former definition (§ 20), of all the left annihilators of R. THEOREM 227. If {a) is a unitary cyclic s-module and a the annihilator left

ideal of a in JP, then we have the J9-isomorphy +/a+.

{a)

(97.5)

COROLLARY. For a skew field.7 every unitary cyclic 7-module other than 0 is isomorphic with Y. For the proof, consider the mapping a --> ax

'-homomorphism of 39+ onto {a). Its kernel consists of the a with as = 0, therefore it is equal to a. Hence, and from the bomomorphy theorem (Theorem 60), Theorem 227 follows. The corollary is self evident. NOTE. Conversely, if a is a left ideal of A, then the factor module in (97.5) is a unitary cyclic .39-module, which is generated by the residue class e (mod a). Accordingly, the determination of all the unitary cyclic ,R-modules is essentially identical with the determination of the left ideals of '. More precisely, we are interested in the determination of all the s-left submodules of ,51+ (these being the left ideals of JP) and of the subsequent formation of the related factor modules, among which the nonisomorphic ones have still to be determined. All this implies the homomorphy problem of the .51-left module J'+. Theorem 227 is the special case n = 1 of the following generated by THEOREM 228. For a unitary 39-module M = {al, ..., finitely many elements and the l'$-vector space V with a basis Q1, ..., Sl of. 39 onto {a}. This is, by (97.12 3), an

V ^' M (a1JQ1 +

... + a,yQn -+ alal + . .. + aaan),

(al, ..., an E e51). (97.6)

Consequently,

M

V/V',

(97.7)

where V' is a proper submodule of V, the kernel of the homomorphism (97.6).

For, since the a1Q1 + ... + an.Q are all the different elements of V, so (97.6) follows from (97.1). Hence the theorem has been proved. Since, conversely, the right-hand side of (97.7) is a finitely generated -submodule V' of V, just as unitary A-module for an arbitrary above we see that the determination of the finitely generated unitary Jpmodules is reduced to the homomorphy problem ofd-vector spaces. On the other hand, Theorem 228 retains its validity for arbitrary (instead of finitely generated) . 9-modules.

OPERATOR MODULES

398

THEOREM 229. Over a skew field .7 every unitary module is an .5"-vector space.

For, let M be a unitary .-module. According to the lemma of TEicHMULLER-TUKEY (Theorem 17), M contains a maximal linearly independent

subset M = . Every element a of M may then be written in the form

a = a1a1 + a2a2 + ...

(a1, a.,,

... E,71

(97.8)

where only finitely many coefficients are different from 0. If a lies in M, then it is obvious. Otherwise, the union set M U a is not linearly independent,

whence the existence of an equation of the form

cm+ c1a1+... +

(c,c1,...,CnE-7)

with c :A 0 follows. After multiplying by c-1 the required equation (97.8) is obtained. This representation (97.8) of the elements of M is, because of the linear independence of M, also unique. Hence and from (97.1) it follows that M is the -vector space with the basis al, a2, .... Consequently the theorem is proved. THEOREM 230. All the submodules of an n-dimensional vector space over a principal left ideal ring R with unity element and without zero divisors are also J2-vector spaces of dimension at most n. Let Vn denote such a vector space with the basis al, . . ., ac.. Similarly

(for n > 1) will denote the ,-vector space with the basis al, ..., a submodule of V. For n = 0, Vo = V' = 0, therefore the theorem is now trivial. For n >-- I we assume the theorem for n - 1. The elements of V' are uniquely written in the form

a = alas + ... + anan

(a1, ..., an E A).

(97.9)

If always an = 0, then V' c V. -1; consequently, on account of the induction assumption, the theorem is then true. If an 96 0 for at least one a, all the possible an in (97.9) evidently constitute a left ideal of A different from 0. This is a principal left ideal Ra for some suitable a (E 2, 96 0). Then V' contains an element a, for which an = a holds in (97.9). By (97.9) the

ca + fi

(c E A ,B E v. _1(1 V')

(97.10)

are then all the different elements of V'. But now V _1 n V' constitutes, according to the induction assumption, an (m - 1)-dimensional vector then it follows space with m < n. If a basis of it is denoted by from (97.10) that the ca + b1N1 + ... + bm-iNra-1

(c, b1, ..., bn_t E.t` )

OPERATOR MODULES AND VECTOR SPACES

399

are all the different elements of W. Consequently V' is the /?-vector space with the basis a, F'1, ..., An -1- Thus the theorem is proved. EXAMPLE 1. The previous theorem is false for arbitrary integral domains !k. If !k = 9[xl and V1 is the Q -vector space with the basis al, then the elements

(2.f (x) + x 9(x)) al

(1(x), 9(x) E "k)

constitute an !k-submodule of V1, which is not an !k-vector space. EXAMPLE 2. If a is an element of an 0-module (i.e., of a module without operators),

then the annihilator ideal of a in J is just the principal ideal (n), where n = o+(a) is the additive order of a. Accordingly the notion of the annihilator left ideal is to be regarded as a generalization of the (additive) order of an element. Therefore, for an element a of a commutative ring R (with regard to the R-module R+) the annihilator ideal of a in R is sometimes also called the order ideal of a ; this consists of those elements

e of R, for which ea = 0. EXAMPLE 3. The submodules of a finitely generated unitary module M over a left principal ideal ring !k are likewise finitely generated. Then M, by Theorem 228, is the homomorphic image of a finite-dimensional ek-vector space V, so that, by Theorem 58, every submodule M1 of M is the homomorphic image of a submodule V1 of V. Now V1, according to Theorem 230, is also a finite dimensional !k-vector space and so finitely generated, whence the same follows for its homomorphic image M,.

§ 98. Determinant Divisors and Elementary Divisors

Here we consider matrices (of an arbitrary type) over a commutative ring

with unity element 1, which will later be assumed to be Euclidean.

The notions to be introduced will then be applied to JP-modules. We consider a matrix A of type m x n over and put p = min (m, n).

The determinant of an arbitrary square submatrix of A is called a subdeterminant of A. The order of such a subdeterminant is then always one of the numbers 0, . . ., p. The maximum r of the orders of all non-vanishing subdeterminants of A is called the rank of A. Accordingly

0
For every k (= 0, . . ., r) the ideal bk of J' generated by the subdeterminants of order k of A is called the kth determinant divisor of A. Since

bo = 1 (namely the unity ideal), bo is often disregarded. On account of Laplace's expansion theorem for determinants every subdeterminant of

order k + 1 of A is an element of the ideal bk (k = 1, ..., r - 1). According to this 14 R.-A.

b1Dh.,=...2br.

(98.1)

OPERATOR MODULES

If necessary we expand the definition of the determinant divisor by the convention b,+t = b,+2 = ... = 0. If,V is an integral domain and at the same time a principal ideal ring, then by Theorem 189 we may take for bk an element of V, namely the greatest common divisor dk of the subdeterminants of order k of A. Then (98.1) yields the divisibility relations dt I d.,

I ... I d, .

(98.1')

(The term "determinant divisor" derives from this special case.) THEOREM 231. Let A' = AB (or A' = BA), where A, B are matrices over a commutative ring with unity element. For the determinant divisors bl, b2, .. .

and b;, b2 ... of A and A', respectively,

bkcbk

(k=1,2,...),

so that the rank of A' is at most equal to that of A. If B is square and invertible, then A, A' have the same determinant divisors, consequently also the same rank. From the definition of the matrix product it follows that with repeated

application of Theorems 166 and 167, every subdeterminant of order k of A' is an element of the ideal bk. This implies the truth of the first part

of the theorem. If B is invertible, then A = A'B-' (or A = B-'A'). Therefore, from what has just been proved it follows that bk c bk

(k = 1, 2, ...).

It follows from this that bk = b'k (k = 1, 2, ...) , thus the theorem is proved. Remember (§ 64) that the rank, i.e., the dimension, of a finite-dimensional vector space is not in general invariant. Nevertheless the following holds:

THEOREM 232. The rank of a finite-dimensional vector space V over a commutative ring 91 with unity element is invariant, i.e., all its bases consist of the same number of elements. We suppose that V has two bases wt, ..., (om and a, ..., a0,,, with m < n. Let (co) = (wt, ..., (um), (e) = (got, ..., be the corresponding basis vectors and (w)', (A)' their transposes. Then

(a)' = A(w)', (w)' = B(e)' with matrices A, B over * of type n x m and m x n, respectively. [Cf. (70.1), (70.2).] Hence (N)' = AB(N)', so that

AB = U,,, where U is the unity matrix of type n2. But this gives rise to a contradiction

with respect to Theorem 231, since the rank of A is at most m and that of U. is equal to n.

DETERMINANT DIVISORS AND ELEMENTARY DIVISORS

401

The case has still to be considered where V has an infinite basis el, 02, .. . as well as the former basis coy, ..., wm. But since the cot, ..., eo,,, may be expressed by finitely many At, .... an, where we may assume n > m, the

previous conclusion may be repeated. Hence Theorem 232 is proved. The first part of the proof might be replaced by a simple direct proof.

We now suppose that A is also a commutative Euclidean ring. In this case we want to get more precise determination of the submodules of an .-vector space than in Theorem 230. Let an n-dimensional,A-vector space V be given with a submodule U, which, by Theorem 230, is an m-dimensional s-vector space (m < n). They then have bases wt, . . ., w and o,, ..., Q., respectively. These are connected by certain equations n

o; = Y=ta wi

(ate E *)

(98.2)

with uniquely determined coefficients a,,. Their matrix of type m x n is denoted by

A = (a,;).

(98.3)

(Later it will be evident that A is necessarily of rank m.) Now, the choice of the bases col, . . ., w,, and et, ..., em may be made in different ways. if we carry out a basis transformation on each of them with the matrices N and M, respectively, then the matrix A, with respect to the new bases. according to Theorem 161, becomes the matrix

MAN".

(98.4)

Here, according to Theorem 159, M, N may be any invertible matrices over ,2 of types m2 and n2, respectively. Thus, in order to make the survey of the relation of U to V easier we want to reduce (98.4) to the simplest form possible by a suitable choice of M and N. Therefore we now discuss.

this problem. We allow somewhat more generality since we admit an arbitrary matrix A (over R). The answer is implied in the following THEOREM 233. For every matrix A of type m x n (m <_ n) over a com-

mutative Euclidean ring R there are invertible matrices M, N over A of t} pes m2 and n2. respectively, such that I

MAN

=I

eI

e2

(98.5)e.,

..0

402

OPERATOR MODULES

is a diagonal matrix (of type m x n) and the divisibility relations el e2 l ... I

e,,1

(98.6)

..., e,,, are, apart from unit factors, uniquely determined. We call the el, ..., em the elementary divisors of the matrix A. More precisely ek is said to be the kth elementary divisor of A. From (98.5),

hold. In addition e1,

(98.6) and Theorem 231 it immediately follows that

(k = 0, .... m),

dk = ei ... ek

(98.7)

where dk is the k`h determinant divisor of A. Accordingly, elementary divisors may be determined only with the aid of determinant divisors. The rank r of A may be determined from the elementary divisors by the rule that only the first r elementary divisors are distinct from 0. The other elementary divisors e,+i = ... = em = 0 are often disregarded. The right-hand side of (98.5) is called the normal form of the matrix A. Of course, in the theorem, the assumption m <_ n is unessential. To prove Theorem 233, we define the so-called elementary tranformations of the matrix A, of type m x n, where for the sake of subsequent applications we admit an arbitrary (also non-commutative) fundamental ring.5P with unity element. We define an elementary row transformation to mean that

either two rows of A are interchanged or a row, multiplied on the left by a scalar factor c, is added to another row (c E-). Both may be carried out so that we multiply A on the left by a suitable invertible matrix of type m2. For example, left multiplication by

cr

1

(the unmarked terms being zeros) causes the interchange of the first two rows or the addition of the first row, multiplied on the left by c, to the second row. The meaning of an elementary column transformation is analogous, where, however, the scalar factors are now applied on the righthand side. Therefore an elementary column transformation is equivalent to multiplication on the right by a suitable invertible matrix of type n2. In order to prove the theorem we will let A undergo a series of elementary

transformations until a matrix of the form (98.5) is obtained, where we

DETERMINANT DIVISORS AND ELEMENTARY DIVISORS

403

postulate that (98.6) also be satisfied. Simultaneously, the M, N, required in (98.5), will be obtained in the form

M= MS...M9Ml, N=N1...N, where MI, ..., MS and N1i ..., Ni denote successively those invertible matrices of type m2 or n2, respectively, which, applied as left or right-hand factors, produce the desired elementary transformations. The elements of A may be designated by aij as in (98.3). We may assume that not all the aij vanish, since the assertion is trivial otherwise. We con-

sider an aij ( 0) with minimal a*, where ° denotes a Euclidean absolute value in tR. By suitable interchanges of two rows and two columns, we ensure that the element all has the above minimal property. With the aid of all we carry out the following Euclidean divisions: ail = gi au + bit , aij = hj all + blj

(i, j > 2),

(98.8)

where gi, hj denote the quotients and bil, blj the residues. We add in suc-

cession - gi times the first row of A to the i`s (i = 2, ..., m) row, then - hj times the first column to the j`s (j = 2, ..., it) column. Then the elements (98.8) are replaced by their residues bil, blj. If among these

residues an element other than 0 occurs, then this has a

smaller

absolute value than all. The above process may then be repeated with this. After finitely many repetitions a matrix of the form all

A

I

(98.9)

I

is obtained from A (which we have again denoted by A), where Al is a

matrix of type m - I x n - 1 and the unmarked terms are zeros. Denoting the elements of A again by aij, we carry out the following Euclidean divisions:

aij = gijail + bij

(i, j >_ 2).

(98.10)

If a residue bij is other than zero, then we add - qij times the first row of A

to the ith row, then the first column to the ja' column. By this the element aij in A is turned into bij. Similarly as above we can replace all by this bij. But since ail, we can in this way decrease ail more and more, until finally in (98.10) all the bij vanish completely, i.e., in (98.9) all l aij

(i, j >_ 2)

.

(98.11)

If Al is a zero matrix, nothing remains to be done. In the other case AI may be carried over to

A1I`( ant

404

OPERATOR MODULES

where a22 I a;, so that thus,

(i, ] ? 3).

an I a221 au

It should be noted that, by (98.9), the elementary transformations of A, are equivalent to those of A. Continuing this process a diagonal matrix finally arises from A where all I a 2 2 1

This means that we have carried out our programme. We have only to prove the last assertion of Theorem 233. This results from (98.7), since the determinant divisors d1, . . ., are, apart from unit factors, uniquely determined. Consequently we have proved Theorem 233.

From this, and from what has been said above we obtain: THEOREM 234. If V is an n-dimensional vector space over a commutative Euclidean ring JP and U is a submodule of V, which, according to Theorem 230, is an m-dimensional vector space over V (m <_ n), then bases w1, .. may be given in V and U respectively, such that and o1, . . ., t/1 = e1co1, ..., :/m = emtom

(e1 I e2 I ... I em) .

(98.12)

NOTE. In this theorem the el, ..., a have the following meaning: if we proceed from arbitrary bases w1, . . ., co,, and o,, ..., om of V and U respectively, which are connected by (98.2), then the e1, ..., em are determined as the elementary divisors of the related matrix (98.3). Hence it also follows

that this matrix - as already noted above - must be of rank m, since e1, .

.

., em,

by (98.12), are distinct from 0.

EXAMPLE 1. The elementary divisors of the matrices

1123

456

(789)

11200

(031

030 006

over I are as follows: 1, 3, 0; 1, 6; 1, 2, 12. EXAMPLE 2. By (98.6), elementary divisors can always be taken as

4'k = A ... h

where f ...,

(k - 1, .., /n).

are elements of E. For the determinant divisors we have d_ ._ fkf!-i A

EXAMPLE 3. The property (98.6) of elementary divisors is equivalent, on account of (98.7), to the property dkIdk_Idk.._,

(k = 1,...,in- 1)

(98.13)

DETERMINANT DIVISORS AND ELEMENTARY DIVISORS

405

of the determinant divisors, for which we may also write [cf. (98.1) ]

(k = 1, ..., in - 1).

bk-Ibk+l a bk

(98.13')

REDEI (1952b) has proved that the weakened form

k*bk-,bk+I S bk

(k = I, ..., m - 1)

(98.14)

of (98.13') holds in all commutative rings /k, where k* denotes the least common multiple [1, ..., k]*. [The factor k* in (98.14) is to be interpreted as multiplying all the elements of the ideal bk_, bk+l by this factor.] EXERCISE 1. We may transform Theorem 234 as follows. Let V be an n-dimensional

vector space over a commutative Euclidean ring (k with the basis co,, ..., 0.). and U a submodule of V with the generating elements ei, ..., If r is the rank of the matrix A = (air) belonging to (98.2), then U is an r-dimensional (k-vector space. If furthermore e ..., e, are the elementary divisors, other than 0, of A, then we may give a basis w,, ..., w in V such that e,w', ..., e,wr' is a basis of U. EXERCISE 2. For an integral domain /k, Theorem 232 may be proved by means of determinants. EXERCISE 3. We consider the system of linear equations n

Y_ aril i=r

(i = l,... , in)

b,

(98.15)

in a ring (k, where the all and b, denote given elements of (k and the solutions it, . . ., t are also to be taken in (k. The matrix A (air) of type m x n is usually called the matrix of the coefficients of (98.15). If, moreover, [t], [b] denote the column vectors

consisting of t,, ..., t,, and b,, ...,

respectively, then (98.15) may be written in

the elegant form

A[t] = [b],

(98.15')

where we have a product of matrices on the left. [The special case nt = n of (98.15') has already been mentioned in (73.14)]. Sometimes the matrix A' = (A,[b1)

is called the completed matrix o1' (98.15). The case of a skew field a = . will be discussed in § 102. For a commutative ring /k, (98.15) is also called a system of linear diophantine equations. We now consider the case where rk is Euclidean. We then denote the kth elementary divisors of A and A', by et and ek respectively. Furthermore let r be the rank of A. If A has the normal form then (98.15) becomes elf, = b,

so that now ergot

(i = 1, ..., iii),

(98.16)

(i= 1,...,nr)

is the necessary and sufficient condition for the solvability of (98.15), for which we may evidently write ek = ek

(k = I, ..., in).

(98.17)

Furthermore all the solutions of (98.15) are given by

t, =

b.

(i

l...., r): t,+,, ..., t arbitrary elements of Ik.

(98.18)

OPERATOR MODULES

406

The general case is, however, easily reducible to this trivial one. To do this we employ the notation of Theorem 233. Since M is invertible, (98.15') is equivalent to

MA[t] = M[b] , for which we may write

MANN-1[t] = M[b].

(98.19)

We now introduce new unknowns u1..... u and new "constants" c,, ..., cm, defining the column vectors [u], [c] composed of them by [u] =1V-1[t]. Then (98.19) becomes

[cl = M[bl

MAN[u] _ [e].

(98.20)

Since MAN is now the normal form of A, the former trivial case is implied in (98.20).

The solutions of (98.15') are obtained there from those of (98.20) by [t] = N[u]. Even the conditions of solvability (98.17) have an invariant character, as may easily be deduced. Consequently the following theorem is valid. THEOREM (FROBEtvivs's theorem). The system of linear diophantine equations (98.15)

in a non-commutative Euclidean ring ' is solvable if, and only if, the matrix of the coefficients A of (98.15) and the completed matrix A' have the same elementary divisors.

(This is also true for determinant divisors as well as elementary divisors. For the typical case 2 = J cf. FROBENIU3 (1878).) PROBLEM. We presume that k* may not generally be replaced in (98.14) by any

lower natural number.

§ 99. The Main Theorem for Finitely Generated Abelian Groups In this paragraph we shall deal in detail exclusively with finitely generated

unitary modules over a commutative Euclidean ring. THEOREM 235 (main theorem for finitely generated Abelian groups). Every finitely generated unitary module over a commutative Euclidean ring is a direct sum of cyclic J-submodules. Denote the given 5-module by M. By Theorem 228 we have M

V/V',

(99.1)

where V is an n-dimensional J-vector space and V' a submodule of V. (For it we can take the number of generators of M.) By Theorem 234 V' is an m-dimensional J9-vector space with m < n and there are in V, V' bases cot, ..., co. and w,, ..., wm such that co = etwt, ...; wm = emwm

(et en I ...

em)

(99.2)

. . ., em (96 0) are suitable elements of , . Because of (99.1) we may assume that (99.3) M = V/V'.

where e1,

MAIN THEOREM FOR ABELIAN GROUPS

407

We denote the residue class a (mod V') for an arbitrary x E V by a. These & are then all the elements of M. If

a = alwl + ... + aw

(a, E J-')

(99.4)

is the basis representation of a, then it follows that

a = alwl + ... +

(99.5)

If we now show in particular that from 0 = alto, + ... + aR6-)

(99.6)

it follows that all the terms of the right-hand side vanish, then M is the direct sum of the cyclicJV-modules ((Al}, ..., so that the theorem is proved. From (99.6) it follows that

alwl + ... +

0 (mod V').

Accordingly, because of (99.2) an equation such as a1w1 + ... +

x1elwl + ... + xme,,,wm

holds, where x1, ..., x, are suitable elements of J. Since the o ...., w constitute a basis, it follows from this that

al=x,e, (i= I,....m); am+I=...=a,,=0. Therefore, by (99.2), aiw; E V', i.e.

a,to; = 0

(i = 1...., n).

Consequently Theorem 235 is proved. We still wish to refine this result. With this end in view we have to study cyclic V-modules in detail, paying special attention to their submodules and direct decompositions, particularly to the indecomposable cyclic modules. We consider a cyclic ,2-module (a}. Since.5P is a commutative principal ideal ring, the annihilator ideal of {a} may be given by an element a of,*

in the form (a). Of course, a may be freely chosen from the associated elements. Moreover it is evident that the annihilator ideal of {a} is identical to that of a, i.e., the order ideal of a. With the annihilator ideal (a) of {a), according to Theorem 227, we have

{a}/(a)+.

(99.7)

The case where a is a unit may be disregarded, since then (a) - 0. 14/a R. - A.

408

OPERATOR MODULES

In particular a cyclic s-module is called a prime power module, if its annihilator is generated by a prime power. Accordingly, the primepower modules are (apart from isomorphism) the factor modules +/(pl)+, where p is an arbitrary prime element of JV and k > 1. Evidently, two primepower modules are isomorphic if, and only if, their annihilators are equal. We now consider the general case in (99.7) in order to determine the sub-

modules of (a). The generating elements of such a submodule are of the form cla, c2a, ... (cl, c2, ... ER). This submodule is then a cyclic module {da}, where d = (ci, c2, . . .). On the other hand, as = 0 and therefore d I a. We have thus obtained the proposition that the submodules of a cyclic J *-module {a} are again cyclic, and of the form {da}

(d E JP, d I a),

(99.8)

where (a) is the annihilator ideal of {a}. Further if d 0 the annihilator ideal of {da} is evidently equal to (d-' a). In the following we distinguish between the cases a = 0, a 0. The case a = 0 is very easy. For, d may now be any element of JP in (99.8). If we have d :A 0, then the annihilator ideal of (da) is equal to 0. like that of {a}. Because of (99.7), {a} and also its submodules other than 0

are now all isomorphic with 92+. Since, furthermore, two submodules {dla}, {d2a} (d,, d2 # 0) have an intersection other than 0, (a) is indecomposable.

For the case a # 0 we proceed as follows. If

a = be

(b, c E'; (b, c) = 1),

(99.9)

then we have the direct decomposition {a} _ {ba} ® (ca).

(99.10)

For, the equation

xb+yc= 1

(x,yE.

)

is solvable, so it follows that at = xba + yca, so that firstly we have

{a} _ {ba) + {ca}. Further, the intersection of the modules {ba}, {ca} consists of the za with z E JP and b, c I z. Hence by (99.9), a I z and so za = 0. Thus we have proved (99.10). By repeated application of (99.10) it follows that {a} may be decomposed into the direct sum of prime-power modules and also the

product of the annihilator ideals of the direct summands is equal to the annihilator ideal of {a.}. In addition to this we still have the obvious consequence from (99.8) that prime-power modules are indecomposable.

MAIN THEOREM FOR ABELIAN GROUPS

409

Hence, from Theorem 235 and the theorem (Theorem 146) we obtain the following THEOREM 236 (theorem of uniqueness for finitely generated Abelian groups). Every finitely generated unitary module M over a commutative Euclidean ring,`? is a direct sum of cyclic.-submodules, which are partly isomorphic with J1 , partly prime power modules, and, apart from order and isomorphy, uniquely defined.

We also have the following trivial SUPPLEMENT. If Mo and MP denote the sum of those direct summands of M,

which are isomorphic with .-,9+ or are annihilated by an ideal (p") (p a prime

element of -2, k = 1, 2.... ), respectively, then M has the direct decomposition

M=MoeIMP,

(99.11)

P

where p has to run through finitely many different prime elements. The direct summands MP are uniquely defined; furthermore, because

M0 ., M/IMP,

(99.12)

P

Mo is, apart from isomorphy, uniquely defined. EXAMPLE. Since .1 is Euclidean, all that has been said above also refers to the finitely

generated modules without operators. In particular, the additive finite Abelian groups are included here, so that some of the results in the §§ 90, 91 may also be obtained by particularizing from the above. If P1, . . ., P, are the invariants of a finite Abelian group, then (PI), ..., (P,) are exactly the annihilator ideals of those prime-

power groups, into which this group is decomposed by Theorem 236 or Theorem 214. Accordingly, in the general case the annihilator ideals of the prime-power modules in Theorem 236 and the number of the direct summands isomorphic with (R+ are called the invariants of this module. These again constitute a complete independent system of invariants.

§ 100. Linear Dependence over Skew Fields

In this paragraph we define the notion of "linear dependence" and set

forth in detail the fundamental facts referring to it. The concept of "linear independence" introduced previously will now appear to be a derived one. We follow VAN DER WAERDEN'S presentation, in such a

manner that the results may also be applied to certain concepts to be introduced later, e.g., to that of "algebraic dependence". Let a skew field 3 and a unitary module M over.7, which, according to Theorem 229, is then a vector space over .7, be given. The whole paragraph refers to this. Small Roman and Greek letters (perhaps with indices) always denote elements of Y and M, respectively.

410

OPERATOR MODULES

An element o is said to be linearly dependent on the elements wt, ..., co,., (with respect to .7), if an equation such as (100.1)

holds. In particular, this linear dependence will mean that o = 0 for n = 0. We prove that the following three fundamental properties hold: FUNDAMENTAL PROPERTY I. Every wi (i = 1, . . ., n) is linearly dependent on w1, ..., co.. FUNDAMENTAL PROPERTY II. If o is linearly dependent on co,, ..., wn, but not linearly dependent on col, . . ., co,,,, _then co,, is linearly dependent on Lo, w1, ..-,(on -t-

FUNDAMENTAL PROPERTY III. If a is linearly dependent on 01, ..., Lon, . ., m) is linearly dependent on co,, ..., co,,, then a

and every p; (i = 1, .

is linearly dependent on w), . . ., con. Of these, I and III need no proof. 11 is proved as follows: By hypothesis, (100.1) holds with an # 0. Thus

wn = ant (O - a1w, - . . - a,, .

-1w. - ,).

Therefore the assertion is true. The proofs of Theorems 237 to 242 do not rely on the above definition

of linear dependence, but only on the fundamental properties I to III. by which the more general applicability of these theorems is guaranteed. From I and III immediately follows THEOREM 237. If a is linearly dependent on ..., om and every ei (i = 1, ..., m) occurs among the w,, ..., to,,, then a is linearly dependent On w1, ..., w,. It is worthwhile to note the special case of this theorem, when 01, agree apart from order with the wt, . . ., w;,. For this case the theorem

means that in the proposition "a is linearly dependent on cot, ..., (A)n" the succession of the elements wt, ..., co. is immaterial. The elements wl, ..., w,, are said to be linearly independent, if none of them is linearly dependent on the others. In particular the empty system is said to be linearly independent. From Theorem 237 it follows that every partial system of a linearly independent system is also linearly independent. This definition is not new, but agrees in content with the special case for

Q = .7" of our former definition [(97.2), (97.3)], as it results from the note made there [(97.3')]. THEOREM 238. If w1, .... are linearly independent, but w1, ..., (') are not linearly independent, then w is lin?arty dependent on w1, ..., wn_t.

LINEAR DEPENDENCE OVER SKEW FIELDS

411

Otherwise an w; (i # n) would be linearly dependent on the other w1, ..., w,,. We may suppose that con -1 is linearly dependent on w1, ..., (On-21 CO.- On the other hand, wn_I is not linearly dependent on wl, ..., wn_2, so that wn, by II, is linearly dependent on 0 ) 1. . .., wn_I. Consequently Theorem 238 is proved.

THEOREM 239. Among arbitrary w1, ..., wn there are always linearly independent elements o1i ..., em (m ? 0), such that all the elements are linearly dependent on them. Let us choose from among the w1, . . ., con a maximal number of linearly independent elements CI, ..., Cm. Every w; different from these is linearly

dependent on them by Theorem 238, and, according to 1, also every e . This proves Theorem 239. Two systems of elements 01, ..., qm and w1, ..., w are called linearly equivalent, if every t), is linearly dependent on the w1, ..., con and every w; is linearly dependent on the 91' , Pn,.

By this we have defined an equivalence relation, since the symmetry follows immediately from the definition, and the reflexive and transitive laws follow from I and III. If an element is linearly dependent on one of two linearly equivalent systems, then it is also linearly dependent on the other, from III. From Theorem 239 it also follows that every system of elements w1, ..., con is linearly equivalent to a linearly independent partial system of these elements. THEOREM 240 (STEINITZ'S exchange theorem). If OI, ..., Pm are linearly independent and every Qi is linearly dependent on w1, ..., wn, then m < n. and one can suitably order the w1, ..., co,, such that the two systems

w1, ..., co,,; QI... , Cm , wm+I , ..., co,

(100.2)

are linearly equivalent. We say that the system (100.22) originates from the system (100.21) by

the exchange of the elements w1, ..., w,n (for the elements e1, . . ., on,). The theorem is trivial for in = 0. If m z 1 we assume its truth for m - 1. Since e1, ..., CC.-1 are linearly independent, because of the induction assumption, m - 1 < n, and after a suitable rearrangement of the elements (o1, ..., co,, the two systems 0) I, ..., (0n; 01, .... gm - I

,

Loon, ..., wn

(100.3)

are linearly equivalent. Since Ln, is linearly dependent on (100.31), it is also linearly dependent on (100.32). Since, on the other hand, nom is not linearly dependent on eI, ..., 2m_1, it follows that m < n. Further, by

OPERATOR MODULES

412

suitably ordering the wm, . . ., co,,, we can say that for a suita ble t (m < t < n)

em is linearly dependent on 91, -

, Um -1 ,

wm.... 1011,

(100.4)

but is no longer so, if any co, (i = m, ..., t) in (100.4) is eliminated. From 11 it follows that wm is linearly dependent on

01,..., nr

wmfl,-..,IO,.

and so, by Theorem 237 on 601...

wm+1, ... w .

(100.5)

But since, as stated above, 9m is linearly dependent on (100.32), it follows that (100.32), (100.5) are linearly equivalent, as are also (100.31), (100.5). This implies Theorem 240. THEOREM 241. If L01, ..., Am and co], ..., co,, are linearly independent and linearly equivalent, then m = n. This follows from the fact that, according to Theorem 240, m:!5; n and n _< m.

We say that a subset M of the .%--module M is of finite rank (over 3), if in M there are finitely many elements, on which every element of M is linearly dependent. Then there are in M, according to Theorem 239, linearly independent elements w1, . . ., co,, such that every element of M is linearly depend-

ent on them. In addition, by Theorem 241, n is uniquely defined. This n is said to be the (linear) rank of the set M (over or with respect to .-".) Since in M, by Theorem 240, there are then at most n linearly independent elements, we may define the rank of M as the maximal number of linearly independent elements of M. THEOREM 242. If a set M is of finite rank, then so is every subset M' of M. Moreover, the rank of M' is at most equal to that of M. In M'there may be at most as many linearly independent elements as in M. Therefore the theorem is true.

§ 101. Vector Spaces over Skew Fields Let . " be a skew field. As is known from Theorem 229 every unitary.? module is an .7"-vector space. We now wish to study .f"-vector spaces of finite rank in fuller detail. We have defined the rank of such a vector space V in two ways. According to the definition of the previous paragraph, it

means the maximal number of linearly independent elements of V. According to a former definition (¢ 64) it means the number of elements of an arbitrary

VECTOR SPACES OVER SKEW FIELDS

413

basis of V. This second notion of rank refers to vector spaces over any ring with unity element and in general it has no invariant meaning; its invariance

has been proved, by Theorem 232, only for commutative fundamental rings. We now prove the following THEOREM 243. In a vector space V over a skew field" the elements w],. . .. to,, constitute a basis if, and only if, n is the maximal number of linearly independent elements in V and col, . . ., co. are linearly independent. Consequently for finite-dimensional vector spaces over skew fields the rank defined in § 64 is identical with that defined in § 100, and so is invariant.

The assertion "if" can be proved in the same way as Theorem 229, or even somewhat more easily. In order to verify the assertion "only if", we suppose that wl, . . ., w is a basis of V. Evidently o , . . ., co,, are then linearly independent; furthermore every element of V is linearly dependent on them, whence it follows by Theorem.240 that n is the maximal number of linearly independent elements in V. Hence the theorem has been proved. The (admissible) submodules of a vector space we briefly call its subspaces provided that they are also vector spaces over the same fundamental ring (this is always valid for fields). THEOREM 244. If V is a finite-dimensional vector space over a skew field.? and U is a subspace of V, then every basis of U may be completed to one of V.

For the proof we denote bases of U and V by ol, ..., a,, and w 1 , . . ., w,,, respectively. Then Theorem 240 is applicable, so that m suitable elements of the w1, . . ., co,, may be replaced by the ol, ..., om in such a manner that again there arises a basis of V. Consequently Theorem 244 is proved. By applying the results on.7-vector spaces we obtain the following THEOREM 245. In the full matrix ring .-` over a skew field .rte' every matrix

is either invertible or a zero divisor. Consider a matrix

A=(aid)(E n)

(aiiE. )-

We make use of an n-dimensional.7-vector space V with the basis wl, .... n0,,. The necessary and sufficient condition that the

also constitute a basis of V may be given in two ways. On the one hand, this

condition consists, according to Theorem 159, in the invertibility of A, on the other hand, in the linear independence of ol, ..., o,,. This independence means, furthermore, that the equation Y_ cioi = i=1

J

i cia, cot = 0

i=1;=1

(c, E -t)

(101.1)

OPERATOR MODULES

414

can be satisfied only with c1= ... = cn = 0. Since the w1, ..., con constitute a basis, equation (101.1) is equivalent to the system of equations n

I=1

ciaij=0

(j

I... ,n).

This is representable in the form (c1, ..., Cn)A = 0

.

(101.2)

where the left-hand side is to be interpreted as a product of matrices. But, since (101.2) can be satisfied with a vector (cl, . . ., cn) other than 0 if, and only if, XA = 0 has a solution X other than 0 in ., we have obtained the following: the invertibility of A is equivalent to the fact that A is not a right zero divisor (in .5 ). Since invertibility is a self-dual property, Theorem 245 has been proved. EXAMPLE 1. In an 'k-algebra of finite rank n, and so also in the full matrix ring Q. of rank n2, these ranks have an invariant meaning, if 'k is a commutative ring with unity element or skew field. EXAMPLE 2. From Theorem 245 it follows that if r,Q is an integral domain, then in the full matrix ring Q. every left zero divisor is a right zero divisor, and conversely. EXERCISE. Theorem 244 also holds for infinite-dimensional . '-vector spaces. Prove this using the lemma of TEICHMULLER-TUKEY (Theorem 17). PROBLEM. Does Example 2 hold for subrings rk of skew fields?

§ 102. Systems of Linear Equations over Skew Fields Let us consider a system of linear equations over a skew field Y n

E ajtj=bt j=1

(i= 1,...,m),

(102.1)

where the "coefficients" aij and the "constants" bi are arbitrarily given elements of.7 and the tj denote the unknowns, which are likewise to be taken in Y. The number of equations and unknowns is designated by m and n.

respectively. In order to determine the solutions we use the linear forms n

Li = E aijxj j=1

(i=

(102.2)

from the polynomial ring _ [x1, . . ., xn]. In this all the linear forms, i.e., the homogeneous linear polynomials, constitute an."-vector space of rank n. The linear forms (102.2) have rank r, which we also call

415

LINEAR EQUATIONS OVER SKEW FIELDS

the rank of the system of the linear equations (102.1). Accordingly r is the maximal number of linearly independentelements among the L1, ..., Of course

r < min (m, n) .

(102.3)

We prove the following THEOREM 246.

The system of linear equations (102.1) of arbitrary

rank r over a skew field 3 is solvable if, and only if,

k,Ll+...+k»,L,,,=0=k1b, +... +kn,b,,,-0

(k i E .7). (102.4)

When the equations ([02.1) are solvable, these equations and the unknowns

tl, ... , t may be numbered so that LI,

,Lr,.Yr+l,.

.

,x

(102.5)

are linearly independent, whence, with uniquely determined c,.. d,j (E 7), we have n

r

xi = I cl Lj + Y d,jx; j=r+l

j=1

(1 = 1,

..., r).

(102.6)

Then all the solutions t1...., t,, of (102.1) are given by the formula n

r

t, = E cjbj + Y d,jtj j=1

i=r+1

(i = 1, ..., r)

(102.7)

. . ., t we have to substitute arbitrary elements of 2-. The condition of solvability (102.4) may be expressed by saying that

where for tr+1

(102.1) is solvable if, and only if, every linear relation of the form (102.41) remains valid after the substitution L; -* b, (i = 1, . . ., m). The equations

(102.7) may be called the solution formulae for the system of linear equations (102.1). (Of course these need not be uniquely determined, since the L1, . . ., Lr may generally be chosen in several ways.) For the proof we assume first of all that (102.1) is solvable. We consider a linear relation (102.41). This remains true if the indeterminates x1, . . ., x are replaced by arbitrary elements of t We perform the substitution xj -+ tj

(j = 1, .

.

.,

n), where t1,

. .

., t,,

is a solution of (102.1). Then (102.41),

because of (102.2), turns exactly into (102.42), wherefore (102.4) is, in fact, a necessary condition for the solvability of (102.1).

In the following proof of the other assertions of the theorem we may assume that the condition (102.4) is satisfied. We number the linear forms

OPERATOR MODULES

416

(102.2) such that L1, ..., Lr are linearly independent. First of all we show that if the first r equations n

(1=1,....r)

Yaljtj=b,

(102.8)

j=t of the system (102.1) are satisfied, the other m - r equations (102.1) are automatically fulfilled.

We consider an Lp (p = r + 1, ..., m). Then r

LP=>11L1

(102.9)

l=t

with suitable coefficients 11, ..., 1, in 7". Because of (102.4), r

l;h1,

by

i.e., according to (102.8), d

r

by = Y Y l,aljtj .

(102.10)

1=1j=1

On the other hand, it follows from (102.2) and (102.9) that n

r

E avjxi-

j=1

apl=

E Ilauxi 1=1j=1

(j= 1,...,n).

hall

l=j Hence and from (102.10) follows the validity of thep`h equation of the system (102.1), i.e. the validity of our assertion. We apply STEtrnTZ'S exchange theorem (Theorem 240) to the linearly independent Ll, ..., Lr and to the likewise linearly independent x1, ..., xn.

It follows that for a suitable order of the latter (which is practically equivalent to the renumbering of the unknowns t1, . . ., 1,,) the linear forms (102.5) are linearly independent, as was asserted. It still remains to be proved that (102.7) exactly yields all the solutions of (102.1). It will suffice if we show that the equations (102.8) follow from (102.7). To do this we substitute (102.6) in (102.2): r

Ll =

f

I aj

(t ejkLk + k =rt1 n

r

n

djkxk

aljxj

+

(i = 1..... !') .

j=r+t These equations remain true because of the linear independence of the elements (102.5) when they are replaced in order by j=1

k= L

b1i

, ,

., br,

tr+1, ..., to

.

Thus we obtain (102.8), which is what was required to be shown.

417

LINEAR EQUATIONS OVER SKEW FIELDS

Secondly, we show that (102.7) is satisfied for every solution of (102.1), and even for every solution of (102.8). For this purpose we substitute (102.2)

in (102.6). By the substitution x, --+ ti (i = 1, ..., n) we obtain (102.7) because of (102.8). This completes the proof of Theorem 246.

It is both theoretically and practically important that the rank r of the system of equations (102.1) as well as the solution formulae (102.7) may be determined by the following process, which is called successive elimination. We choose an x;, which occurs (with a coefficient other than 0) in one of the equations (102.2). We solve this equation for x? and substitute the expression so obtained for x; in the other equations (102.2). (This means that in the

basis x1...., .r the element x; is replaced, in the sense of Theorem 240, by a suitable L,.) We proceed similarly with the system of m - I equations obtained in this way and continue this process until finally the x1...., x no longer occur in the remaining equations, but at most the LI, ..., L,,,. After renumbering the equations (102.1) and the unknowns t; (i.e., the equations (102.2) and the indeterminates x;) the equations, with the help of which the above elimination took place, will be of the form

(i = I,

xi=f,(L1,.. ,L,,

,r),

(102.11)

where the right-hand sides are linear expressions in all the elements (with coefficients from ") and the coefficient of L, in f,. is different from 0 (i = 1, ..., r). From (102.11) it is evident that the systems

are linearly equivalent, whence it follows that LI, . . ., L, are linearly independent. If in the last m - r equations of the system (102.2) the xj, . . ., x, are successively replaced by the right-hand sides of (102.11), then all the xI. ..., x,, must disappear, since otherwise the process could be continued.

Consequently equations are obtained of the form

L,=f(LI,....L,)

(i=r+ 1, . . ., m) .

(102.12)

In conformity with this the L1, ..., L. are linearly dependent on LI...., L,.. Therefore r is necessarily the rank of (102.2), i.e. of (102.1). Every linear dependence between L1, . . ., L,,, is then a consequence of (102.12), otherwise L1, ..., L, would not be linearly independent. Consequently the condition of solvability (102.4) is expressed by the equations

b,=f,(b1.....b,)

(i=r + 1,...,m).

Finally we see that the equations (102.11), after the substitution

x;--+t;, L,-*b,

(j=

i= 1....,r)

OPERATOR MODULES

418

become the solution formulae for (102.1). ((102.6) may be obtained by replac-

ing the indeterminate x1 for i = r, r - 1, ..., 2 in the first i - I equations of the system (102.11) by the right-hand side of the 1`h equation.) Since the process of elimination above is based exclusively on the coeffi-

cients aij, b1 in (102.1), the following extension of Theorem 246 holds. THEOREM 247. The coefficients cij, d1j in the solution formulae (102.7) for (102.1) lie in the sub-skew-field of 3 generated by the coefficients aij, b;. Thus if (102.1) is solvable in Y, then it is solvable already in this subskew-field.

Theorem 246 is much simplified for a homogeneous system of linear equations.

(i = 1, ..., m) .

a;jij = 0

(102.13)

Now the condition of solvability (102.4) is trivially satisfied, wherefore (102.13) is always solvable. Indeed we see immediately that t1 = ... = t == 0 is a solution, known as the trivial solution of the homogeneous system of linear equations. We see moreover that on the right-hand side of (102.7) the first sum is zero, so that this right-hand side is now homogeneous in the

tr+l...., t.-

For a solution 11, ..., t of (102.13) we call (tl_, . . ., of this system. According to (102.7) all the solution vectors are: rn

(ti, ..., t,,) = (

dijtj, j=r+1

a solution vector

rr

'

.,

j=r+l

clrjtj ir+1 ,

, tn)1

(tr+1... ,t,,E, ).

if we replace one of the tr+1, ..., t here by the unity element I of.7 and the others by 0, then we obtain the special solution vectors r+i

at-(dl,r+i,

1-1

1 ...,0)

(i=1,.. (102.14)

All the solution vectors are obtained from this in the form (102.15) als1 + ... + a,, rsn- r , where the scalar factors s1, ..., are arbitrary elements of.i Accord-

ingly all the solution vectors constitute an (n - r)-dimensional Y-rightvector space with the basis al, . . ., Therefore the solution vectors (102.14) are called the fundamental solutions of the homogeneous system of the linear equations (102.13). In conformity with this we have the following theorem and corollary.

LINEAR EQUATIONS OVER SKEW FIELDS

419

THEOREM 248. A homogeneous system of linear equations with n unknowns

of rank r over a skew field has exactly n - r fundamental solutions. Consequently the trivial solution is the only one if, and only if, r = /I. COROLLARY. A homogeneous system of linear equations over a field with an equal number of equations and unknowns has at least one non-trivial solution

if, and only if, the determinant of the system is 0.

We now consider a matrix A of type m x n over a skew field 3. We shall always consider its rows and columns as elements of an n-dimen-

sional .7-left-vector space and an m-dimensional .7-right-vector space, respectively. Then both the row and column vectors of A have a (linear) rank r' (< m) and r" (- n), respectively, called the row rank or the column rank of this matrix. THEOREM 249 (rank theorem for matrices over skew fields). The row rank and the column rank of a matrix A over a skew field.7 are equal, and, if." is commutative, then they are equal to the rank of A. Because of this theorem we may talk of the rank of the matrix A also

in the case of a skew field .7 and understand by it the row rank (or the column rank) of A. In order to prove the theorem, we denote the row and column rank of A by r' and r", respectively. If A is diagonal, then the assertion r' = r" is trivial.

On the other hand, by the help of elementary transformations A may be changed into a diagonal matrix. Thus to prove r' = r" in general, it suffices to show that r' and r" are invariant for example with respect to elementary row transformations. For the interchange of two rows, this assertion is trivial. For the proof of the remaining assertion, we denote the row vectors of A by aI, .., am

(102.16)

If, e.g., ca.. is added to at (c E .7), then the new row vectors are

aI + ea.,, a2,...,a,,,.

(102.17)

The systems (102.16), (102.17), conceived as systems of elements of a left vector space, are linearly equivalent, from which the asserted invariance for r' follows. Furthermore the change from (102.16) to (102.17) for the column vectors of A, which we denote by bt, . . ., b,,, means that in every bI the second component, multiplied on the left by c, is added to the first one. Every linear dependence is then evidently still true, whence the invariance of r" follows. This proves the first part of the theorem.

OPERATOR MODULES

420

Now let .7 be commutative and denote the rank by r, i.e.. the maximum of the orders of the non-vanishing subdeterminants of A. It is sufficient to prove that r = r'. Again this is trivial if A is a diagonal matrix. So it will suffice to show that r is invariant with respect to elementary transformations. On account of the duality principle for determinants and the invertibility

of the above transformations, it is sufficient to show that r will not be less, if we carry out on A, e.g., the row transformation considered in (102.17). (For row interchanges even the invariance of r is evident.) Let d run through those subdeterminants of order r of A, in which the first row of A participates, and the second row does not participate; moreover let d'run through

the other subdeterminants of order r of A. In our row transformation. according to Theorems 166, 167, every d is changed into a d + cd', while all the d' (partly by Theorem 168) remain invariant. Thus if all the d' vanish,

then all the d remain invariant. But since, by hypothesis, not all the d and d' vanish, so the same follows for the subdeterminants of order r after the row transformation. Consequently Theorem 249 is proved. This theorem permits the replacement of the conditions of solvability (102.4) for the system of linear equations by one condition only. This is the content of the following THEOREM 250. For the system of linear equations (102.1) over the skew field,t"let A denote the matrix of coefficients a;j and A' that matrix which

arises from A by adding a (last) column, consisting of the constants bl, . . ., b,,,. Then the necessary and sufficient condition for the solvability of (102.1) is that A and A' are of equal rank. If a,, . . ., a,,, b denote the column vectors of A', then the system of equations (102.1) may be condensed into the single equation

alts+...+at,;=b. The solvability of this equation is equivalent to A and A' having the same column rank. Hence, and from Theorem 249, Theorem 250 follows. THEOREM 251. A square matrix A over a field Y is invertible if, and only if.

its determinant is not equal to 0. By Theorem 245 the invertibility of A is equivalent to the solvability

of the equation

AX= U

(X, U E Y ; U the unit matrix).

This means that in Y" all the systems of linear equations (with the same number of equations and unknowns) with the matrix of coefficients A are solvable. By Theorem 250, this is true if, and only if, the rank of A is the greatest possible, i.e., equal to n. Hence Theorem 251 follows.

LINEAR EQUATIONS OVER SKEW FIELDS

421

Another method of considering the above topics is given by EGERVARY (1955), where he takes practical applications into consideration as far as possible. An extension of Theorem 246 to the case of infinitely many unknowns and equations has been accomplished by GACSALYI (1951-52) and SZELE (1951-52b). A gener-

alization of Theorem 246 for semisimple rings (see § 108) instead of skew fields is given by KERTFSZ (1955-56). EXAMPLE 1. If a solution t, = c, (j = 1, ..., n) of (102.1) is known, then the question

of the determination of all the solutions may be reduced to the homogeneous system (102.13). Because

aic,=b,

(i = 1,

M),

f=1

(102.1) is representable in the form

ault; - c,) = 0

(i = 1, .... m).

This system of equations is homogeneous in the unknowns t; - c (j = 1, ..., n). If, moreover, we introduce the notation

c = (cy...., then, by making use of (102.15), it follows that the solution formulae (102.7) appear in the form

Thus, the solutions of the system of linear equations consist of a special solution of the general system of equations and of the general solution of the special (viz., homogeneous) system of equations. EXAMPLE 2. It is essential in Theorem 249 that in the definition of the row and

column rank of a matrix, we have regarded its rows and columns as elements of a left and right vector space, respectively. In order to prove this we consider the matrix bb

I ca

'

where a, b, c ( : A 0) are elements of a skew field. Because

c(a, b) = (ca, cb) the row rank is now equal to 1. The column rank must be likewise equal to I. and we indeed find that (a, ca) a- 'b = (b, cb).

But if we considered the column vectors as elements of a left vector space, too, then

the new "column rank" of the matrix would, in general, no longer be 1, since the equation

q(a, ca) = (b, cb)

need not have a solution q in the given skew field. For, it would have to be q = ba-

then ba 'ca = cb, i.e., ba 'c = cba ', and this does not always hold. EXAMPLE 3. For the solution of the system of equations (102.1) for a commutative fundamental field JK - after determining the rank r of the system - we can make use

of Cramer's rule. For this purpose we renumber the equations and unknowns in

OPERATOR MODULES

422

such a manner that the determinant i ail 1r of order r, formed from the ai, (i, j 1, ..., r). is not 0. If we then write the first r equations of (102.1) as n

r

i=t

aiiti

b, - F, atiri

(r = 1, ..., r),

1=r--l

then Cramer's rule yields the solution formulae of (102.1). EXAMPLE 4. If A is a matrix of type in x n over a skew field .7-, then there are (square) invertible matrices M, N of type m' and n', respectively, over .F-, for which

MAN is a diagonal matrix all of whose elements are 1 or 0 and such that in the diagonal the elements 1 precede the elements 0. This matrix is called the normal form of the matrix A (over the skew field .F). It is uniquely determined, and the number of diagonal elements equal to I is the rank of A. The normal form of a matrix over a field was defined differently on p. 402, but the difference is immaterial.

§ 103. Kronecker's Rank Theorem if one of the matrices At, A.,, say At, is a submatrix of the other, then we

call A2 a hypermatrix of A,. THEOREM 252 (KRONECKER'S rank theorem). 1j' a matrix A over a skew field .7 contains a square submatrix B of type r2 and of rank r, which has the property that (in A) all the square hypermatrices of B of type (r + 1)2 are of rank r, then r is the rank of A. For the proof we may suppose that B lies in the first r rows and columns

of A. Let A be of type m x n. We assume that the theorem is false, i.e. the rank of A is greater than r and prove that B (in A) has two hypermatrices

of types (r + 1) x is and m x (r + 1), which are of rank r + 1. This proves the theorem, since by applying this fact twice, the existence of a (square) hypermatrix of B (in A) of type (r + 1)'2 and of rank r + I follows, in contradiction to the supposition. It is sufficient to prove the first of the assertions, as the second may be similarly proved. In conformity with the supposition A certainly contains r + 1 linearly independent row vectors. But since the first r rows are linearly independent, it follows from STEINITZ's exchange theorem (Theorem 240) that the first r rows and a further row of A are linearly independent. These rows then constitute a submatrix of A which is of type (r + I) x n and has

rank r + 1. Consequently the above theorem is proved. § 104. Schur's Lemma We mention by way of introduction that a module M may be considered as an 3'-module, where 8' is the full endomorphism ring of M or a subring of this ring. If 8' is a skew field, which contains the identical endomorphism of M, then M is unitary over 8'. and thus, according to Theorem 229, an

'-vector space.

SCHUR'S LEMMA

423

The simple modules other than 0 (without operators) are those of prime order p. If M is such a module, then it is immediately obvious that its full endomorphism ring is a field r " with 0(Y) = p, for which so that M is now an Y-vector space. However, this is not an isolated fact, since for every simple operator module over an entirely arbitrary ring,

which need not even contain a unity element, we have the following: THEOREM 253 (SCHUR'S lemma). The R-endomorphism ring .7 of a simple

,R-module V (0 0) is a skew field,V and itself is therefore anY vector space. The identical endomorphism of V belongs to Y and is the unity element

of it. Let k be an element of Y other than 0. It will suffice to show that k is an automorphism of V, for then the inverse mapping k-1 likewise belongs

to Sr. Since k is an 9-endomorphism of V, so

k(ax) = a(ka)

(104.1)

for all a Et and a E V. So, if ka = 0, then k(am) = 0. According to this the a with ka. = 0 constitute an R-submodule of V, which is, because k:0 0, different from V. But since V is a simple .52-module, this submodule

is equal to 0. On the other hand it follows from (104.1) that the image kV of V under k is an. *-submodule of V other than 0, i.e., kV = V. This, together with what has gone before, means that k is an automorphism of V. So the theorem is proved.

§ 105. The Density Theorem of Chevalley-Jacobson

To begin with we give the following definition: for a semiordered set S (or for its elements) the minimal condition is satisfied (or C5 is a semiordered set with minimal condition), if every descending chain a1 > a2 > ...

(a,, a..... E C5)

(105.1)

is finite. So, e.g., a ring with minimal condition for the set of left ideals means that every descending chain of its left ideals is finite. The minimal condition admits an equivalent formulation: for the semiordered set 6 the minimal condition is satisfied if, and only if, every nonempty subset of C5 contains at least one minimal element. (Hence the name "minimal condition".) If namely C5' is a subset of G without minimal elements, then if we pro-

ceed from an arbitrary element a, (E s'), an infinite chain of the form (105.1) may be constituted from the elements of G'. Conversely, if e con-

tains an infinite chain (105.1), then there is no minimal element in the subset
424

OPERATOR MODULES

We say, entirely analogously, that for the semiordered set condition is satisfied, if every ascending chain

a1 < a2 < ...

the maximal

(a1, a.,, ... E S)

is finite. This again means that every non-empty subset of C contains at least one maximal element. in a vector space V (of finite or infinite dimension) over a skew field, we call a set S of mappings of V (into itself) dense, if for any linearly independent elements wl, ..., w, (E V) and any further elements al, ..., a (E V) there is always a mapping a (E S), which maps w; into a, (i = 1, ..., n). An extension of SCHUR'S lemma (Theorem 253) is the following: THEOREM 254 (density theorem of CHEVALL.EY-JACOBSON). Let V ( : A 0)

be a simple a-module, and so, by Theorem 253, a vector space over the skew field.7" consisting of all the R-endomorphisms of V. Moreover assume that 0. Then the ring P' of the (distinct) .Y -endomorphisms .59V

a -> as

(a E .')

(105.2)

of'V is dense (with respect to V as a rector space over .3). If, furthermore, the minimal condition is satisfied for the left ideals of J', then V is finitedimensional (over Y), so that S' is then the full .7-endormorphism ring of V. For the proof we need the following PROPOSITION. If'wi, ... , co. (n >_ 1) are linearly independent elements of V, then there is an element a of -.99 with

awl =...=awi_1=0,

(105.3)

For n = I we have only to prove that 2wl # 0. With this end in view we suppose that Jiwi = 0. Then the set of a (E V) with Pa = 0 is an ,

-submodule other than 0 of V, hence -5Ia = V. From this it follows that

.V = 0 which contradicts the supposition that PV # 0. For n > 1 we assume the assertion for n - 1 instead of n. We denote by J9P' the set of elements a of. -*with

awl = ... =

(105.4) z = 0. Evidently 2' is a left ideal of }', so is an - -submodule of V. On the other hand, according to the induction assumption, 0, so that 1

i = V. We have only to show that there is an a (E P') with We suppose this to be false. Then for the elements a of .1' awn- ,

= 0 > a(o = 0.

(105.5)

aw

0.

(105.6)

DENSITY THEOREM OF CHEVALLEY-JACOBSON

425

In order to derive a contradiction from this, we show that (a E,.*')

awn_ 1 -* awn

(105.7)

defines anR-endomorphism of V. Because of (105.5), (105.7) is a (possibly many-valued) mapping of V into itself. Furthermore, this mapping is unique, since if awn_1 = a'wn_1 for two

elements a, a' of 2', then (a - a') w.,_1 = 0. By (105.6), (a - a')wn = 0 and hence awn = a'wn. Since 2' is a left ideal of .-.*, for arbitrary a, b (r 2') and c (E 19) the elements a + b, ca belong to *', whence it follows that the elements

awn-1 + bwn-1 (= (a +

e.awn_1( = cawn-1)

are mapped by (105.7) onto ((a + b)wn =) awn + bw,,,

(ca(On =) c awn,

respectively. Accordingly (105.7) is in fact an '-endomorphism of V. Consequently, there is a k (EY), for which (105.7) agrees with the mapping

cc -' ka

(aE V).

This implies that acon = k(awn_1)

for all a (Es'). But since k is an R-endomorphism of V, the right-hand side is equal to a(kwn-1), thus

a(w - kwn-1) = 0 for all a E JP'. Hence it follows, by (105.4) and the induction assumption. that wl, ... wn_n, wn - kwn-1

must be linearly dependent. Then w ...., co,, are also linearly dependent. This contradiction proves the proposition. Now, in order to prove the first assertion of the theorem, let us also denote ?" be the left ideal

by w1, . . ., wn linearly independent elements of V. Let

of V consisting of the a with

Because of the proposition

c.

and

"w,, are an .. V-submodule ( : A 0)

OPERATOR MODULES

426

of V and so equal to V. Consequently, there is, for every en (E V), an an (E R) with anwl = ... = anwn-1 = 0, anwn = en

Apply this for 1, ..., n instead of n. It follows that for a = al + ... + a,, the equations awl = 21, ... aw,, = en

bold, where el,. .., en are arbitrarily given elements of V. This implies that the ring 9 of the endomorphisms (105.2) of V is dense.

Finally, in order to prove the second assertion of the theorem, we suppose that V is of infinite dimension over Y. Then we take countably infinitely many linearly independent elements w1, w2, ... of V and denote by Jan the left ideal of consisting of those elements an for which From the proposition it follows that,55'1 D 1*2 .... But this is in contradiction to the supposition that for the left ideals of . the minimal condition is satisfied. Consequently the theorem is proved.

§ 106. The Structure Theorems of Wedderburn-Artin Before considering important applications of the results obtained above, we need the following theorem, important in itself. THEOREM 255. The full matrix ring f n of rank n2 over a skew field .7 is simple. Further the ring (.7 )° opposed to Y,, is isomorphic with the full matrix ring (.f°)n over the skew field9" ° opposed to Y.

We consider an ideal a other than 0 of .Y,,. We have to prove that a = Y,,. To this end, we take an arbitrary element, other than 0, n+

a = L aijE,i i.j=1

of a, where the Eij denote the matrix units of Y,,, and at least one coefficient, say au, is different from 0. Because a,j' E,,c'-Ejs = E,.,

(r, s = 1, ..., n),

a contains all the E,,s, consequently all the elements of Y . So the first part of the theorem is proved. The second part of the theorem is easy to prove, namely,

(T,)°

(-170). (A -A),

where A' denotes the transpose of the matrix A (E F,,).

427

STRUCTURE THEOREMS OF WEDDERBURN- ARTIN

As a generalization of the concept of nilpotent elements, a left ideal (right ideal or ideal) a of an arbitary ring is called nilpotent, if there is a natural number n with a" = 0. A ring with minimal condition for left ideal is called semisimple, when it has no nilpotent left ideals different from 0. This nomenclature is justified by the fact that every semisimple ring is a direct sum of finitely many simple rings. We prove much more than this assertion. Namely the semisimple rings constitute a fully discovered structure class for which first of all the following extremely important theorem holds. THEOREM 256 (first structure theorem of WEDDERBURN-ARTIN). Every semisimple ring ( 0) is a direct sum of finitely many full matrix rings over skew fields. Let R (# 0) be a semisimple ring and V a minimal left ideal of 'R. Then V is a simple R-leftmodule. Also R V 0, since if R V = 0 then, a fortiori, VZ = 0, while V contains no nilpotent left ideals different from 0. Therefore we can apply the density theorem Of CEV P ALLEY - JACOBSON (Theorem

254). If we take into consideration the second parts of Theorems 158 and 255, then it follows that V is a vector space of finite dimension n over a skew

field 7, and the ring, which consists of the distinct .7-endomorphisms

a -* as

(a E -*)

(106.1)

of V, is isomorphic with the full matrix ring of rank n' over the skew field 3° opposed to.7. But this endomorphism ring is obviously a homo-

morphic image of S'z, because it is isomorphic with .9/a, where a is a left ideal (and so an ideal) of .

which annihilates V. Consequently ._`P/a

(70)..

(106.2)

We show that 5z' has the direct decomposition

=V ®a.

(106.3)

Next, it is evident that VV is an ideal of _. For the intersection

fl a, b2=bb

1=0.

=0.

Since S'z contains no nilpotent left ideal different from 0, it follows that b = 0. Therefore we have only to show that

-92 =V + a. We suppose this to be false. Since R/a, according to (106.2) and the first part

of Theorem 255, is simple, it follows that idthe eal (VV + a)/a of *la

OPERATOR MODULES

428

is equal to 0. This means that VR S a. Because b = 0, then V'* = 0, consequently V2 = 0. This contradiction proves (106.3).

Now for the first term of the right-hand side of (106.3), we have, by (106.2), the isomorphy

Vt

(3°)" .

(106.4)

But because of (106.3) every left ideal of a is also one ofd. Accordingly, together with JP, a (c 2) is a semisimple ring, with which (instead of '2) we can repeat the above process. But because of the minimal condition this must be broken up into finitely many steps. Hence, and from (106.3) and (106.4), follows the truth of Theorem 256. THEOREM 256' (second structure theorem Of WEDDERBURN-ARTIN). Every simple ring ( 0) with minimal condition for left ideals is isomorphic

with a full matrix ring over a skew field, or is a zero ring of prime order. Let 2 denote a ring, for which the conditions are satisfied. If it has no nilpotent left ideal other than 0, then it is semisimple and simple, therefore on account of Theorem 256 the theorem is true. In the other case let a denote a nilpotent left ideal of V different from 0.

If a

_ W, then a"2 = * (n = 1, 2, ...). This is impossible as a is nil-

c A. But since the left-hand side is an ideal of the right-hand side and this is simple, it follows that a 2 = 0. Thus the ideal consisting of the left annihilators of R is different from 0 and consequently equal to V. Thus 22 = 0. Since in conformity with this J` is a zero ring, the module '+ is itself simple, thus of prime order. Therefore the theorem potent. Accordingly a

is proved.

Theorems 256 and 256' with their converses, to be considered later, make a complete characterization of simple rings with minimal condition for left ideals or of semisimple rings possible.

THEOREM 256" (converse of Theorem 256'). The full matrix ring .3" of rank n2 over a skew field .7 (which is simple, according to Theorem 255), is also

semisimple. Also the .9,, left module .7 is the direct sum (106.5)

of minimal left ideals 11, ...,1", where l; consists of those matrices A (E which have only elements 0 outside the j`h column.

THEOREM 256" (converse of Theorem 256). Every direct sum of finitely many full matrix rings over skew fields is semisimple.

For the proof we need the following THEOREM 257. In every group with composition series for the normal divisors

the minimal condition is satisfied.

STRUCTURE THEORLMS OF WEDDERBURN- ARTIN

429

This theorem also holds for operator groups, and the corresponding theorem holds for rings with operators. If n is the length of the composition series of the group to be considered, then, from the theorem of JORDAN-HOLDER (Theorem 136), it follows that

every normal divisor of it has a composition series of at most length n, so Theorem 257 is proved. We now prove Theorem 256". It is obvious that 11, . . ., 1 are left ideals of .7; and that (106.5) holds for them. To prove the assertion that 11, . . ., 1 are minimal, it is sufficient to show it for 11. We have recourse to the matrix units E,1 of .7,,,. Then 11 consists of the matrices n

A=

i=1

(al...., an E .`) .

a.E,1

Of these we take an A different from 0, where e.g., ai is different from 0. It will suffice if we show that t-5rn A = 11. Because

Cra, 1Er,A=CrErl

(Cr E

;r

1,....n),

contains all the C,Erc, and so also the I?

Y c Erl

r=1

whence the assertion .7.A = 11 follows. Moreover we have to show that no left ideal I (# 0) of , , is nilpotent. For that purpose we take an element n

A= of 1. Let, e.g.

(air E 9r)

ai1E,1

ars=c'#0.

The matrix

B = ES,A = : ar,Esj j=1

lies in 1. Furthermore n

CCn

B- _ E ar;arjEs;Esj = L arsarjEsj = cB . 1,j=1

i=1

Since, according to this, B is not nilpotent, then neither is I. Now (106.5) means that -9',+, considered as an .5z-left module, is completely

reducible, since the minimal left ideals of .fin are the simple 5 -left submodules of 7,+,. Consequently it follows from Theorems 137 and 257

OPERATOR MODULES

430

that in .7. the minimal condition for left ideals is satisfied. This completes the proof of Theorem 256". In order to prove Theorem 256", we consider a ring

R=R1®...®R,,,, which is the direct sum of full matrix rings R1, . . ., R,,, over skew fields. By Theorem 256" R is the direct sum of finitely many minimal left ideals. So it

follows, as above, that the minimal condition for left ideals holds in R. Consider further a nilpotent left ideal I of R. The R; component of I is then a nilpotent left ideal of R, (i = 1, ..., m), which, by Theorem 256", equals 0, whence I = 0. Consequently Theorem 256'" has been proved. More generally, we can obta inimportant information concerning the socalled Artin rings, i.e. rings with minimal condition for left ideals, from the first structure theorem of WEDDERBURN-ARTIN (Theorem 256). The follow-

ing theorem is a necessary preliminary. THEOREM 258. The sum of two nilpotent left ideals of a ring is again nilpotent;

further, every nilpotent left ideal is contained in a nilpotent ideal. For the proof, let a denote a left ideal of a ring R with as = 0 for a natural

number a. Let Pk(a) = ... OC1 ... OC2 ...... OCk ...

((C1,

..., Sc, E a)

be an arbitrary product of elements from R with at least k factors from a. Since Pk(a) = (... OCl)

(... y.,) ... (... OCk) ...

and the partial products, included within parentheses, lie in a, it follows from as = 0 that all the PP(a) vanish. From this observation it is easy to derive the theorem. For, if b is a further ideal of R with b° = 0, then every element of (a + b)at8-1 is a sum of products Pa(a) or Pb(b), and thus equal to 0. Therefore a + b is nilpotent. Moreover, every element of (aR)a is a .Pa(a), thus the left ideal aR is nilpotent, consequently the ideal a + aR (? a) is also nilpotent. So the theorem is proved. A left ideal consisting entirely of nilpotent elements is called a nil left ideal.

Every nilpotent left ideal is then, afortiori, a nil left ideal. The following definition is of great importance : the sum of all the nil-

potent left ideals of a ring is called the radical of the ring. A ring with radical 0 is called radical ,free. THEOREM 259. The radical of a ring is always a nil ideal, that of an Artin ring is a nilpotent ideal. Let R denote a ring and it its radical. From the second part of Theorem

258 it follows at once that n is the sum of all the nilpotent ideals of R,

STRUCTURE THEOREMS OF WEDDERBURN-ARTIN

431

consequently it is itself an ideal of. Furthermore, every element of it lies in a sum of finitely many nilpotent left ideals of R, and so, because of the

first part of Theorem 258, is nilpotent. This proves the first part of Theorem 259.

Then let R be Artinian. We even prove that every nil left ideal a of R is nilpotent. Since a a2 ? ... is a descending chain of left ideals, there is a natural number a with aa = as+I (= as+z We have to show that a° = 0. With this end in view we suppose that the left ideal b = a° is not 0. Then b2 = b 96 0. From among the left ideals m of R, contained in b, with bm # 0, we take a minimal one. This m contains an element ,u with bu 0. But since by (C m) is a left ideal of R and bb,u = b,u 0, it follows that by = nt. Accordingly there is a (3 (E b) with fla = p. So ,u = pit = l32µ = = ..., i.e., µ = 0, since (3 is nilpotent. This contradiction proves Theorem 259. THEOREM 260. If the radical it of a ring R is nilpotent, then the factor ring R/n is radical free. According to the Supplement of Theorem 128', every left ideal of R/n may be taken as I/n for some left ideal f (;? n) of R. We suppose that I/n is nilpotent, i.e., for some natural number I (I/n)' = 0

.

We have to prove that I = it. The product of 1 arbitrary residue classes mod it of I is 0, so the product of 1 arbitrary elements of I lies in it. This implies that I' c it. Therefore from the nilpotence of it, that of I follows. Since accordingly 19 it, we have I = it as required. For this paragraph cf. SZELE (1954). EXAMPLE. Theorem 130' may be obtained from Theorems 256' and 256" as follows:

Let R (96 0) be a ring without proper left ideals. If R is a zero ring, then O(R) must be a prime number. Otherwise, it follows that R is a full matrix ring F. of rank n$ over a skew field F and necessarily n = 1, so that R = F. PROBLEM. Give for a skew field F a complete system of non-isomorphic F-subrings of the full matrix ring FA.

CHAPTER VII

COMMUTATIVE POLYNOMIAL RINGS Some fundamental facts on commutative polynomials of one indeterminate will be discussed here. Polynomials of several indeterminates will be examined partly by the use of results obtained here and partly by studying more closely the so-called symmetric polynomials among them. The fundamental domain will mostly be an arbitrary ring with unity element, but some-

times an integral domain or field, in connection with special problems. § 107. McCoy's Theorem

Polynomial rings with zero divisors have, so far, been given far less attention than those without zero divisors. The following theorem is very important. THEOREM 261 (McCoy's theorem). If the polynomial f(x) (# 0) is a zero divisor in the polynomial ring R[x], then there is an element c (# 0) of J with cf(x) = 0.

For the proof we take f(x) in the form f(x) = aox" + ... + a,,

(n >_ 0; at E .T; ao # 0) ,

(107.1)

and denote by g(x) (A 0) a polynomial in fi[x] with

fix) g(x) = 0 .

(107.2)

If g(x) is constant, then the theorem is proved. From now on we assume that

g(x) is of degree at least 1. It will suffice to prove that in R[x] there is a polynomial h(x) (# 0) of smaller degree than g(x) with the property that f (x) h(x) = 0 , since from repeated application of this the theorem will follow. Let us consider the polynomials aog(x), .

. .

432

a"g(x)

(107.3)

McCOY's THEOREM

433

If these all vanish and c (# 0) denotes the leading coefficient of g(x), then

aoc = ... = ac = 0. Because of (107.1) we then have f(x)c = 0, therefore the theorem is now true. Otherwise there is an r (0 5 r S n) with

aog(x) _ ... = ar-1 g(x) = 0,

(107.4)

a g(x) s 0 .

(107.5)

Now (107.2) may be written, by (107.1) and (107.4), as:

(arx"-' +

... + a,Jg(x) = 0.

Hence it follows that the polynomial

h(x) = arg(x)

(107.6)

is of smaller degree than g(x). By (107.5), h(x) # 0. Finally from (107.2) and (107.6) we obtain (107.3). Hence the theorem is proved. Cf. McCoy (1948). EXAMPLE. Theorem 261 is false for non-commutative polynomial rings 12[x]. Consider in this respect the polynomial 1

f (x) = -1

-1, x- (1 l)

in 9$[x]. Although for

g(x) =

1 -1

11 1,1

l) x + (l -1)

the equation g(x)Ax) = 0 holds, we have (e d

for every element (a dl ( c

f(x) # 0

0) of 9=, as is obvious.

§ 108. Differential Quotient As is well known, in function theory the differential coefficient of a function is defined as a limit. The following definition of the differential quotient of polynomials is not based on the notion of a limit. First of all let R be an arbitrary ring with unity element, for which we shall later also postulate commutativity. We consider an arbitrary poly-

nomial f(x) in R[x]. Form by the help of a further indeterminate y the polynomial f(x + y) (in the polynomial ringR [x, y]) and write it in the form

Ax + y) =.fo(x) + Yfi(x) + Ysf2(x) + ... ,

(108.1)

COMMUTATIVE POLYNOMIAL RINO3

434

where the coefficients fi(x) are uniquely defined polynomials in '*[x]. If we put y = 0, then fo(x) = f(x). Hence, and from (108.1), we have the congruence

f(x + y) = f(x) + yf1(x) (mod y2) with respect to the ideal (y2) of R[x, y]. The polynomial f1(x), which we call the differential quotient of f(x) and denote by f'(x), is uniquely defined

by f(x). Therefore its definition is contained in the congruence J (x + y) = f(x) + yf'(x) (mod y2) .

(108.2)

We denote a further polynomial from R[x] by g(x). Then g(x + y) = g(x) + yg'(x) (mod y2) . Hence, and from (108.2), after addition and multiplication we obtain

f(x + y) + g(x + y) =.f(x) + g(x) + y(f' (x) + g'(x)) (mod y2) , J (x + y) g(x + y) = f(x) g(x) + y (f'(x) g(x) + f(x) g'(x)) (mod y2) . Comparison with definition (108.2) yields the formulae for the differential quotient of the sum and product of two polynomials:

(f(x) + g(x))' = f'(x) + g'(x) ,

(108.3)

(Ax) g(x))' = f'(x) g(x) + f(x) g'(x) .

(108.4)

Of these (108.3) means that the mapping f (x)

f' (x) is an endomorphism

of the module A[x]+. Formula (108.4) is known as Leibniz's rule. Both formulae (108.3), (108.4) may be immediately generalized to k

ll'

i=1

k

(108.5)

i=1

Ma1 ... ak I, _ E a1 ... ai-1 ai ai+1 ... Mk ,

i=1

/

i=1

(108.6)

where al,..., ak are arbitrary polynomials of Jp[x] (k >_ 1). We now wish to determine f'(x) explicitly. From (108.2) we obtain

c'=0, x = 1

(cE, ).

Hence, and from (108.6), first (xk)' = kxk-1

(k >_ 1) ,

(108.7)

435

DIFFERENTIAL QUOTIENT

then, by (108.4), (cxk)' = kcxk'1 (c E 2). So, because of (108.5), the explicit formula required for the differential quotient of an arbitrary polynomial of i[x] reads: (ao + a1x +

... +

ax")' = a1 + 2a2x +

... + nax"-1.

(108.8)

We see that (108.8) agrees formally with the rule from function theory for the differentiation of polynomials. However, formula (108.8) means much more, as it refers to an arbitrary polynomial ring rk [x], whereas in function theory usually only the field of real or complex numbers is admitted for the fundamental ring rk. Of course we could have defined the differential quotient immediately by (108.8)

but the idea of the original definition (108.2) is clearer, and the application of (108.2) is often more convenient than that of (108.8).

We gather from (108.8) that the differential quotient of a polynomial

f(x) = ao +

... +

(108.9)

vanishes if, and only if,

o+(a,)Ii

(i=1,...,n).

(108.10)

Accordingly we may even have f'(x) = 0, when f(x) is non-constant. The polynomials f(x) for which f(x) = 0 are called polynomials with vanishing differential quotient. These are thus characterized by condition (108.10). It is in the case of a zero-divisor-free fundamental ring .2 that the question of the vanishing of f'(x) is of the greatest importance. By Theorem 38,

the characteristic of R is then equal to 0 or to a prime number p. In the first case (108.10) implies that a1 = ... = a" = 0, i.e., that f(x) is a constant. Since, in the second case, o+ (a;) = p for a1 0 0, (108.10) now implies that in the sequence ao, al, ... at most the terms a0, ap, a2p, .. .

are different from 0, i.e., f(x), according to (108.9), is of the form

ap + apxp + a2p x2p + ... ,

i.e., a polynomial in x". This result, for both cases, is contained in the following

THEOREM 262. In a zero-divisor free polynomial ring '[x] of characteristic

p (= 0 or a prime) all the polynomials with vanishing differential quotient are the f(xp) (f(x) E 2[x]). From (108.8) we also see that for a non-constant f(x) the differential

quotient.f'(x) is of smaller degree than f(x). If f(x) is of degree n with the leading coefficient c, then f'(x) is of degree n - 1 if and only if, nc

0.

COMMUTATIVE POLYNOMIAL RINGS

436

We define the ia` differential quotient (or the differential quotient of order 1) f te(x) of f (x) by

(i = 0, 1, ...) ,

(f 0(x))'

where it is understood that f) = f. In particular f(l) = f'. For f 2), f« we also write f",f"'. From function theory we borrow the following notations: ddxr)

=

dd-x'f(x) =.f"(x)

and in particular df(x) dx

=

d .l(x) =f'(x)

dx

Since a polynomial f (x1, ..., x,,) (E [xi, ..., x"]) may be considered as a polynomial in xk over 9[xl, ..., xk-v xk+v ..., we can correspondingly form the differential coefficient

dk f(xI,...,xa, which is called the (partial) differential quotient of f (x1, ...,

with

respect to xk, and to avoid possible misunderstanding, we denote it by a

axk J XII .

.

., x") .

We also write frr(xj, ..., for this or, sometimes, fk(xj, . . ., x.). 263 (EuLui 's theorem). For a homogeneous polynomial f (x1, .,

of degree r in an arbitrary polynomial ring tR [xI, . . ., x"] k

i=1

a

y

xr - (xl, ..., xa) = rf (x1, ..., x,,) ax,

It suffices to prove the theorem for a one-termed polynomial of the form

(k1+... +k"=r). Since we now have xI

x f (x1, ..., x,,) = krf(xI, ..., x,,) , f

we obtain the theorem by summing for i = 1, ..., n.

DIFFERENTIAL QUOTIENT

437

If in the polynomial P'(x) the indeterminate x is replaced by c (ER) or by g(x) (E 51 [x]), then we denote the result of this replacement by

f (')(c) or by f tn(g(x)). A similar notation is applied in connection with several indeterminates.

From now on let us suppose that * is commutative. If f(x), g(x) are arbitrary polynomials in,* [x], then we obtain by repeated application of (108.2) :

f(g(x + y)) m f(g(x) + yg'(x)) = f(g(x)) + yg'(x)f'(g(x)) (mod y2 . Again, because of (108.2), this implies that (108.11)

(f (g(x))' =f'(g(x)) g'(x) ,

where the differential quotient of the iterated polynomial f(g(x)) is denoted on the left. Especially we have

(f (x + c))' =Ax + c)

(108.12)

(c E JT) .

We prove the more general rule d

f(gI(x),

, gn(x)) = E fi(gl(x)

, gn(x))

,

t=1

dg,{x) dx

,

(108.13)

where

a

(i = 1, ..., n).

f (xl, ..., x,. ) = ax, f ( x t , ..., x n ) Write

F(x) = .1(gl(x), ..., gn(x)) then according to (108.2)

F(x + y) ° .1(gl(x) + ygi(x), ..., gn(x) + ygn(x)) (mod y2) .

(108.14)

But from (108.2) we obtain the congruence

f(x1 + ti, ..., xn + tn)

f(x1, x2 + t2, ..., X. + t,.) + tlfl(x1, x2 + t2i ..., X. + t .)

(mod ti)

in the polynomial ring S[x1, ..., xn, t1i ..., tn]. If a denotes the ideal generated by all the tits, then we obtain from the above congruence by continued application of (108.2): n

.1(x1 + 1 1,--.

E tif, (xl, ..., x,) (mod a) . .Xn + tn) =.1(x1, ..., xn) + =1

438

COMMUTATIVE POLYNOMIAL RINGS

After that we carry out the substitutions xr = 9r(x), ti = yg (x)

Since every element of a becomes an element of the ideal (y' and because of (108.14) we obtain

F(x + y) = F(x) + y

r=1

9i (x) ff (91(x), ..., 9"(x)) (mod ys).

This, together with (108.2), proves (108.13). THEOREM 264 (TAYLOR'S theorem). For every polynomial f(x) of degree n

over a field .7 of characteristic 0 or p > n

AX + c) = E =o

i!

x'

(c E .7)

(108.15)

.

[In (108.15) the denominator i! is to be regarded as an element of J r, namely as i t times the unity element.] By hypothesis

J(x+ c)=bo+... +bx"

(bo,...,b"E. ').

After differentiating i times, we obtain for x = 0

f(O(c)=i!b,

(1=0,...,n); -

then bi = (fl)-'f() (c). Thus the theorem is proved. EXERCISE. In an arbitrary ring R following BOURBAKI (1939) we understand by

differentiation any endomorphism a -- a' of R+ with the additional property that (a3)' = a'/3 + afl' ((X, fl E R). This yields a generalization of differentiation of polynomials. Rules (108.5) and (108.6) also hold here. If, moreover, R is an integral domain, then a - a' may be extended to give a differentiation in the quotient field F of R as follows a

a l'

T

a'p - a#' #2

(a, P E R ;

0)

In the ring J the zero endomorphism of J+ is the only differentiation.

§ 109. Field of Rational Functions We denote a field by -9'. Since the polynomial ring . '[x1, ..., x"] is an integral domain, it has a quotient field, which we call the field of rational functions (or rational function field) of the indeterminates x1, ..., x" over and denote by ,7(x1, ..., x"). Its elements

r(x1, ..., x") =

.(x1, . . ., x,)

9(x1, ..., xa

(9(x1, ..., x') # 0)

(109.1)

FIELD OF RATIONAL FUNCTIONS

439

are called the rational functions of the indeterminates x1, . . ., x over Y, where numerator and denominator are elements of -'X[x1,..., xn]. If we consider the polynomial ring .7 [x1, x2, ... ] with infinitely many indetermi-

nates, then we obtain in the same way a field of rational functions of infinitely many indeterminates, which we correspondingly denote by '(x1, x2, . . .). Each of its elements contains only finitely many indeterminate,, so it is likewise a rational function. The term "rational function", borrowed from function theory, is, admittedly, not very suitable as it has nothing to do with functions as such. On the contrary, (109.1), after replacing the x1, ..., xn by variable elements t1, ..., t,, of f, becomes a proper function .

g(t1,

.., Q

where, of course, only the t1, . . ., t such that g(t1i . . ., t,,) # 0 are admitted. Since, according to GAUSS'S theorem (Theorem 207), -9'[x1, ..., xn] is a ring with prime decomposition, we may suppose in (109.1) that the numerator and the denominator have no common divisor. Then (109.1) is called

the reduced form of the rational function or a reduced rational function. In this the numerator and the denominator are, apart from a common

constant factor c (E"

0), uniquely defined. We occasionally refer to them

as the numerator or denominator, respectively, of this rational function. The maximum of their degrees is said to be the degree of the rational function. In particular, a rational function a1x1 + ... - anx,, + a b1x1 + ... + bnxn + b

of the first degree is called a linear rational function and, if a = b = 0, a homogeneously linear rational function.

In (109.1) numerator and denominator may also be considered as polynomials in xn over .52'[x1, ..., xn_1]], whence it follows that

-7'(X1, ..., xn) _ (.7[X1..... xn-1J)(xn)

According to this Y(x1, ..., xn)

(_7(X11

..., xn-))(Xn) ``

Since this also holds trivially with Q instead of S, it follows that

J,(x1, ..., x,.) _ (7(x1, ..., Xn-1))(Xn) Because of this recurrence formula it will be sufficient to restrict ourselves,

in everything concerning fundamental questions, to the consideration of .7(x). 15/a R.- A.

440

COMMUTATIVE POLYNOMIAL RINGS

§ 110. The Multiple Divisors of Polynomials

First we consider an arbitrary commutative ring R with unity element. Of two elements a, j9 (E R) we call a a k ple divisor or a k ple factor of f

(k=0, 1,...), if

ak

Y

(110.1)

ak+1

and we use for this the notation (110.2)

akIIP

If k >_ 2, then we say that a is a multiple divisor or multiple factor of P. If, moreover, R is zero-divisor-free, and so an integral domain, then (110.2) evidently means that there is an element y of R with # = aky , a,f'y

If a is a unit, or P = 0, then (110.2) is not satisfied by any k. In what follows we are concerned only with the case where R is an integral domain with prime decomposition. Apart from the exceptional cases just mentioned, there is exactly one k for each pair a, f (E R) for which (110.2) holds. The case where a is a prime element is important. In this respect we note the following: if e?CT

... 79r

is the prime-power decomposition of f, where N is a unit and

v1, .. y 7cr

are the distinct prime factors of j9, then

r) . We now consider a commutative polynomial ringR[x]. We know from the corollary of Theorem 204 that an element c (E ') is a zero of a poly-

nomial f(x) (E 2[x]) if, and only if, x - c I f(x). More generally, if (x - c)k I I f(x)

(110.3)

we call c a k ple zero of the polynomial f (x) or a k ple root of the equation f(x) = 0. If k Z 2 we use the terms multiple zero or multiple root. THEOREM 265. If, for two elements f (x), g(x) of a commutative polynomial

ring 2[x] and an integer k (>_ 1), (g(x))k I fix)

,

(110.4)

then (g(x))k-I If(X)

(110.5)

441

MULTTPLE DIVISORS OF POLYNOMIALS

According to the given supposition f(x) _ (g(x))kh(x)

(110.6)

for some h(x) E .R[x]. Hence

f'(x) = kg'(x)(g(x))k-I h(x) + (g(x))kh'(x),

(110.7)

and, consequently, the theorem is valid. It is important that under certain circumstances this theorem should hold

with "! l" instead of "I". The following theorem brings out this point. THEOREM 266. Let -W be an integral domain of characteristic p (= 0 or a prime number) with prime decomposition. If, for an irreducible polynomial g(x) and for a further polynomial f (x) in R [x], (g(x))k I I f(x)

,

k ? 1 , p , k , g'(x)

0

,

(110.8)

then

(g(x))k -' ! ! f'(x)

(110.9)

According to GAUSS'S theorem (Theorem 207) R[x] is also an integral domain with prime decomposition. Thus if we assume that f(x) has the form (110.6), where now, because of (110.8), Ax) 'f' h(x)

then (110.7) reveals the absurdity of (g(x))k f'(x). Hence, and from Theorem 265, Theorem 266 follows. From this follows immediately

TimOREM 267. If ii (and so also fi[x]) is an integral domain with prime decomposition and j (x) a product of irreducible linear polynomials in _ [x], then (f(x), f'(x)) = 1 is a necessary and sufficient condition for f (x) to have no multiple divisors. § 111. Symmetric Polynomials We take an arbitrary ring with unity element. A polynomial f (x1, ..., xA) is called symmetric if it is invariant with respect to all permutaover

tions of the indeterminates x1, . . ., With the help of a further indeterminate z we form the polynomial

2-...+(-1)"s. , (111.1)

COMMUTATIVE POLYNOMIAL RINGS

442

where s1, .

.

., s belong to * [x1,

. .

It is evident that by this the

., xn].

s1i ..., s are uniquely defined and symmetric. We call s; the ith elementary symmetric polynomial in the indeterminates xl, .... xn (i = 1, . . ., n). Since the left-hand side of (111.1) is homogeneous in the indeterminates z, x1, . . ., xn, the right-hand side shows that 1 is at the same time the degree of s,. By (111.1), the explicit definition of the elementary symmetric polynomials is as follows:

S1=x1+x2+...+xn, S2=XIX2+X1X3+...+XJxn+...+Xn-lXn, S3 = X1X2X3 + X1X2X4 + ... + Xn-2Xn-1Xn,

TtnoRa1268 (main theorem for symmetric polynomials). Iff(x1i ..., x,,) is a symmetric polynomial over a ring R with unity element, then there is exactly one polynomial F(x1i ..., xn) over *such that -

f(x1, ..., x,,) = F(s1, . . ., s,) ,

(111.3)

where si is the ith elementary symmetric polynomial of x1, ..., xn (i = 1, ..., n). Furthermore the coefficients of F(x1i ..., x,,) belong to the submodule of $ generated by the coefficients of f(x1, ..., xn). The first part of the proof, in which we verify the existence of F, provides at the same time a process for determining the coefficients of F. We order f(xl, . . ., x,) lexicographically and denote by ax,k,

...

X,k,n

(aE

_

), #0)

(111.4)

the lexicographically last term of it, which we call its principal term. Since . ., x,.) is symmetric, it contains all the terms resulting from (111.4) by the permutation of the x1, . . ., x,,, whence

fxl, .

k1 ?

> kn.

We then consider gSki-k'S2'-k..

S,k,-1_knsnn

By (111.2) the principal term of (111.5) is obviously axi`_k' (x1 x2)k'-k'... (X1

...

xn-1)ka-I-kn (x1

... xn)kn

(111.6)

SYMMETRIC POLYNOMIALS

443

Since this is equal to (111.4), it follows that all the terms of

f1(x1, ..., x,.) = (x1, ..., x") -

asi'-k2

f

... sn"

(111.7)

in the lexicographical sequence precede the term (111.4). Now f1(x1, ..., is also symmetric in x1, ..., x", wherefore the process may be repeated with it instead of f(x1i ..., x"). The principal terms, occurring at each step, constitute, according to what was said above, a lexicographically decreasing sequence and they are all of the form

bxl'...xt

(b ER)

with

11>...z1", 115k1. Since there are only finitely many possibilities for these exponent systems 11, ...,1., it follows that the process terminates after finitely many steps. If we write (111.7) in the form

f(x1, ..., x,,) = fl(X1...., x,,)

asi'-k' ... Jrs"

we see that it leads to the determination of a required polynomial F(x1,..., x,). It is evident that the last assertion of the theorem is satisfied by this. We have still to prove the assertion of the theorem regarding uniqueness. Evidently it will suffice to show for a polynomial G(x1, . . ., over.2 other than zero that G(s1i . . ., s") is also different from zero. With this end in view we consider an arbitrary term of G(x1, ..., x,,.), which we write in the form axi'-k' ... --k" xn" (a E R; k1 >_ ... z k") . (111.8) If we replace x1, ..., x by s1,.. ., s", then according to (111.2) we obtain a polynomial in the x1, . . ., x" whose lexicographically last term is evidently (111.6), i.e., (111.4). If moreover (111.8) is the principal term of G(x1, ...,

x"), then the term (111.4) of G(s1, ..., s") is not eliminated. This completes the proof of Theorem 268. ExAMPLE. We have (cf. § 115)

xi+xq=s1-2s2, xi+x$=sj-3s,s2 (s,=x,+x2; s2=x,x1) § 112. The Resultant of Two Polynomials To decide whether two polynomials have a non-constant common divisor is a fundamental problem. We shall deal here with the simplest case, where

there are two polynomials f(x), g(x) over a field. The question can be answered by means of the Euclidean algorithm, which immediately gives the

444

COMMUTATIVE POLYNOMIAL RINGS

greatest common divisor; on the other hand, the above question of existence is, as we shall see, much easier to answer. First of all we take an arbitrary commutative ring R with unity element and in it form a so-called Sylvester determinant

ao...am rows

R =

ao

bo ..b

.. am

(112.1) rows

bo

b

of order m + n (m, n > 0), where the a,, b, denote elements of R and with only zero elements in the empty spaces. By virtue of the Theorem 270 below, R is called the resultant of the polynomials

f(x)=aox'"+...+am g(x)=box" +... +b

(m > 0),

(n > 0),

(112.2) (112.3)

so that we also use the notations R = R(f(x), g(x)) = R(f, g) .

(112.4)

We often consider R to be a polynomial in (,R =) _7 [ao, ..., am, bo, ..., The product as bn of the diagonal elements of (112.1) is called the leading

(principal) term of the resultant R. It should be noted that R(f, g) is not uniquely defined by the polynomials f, g, since, in general, m, n, denote the formal degrees of f and g. Thereby, if necessary, we have to use the notation R,,(f, g) for the resultant. One could ensure uniqueness by demanding that ao 96 0, bo 96 0, but this would detract from the general validity. From now on, m, n will denote fixed chosen numbers, so that we may keep to the shorter notation (112.4).

In the resultant R(f, g) the order of succession of the polynomials f(x), g(x) makes a difference, but by (112.1)

R(g,f) = (-1)m"R(f, g).

(112.5)

THEOREM 269. The resultant R of the polynomials f(x), g(x) given in (112.2) and (112.3) over a field 9 is equal to 0 if, and only if, these polynomials have a non-constant common divisor, or when ao = bo = 0.

445

RESULTANT OF TWO POLYNOMIALS

If ao = bo = 0 then, by (112.1), R = 0. Thus the theorem is true for this case. Therefore in the following we suppose that a0, bo do not both vanish. We have to prove that the greatest common divisor d(x) _ (f(x), g(x))

(112.6)

is not constant if, and only if, R - 0. First of all we show that d(x) is not constant if, and only if, the equation (112.7)

f(x) g1(x) -I- g(x)f1(x) = 0

can be satisfied by two polynomials fl(x), g1(x), which do not both vanish and are of smaller degree than m or n.

For that purpose we first assume that d(x) is not a constant. Then, because of (112.6)

g(x) _ -d(x) 91(x) J(x) = d(x)f1(x), for two polynomials fi(x), g1(x), which possess all the required properties [including also (112.7)]. Conversely, we now suppose that the required polynomials f1(x), g1(x) exist. It has then to be proved that d(x) is not constant. We may suppose that ao # 0, whence it follows that f(x) is not constant. If g(x) = 0, then the

statement is self evident, wherefore we have only to consider the case g(x) # 0. Since f1(x) = g1(x) = 0 is not true, it follows from (112.7) that neither f1(x) nor g1(x) is equal to 0. It will suffice for the following that f1(x) s 0. Since according to (112.7) we have .f(x) I g(x)f1(x)

and f1(x) has a smaller degree than f (x), it follows that g(x) is divisible by at least one prime factor of f (x). This means, by (112.6), that d(x) is not a constant. By this means we have proved the statement concerning (112.7). We now take

f1(x) = uox-1 + ... + um-1 91(x) = vpxn-1 + ... + vn-1

(u, E .}) ,

(v. E 9)

.

Equation (112.7) is then identical to +bou0 + blu0 + b0u1 + b2uo + blur + b0u2

a0vo

a1v0 + aovl a2v0 + a1v1 + aov2

amen-2 + am-lvn-1 amen-1

+ bnum-2 + bn-lum-1 + bnum-1

=0, =0, = 0,

= 0, =0,

446

COMMUTATIVE POLYNOMIAL RINGS

where the number of equations and that of the unknowns is m + n. By what was stated above d(x) is not constant if, and only if, this homogeneous

system of equations is not trivially solvable. For this, on account of the corollary to Theorem 248, it is necessary and sufficient that the determinant of this system of equations should be equal to 0. Since this determinant is exactly the transpose of the Sylvester determinant (112.1), and so is equal to this, Theorem 269 is proved. THEOREM 270. For the resultant R of two polynomials f(x), g(x) we have

f(x) F(x) + g(x)G(x) = R

(112.8)

with two suitable proper polynomials F(x), G(x) whose degrees are smaller than the given formal degrees of g(x) andf(x), and whose coefficients lie in the ring generated by the coefficients off(x) and g(x). For the proof we add, for j = 1, . . ., m + n - 1, the j`h column of the determinant (112.1), multiplied by xm+n-', to the last column. This leaves

the value of this determinant unchanged but its last column, because of (112.2) and (112.3), will consist of the elements xn - I.f l x ) , ..., I ( x ) , X, - Ig(x), ..., g(x)

Thus if it is expanded with respect to the last column, the theorem follows. In general, by the weight of a one-term polynomial cxp

... xr' ... ... zl r zQ°

(c # 0)

in .xo...., x ..., zo, ..., zf we mean the sum

it+2i2+...+rir+... +11+21.,+...+tlj. A polynomial f(xo, ..., x,., ..., zo, . . ., zr), is called isobaric if all its terms are of the same weight, which is then called the weight of the polynomial. It should be noted that, if st, . . ., sn are the elementary symmetric polyis a homogeneous symmetric polynomials in x1, . . ., x and f(xt, . . ., nomial of degree k, and so, according to Theorem 268, equal to a polynomial F(st, ..., sn), then it is isobaric with respect to s1, . . ., sn and of weight k. THEOREM 271. The resultant R given in (112.1) is a homogeneous isobaric polynomial of the ao, ..., am, b0, . . ., bn of degree m + n and of weight mn. With respect to ao, ..., am and with respect to bo, . . ., b", R is likewise homogeneous of degree n and m, respectively.

The assertion on the homogeneity and the degrees are trivial. We have

only to prove that every term of R is of weight mn. For that purpose

RESULTANT OF TWO POLYNOMIALS

447

we first consider an arbitrary determinant

D=IC1,I of order m + n, and note at once that under the replacement C

(1 < i < n) ,

= { a,_1

(n+1
bn+i-i

(112.9)

D becomes R, if we put

ak = 0 (for k<0 or k > m) , bk = 0 (for k<0 or k > n) . We now consider an arbitrary term such as ± C11' C22'

of D, where 1',

Cm+n, (m+n)'

..., (m + n)' is a permutation of 1, ..., m + n. It will

suffice to prove that in this term after substituting (112.9) the sum of the indices will be equal to mn. This sum is n

m+n

m+n

E(i'-i)+ 1=n+1 E (n+i'-i)=mn+ E(i'-O.

1=1

1=1

But since the last sum vanishes, Theorem 271 has now been proved. THEOREM 272. For the resultant R of two polynomials . (.Y - Ym)

(m > 0) ,

(112.10)

g(x) = bo(x - z2) ... (x - zn)

(n > 0),

(112.11)

f(x) = ao(x - yl) .

.

we have the three product representations m n

R=aobo fi l(y, -z1),

(112.12)

R = ao 11 g(Yk) ,

(112.13)

k=11=1

k=1

R = (-1)'nn bi fl f (z,)

.

(112.14)

/=1

First we prove (112.12). Here, we may assume that ao, bo, yl, Ym, z1, ..., zn are indeterminates over.5ro, so that after suitable substitutioila the general validity of (112.12) will follow.

448

COMMUTATIVE POLYNOMIAL RINGS

We write the given polynomials (112.10), (112.11) in the form (112.2), (112.3), where then

ar = (-1)'aosr

(1 = 1, ..., m) ,

(112.15)

br = (-1)'botr

(i = 1, ..., n),

(112.16)

while the elementary symmetric polynomials of yl, ..., y, and are designated by s1, ..., sm and t1, . . ., tn. The divisibility

z1, ..., z, (112.17)

ao bo I R

follows from (112.1), (112.15) and (112.16).

We show further that

Yk - zlI R

(k= 1,...,m; 1 = 1,...,n).

(112.18)

If yk is replaced by z1, then by (112.10) and (112.11) f(x), g(x) have the

common divisor (x - yk =) x - z1. Hence, because of Theorem 269 it follows that R vanishes after this substitution. This implies, by the corollary to Theorem 204, the divisibility (112.18). Since, by Gauss's theorem (Theorem 207) 9ro [ao, bo, Yl, Ym, Z1, Zn] is now a ring with prime decomposition, it follows from (112.17) and (112.18) that R is divisible by the right-hand side of (112.12). But it follows from Theorem 271, having regard to (112.15) and. (112.16),

that the polynomial R is homogeneous of degree n with respect to ao and also homogeneous of degree m with respect to bo. Furthermore it is of weight mn with respect to sl, ..., s,, t1i . . ., t,,. This means that R is homogeneous

of degree mn with respect to y1i ..., ym, z1, ..., zn. From the divisibility above m

,

k=1r 1 for some integer q. In order to determine this we substitute y1 = ... = ym = 0. By (112.15) a1= ... = am = 0, thus our equation, on account of (112.1), becomes then m

n

aob. = gaobo

(- zr)I r= I

The right-hand side is equal to (-1)gaobo t'", and so, according to (112.16), equal to gaob'. Hence it follows that q = 1, implying the truth of (112.12). From (112.12) because of (112.11) we obtain equation (112.13) and also,

by (112.10), equation (112.14). Consequently Theorem 272 is proved.

RESULTANT OF TWO POLYNOMIALS

If JV [x1, .

.

.,

449

xn], Y[x1, ..., xn] are two polynomial rings, in which 5°

is an overring of '9, then we say that 9'[x1, ..., xn] is obtained from .51+[x1, ..., xn] by an extension of the fundamental ring. A non-constant of a commutative polynomial ring .Ji[x1, ..., xn] polynomial f(xl, . . ., is called absolutely irreducible, if there is no commutative polynomial ring So[x1i ..., xn] arising from *[x1, . . ., xn] by an extension of the fundamental ring with the property that f(x1, ..., x,,) splits into a product of two non-constant polynomials in ..?[x1..... xn]. THEoREM 273. The resultant (112.1) is an absolutely irreducible polynomial

of ao, ..., a., bo, ..., b,,. We take a commutative polynomial ring .9'[a0..... am, b0..... arising from .51[a0, ..., am, lo..... bn] by an extension of the fundamental ring, and consider in it a decomposition of (112.1) into two factors F, G:

R=FG.

(112.19)

We subtitute (112.15) and (112.16), where s1, ..., sm and t1, ..., t again denote the elementary symmetric polynomials of the indeterminates Ym and z1, ..., z,,. Then R becomes the right-hand side of (112.12), Y1, while F and _G become symmetric with respect to y1, ..., ym as well as to z1, ..., z,,. From (112.19) it follows that YI - z1 is a divisor of F or of G. We may suppose that y1 - z1 I F. Because of the above symmetry F must then be divisible by all the factors Yk - zr in (112.12), which we denote irrespective of order by 11, ..., l,,, and show that F = 11... 1mnFo

f o r some polynomial F 0 from R = Y[ao, bo, y1.

(112.20) Ym, z1.

In addition to this we assume that

F= 11... 1,F1, for some r such that 1 5 r < mn and for some F1 from R. It is sufficient to prove that this is also valid for r + 1 instead of r. Let 1,+1 = Yk - z1. On account of the residue theorem for polynomials (Theorem 204), F* = 0 follows from 1,+1 F, i.e., If ... 1; Fl = 0 where "*" denotes the substitution yk -+ z1. But since the li, ...,1*, according to McCoy's theorem

(Theorem 261), are not zero divisors, it follows that F; = 0. Again by Theorem 204, it is implied that 1,+1 1 F1. This proves (112.20). Since now, according to (112.12), R = 11... lmnaobo and the 1, are not zero divisors, it follows from (112.19) and (112.20) that aobo = F0G. On the other hand, because of (112.1) ao, ho (before the substitution), are not divisors of R. Thus, by (112.19), neither are they divisors of G. Hence it follows immediately that G is a constant. This proves Theorem 273.

450

COMMUTATIVE POLYNOMIAL RINGS

EXAMPLE 1. The resultant of the polynomials ax + al, box + bl is ao a, bo b,

EXAMPLE 2. The resultant of the polynomials aox$ -,- alx + a2, box2 + blx + b2 is ao al a2 0 0 ao al as bo bl b2 0

0 bob, b2

ao a, a20 ao as

bo bl b2 0

= - 0 ao al a2 =

I

bo b2

(-

ao a,

al a.

bo bI

b, bs

0 be b, b2

EXAMPLE 3. Theorem 269 is false if, instead of the field .F', we take as a basis the integral domain 9[ J- 5], as in this the polynomials (cf. § 88, Example 2)

3x2+4x+3,

3x+2+V=5-

have no non-constant common divisor, although their resultant is 0. EXAMPLE 4. The proposition in Theorem 269 that the resultant R is equal to 0, if f(x), g(x) have a non-constant common divisor, also follows immediately from Theorem 270, even for an arbitrary integral domain instead of the field 31' . EXAMPLE 5. For the resultant R(f, g) of the polynomials given by (112.2) and (112.3) use the more precise notation R,,,,,(f, g). Then it follows from (112.1) that Rm.n+.t(f,g) = au Rmn(f, g)

(k = 1, 2, ...).

EXAMPLE 6. From the product formula (112.14) it follows that Rm+n. p(fg, h) = Rmp(f, h) R,(g, h)

(112.21)

for three polynomials f(x), g(x), h(x) of formal degrees m, n, p (> 0) primarily for the case where they may be decomposed into a product of linear factors, but also generally, by Theorem 299, (to be discussed later). We call (112.21) the multiplication theorem for resultants. EXAMPLE 7. The polynomial x2 + y3 is absolutely irreducible, but x2 + y2 is not

absolutely irreducible, since in the field of complex numbers x2 + y2 = (x + yI)

(x - A.

§ 113. The Discriminant of a Polynomial Let a polynomial

f(x) = aox" + ... + aA

(n > 0)

(113.1)

be given and suppose for the time being that its coefficients ao, ..., a,, are indeterminates (over Y). Its differential quotient is

f(x) = naox"-I + ...+ a"_I.

(113.2)

In the Sylvester determinant (112.1) defining the resultant R(f, f'), the elements of the first column are divisible by ao, i.e.,

in .7[ao,

. .

.,

1 R(f, f') is a polynomial ao

a"]. For certain reasons to be clarified later, we assign the

451

DISCRIMINANT OF A POLYNOMIAL

factor (-1)(2) to this polynomial; we call it then the discriminant of the polynomial J (x) and denote it by

D = D(f(x)) = D(f) _ (- G) 1 R(ff').

(113.3)

ao

We immediately extend this definition to the case where f(x) lies in an arbitrary commutative polynomial ring R[x], so that then 40, ..., a, are to be replaced by the coefficients of f(x). Definition (113.3) fails if n = 1,

but in this case we write D(f) = 1. In order to transform (113.3) we perform the permutation

n=

1

3... n-1 nn+1 n+2... 2n-1

2

24

6

2n

2

2I

5

3

1

n -

on the rows of the determinant D. Since sgn n = (- 1)(2 it follows from 113.3) (after dividing by ap) that I

(n - l) a1... a1...

n

an-1 an-1 an

... an-1

nao (n - 1) a1

D=

a0

... an-1 an

a1

...

nap

2n - 1 rows

an-1

To explain this formula we compute the discriminant of apx2 + alx + a2: 2

a1

01

2ap

0

= a2 - 4apa2

a1

(113.4)

a1 a2

and that of a0x3 + alx2 + a2x + a3: 13

2a1

a2

0

0

1

a1

a2

as

0

0

3a0

2a1

a2

0

0

ap

a1

a2

a3

0

0

3a0

2a1

a2

= ai a2 - 4ai a3 _4a04 + + l8a0a1 a2 a3 - 27ap a3 . (113.5)

Both discriminants are lexicographically ordered. If necessary, we use the more precise notations

Dn(f) or D(ap xn + for D(f).

... + an)

COMMUTATIVE POLYNOMIAL RINGS

452

We now prove the formula

D(Ox" + alx"-1 + ... + a") = a,D(a1x"

+ ... + a") .

(113.6)

The left-hand side is obtained from the determinant (113.3') by replacing ao by 0. If we apply Laplace's expansion theorem to the first two columns, we then obtain (113.6). THEOREM 274. The discriminant D of the polynomial f(x) (& 0) in (113.1)

over a field,r "is equal to 0 if, and only if, f(x) has a non-constant multiple factor or an irreducible factor with vanishing differential quotient, or when ao = a1 = 0. If the formal degree n of f(x) is equal to 1, then D = 1, so that the theorem

is then trivially true. It is also true if a = a1 = 0, since then, according to (113.3'), D = 0. We now assume that n 2 and ao, a1 do not both vanish. If ao

0, then according to (113.3), D is equal to 0 if, and only if,

R(f,,f') = 0. On account of Theorem 269, this means that f(x), f(x) have a common irreducible divisor g(x). Furthermore this is the case if, and only if, g(x) is a multiple factor of f(x) or is a factor of Ax) and g'(x) = 0 [see (110.7)]. Thus the theorem is true for ao 0 0. Finally, let ad = 0, then a1 0 0. This may be reduced to one of the cases already discussed, since

now fl x) = a1x"-' + ... + a,, and, according to (113.6), D = 0 means that D(a1x"-1 + ... + a") = 0. Consequently Theorem 274 is proved. THEOREM 275. For the discriminant D of a polynomial f(x) of formal degree

n (> 2) we have

f(x) F(x) + f'(x)G(x) = D

(113.7)

with two proper polynomials F(x), G(x) whose degrees are at most n - 2 and n - 1, respectively, and whose coefficients lie in the ring generated by the coefficients off (x). Add the first column of the determinant (113.3'), multiplied by

aox2"-2,

and, for i = 2,..., 2n - 2, the 1th column, multiplied by one. Since this then consists of the elements

'x"-1f'(x), x"-2f (x),

x"-2./

'(x),

x"-3./

x2"-1-I, to the last

(x), ...,./ '(x),

the theorem follows from (113.3'). THEOREM 276. The discriminant of the polynomialf(x) in (113.1) is a homogeneous isobaric polynomial of the ao, . . ., a,, of degree 2n - 2 and of weight

n(n - 1). This follows at once from (113.3) and Theorem 271. THEOREM 277. The discriminant D of the polynomial

f(x)=ao(x-x1)...(x-x")

(n>_ 1,ao#0)

(113.8)

453

THE DISCRIMINANT OF A POLYNOMIAL

has the following two product representations D = ao"-2 D=(_

fl

(Xk

15
- xI)2,

(113.9)

1)(2ao-2 flf'(xk).

(113.10)

k=1

If f' (x) = nao(x

- Y) ... (x - yn-I) ,

(113.11)

then D also has the product representation

D = (-

1)G2 n"ao-1

U f(yk)

(113.12)

k= 1

The formulae (113.10) and (113.12) follow at once from definition (113.3) and formulae (112.13) and (112.14) in Theorem 272. Further, (113.9) follows

from (113.10) by applying Leibniz's rule. Consequently Theorem 277 is proved. EXAMPLE 1. The proposition in Theorem 274 that the discriminant D is equal to 0, if f(x) has a non-constant multiple divisor, also follows immediately from Theorem 275, for every commutative integral domain instead of the field K. EXAMPLE 2. From the product formula (113.9) and from the formula (112.12) we obtain the multiplication theorem for discriminants

D(fg) = D(.f)D(g)R(f g)2, where f, g are polynomials of formal degree Z 1. EXERCISE. A theorem analogous to Theorem 273 also holds for discriminants.

§ 114. The Newton Formulae Call the symmetric polynomial

pk=A+...+xn

(k=0,1,...;p0=n)

(114.1)

the k h power sum of the indeterminates x1i . . ., x,.

THEOREM 278 (NEWTON formulae). If sk (k = 1, ..., n), Pk (k = 0, 1,

...) denote the k' elementary symmetric polynomial and the ktb power sum of x1, . . ., x", respectively, and

ak = (- 1)ksk

(k = 1, ..., n),

(114.2)

then

P1+a1=0,

P2+aLp1+2a2=0,

(114.3)

Pn-I + a1Pn-2 + ... + an-2 P1 + (n -- 1) an-1= 0 Pk+n + a1Pk+"-I + ... + anPk = 0

(k = 0,1,...).

454

COMMUTATIVE POLYNOMIAL RINGS

For the proof we employ the polynomial

f(x) = (x - xl) ... (x - xn) . Then, by (114.2),

n

f(x)=x"+a1x"-1+... +an= Z arxn-r

(114.4)

(ao= 1).

(114.5)

r=0

Since, according to (114.4), f(xt) = 0, so n

xkf(xt)=0

(k = 0, 1, ...) .

1=1

Hence, because of (114.5) and (114.1), we obtain the last of the equations (114.3).

From (114.4) we obtain by Leibniz's rule and (114.5) " -1 .f' (x) = E f(x) _ " f(x) .f(xt) _ 1=1 t=lr=o x - xt t=1 X - xt

axttx"-r-t-1 i-1 F=O

t=o

- xi-r

x - xt

n-In-r-1

n n-1 n-r-1 f'(x) _

xn-r

= r=o

t=o

A f t e r comparison of the coefficients of x"-k-1 we obtain f o r k = 1, ...,

n - 1:

(n-k)ak=

arpt=pk+a1Pk-1+... +ak_lpl+nak.

r+t=k

These result in the first n - 1 equations of (114.3). § 115. Waring's Formula TxnoREM 279 (WARING's formula). With the notations of Theorem 278 Pk

=

k (k)

(-

1)t k

(a1 + ... + a n ) t

( k = 1, 2, ...)

,

(115.1)

indicates that only the terms where the "(k)" beside the summation sign of weight k with respect to a1i . . ., a,, are to be retained on the righthand side after the expansion of the power term. Nom. Waring's formula (115.1) may be regarded as the solution formula

for the Newton formulae (114.3). Hence it follows that the coefficients in (115.1) are integers, as may also be proved directly. (115.1) can be obtained

WARING'S FORMULA

455

from the Newton formulae, but we shall use a shorter method, where (115.1) will be verified in the alternative form Pk

1(k)

=-

(- 1)'(a1+2a2+...+na)(al+...+a,Y. (115.1')

From this, too, it follows that the coefficients in (115.1) are integers. In the proof we can suppose that x1, ..., x,, are indeterminates over 70. Next we show that (115.1) follows from (115.1). For this purpose, we

perform the substitution x, -> zx, (r = 1, . . ., n) in (115.1), by which Pk, a1, ..., an become zkpk, zal, ..., z" a,,. Then we differentiate with respect

to z and put z = 1. Thus we obtain kpk =

(k)(- 1)'k(a1 J=1

+ 2a2 + ... +

(a1 + ... +

(115.1') is obtained after division by k. For the remainder of the proof we require the formula

aaj

.=1

-axr -=

- (n - j+ 1)a;-1

(j= 1,...,n; ao= 1).

(115.2)

Here the left-hand side is symmetric in the x1, ..., x,,, and so is an integral

multiple of aj-1. It suffices to show that, after the substitution x1= ... = = x = 1, (115.2) becomes a proper equation. After this substitution (115.2) is as follows:

n (-1)' (n - 11= - (n - f + 1) (- 1)i-1 ( j- 1

n 1

Since this equation is true, (115.2) is also true. PROPOSITION. A polynomial f(al, .

. .

with

af(al, ..., a constant. We suppose that this is not true and denote the lexicographically first term of f(al, . . ., by ca' ' ...a.n (cE7, #0) Then for some j (1 < j S n)

i1 = ... = i;_1 = 0 , ii > 0. By (115.2) it follows that the left-hand side of (115.3) contains the term

- c(n - j + 1)i a;-la -'a +i ... a.'m

(ao = 1) ,

COMMUTATIVE POLYNOMIAL RINGS

456

as the lexicographically first term, which is consequently not cancelled out by any other terms. This contradiction proves the proposition. Now (115.1) is true for k = 1, since the right-hand side is equal to -a1 (= s1= p). We suppose that (115.1), and so also (115.1'), is true for some k and prove it for k + 1, thus proving the theorem. We write (115.4)

and denote the right-hand side of (115.1) by Ak. We have to prove that (115.5)

Pk+1 = Ak+1

It will be sufficient to prove that

' aAk+1 = (k + 1)Pk

(115 . 6)

axr

r=1 Since n

aPk+1

axr

r=

= (k + l)Pk ,

it follows that aX

r

(Pk+1 - Ak+1) = 0

Therefore, because of the proposition, Pk+1 - Ak+1 is a constant. But

since this polynomial is homogeneous of the (k + 1)' degree (with respect to x1, . . ., x,,), it must be 0. According to this (115.5) is a conse-

quence of (115.6). The equation (115.6) to be proved becomes, after dividing by k + 1, n

a

k+1(k+1) (- 1)i

F, F, r=1 eXr 1=1

st-- Pk

I

By (115.2) and (115.4), the left-hand side is equal to k+1

- f=1(k)(-1)' [n + (n - 1)a1 + ... + an_l] S1"1. This may be transformed first into n

k (k)

r

n + i (n -

j)aj

S' ,

ll

and then into k

k

n Z(k) (- 1)' (S' + S'+1) - F(k)E(_ 1)' ja,S. J =0

=0

1==1

(115.7)

WARING'S FORMULA

457

The first sum consists of those terms of n(1 + (- 1)k+I Sk+I), which are of weight k with respect to al, . . ., a,,. But since there are no such terms, this sum vanishes. Accordingly, (115.7) agrees with the right-hand side of (115.1'), and so by induction is equal to pk. This means that (115.6) holds. Consequently the proof of Theorem 279 is completed. As to the above proof cf. Ran$I (1952c). For another proof cf. § 167, Exercise 3. EXAMPLE 1. We obtain from (115.1) for k = 1, 2, 3, 4:

Pi=-a,, pe = alz - 2a2, Ps = -as, + 3a1a2 - Sae , p4 = ai - 4a,2a2 + 2a22 + 4alas - 4a4,

where we have to suppose that the ak such that k > n are equal to 0. EXAMPLE 2. We call

1 xl ... xi-

V.=I -' I =

(115.8)

11X....4' the Vandermonde determinant of the elements x1, . . ., x,,, which may belong to an arbitrary commutative ring with unity element. We prove that

V. = [I

(Xk - x,). (115.9) lei
xk - x,IVn

(15i
Hence it follows that the left-hand side of (115.9) is divisible by the right-hand side.

Both sides are homogeneous of degree 2 n(n - 1), therefore their quotient is a con-

stant. But since both sides contain the term x24... xn-', they are in fact equal. We write the formula (115.9) for the discriminant obtained in Theorem 277 in the form

D=ao-'V!.

By applying the multiplication theorem for determinants we obtain

D=aom-2IP,-,-2I

n

p1 ... Pn-1

Pi

P2 . . . P.

=ao-2

P.-i P. . . . Pen-2

458

COMMUTATIVE POLYNOMIAL RINGS

EXERCISE 1. Let V( m) be the determinant of order n with-it" row 1, ..., xj -' , ...,

x;"+', ..., xi (i = 1, ..., n). Then we have (m)

where V. denotes the Vandermonde determinant (115.8) and sk the kut elementary symmetric polynomial of x1, ..., x (so = 1). EXERCISE 2. The "reciprocal" formula to (115.1) also holds:

(k = 1, 2, ...) .

ak

§ 116. Interpolation

If a polynomial f(x) in a polynomial ring JY[x] is given, then, as mentioned above, a function f(t) (t E,2) is defined in R. The question is whether this function may be arbitrarily given. In general it cannot, since because of Theorem 205 no polynomial f(x) (# 0) exists with an infinite

number of zeros over ,5, so that all the functions in . cannot be given (It holds moreover quite generally that by the polynomials f(x) over over a ring with a countably infinite number of elements there are only a countably infinite number of polynomials, but in such a ring the set of functions is not countable.) An approximate answer to the above question is contained in the following THEOREM 280 (LAGRANGE'S formula of interpolation). Over a field .7 there is exactly one polynomial f(x) of degree at most n, which assumes the given values wo, ..., w (E .5) at n + 1 given different places co, ..., c (E.7'), and this polynomial is given by the formula

f(x) =

n

w'

F(x)

i=c F (ct) x - cI

(F(x) = (x - co)

... (x - cn)) .

(116.1)

For the proof it will be noticed that because the co, ..., c are distinct Y(cc) # 0

(i = 0, ..., n).

Since further F(x)

(i = 0, . . .,n)

(116.2)

x - Ct

is a polynomial of degree n, it follows from both that the right-band side of (116.1) is meaningful and is a polynomial of degree at most n over Jr. The value of (116.2) for x = ck (k = 0, ..., n) is, according to Leibniz's formula, equal to 0 or F'(c,), according as k i or k = i. Therefore the right-hand side of (116.1), for x = ci, assumes the value w1, so that the polynomial f(x) defined by (116.1) satisfies the requirements of the theorem.

459

INTERPOLATION

We suppose that there is a further such polynomial e.g. g(x). Then f(x) -

- g(x) is of degree at most n and vanishes at the n + 1 places co, ..., c,,. Hence, and from Theorem 205, it follows that f(x) - g(x) = 0, i.e., f(x) _ = g(x). Consequently Theorem 280 is proved. THEOREM 281 (NEWTON'S method of interpolation). The polynomial defined in the preceding theorem may be given (instead of (116.1)) in the form

.f (x) = ao + a1(x - co) + a2(x - co)(x - c1) + .. . (116.3)

where the coefficients ao, ..., a are defined by the recurrence formulae

.

f o ( x ) = w(x),.f,(x) _

l

-1

(x) - 1- 1 (c7 - I)

x-c,-1

a; =

1

, . . . , n) ,

(i = 0, . . ., n) ,

(116.4)

(116.5)

n which w (x) denotes an arbitrary function in.7 with

(i = 0,..., n).

w(c,) = w;

(116.6)

NoTE. The indeterminateness involved in the arbitrary choice of the functions w(x) is apparent, since (116.5) depends only on the values (116.6).

The advantage of Lagrange's formula of interpolation lies in its being simple, that of Newton's method in the fact that the coefficient a, depends only on the c o , ..., c wo, ..., w; (i = 0, ..., n). For the proof of Theorem 281 let f(x) denote the polynomial of degree at most n over 7 with

(i = 0, . . ., n).

f(c,) = w,

(116.7)

This f(x), by Theorem 280, is uniquely determined. We can state it with uniquely determined ao, . . ., a,, (E -7) in the form (116.3). It has only to be proved that these a; agree with those in (116.5). By (116.3) and (116.7)

No=ao, w1=ao+a1(c1-co)

w2 = ao + a1(c2 - co) + a2(c2 - CO) (c2 - CO co)

+ a (c - c0) .

.

.

(c -

c0)(cnc- 0 .

cl)+...+

COMMUTATIVE POLYNOMIAL RINGS

460

Since the ao, ..., a are determined by these equations, it will suffice to prove that these equations also follow from (116.4), (116.5) and (116.6). From (116.4) and (116.5) we obtain

w(x) =.fo(x), f-1(x) = at-1 + f (x)(x - ci-1)

(i =1, .

. .

n) .

Hence we obtain w(x) = fo(x)

,

w(x)=ao+fi(x)(x- co), w(x) = ao + ai(x - co) + f2(x) (x - co) (x - c)

,

w(x)=ao+a1(x-co)+a2(x-co)(x-c1)+...+ + fn(x) (x - CO) ... (x -

If, in these equations, we insert x = co in the first, x = cl in the second, ..., x = c in the last, then because of (116.5) and (116.6) the equations (116.8) are obtained exactly. Thus Theorem 281 is proved.

§ 117. Factor Decomposition According to Kronecker's Method

Let,R be an infinite integral domain for which every element other than 0 has only a finite number of divisors, whence in particular it follows that the group of units of R is finite. Our aim is to determine the factor decompositions f(x) = g(x) h(x) (117.1)

of a given non-constant polynomial J(x) over *, where we restrict ourselves to non-constant factors g(x), h(x) over *. The method of determining the above-mentioned factor decompositions, to be discussed here, is due to KRONECKER. First consider a factor decomposition

(117.1). Let n be the degree of f(x). Without loss of generality we may then assume that the degree of g(x) is not greater than r, where r shall denote the

largest integer which is not greater than n Now g(x) I f(x)

on account of (117.1).

(117.2)

FACTOR DECOMPOSITION

461

Kronecker's method is essentially based on the fact that all the g(x) with this property (117.2) are determinable as follows. From tR take any r + I distinct elements co, . . ., c with the sole restriction that they are not zeros of f(x), which is possible according to Theorem 205, since V is infinite. Then

g(c,) I f(ci)

(i = 0, ..., r),

from (117.2). For every i = 0, ..., r we let d, run through all the divisors of f(c,): di I.f(c)

(i = 0, ..., r; d, E R),

(117.3)

where we now also admit associated divisors. For one of these systems do,

..., d,, we then have, by (117.2), g(ci) = di

(i = 0, ..., r).

(117.4)

Thus if we define for every do, ..., d, the polynomial of degree at most r with the property (117.4) by (Lagrange's or Newton's) interpolation, then we find among this finite number of g(x) all those polynomials which may occur in (117.1). Of course, the g(x) obtained can lie partly or complete-

ly outside ,A[x], since for their coefficients it is, in general, only certain that they are elements of the quotient field of JP; such g(x) are disregarded. Those g(x), which do not satisfy the condition (117.2), are similarly eliminated. But the g(x) which are left constitute all the solutions, since (117.2) implies that (117.1) is satisfied by the polynomial h(x) = g(x)-' f(x). The main point of Kronecker's method is, as we see, that the determination

of the divisors in the polynomial ring 3P[x] may be reduced to that in the fundamental ring V. An essential point is that the process is practicable in a finite number of steps, provided that one is able to define all the divisors of an element (# 0) inside,* in a finite number of steps. Again, we see that the process is then more generally applicable to V [x1, . . ., so that here we successively reduce the determination of the divisors to that in the ring ,9z'[xj, ..., xk] (k = n - 1, . . ., 0). It should also be noted that Kronecker's = L7. method is especially applicable in the important special case when EXAMPLE. We wish to determine the factor decompositions of

f(x) = 2x5 + 8x4 - 7x3 - 35x5 + 12x - I in the polynomial ring 9'[x]. Since r = 2, we wish to interpolate on three places co, cI, c2. We have

f(0) = -1, f(1) = -21, f(-1) _ -35, f(2) = 19, f(-2) _ -45, f (3) = 665, f (- 3) _ -1.

COMMUTATIVE POLYNOMIAL RINGS

462

Since, in general, the values of f(c) with smaller numbers of divisors are more manageable, we choose

co=0; c,=2, c2=-3 for which the corresponding substitution values are -1, 19, -1. Thus we have to take into consideration the divisors

do=1; d,=± 1, ±19, d2=±1. [We ought also to admit do = -1, yet it will always suffice to take only one of the two systems do, dl, d2 and -do, -d,, -d2, since the two corresponding polynomials g(x) differ from each other only in the factor -1, i.e., they are associated.] We carry through the interpolation according to LAGRANGE. For this purpose we put

F(x) = x (x - 2) (x + 3) and then have

g(x)

F

xx

6° (x - 2) (x + 3) +

10 x(x + 3) + 155 x(x - 2), i.e., (because do = 1)

30 g(x) = (- 5 + 3d3L + 2d2) x2 + (-5 + 9d, - 4d2) x + 30.

Since only the g(x) E .[x] are to be considered, we have only to deal with those dl, d2, for which

301-5+3d,+2d2f

-5+9d,-4d2,

so that only d2 = 1 and d, = 1, -19 are left. Correspondingly we obtain

g(x) = 1,

- 2x2 - 6x2 + 1.

The first of these is to be disregarded but the second gives

f(x) = (2x2 + 6x - l) (x2 + x2 - 6x + 1). NOTE. For a modification of Kronecker's method in a special case see SERES (1956).

§ 118. Eisenstein's Theorem

Tlioi

i 282.

If

f(x) = aox" + ... + a"

(118.1)

is a polynomial over an integral domain J2 and a a maximal ideal of * with

ao * 0 (mod a),

aim...

an

0 (mod a);

an * 0 (mod a2) ,

(118.2) (118.3) (118.4)

then f(x) cannot be decomposed in R [x] into a product of non-constant factors.

EISENSTEIN'S THEOREM

463

Suppose that in f(x) = g(x) h (x) (g(x), h(x) E R[x])

(118.5)

the factors on the right-hand side are not constant. Then aox" __ g(x) h(x) (mod a)

(118.6)

follows from (118.1) and (118.3). Since, according to Theorem 130, the factor ring JP/a is a field, (s/a) [x] is a ring with prime decomposition. So if we write

g(x) =box'+ ... + b,, h(x)=coxs+...+cs

(118.7)

(r,s>0; r+s=n), then bo, co * 0 (mod a) follows from (118.1), (118.2) and (118.5). Then from (118.6),

g(x) = box' (mod a), h(x) - coxs (mod a), thus in particular br , cs m 0 (mod a), consequently

brc, - 0 (mod a2)

.

But, because of (118.1), (118.5) and (118.7), a" = b,cs, so that we obtain a contradiction with (118.4). This proves the theorem. Hence, and from GAUSS'S theorem (Theorem 207), we obtain as a special case the following THEOREM 283 (EISENSTEIN'S theorem). If ` is an integral domain with

prime decomposition and the conditions

p,f' ao; p I aI,

, an-1; p

(118.8)

II an

are satisfied, for the coefficients of the polynomial f(x) in (118.1) and for a prime element p of JP, then f(x) is irreducible over the quotient field of EXAMPLE 1. The so-called ptu cyclotomic polynomial (cf. § 135)

FP(x) = z _ is irreducible over

1

xP_= +

... + I

(p prime)

"o, since for

F'P(x + 1) =

(x + Ix )P _ 1

conditions (118.8) are satisfied.

16 R.-A

= xP-' +

=

xP_1 + ( p ) x- +

... + (p

p

1

464

COMMUTATIVE POLYNOMIAL RINGS

EXAMPLE 2. If a is an integer containing a simple prime factor, then because of Theorem 283 x" + a is irreducible over . a. On the other hand:

x4+4=(x2+2x+2)(x2-2x+ 2). (Cf. Theorem 428.)

§ 119. Hilbert's Basis Theorem THEOREM 284 (H1LBERT's basis theorem). If every ideal of a ring J" with unity element is finitely generated, then this is also true for the polynomial ring

J'[x]. (The term "basis theorem" dates back to the time when the generating systems of an ideal were called its bases.) For the proof, let us consider an ideal SIX of R[x]. Let 1 (n z 0) denote the set of those elements of K whose degree does not exceed n. Furthermore let a and a denote the set of leading coefficients of the polynomials belonging to W and ul", respectively.

We prove that both a and a are ideals of V. With this end in view we denote two elements of a by a, b and an element of R by c. There are two elements of 21 of the form

f(x)=ax'+..., g(x)=bx' + ...

(119.1)

We can suppose that r < s. Then

xs-'f(x) - g(x) = (a - b)e + ...; also belongs to 2l, whence it follows that a - b belongs to a, thus a constitutes a module. Since moreover

cf(x) = cax' + ...,

f(x)c = acx' + .. .

also belong to 21, it follows that ca, ac E a. Thus a is in fact an ideal. The proof of the same assertion regarding a" is parallel, with the only difference that only the cases r, s < n are admitted in (119.1).

We assign an ideal 3 of fi[x] to the ideal a in the following manner. By hypothesis

a = (a,, ..., a")

(119.2)

for a finite number of a,, ..., a, (E 9'). Moreover 21 contains elements of the form

f1(x) = a, x"' + ...; ...;f"(x) = ax"" + ..., for which we put

58 = (fi(x), ..., f"(x))

HILBERT'S BASIS THEOREM

465

Further, we write

n = max (n1, ..., and denote certain ideals of R [x] assigned in a corresponding manner to the ideals ao, . . ., ai_1 by F80,..., respectively. All the 18, It (i = 0, .. .

..., n - 1) are generated by a finite number of elements of W. We shall prove that K = (^', 8n-1, ...1580), whence the theorem will follow. It is sufficient to show that every element of the left-hand side also belongs to the right-hand side. For this purpose we consider an arbitrary element

F(x) = ax" + .. . of K and show that there is an element F.-I(x) of ^n-1 with F(x) = F.-I(x) (mod 18) . (Here we understand that S2C_1 = 0.) It will suffice to consider the case N >-- n. Since a E a, then by (119.2) a is the sum of a finite number of terms of the form kat, cat, at c', dat d' (k E 7; c, c', d, d' E (119.3) where, of course, k, c, c', d, d' change from term to term. We see that the polynomials

kx'-1f(x), cx"-nf(x), xN-nf(x)c'

,

dx-°`fi(x)d'

have as leading coefficients just the elements (119.3) and are all of degree N. So if they are all subtracted from F(x), then we obtain a polynomial of

degree at most (N - 1). After a finite number of repetitions we obtain the required polynomial Fn-1(x). In the case n z 1 one proves similarly that Fn-1(x) = Fn-2(x) (mod Zn-1) ,

where Fn_2(x) is a suitable element of `1n_2. We then have

F(x) = Fn-2(x) (mod (8, After sufficient repetitions we finally obtain

F(x) = 0 (mod (Z,n_1, ..., 58o)) The theorem is now proved.

466

COMMUTATIVE POLYNOMIAL RINGS

§ 120.* Szekeres's Theorem To obtain all the different ideals of a ring is an important problem which has only been solved for a few cases. The solution for a polynomial ring %R[x] over a (commutative) principal idealring -Rwith prime decomposition will be given here.

In general an ideal (# 0) of a commutative ring R is called primitive, if it may not be written as the product of a principal ideal different from 1 and a further ideal of R. It is obvious that in the polynomial ring R[x] every ideal different from 0 may be uniquely decomposed into the product of a principal ideal and a primitive ideal. Therefore it is sufficient to give the primitive ideals of fi[x]. With this end in view we introduce a norming of the elements in the fundamental ring R by singling out of each class of associated elements a representative as normed element (§ 79). Since 'R is a ring with prime decomposition, we arrange the norming so that the normed elements constitute a free commutative semigroup with the same unity element as 9P. It is also necessary to give to every normed element a (# 0) of a system of representatives RR(a) of the residue classes mod a, so that all the 91(a) contain the element 0 [which implies that every class 0 (mod a) is represented by 0]. In particular then l1(a), for the unity element a = 1, consists only of the element 0. THEOREM 285 (SZEKERES'S theorem). Let a system of representatives 1(a), containing the element 0, of'the residue classes mod a be given to every normed element a (# 0) in a principal ideal ring li with prime decomposition. Then if we take an arbitrary, but finite, number of the normed elements other than 0 ai, ..., a m (E JI) (m ? 0; am # 1) (120.1) and, to every ak, further k elements

rik , ..., rkk (E R (ak)) then from

go(x) =

al

(k = 1, ..., m),

(120.2)

k

... am , akgk (x) = xgk-I(x) +

i=I

rikgi-I(x)

(k = 1, ..., m)

(120.3)

(after cancelling by ak) we obtain further polynomials go(x), ..., gm(x) in _V[x] and if a = (go(x) , ..., 9m(x))

then a gives all the primitive ideals of 92 [x], each exactly once.

(120.4)

SZEKERES'S THEOREM

467

Let us call (120.4) the Szekeres normal form for the primitive ideals of ,R[x]. Before beginning the proof, it should be noted that, by our theorem, the elements (120.1) and (120.2) constitute a complete system of invariants of the primitive ideals of.$[x]. The elements (120.1) depend only on t-*, a and on the norming in * [the ideals (a), . . ., (am) thus depend only on and a] but the elements (120.2) also depend on the choice of the systems of representatives 91(a) (a E A 0 0). We begin the proof by verifying that go(x), ..., gm(x) are in fact polynomials in R[x]. We show from (120.3) that for the polynomials defined by

f0(x) = 1, fk(x) = xfk-1 (x) +

/

i=1

rikai ... ak_lf-1(x)

(k= 1, .., m)

(120.5)

/(k = 0, ..., m).

9k(X) = at +1 ... am fk(x)

(120.6)

This is obvious because the equations (120.5) are obtained from the equations

(120.3) by substituting (120.6) and cancelling a1... am or ak ... am. Then we show that by (120.4) only primitive ideals of R[x] are given. By (120.5), fk(x) is evidently a principal polynomial of degree k. By (120.6), gk(x) is a polynomial of degree k. Since gm(x) = fm(x) is a principal polynomial of degree m and, according to (120.31), go(x) is a constant other

than 0, it follows that for m > 0 we obtain by (120.4) primitive ideals of JP[x]. The same holds for m = 0, since then the polynomial sequence is reduced to go(x) = 1, so that (120.4) now yields the unity go(x), ..., ideal.

From now on, let a denote an arbitrary primitive ideal of _ [x]. It still remains to be proved that a is given exactly once by (120.1), (120.2), (120.3), and (120.4). As a preliminary, we define certain invariants of a. Let Mk be the 'R-module

consisting of those elements of a whose degree is at most k. Evidently Mo c M1 c

..

(120.7)

The leading coefficients of the elements of Mk constitute an ideal of,2, which, since 2 is a principal ideal ring, we take in the form (ck). Moreover it may be supposed that the co, c1, ... are normed elements of so that these become invariant. Evidently

(co) c (c) c ...,

(120.8)

i.e., Ck I Ck-1

(k = 1, 2, . . .).

(120.9)

468

COMMUTATIVE POLYNOMIAL RINGS

We obtain a further invariant in the following manner: since each ideal of may be generated by one element, so, according to HILBERT'S basis theorem (Theorem 284) a is finitely generated. Consequently there is a minimal k, for which a is generated by the elements of Mk. We denote this k by m(a)

(120.10)

.

Now, we temporarily disregard postulate (120.2) [thus we provisionally admit in (120.3) arbitrary elements rik of 92] and show that for the ideal a the conditions (120.1), (120.3) and (120.4) can be satisfied. For this purpose we prove that

M0 00,

(120.11)

i.e., that a contains constant elements other than 0. If this is not correct, then we suppose that

h(x)=ax'+...

in a is a non-constant element of minimal degree. Every element f(x) of a other than 0 is then of degree n (>_ 1). For this a"_'+l f(x) = q(Y) h(x) + r(x)

with two polynomials q(x), r(x) in R [x], of which the latter is of smaller degree than h(x). Since evidently r(x) E a, it follows that r(x) = 0. Hence h(x) I

an-,+ If(x) which, by GAUSS'S theorem (Theorem 207), gives ho(x) I f(x) ,

where ho(x) is a primitive polynomial of degree 1. We have now obtained

the proposition that the elements of a have the common divisor h0(x), while, however, a is a primitive ideal. This contradiction proves (120.11). Now (120.11) implies that co 0 0, whence by (120.9)

co,c1,... 00.

(120.12)

Since, furthermore, Mk contains polynomials with the leading coefficients ck, it also contains those of degree k. We introduce for (120.10) the notation m = m(a)

(120.13)

and choose, in one way or another, from among each of the M0, a polynomial

gk(x) = ckxk + ...

(k = 0, ..., m).

..., M. (120.14)

Since then, for every element f(x) of Mk (k > 0), there is an a (E R), for

SZEKERES"S THEOREM

469

which f(.x) - agk(x) lies in Mk_ 1, it follows by induction that the go(x), . . ., gk(x) constitute an 2-basis of the module Mk:

(k = 0, ..., m) .

Mk = { 90(x), ..., A(x)}

(120.15)

The validity of (120.4) follows from the special case k = m because of the definition of m = m(a). It still remains to be shown that go(x), ..., gm(x) can be defined by (120.1) to (120.3).

To do this, we define the elements a1, . . ., am (E,59) on the basis of (120.9) by akck = Ck-1

Then, by (120.12), a1,

. .

(k = 1, . . ., m).

(120.16)

., am are not 0 and are normed so that

Ck = ak+1 ... amCm

(k= l...., m)

(120.17)

follows from (120.16). By (120.14) and (120.16)

akgk(x) - xgk-1(x) E Mk-1

(k = 1, ..., m) .

(120.18)

Hence, and from (120.15), the validity of (120.32) for the elements r,k of follows. From (120.14) and (120.17)

9o(x) = a1 ... a,,, c,,, .

(120.19)

By a similar proof to that of (120.5) and (120.6), we obtain the divisibilities ak+1

amC, 1 9k(x)

(k = 0, ..., m)

from (120.19) and (120.32). Since then we have cm I go(x), ..., 9m(x), it follows that cm = 1 from (120.4). By this (120.19) becomes (120.31). Taking all these statements together it has now been proved that all the equations (120.3) are valid. It still remains to be shown that the condition am 1 in (120.1) is also

satisfied. (This assertion concerns only the case m > 0, since for m = 0 the system (120.1) is empty.) If am = 1 (thus m > 0), then it would follow from (120.18) for k = m that, because of (120.15), gm(x) is contained in the ideal (9o(x), ..., 9m-1(x))

But then because of (120.4) this ideal would be equal to a. Because of (120.13) this contradicts the definition of m(a). Consequently the assertion is proved. Conversely, we consider in the primitive ideal a of ,A[x] all the systems go(x), ..., gm(x) (for any m), for which the conditions (120.1), (120.3)

COMMUTATIVE POLYNOMIAL RINGS

470

and (120.4) [consequently also (120.5) and (120.6)] are satisfied. We have

already seen that there is at least one such system. It still remains to be proved that among these systems there is exactly one, for which conditions (120.4), too, are satisfied. For this purpose we show that every element of a is uniquely representable

in the form .f(x)9m(x) + bm-19m-1(x) + ... + bogo(x)

(120.20)

(f(x) E fi[x]; bo, ..., b,,,_1 E 39) If we write (120.32) in the form k+1

xgk(x) = ak+19k+1(x) - , ri,k+19,-1(x) i=1

(k = 0, ..., m - 1)

,

we can transform the elements xigk(x) of a into an expression of the form (120.20) by repeated application. But since, according to (120.4), every element of a is a sum of terms of the form axlgk(x), the same follows for all the elements of a. The uniqueness of this representation (120.20) is evident, since the element 0 is only representable by f(x) = b,"_1 =

=b0=0.

Hence the validity of (120.15) necessarily follows. We show also that (120.13) necessarily holds (so that the number of the

9k(X) is invariant). By (120.4) and (120.15) m(a) 5 m. But, according to (120.6), we have am 19o(x), . , 9n,-1(x) and, by (120.1), am 1, so the ideal (go(x), ..., g,,,-1(x)) is not primitive. Therefore, because of (120.15), it follows that m(a) > m - 1. This proves the validity of (120.13). From (120.5), (120.6) and (120.15) it follows that (ck) = (ak+,... am) and so, since all the ck, ak are normed, ck = ak+1 ... am

(k = 0, ..., m) .

Since the ck are invariants of a, the uniqueness of the system (120.1) follows. It still remains to be shown that the additional postulates (120.2) can also be satisfied, thus uniquely defining the go(x), . . ., g,,,(x) (and system (120.2)). This holds for g0(x) because of (120.31). We suppose that for some k (= 1, . . ., m) we have already chosen the go (x), . . ., gk_1(x) so that the coefficients ril (i = 1, . . ., 1; 1 = 1, ..., k - 1) in (120.32) satisfy conditions (120.2). Because of (120.15) the

9k(x) = 9k(x) + dk-19k-1(x) + ... + dogo(x) (d0, ..., dk-1 E are all the polynomials which may replace gk(x). (It should have been taken into consideration here that, by (120.5) and (120.6), gk(x) has the leading

SZEKERES'S THEOREM

471

coefficient ak+l ... am and this is an invariant of a.) From (120.32) we get k

akgk(x) = xgk-1(x) + Y_ (rik + akdi-)gi-l(x) i=1

This shows that we can determine the do, ..., dk_l in exactly one way so that for the new coefficients

(i = 1, ..., k)

r k = rik + akdi-l

conditions (120.2) are satisfied. Hence by induction the proof of the theorem is complete. As regards the case /k=J of Theorem 285 cf. SZEKERES (1952). EXAMPLE 1. The above theorem may, in particular, be applied to the case rk = 9.

For the normed elements we then take the non-negative integers and as a residue system W(a) (a > 0), the numbers 0, . . ., a - 1. In this way we obtain all the different ideals of 9'[x]. EXAMPLE 2. In the polynomial ring

[x, y] over an arbitrary field J%', all the different ideals may be obtained by means of Theorem 285, since .59x] is a principal ideal ring with prime decomposition.

§ 121. Kronecker-Hensel Theorem

Szekeres's theorem was (for the case

2=,7) formerly obtained by

KRONECKER - HENSEL (1901) in the following less simple but somewhat more explicit form: THEOREM 286 (KRONECKER-HENSEL theorem). Let JI, 91(a) mean the same as in Theorem 285. If we take natural numbers

n1<...
(r>0),

(121.1)

(E A 96 1),

(121.2)

(1 Sk<1!9 r)

(121.3)

Wormed elements

d1i ..., d, and polynomials

fki(x)

with coefficients from l1(di) and of formal degrees nk - nk-1 - 1 (no = 0), then

Fo(x)=dl...dr, d1F1(x) = (x"' + fil(x))Fo(x) , d2F2(x) = fit (x)Fo(x) + (x"' - "' + f22(x))Fi(x) ,

drFr (x) = f1r(x)Fo(x) + f2r(x) 'i(x) + ... + + (xnr-"'-t + frr(x)) Fr-1(x) 16/a R.-A.

(121.4)

472

COMMUTATIVE POLYNOMIAL RINGS

give (after cancelling by d1, ..., d,) only polynomials Fo(x), ..., F,(x) in .fl [x], and the a = (Fo(x), . . ., F,(x))

(121.5)

yield all the primitive ideals of *[x], each exactly once. Call (121.5) the normal form for the primitive ideals

of V [x]. In this, in general, less generators appear than in the Szekeres normal form for the same ideals. This theorem essentially arises from that of SZEKERES (Theorem 285) by changing the notations. For this purpose we consider from Theorem 285 the elements

a1, ..., am (E,*); and the polynomials

r1k, ..., rkk (E y(ak))

go(x), ..., g,n(x)

We denote by (121.1) the indices of those elements a1,. .., am which are different from 1. (Because am 0- 1, we then have m = n,.) We now denote the above-mentioned elements themselves by

(j= 1,...,r);

an,=dj

further we write

(j= 0,..., r; no = 0).

g1(x) = F)(x)

(121.6)

Since, if ak = 1, J(ak) consists only of the element 0, we have

r,k=0

(k0nl,...,n,; i= 1,...,k).

(121.7)

By (121.7) and from the equations defining our gk(x) [Theorem 285, (120.3)] we obtain g,(x) = x`-"jg",(x) = x'-"' F1(x)

(n1 < i < nj+1; j = 0, ..., r - 1). (121.8)

From (121.6) and (121.8) it follows that (go(x), ..., gm(x)) = (Fo(x), ..., F,(x).

Finally, we write the equations defining the gk(x) with k = nj (j = 0, ., r), taking into account (121.6), (121.7) and (121.8), as follows:

Fo(x) = d1..., d, djFj (x) =

j + F, F,_1 (x) 1=1

x",-n,-1 Fj_1(x) +

n,-1

Y, 1=n1-1

r1,,,

x1-nl-1

(1= 1, .., r) .

KRONECKER-HENSEL THEOREM

473

The inner sum is a polynomial of formal degree nl-n1_1- 1 with coefficients

from R(aj). If we denote this by f j(x), then we obtain the equations of . . ., F,(x). Hence we see that Theorem 286 is true.

(121.4) defining Fo(x),

NOTE. The unique representation

,fo(x) Fo(x) + ... +f,(x)F, (x)

(o(x),...,f,(x) E /k[x])

(121.9)

for the elements of the ideal a in (121.5) follows from (121.8) and (120.20), in so far as we prescribe that the degree of f,(x) F,(x) < the degree of F,+1(x) (i = 0, ..., r - 1). From (121.4)

(i = 0, ...,r)

F,(x) = d1+1 ... d,G,(x)

(121.10)

with principal polynomials G,(x) from r2[x]. EXAMPLE. The ideal a in (121.5) may, in general, be represented by less than r + 1 elements. We consider, e.g., the primitive ideal

a = (x5, -0 - p3)

(p prime)

of [x]. It is obvious that for this the KRONECIER-HENSEL normal form is a = (Po, P3xz, x3 - P3).

It would be desirable to set up, in addition to the SZEXERES and the KRONECXERHENSEL normal forms, a third one with a minimal number of generators. EXERCISE. In the ring 9[x] a primitive ideal means the same as an ideal of finite

index. Compute for this case the index of the ideal (121.5) in terms of the numbers (121.1) and (121.2).

§ 122. Tschirnhaus Transformation of Ideals

In a commutative polynomial ring ,*[x] with unity element we select a subring R with the property that, for any two elements f(x), g(x) of R, the iterated polynomial f(g(x)) also lies in R and then we say that R is closed with respect to iteration of polynomials. We consider an ideal a and an element ip(x) of R and prove that the elements f(x) of R with the property

f(&)) E a

(122.1)

constitute an ideal of R, which we denote by a9'1 or by ag', and say that this ideal a" arises from the ideal a by the Tschirnhaus transformation with the polynomial T(x). (Tschimhaus himself has only considered the case

where . * is a number field and R = .9[x]. Cf. § 175.) In order to prove the assertion let us consider, besides the above f(x), a further polynomial g(x) from af, for which 9(9(x)) E a .

474

COMMUTATIVE POLYNOMIAL RINGS

For the difference a(x) = f (x) - g(x) a(q (x)) = f(q (x))

- g(,p(x)) E a,

i.e.

a(x) E a"

so that a' constitutes a module. If furthermore h(x) is an arbitrary polynomial from R, then, because of the ideal property of a,

b(&)) = h(gq(x)) f(P(x)) E a, i.e. b(x) E a" for the product b(x) = h(x) f (x). Thus all is indeed an ideal of R. We sum up the definition of the ideal a" as follows:

f(x) E a,' o f (p(x)) E a

(f(x) E R) .

(122.2)

Moreover we note that p(x) _- V(x) (mod a)

a9= av (&), v(x) E R) ,

(122.3)

the validity of which follows from f(gq(x)) - f (tp(x)) (mod a)

(f(x) E R)

.

We shall see that Tschirnhaus transformations can be used in connection with the homomorphism problem of the ring R. This depends on whether two ideals a, b of R are related, i.e., whether R/a R/b. However, we shall take into consideration only certain special isomorphisms, and also subject the ring R to a further restriction. TmroxnM 287. Let a subring R, containing the element x and closed with respect to iteration of polynomials, of a commutative polynomial ring V [x] be given. Then, in order that the isomorphism

R/a : R/b (f(x) (mod a) -f (-p(x)) (mod b))

(122.4)

should hold for two ideals a, b and an element qq(x) of R, it is necessary and sufficient that there be a further polynomial ip(x) (E R) with b4' = a ,

tp(gq(x)) _- x (mod b)

.

(122.5)

First of all let us assume (122.4). The inverse image of the class x (mod b) in this isomorphism may be given as tp(x) (mod a) where ?P(x) is a polynomial from R. By (122.4) and (122.2) we have:

f(x) E a as f(g2(x)) E b e*.f(x) E b' . By this (122.51) is proved.

TSCHIRNHAUS TRANSFORMATION OF IDEALS

475

Because of the meaning of V(x), the class tp(x) (mod a) is reduced by the isomorphism (122.4) into the class x (mod b). This implies the validity of (122.52). Accordingly condition (122.5) is necessary. Now assume (122.5). We have to prove that the mapping f(x) (mod a) -*f(gq(x)) (mod b)

(122.6)

of R/a is the isomorphism (122.4). Because of (122.52), for an arbitrary g(x) (E R) the class g(tp(x)) (mod a) is transferred by the mapping (122.6) into the class g (x) (mod b). Accordingly R/a is mapped by (122.6) onto R/b. By (122.51) and (122.2)

f(x) - g(x) E a' .f(T(x)) - 9(99(x)) E b This means that the mapping (122.6) is one-to-one. Finally the homomorphism property of this mapping is self-evident, so the proof is complete. EXAMPLE 1. The subrings of .7[x] are evidently closed with respect to iteration of,polynomials, so that Theorem 287 is applicable to the subrings of 3[x] containing the element x. EXAMPLE 2. ax = a, so that, by (122.3), we have more generally ao' = a, if gp(x) = x (mod a). In the ring J[x] we have (x3)x+x' _ (x3), although x + x2 * x (mod x3), EXERCISE. We obtain Theorem 287 as an application of the theorem in § 46, Exercise 2.

§ 123. Rings Generated by a' Single Element Structures of every kind generated by a single element are of great importance, since every structure contains substructures of such a kind. In particular the groups generated by one element, i.e. the cyclic groups, are known precisely (§ 90). An incomparably more difficult problem is the determination of the rings generated by one element which may be treated, according to the results of the last three sections, as follows. By Theorem 100 the above-mentioned rings are identical with the homomorphic images of the ring 7 [x]o (cf. § 67, Example 4), so that our problem becomes one of determining the homomorphisms of this ring. We now show that all the different ideals of 7 [x]a are given by the pro-

duct xa, where a has to run through the ideals of 7 [x]. For, on the one hand, it is evident that in this way different ideals of 7[x]o are obtained. On the other hand, we can give every ideal from this ring in the form xa, where a is a complex of ,Y[xl. But since xa is a module, the same holds over a. Further let f(x)Ea, 9(X) E -'Y [X Then

xf(x) E xa , g(x) = c + go(x)

476

COMMUTATIVE POLYNOMIAL RINGS

for some c E .7 and go (x) E .7 [x]0. Thus, because of the ideal property of xa,

g(x) - xf(x) = cxf(x) + go(x) xf (x) E x a . Hence g(x) f(x) E a, so that a is in fact an ideal of .7[x].

This result is also expressible in the form that all the different ideals (# 0) of .7[x]o are the xd(x)a, where d(x) is a polynomial from .7 [x] with positive leading coefficient and a is a primitive ideal of .7[x]. We also know that these have been uniquely given by the KRONECKER-HENSEL theorem (Theorem 286) or that of SZEKERES (Theorem 285).

From the corresponding factor rings .7 [x]o/xd(x)a

the non-isomorphic ones still have to be selected. This can be done by means of the Tschirnhaus transformation in the ring.7[x]o, of which we give a more detailed explanation. First of all, Theorem 287 is applicable to

this ring (R =) 9 [x]o, since we have x E .7[x]0, and for any two polynomials f(x), g(x) from .Y[x]o, the polymonial f(g(x)) also belongs to this ring.

Furthermore, for ideals a, b of the ring .7[x]o every isomorphism of the corresponding factor rings may be given in the form f(x) (mod a) ->.f(gq(x)) (mod b) with 99(x) (E .71x]o). If qq(x) is chosen so that 99(x) (mod 6) is the image of

x (mod a), then cx' (mod a) is transform into c(gq(x))' (mod b) (c E 7, i >_ 1), whence the assertion follows. On account of all this the following holds :

THEOREM 288. All the rings generated by one element are (to within isomorphism) .Y[x]o and its factor rings .7 [x]o/x d(x)a

,

where d(x) runs through the polynomials from .7 [x] with positive leading coefficients and a runs through the primitive ideals (determined by Theorem 286 or Theorem 285) of .7 [x]. For two such factor rings we have the isomorphism

.7 [-vb/xdi(x)a1

.7 [xb/xd2(x)a2

if, and only if, there are two polynomials cp(x), ip(x) (E JI[x]o), for which (xdt(x)a1)11= xd2(x)a2 , (cp(x)) = x (mod xdl(x)al) , where the "exponent" q) denotes the Tschirnhaus transformation (with q)(x)). Of course, this theorem offers a large scope for further studies.

CHAPTER VIII

THEORY OF FIELDS In this. chapter we will discuss the classical theory of fields founded by STEINITZ (1931). This supplies detailed information on all possible fields,

so that the theory of fields is to a certain extent a complete chapter of algebra. Occasionally skew fields will also be discussed.

§ 124. Prime Fields

The following definition is of fundamental importance. Every field which has no subfield other than itself is called a prime field.

We are already acquainted with prime fields, e.g., the rational number

field Y0, and every factor field t7/(p) (p prime) which we denote by Jr,. Since, in compliance with the corollary to Theorem 39, the characteristic of a field can be only 0 or a prime number, the above examples allocate a prime field to every possible characteristic. We show that these are essentially all the prime fields, i.e. that every prime field is isomorphic with some

Jrp (p = 0 or a prime number). Let F. denote an arbitrary prime field of characteristic p with the unity element e.

First, let p = 0. Then the elements as

(a, b E .7; b

be

0)

(124.1)

obviously constitute a subfield of F. But, since FA is a prime field, their elements are already exhausted by (124.1). Two such elements are equal, e.g. as

cs

be - de

if, and only if, ads = bce, ad = bc, i.e. b = d . Thus the isomorphism Fo_

,., l" 0

as

a

bE

b

holds, since the homomorphism property of this mapping is trivial. 477

478

THEORY OF FIELDS

Secondly let p > 0 (i.e., p is a prime number). Evidently

0,e,...,(p - 1)e

(124.2)

constitute a subring of F,,. We now show that this subring is a field, and so is F,, itself. The assertion already follows because the ring is finite and zero-

divisor-free. A direct proof is made possible by the fact that, because o+ (e) = p, the equation

(p, a; a,bEJ) is equivalent to the congruence ax = b (mod p) which has a solution. Now it is also evident that the isomorphism F, ,:; .gyp (as -* a (mod p)) holds. This proves : THEOREM 289. To every possible characteristic there is, apart from isomorphism, exactly one prime field.

In conformity with this theorem we may henceforth briefly speak of the prime field of characteristic p. If F is an arbitrary skew field, then its unity element, and so also the prime field generated by it, is contained in all the sub-skew-fields of F. Therefore we may speak of the prime field of a skew field (field), which is the only

prime field contained in it and is evidently also definable as the intersection of all its sub-skew-fields.

Thus, after imbedding, every skew field will become an extension of some _57 ,, (p > 0), but we do not always carry out this embedding. In accord-

ance with this, Fp will often designate an arbitrary prime field of characteristic p. THEOREM 290. A prime field has no automorphism except the identical one.

Since in every automorphism of a field its unity element is mapped into itself and a prime field is generated by the unity element, the theorem is true.

§ 125. Relative Fields

We have already defined the notion of the relative field F I

sue' (§ 62,

Example 13). This means that F is a field and .f a subfield which acts as an operator domain of F, where the operation is defined as the multiplication valid in F. We also say that F is a relative field over.? (or with respect to .7). We often omit the attributive "relative" and speak briefly of the field F I.5r. Jr itself is called the fundamental field of F 1 .7.

RELATIVE FIELDS

479

For a field F I Y we usually denote the elements of F and Y by small Greek and Roman letters, respectively, as is also usual in connection with operator structures, but we denote the common unity element of these fields by 1. Since the operator product as is a special case of the product in F, we have the rules:

1a=a,

(a+b)a=as+ba, aba=a(ba), (125.1)

aa# = (am) P = a(a9).

(125.2)

Of these (125.1) means that 7+ is a unitary .i-module, i.e., an -7-vector space. Further (125.1) and (125.2) together mean that F is hypercomplex over Y. Thus in a relative field the usual kinds of operations are involved. Having clarified this we wish to establish the further consequences arising from the field F being treated as a relative field F Y. Namely we have to examine the meaning of (admissible) subfields and Sr-isomorphisms of F 1'.7. (As we are considering fields we need to consider only isomorphisms from among the homomorphisms; however, everything regarding these also refers to automorphisms and meromorphisms.) A subfield G of F I.5X means a subfield of F for which

aaEG

(125.3)

In particular, for a = 1, (125.3) means that all the a belong to G, i.e., that .9"is a subfield of G. Conversely, if this is true, then (125.3) is satisfied. Accordingly the subfields of F 1.9r are those subfields of F which are in turn

overfields of .7:

r9G9; F.

(125.4)

The subfields G with this property are sometimes called between-fields (namely between .9 and F). Of course we understand every subfield G of F 1.7 to be a relative field G I r. An.7-isomorphism of two fields F 1.5', F 1.7 (according to our general convention for operator structures) will be denoted as F I .-7

F I Jr (a -+ a') .

This means that the usual isomorphism F additional condition (aa)' = aoc'

(125.5)

F' (a -p. a') is subject to the

(a E .7, a E F)

.

(125.6)

480

THEORY OF FIELDS

But since (ax)' = a'a', (125.6) means that

a' = a

(a E 7) .

(125.7)

According to this (125.5) implies an isomorphism F Zt: F' (a - a') in' which the elements of the fundamental field . " are fixed elements. We also call the ,7-isomorphism (125.5) a relative isomorphism of the fields F, F' (over.}') or an isomorphism of F 1,91- with F I X. In conformity with this a relative automorphism of the field F over its subfield Y - briefly an automorphism of F 1-9- - means an automorphism of F in which the elements of 7 are fixed. A relative meromorphism of F over . " or a meromorphism of F 1.7 is correspondingly explained as an isomorphism of F I Jr with an (admissible) subfield.

Moreover, in a field F 1.7 the fundamental field Jr will play a further important role inasmuch as, in general, we consider the notions connected with

F in relation to J1. (Hence the term "relative field".) This principle is called the principle of relative fields. We shall pay special attention here to some of its important applications. Two extension fields F, F of _7 are called equivalent (over _91') when there exists a relative isomorphism (125.5). An element at of a field F 1.7 is called algebraic or transcendental, respectively, according as to whether there exists or not a non-constant polynomial f(x) over -9' with the zero at (cf. § 74). We also say that at is an algebraic or

transcendental element, respectively, of F over (or with respect to) Sr. Further, we say that F 1.7is an algebraic field (or F is algebraic over,7), when all the elements of F are algebraic over.'. Otherwise F 1 Y is called a transcendental field (or F itself is called transcendental over."); this means t hat F has at least one transcendental element over .7-. If at is an algebraic element of F 1 ,7, there is a principal polynomial f(x) over.rte" of minimal degree with the zero at. This minimal degree is called the degree of the element at (over .7) (cf. § 74). We now show that this polynomial f(x) is uniquely defined, therefore we call it the minimal polynomial of the element at (over .F). If f(x), g(x) were two distinct principal polynomials, so that they were both of the same degree, then f ( x ) - g(x) ( : A 0) would be a polynomial of smaller degree with the zero at. This is impossible, thus the assertion is true. The following theorem is also valid. THEOREM 291. The minimal polynomial f(x) of an algebraic element at of a field F 1 .7 is irreducible over .7 and is also definable as the only irreducible principal polynomial in F [x] with the zero at. Furthermore at is the zero of a polynomial F(x) over." if, and only if, f(x) I F(x). Of the three assertions of this theorem the second is a consequence of the other two; consequently it is sufficient to prove these.

RELATIVE FIELDS

481

We consider a decomposition f(x) = g(x) h(x) such that g(x), h(x) E 9' [x]. Then 0 = floc) = g(oc)h(a) .

Accordingly either g(a) = 0 or h(a) = 0. Because of the minimal property of f(x) it follows that one of the factors g(x), h(x) is of the same degree as f(x). Consequently f(x) is in fact irreducible. Furthermore we carry out on F(x), f(x) the Euclidean division:

F(x) = q(x)j(x) + r(x)

.

Then F(a) = r(oc). But as r(x) is of smaller degree than f(x), the equation

r(a) = 0 implies r(x) = 0, i.e., f(x) I F(x). This completes the proof of the theorem. In order to distinguish the relative fields we call a field without an operator domain an absolute field. If Jr is an arbitrary field, then its prime field is

- as noted above - contained in all the subfields of F. Therefore, and because of Theorem 290, "absolute field" and "relative field over a prime field" are identical concepts. Correspondingly, an element a of a field F is called algebraic or transcendental, according as a is algebraic or transcendental over the prime field of F. If necessary a is called an absolute algebraic or absolute transcendental element. Similarly, the degree, or more precisely the absolute degree, of an algebraic element from a field .7' means the degree

of this element over the prime field of F. Likewise an absolute algebraic field (briefly, an algebraic field) or an absolute transcendental field (briefly, a transcendental field) means an algebraic or transcendental field over its prime field. These examples may be sufficient to explain the meaning of

the adjective "absolute" in other similar cases, some of which will occur later.

Finally, it should be noted that in the course of our considerations, if a fundamental field 9r is already established, then every extension field F of .7 is usually automatically regarded as a relative field over `" without calling attention to it by the use of the notation F I -'/- or in any other way. the indeterminates x1, ...,x EXAMPLE 1. In the rational function field F(xl, ... , are transcendental over F. EXAMPLE 2. The complex field over .moo is algebraic, and its elements are all of degree I or 2.

§ 126. Field Extensions If G is an overfield of the field F and S an arbitrary subset of G, then we say (in the general terminology of § 35) that the subfield { F, S) of G, generated by the union set F U S, arises from F by the adjunction of S (or of the

THEORY OF FIELDS

482

elements of S) to F and use the notation F(S) for this field {F, S}. The elements of S are then called the adjoined elements. When S = , we write F(al, a2, ...), instead of F (S). We also write F(S1, S2, ...) = F(S1 U S2 U .. .) ,

where S1, S2 ,... are arbitrary subsets of G. Since every element of F(S) may be written as an expression with a finite number of elements of F and S, it is obvious that F(S) is the union of all the F(S1), F(S2), ..., where Si, S2, ... denote all the finite subsets of S. For this reason many questions may be reduced to the case where only a finite number of elements are adjoined at one time. The rule F(S, T) = (F(S))(T) is trivial; for this we write, with simpler notation on the right-hand side, F(S, T) = F(S) (T) (S, T S G). By repeated application we obtain F(aj,

. .

., an) = F(a) ... (an) ,

where al, . . ., a,, denote the adjoined elements. Accordingly the adjunction of a finite number of elements may be reduced to the successive adjunction

of each of these elements. § 127. Simple Field Extensions The extension fields F(i) arising from a field F by the adjunction of only one element t9 are called simple extension fields of F. We also say that these arise from F by a simple adjunction. The element 0 itself is called a primitive

element of F(t). The study of simple field extensions is the key to the whole theory of fields. We now give some important definitions as an introduction to their study. We say that a skew field G is of finite degree over its sub-skew-field F, if the

F-module G+ is of finite rank. The rank of this module is then called the degree of the skew field G over F and is denoted by

[G : F].

(127.1)

Further, every basis of the F-vector space G+ is called a basis of the skew field G over F or an F-basis of G. In the commutative case, we speak of the degree or the basis of G I F.

SIMPLE FIELD EXTENSIONS

483

THEOREM 292. Let F, G, H be three skew fields such that F S G S H. H is of finite degree over F if, and only if, H is of finite degree over G and G s of finite degree over F, and then

[H : F] =[H : G] [G : F] . If furthermore aL, ..., a; is an F-basis of G and A1,

(127.2) . .

., A. a G-basis of H,

then

(i=

a;Ai

(127.3)

is an F-basis of H. If H is of finite degree over F, then, according to Theorem 242, G is likewise of finite degree over F. Further, it is evident that H is of finite degree over G. Accordingly, we have only to prove the last assertion of the theorem, since the other assertions follow from this at once. We consider an arbitrary element C of H. This may be written as m

C = Y_ yjAj

(y; E G).

1=I

Moreover I

Y,

i=1

(c1 E F) .

cijai

Therefore, it follows that /

C=

m Y_ c,ja;Aj .

i= L j=1

This representation is unique, for if the right-hand side vanishes : m

l

ciia; Aj=0

then

i=1

cija;=0

(j= 1,...,m)

and further

ci.i=0

(i= 1,...,1; j= 1,...,m).

This completes the proof of the theorem. We now wish to examine simple field extensions. There are two essentially different cases to be discerned according as the adjoined element is algebraic or transcendental. It will later come to light that in the first case the extension field itself is also algebraic so that in these two cases we are justified in

using the terms simple algebraic extension field or simple transcendental extension field. The transcendental case is much easier, therefore we deal with it first.

THEORY OF FIELDS

484

THEOREM 293 (simple transcendental field extensions). Any field F has

only the rational function field F(x) (apart from equivalent extensions) for simple transcendental extension field and all the elements of F(x) - F are transcendental. In order to prove this we consider an extension F(t) with a transcendental element 1. The elements of F(x) may be taken in the form

r(x) = g(x) h(x)

(g(x), h(x) E F [x]; h(x) # 0) .

(127.4)

Since F(O) is transcendental, so h(i9) # 0, thus (127.5)

exists and is an element of F(t9). It is evident, too, that conversely, all the elements of F(9) are furnished by (127.5). The mapping (127.6)

r(x) -+ r(9)

is then one-to-one since, for two elements taken in the form (127.4),

r;(x) =

(i = 1, 2)

rl(x) = r2(x) if, and only if, gl(x)h2 (x) - g2(x)hl(x) = 0, which, by the transcendence of 0, is equivalent to g,(8) h2(9) - g2($) hl(O) = 0, i.e., to rl(t) = r2(0). Further, the mapping (127.6) is obviously homomorphic, and so - since F(x) is a field - isomorphic. Moreover the elements of F are fixed elements,

from which we obtain the relative isomorphism F(x) F s F(t) I F, i.e., the equivalence of the extensions F(x), F(O). It still remains to be shown that an algebraic element (127.4) necessarily lies in F. We may suppose that (g(x), h(x)) = 1. From these suppositions the

existence of an equation (after eliminating the denominator) of the form k

i=0

ci(g(x))`(h(x))k-` = 0

(c1 E F; co, ck 0 0),

follows. Consequently g(x) I h(x) and h(x) g(x), i.e., r(x) E F. Consequently Theorem 293 is now proved.

As a further preliminary to the next case we prove the following easy theorem.

SIMPLE FIELD EXTENSIONS

T.

485

oxns 294. If S is a subring and a an ideal of a ring R with

S fl a= 0,

(127.7)

then in the factor ring R/a the residue classes a (mod a) represented by the

elements a of S constitute a subring S' isomorphic with S : S

S' (a -s a (mod a)),

(127.8)

so that after the corresponding embedding R/a becomes an extension ring of S.

For, it follows from (127.7) that the mapping (127.8) is one-to-one and, furthermore, it is homomorphic, too. This theorem will later prove to be of importance in dealing with the factor ring F[x]/(f(x)), (127.9) where F denotes a field and f(x) a non-constant polynomial over it. Since the condition (127.7) has been satisfied by S = F, a = (f(x)), after embedding (127.9) becomes an extension ring of F, which in future we shall always assume.

THEo1 M 295 (simple algebraic field extensions). Every field F has for every principal irreducible polynomial f(x) of degree n (apart from equivalent extensions) only one extension field F($) which has the property that 0 is algebraic and has f(x) for its minimal polynomial. Moreover F(O) is of finite degree: [F(b) : F] = n, (127.10)

i.e., of the same degree as 1. Furthermore 1, 19, ..., ' i.e., its elements are uniquely given by

ON

is a basis of F (r?)'

(127.11)

where g(x) denotes the polynomials over F of formal degree n - 1. All the elements of F(O) are algebraic and their degrees are divisors of n. In particular,

the elements of degree n are the primitive elements of F(t9). F(t) can be effectively given by F [xJ/(f(x))

(127.12)

For the proof we first suppose the existence of a field F(O) with f(i) = 0, and consider the homomorphism F[x] - F[t]

(h(x) -> h(t9)),

(127.13)

THEORY OF FIELDS

486

where F[z] denotes the ring consisting of the substitution values h('). Thus, in any case, F[t9] S F(t9).

(127.14)

The kernel of the homomorphism (127.13) is constituted by those h(x) for which 0. This condition, according to Theorem 291, is equivalent to f(x) I h(x), therefore the above-mentioned kernel is the ideal (f(x)). Thus from (127.13), we obtain the isomorphism F[x]/(f(x)) & F[i]

(h(x) (mod f(x)) --o. h(ad)).

(127.15)

But since f(x) is irreducible, the ideal (f(x)) is maximal, and so, according to Theorem 130, the left-hand side of (127.15) is a field. This is also true

for the right-hand side. These must contain, together with 0, the field F(a9), i.e. F[a9] ? F(0). Hence and from (127.14) it follows that F(O) = F[0],

(127.16)

and, because of (127.15), that F [x]/(f(x)) s F(a9)

(h(x) (mod f (x)) ->- h(9)).

(127.17)

If, in particular, h(x) = c is a constant, then the class c (mod J (x)) is mapped

onto the representative c. More precisely, this means (because of the embedding) that the elements of F are fixed, i.e., that (127.17) is an F-iso-

morphism. In other words F(8) is an extension field of F equivalent to (127.12).

Furthermore (127.16) means that the elements of F(19) may be given in the form h(3) (h(x)E F[x]). If Euclidean division is carried out on h(x),

f(x), then, because J (O) = 0, it follows that for this purpose one only needs polynomials h(x) of formal degree n - 1. The corresponding h(0) are already different from one another, otherwise 19 would be the zero of a polynomial over F of smaller degree than n. This proves the assertion

in (127.11) and that concerning the basis property of 1, a9, ..., on-1. Hence (127.10) also follows.

We consider another arbitrary element a of F(O). Then F(a) is a subfield of F(0) with F S F(a) S F(O).

According to this, F(a) I F is of finite degree, so a is in any case algebraic. From (127.10) and Theorem 292 we get n = [F(9) : F] = [F(t) : F(a)] [F(a) : F].

(127.18)

SIMPLE FIELD EXTENSIONS

487

Therefore if (127.10) is applied with a instead of 0, this shows that the degree of a is a divisor of n. Since a is a primitive element of F(t4)) if, and only if, F(a) = F(O), i.e., [F(s) : F(oc)] = 1,

it follows from (127.18) that for this it is necessary and sufficient that

[F(a) : F] = n, i.e., that a is of degree n. The existence of F(t9) still remains to be shown. This is easy to prove. We already know that (127.12) is a field, consisting of the residue classes g(x) (mod./(x)) with g(x)E F [x], for which we use the notation -gT;i). Then

f(x) = 0. On the other hand (because of the embedding) f(z) = f(x), so

f(x) = 0.

(127.19)

But since (127.12) (after embedding) is equal to F(.), the required proof of existence is implicit already in (127.19). This completes the proof of Theorem 295.

Because of the existence and uniqueness proposition contained in this theorem the simple algebraic extension fields of F, not equivalent to one another, and the principal polynomials f(x), irreducible over F, are assigned to each other in a one-to-one manner, where to each such polynomial corresponds a field F(t9) in which the primitive element t9 is a root of the equation

f(x) = 0.

(127.20)

Therefore we call (127.20) the defining equation of this field.

In general one should call an equation (127.20) over a field with irreducible or reducible left-hand side an irreducible or reducible equation, respectively. Thus, by Theorem 295, an extension field in which this equation

has at least one solution can be given to every irreducible equation over an arbitrary field. EXAMPLE 1. It is an important part of Theorem 295 that the elements of an algebraic extension field F(i) may be written "denominator free", namely in the form (127.11).

We wish to prove this directly and at the same time study how one can "free" the elements of F($) of their denominators. We denote the minimal polynomial of 61 by f(x) and consider an arbitrary element of F(g): a = g(o) h(ad)

Here h(i})

(g(x), h(x) E F[x]).

0, thus fix) ,f' h(x). Since f(x) is irreducible, we can solve

f(x)fi(x) + h(x) hi(x) = I (fi(x), h,(x) E F[x]). Because f($) = 0 it follows that h(t9)h,(1) = I. thus a = g(a?)hl(a9).

THEORY OF FIELDS

488

EXAMPLE 2. By virtue of Theorem 295 an algebraic field F(9) I F of degree n is an

F-algebra of rank n with the basis 1, 0, ..., 0'1. By the help of the coefficients of the minimal polynomial =x"+alr"_1+...+a" (a,E F) f(x) of 0 the structure constants are computable. For, since 0119 = $1+L

(i = 0, ..., n - 2); 0-10 = -(a" + ... + a119"^1)

the product of two arbitrary basis elements 01, Of is recursively determined. EXAMPLE 3. Skew fields generated by one element are the simple field extensions F,(8) of the prime fields F,, of arbitrary characteristic p (> 0). EXAMPLE 4. The algebraic field F(O) over F is a special case of rings defined by equations. F(t9) is obviously that ring whose generators are the elements of F and the element 0, and whose defining equations are the equations valid between the elements of F, the equations c$ =arc (c E F), and finally f(ad) = 0 ,

(127.21)

where f(x) is the minimal polynomial of 0. It would, therefore, have been more accurate to call (127.21) the defining equation of F(a9) instead of (127.20). Reducible equations are, of course, not suitable for uniquely defining a field.

§ 128. Extension Fields of Finite Degree THEOREM 296. All the fields G I F of finite degree are algebraic and may be obtained from the fundamental field F by adjunction of a finite number of elements; furthermore the degrees of the elements of G are divisors of [G : F]. Conversely, every field over F, which arises from F by adjunction of a finite number of algebraic elements, is of finite degree (and so algebraic). COROLLARY. If a, j9 (j 0) are algebraic elements of an arbitrary field G I F, then (128.1)

are also algebraic elements over F.

In order to prove the first part of the theorem we write

n = [G : F].

(128.2)

Clearly, [F(a) : F] I n follows from Theorem 292 for every cc (E G).

Accordingly a must be algebraic; since, further, according to Theorem 295,

its degree is then equal to [F(a) : F], this is in fact a divisor of n. From (128.2) it also follows that G has an F-basis cot, ..., co,,. Hence G = = F(col, . . ., co). So the first part of the theorem is proved. For the proof of the second part we consider a field

G = F(0,, ..., 0k),

EXTENSION FIELDS OF FINITE DEGREE

489

where 01, ..., O'k are algebraic. These are then also algebraic over the fields

(i = 0, . . ., k; Go = F, Gk = G). = F(01, ..., ,O,) Since furthermore G,+1= G;(l';+I), according to Theorem 295, every Gi

term of the chain of fields

F9Gl9...9Gk_ISG (from the second term on) is of finite degree over the preceding term. Hence it follows, according to Theorem 292, that G I F is of finite degree. Consequently the theorem is now proved. The corollary is similarly true, since the elements (128.1) belong to the field F(«, fi) which is of finite degree over F. THEOREM 297. If the fields H I G, G I F are algebraic then H I F is also algebraic.

We have to prove that every element 0 of H is algebraic over F. On the basis of the supposition an equation such as

,on + al9"-I +

... + a = 0

(at E G)

(128.3)

holds. Since the a.; are algebraic over F, by Theorem 296, the field

is of finite degree over F. Further, because of (128.3), G'(O) is of finite degree over G'. By Theorem 296, G'(0) I F is then algebraic and so 0 is also algebraic over F. This proves Theorem 297. EXAMPLE. If G I F is a field of finite degree, then the rational function field G(x) is of finite degree over F(x), and [G(x) : F(x)] = [G : F].

(128.4)

It is also true that the three algebras G I F,

G[x] I F[x], G(x) I F(x)

have the same rank and every basis of the first is a basis of the remaining two. For the first two of these algebras the assertion is trivial. To prove the full assertion, it is sufficient to show that the elements of G(x) may be written as quotients with numerators from G[x] and denominators from F[xl. For this it is sufficient to prove that every element g(x) (56 0) of G [x] is a divisor of an element (00) of F[x]. Since among the powers (g(x))' (i = 0, 1,. ..) there are only a finite number of linearly independent ones over F(x), an equation of the form r

at(x)t(g (x))t = 0

(ao(x), ..., ar(x) E F[x] , ao(x) 0 0) .

t=o

holds. Hence it follows that g(x) I ao(x), which proves the assertion.

490

THEORY OF FIELDS

§ 129. Splitting Field

Let a field F be given with an arbitrary (finite or infinite number) of non-constant polynomials f1(x),12(x), .. .

(129.1)

over it. An extension field G of F is called a splitting field of these polynomials f(x) or of the equations f;(x) = 0 (over F), when in G [x] these polynomials split into linear factors and no subfield G' (F c G' C G) of G has this property. If, in particular, (129.1) consists of a finite number of polynomials f1(x),..., fk(x), then we may take the single polynomial

1(x) = fi(x) ... A W,

instead of them, since f(x) splits over a field into linear factors if, and only if, this is valid for all the f1(x), . . .,fk(x). Therefore there are only two essentially different cases, where (129.1) consists of only one polynomial or of an infinite number of polynomials. The first case will be studied here,

the second in the next paragraph. (In the trivial case, where the system (129.1) is empty, by the splitting field we have to understand F itself.) As a preliminary we need the following: whenever an isomorphism a -* a between two fields F, F is given, we understand by the image of a polynomial f(x) over F that polynomial f(x) which arises from it by carrying

out the above isomorphism on its coefficients. Of course, the mapping f(x) -+ f(x) is then an isomorphism of F[x] onto F[x]. So two corresponding polynomials f(x)j(x) are simultaneously reducible or irreducible (over F and F, respectively). THEOREM 298. Between two fields F, F let an isomorphism

F ^ F (cc - a)

(129.2)

be given and an irreducible polynomial f(x) over F. Furthermore denote the corresponding (thus likewise irreducible) polynomial over F by Y (A If the equations

f(o) = 0, f(a) = 0

(129.3)

hold for any two elements 9, a from arbitrary extension fields of F and F, respectively, then (129.2) may be continued to an isomorphism of the fields F (a), F(a), in which the elements Lo, a correspond to each other.

Denote by g(x) an arbitrary polynomial from F[x] and by g(x) the corresponding polynomial from F [x]. We establish the isomorphism F(RO)

F(a) (g(e) -* g(a)).

(129.4)

SPLITFING FIELD

491

Since, by (129.3), the fields F(e), F(a) are algebraic, all their elements are given by g(e), g(a), according to Theorem 295. Thus (129.4) maps F(e) onto F(a) in all cases. This mapping is one-to-one, since the following four propositions are equivalent :

f(x) I g(x), A-0 19(x), g(c) = 0. The homomorphism property of (129.4) is evident. If, finally, g(x) = a g(e) = 0,

(a E F) or g(x) = x then g(e) = a or g(e) = e, and therefore the image element g(a) is equal to a or a, respectively. According to this, (129.4) is a continuation of (129.2), mapping a onto a. Consequently Theorem 298 is proved.

We now prove the following: THEOREM 299. For every field F, every non-constant principal polynomial f(x) (E F [x]) has (apart from equivalent extensions) only one splitting field. If we take an arbitrary extension field of F over which then

f(x) _ (x - 191) ... (x -0 ),

(129.5)

G = F(01, . . ., 1n)

(129.6)

.

is a required splitting field, and all the splitting fields of f(x) arise in this way.

(Since, over a splitting field of f(x), because of its definition a factor decomposition of the form (129.5) holds, the first part of the theorem already contains the existence proposition that there are extension fields of F over which an equation (129.5) holds.) First of all we prove that there is an extension field of F over which (129.5) holds. For that purpose we take an arbitrary extension field F (;2 F) of F and consider the prime decomposition (129.7) f(x) = fi(x) ... fl(X) is not linear, then, according to Theorem of f(x) in F'[x]. If, e.g., 295, take an extension field F" = F'(i9) (thus also of F) with f1(1) = 0. It follows that, over F", the polynomial fl(x) has the factor decomposition f1(x) = (x - 1)g1(x) whence, according to (129.7), it follows that the

prime decomposition of f(x) over F" consists of at least r + 1 factors. The assertion follows by induction. This verifies the existence of the field G defined in the theorem. We show that this is a splitting field of f(x). For this purpose we consider a betweenfield G' (F S G' c G) and suppose that f(x) also splits over G' into linear factors:

f(x) = (x - 11) ... (x - OD

(01 E G').

G'. Thus because This factor decomposition then holds over G since G of the uniqueness of the prime decomposition in G [x] it must agree with

THEORY OF FIELDS

492

(129.5). Since, according to this, 19;, ...,19;, differ at most in the order of succession from 191, ...,19,,, so 19,, ...,19 E G'. Hence, and from (129.6),

G 9 G' follows. Because G' S G we then have G' = G. This means that G is in fact a splitting field of f(x). The final assertion of the theorem is trivial. The assertion of uniqueness still remains to be proved. To prove this, we consider, in addition to G, a further splitting field G' of f(x) which is given by

f ( x ) = (x - 01) ... (x - 0. 0,

(129.8)

G' = F(19;, ...,19;,).

(129.9)

(These "new" 0,, ...,19;, are independent of the former ones.) We have to prove the existence of an isomorphism

GIF,:;G'IF(a(129.10) and show at the same time that with a suitable order of 191,

the

additional condition 19,=19;

(i= 1,...,n)

(129.11)

can be satisfied.

We introduce the notations G, = F(191,

..., 9 ),

G, = F(191,

(i=0,...,n; Go=Go=F;

..., ,) G;,=G')

(129.12)

and suppose that for some k (0 <_ k S n - 1)191, ...,19;, could be rearranged

in such a manner that an isomorphism (129.13)

Gk

exists such that 19; = 0

(i = 1, ..., k).

(129.14)

(It is trivially possible for k = 0.) By (129.5) and (129.121) we have

f(x) = (x - 191) ... (x - 1k) g1(x) ...

(129.15)

for some irreducible principal polynomials gl(x), ..., g,(x) from Gk[X]Applying the isomorphism (129.13) we obtain the equation .f(x) = (x - 19) ... (x - 1O) 91(x) ... 9s(x)

(129.16)

493

SPLITTING FIELD

from (129.15), by using (129.14) and the fact that f(x) E F[x], where gi(x) denotes the image of g.(x) corresponding to this isomorphism. Because of (129.5) and (129.15) it may be assumed that x - Ok+l I gl(x), i.e., (129.17)

g1(('k+1) = 0.

Since, moreover, we are free to rearrange -9'k+l, ..., 79'w, we may, by (129.8) and (129.16), assume that x - 79''k+I 191(x), i.e., we have (129.18)

gl(N+i) = 0. Since now, according to (129.13), the "absolute" isomorphism Gk ^' Gk (a -> a)

holds a fortiori, and, according to (129.12), Gk+t = Gk(ok+1),

Gk +1 = Gk(Ok+1),

so, because of (129.17) and (129.18), the existence of an isomorphism (129.19)

Gk+1 " Gk+1,

which is a continuation of (129.13) and maps tk+1 onto 19k'+, follows from

Theorem 298. This means that the supposition made in (129.13) and (129.14) is satisfied with k + 1 instead of k. Because G,; = G, G , = G' the assertion follows by induction. Consequently, Theorem 299 is proved. EXAMPLE. Let f(x) be an irreducible polynomial over a field F for which

f(x) = (x- 61)... over an extension field of F. The (2 ) fields F(g ,9,) (1 < i < k S n) need not be isomorphic to one another. For example x4 + 2x2 - 2, according to EISENSTEIN'S theorem (Theorem 283), is irreducible over -o and is decomposed over the field of complex numbers into the product

(x+V-I+j

(x-V-1-y3

If the corresponding four zeros are denoted by Y91, ...,194, then F. (61.'82), FO (6u 0a) are evidently not isomorphic.

THEORY OF FIELDS

494

§ 130. Steinitz's First Main Theorem A field F is said to be algebraically closed if every non-constant polynomial f(x) over F has at least one zero in F. This means that every such polynomial

splits into the product of linear factors over F, or that in F[x] only the linear polynomials are irreducible. The prime fields are not algebraically closed, for in the prime field of characteristic 0,

x2 - 2 = 0 has no root, and in the prime field of characteristic p (> 0), x" - x - 1 = 0 has no root, as can be seen immediately.

THEOREM 300 (STEINITZ'S theorem). Every field F has (apart from equivalent extensions) only one algebraically closed algebraic extension field F.

We call this field F the algebraic closure of F. The proof is furnished by the following theorem which is a generalization of Theorem 299 and is obtained from it. THEOREM 301 (STEINITZ'S

first main theorem). For every field

F,

every system of non-constant polynomials over F has (apart from equivalent extensions) only one splitting field and this is algebraic over F. For the proof let us put the given polynomials in the form

f, (x) = xn- a,,

xn_1 +

... + (-1)na,.n (a, E F),

(130.1)

where v runs through the elements of an index set. (Of course n depends on v.) We assign to each f,(x) as many indeterminates xv1, ..., x,n over F

as its degree indicates, and denote the polynomial ring of these indeterminates by

R = Q.

.

., x,.I, .

.

., x , . .

. ].

(130.2)

Let s; denote the ia` elementary symmetric polynomial of xv1, ..., xn (i = 1, . . ., n). Then we form the ideal of R: a = (..., svI - a,,, ..., s,, - am, ...). We prove that a tion such as

(130.3)

R. For, if not, then 1 E a. This means that an equa([30.4)

holds, where we have to understand a finite sum and r denotes an element

of R changing from term to term. Since only a finite number of indeterminates x; occur here, only a finite number of polynomials f,(x) belong to them. By Theorem 299 these polynomials have a splitting field G over F. For these polynomials the factor decompositions of the form

.fv(x) = (x - o1) ... (x - m,.) (z,, E G)

495

STEINITZ'S FIRST MAIN THEOREM

.... a,,,, each time then hold. We establish an arbitrary order for and carry out on (130.4) all the substitutions x,.i = a,.i. Since then [in (130.4)] because of (130.1) every si becomes a,,i, the contradiction I = 0 follows from (130.4). Consequently the assertion has been proved. Hence, and from KRULL'S theorem (Theorem 131), there exists a maximal ideal ao of R with (130.5)

a.

a0

Since moreover R is a ring with unity element, so, by Theorem 130, the factor ring R/ao is a field. Fn ao =0 follows from ao T R. The embedding of F carried out according to Theorem 294 (with F, ao instead of S, a) changes R/a0 into an extension field of F.

By (130.1) and (130.3), for all r.

fv(x) = x" - s1 xn -' + .

.

. + (-

l )"sr"

(mod a),

(x - x,,,) ... (x - x,,,,) (mod a). thus, by (130.5),

fi(x) _ (x - x,,,) ... (x - x,.,,)

(mod ao).

This means that, in the field R/ao, all the f;,(x) are decomposed into linear factors and the residue classes xi (mod ao) are all the zeros of all thef,(x) in R/ao. Since, by (130.2), R/ao arises from F by the adjunction of these residue classes, R/ao is a splitting field of the polynomials (130.1). Since, in addition to this, the adjoined residue classes are algebraic over F, R/a is also algebraic over F, because of the corollary of Theorem 296. Accordingly, only the proposition of uniqueness in Theorem 301 still remains to be proved. Instead of that we prove more generally the following THEOREM 302. If G, G' are two splitting fields of the same polynomials over afield F, every F-isomorphism between two subfields of G I F and G' may be continued to one between G and G'. Since, in particular, the identical mapping of F is an F-isomorphism, the required proof of the proposition of uniqueness in Theorem 301 follows from this theorem. With a view to proving Theorem 302, we call an F-iso morphism between two subfields of G I F and G' I F a partial isomorphism. If, of two different partial isomorphisms g, a the second is a continuation of the first, then we write g < a. Let ,u denote a fixed partial isomorphism and :, the set of those partial isomorphisms e for which a >_ p. It will suffice to prove that C contains an element which maps G onto G'. 17 R.--A.

THEORY OF FIELDS

496

The set G`7 is semiordered. We consider an ordered subset

x
(130.6)

of C, where a, fi.... are, in order, the partial isomorphisms A

A',

B ,: B',

and A e B e ... and A' a B' c ...

..., are

subfields of G and G',

respectively. According to Theorem 47, A U B U ... and A' U B' U ... are subfields of G and G', respectively. These are isomorphically mapped by a common continuation of the mappings (130.6) onto one another. This

means that every ordered subset of Z has an upper bound in G, consequently S has, according to the KLRATOWSKI-ZORN lemma (Theorem 15), a maximal element. This implies the isomorphism H

H'

(130.7)

where H, H' are certain subfields of G and G', respectively. It is sufficient if we show that

H=G, H'=G'.

If it is false, then we can suppose, that H c G. Hence it follows that of the polynomials submitted in Theorem 302 there is one such polynomial which has a non-linear irreducible factor over H. Applying Theorem 298, we establish the existence of a partial isomorphism

G' (HcGacG, H'cGocG) which is an extension of (130.7). By this contradiction we have proved Theorem 302, and so Theorem 301 also. Finally, we also prove Theorem 300. A field which is algebraic over F and is algebraically closed, must necessarily be a splitting field G of all the non-constant polynomials f(x) (E F[x]). By Theorem 301 it is thus sufficient to prove that G is necessarily algebraically closed. To do this we consider a non-constant polynomial g(x) over G. This has a zero i in an extension field of G. Since G(A) I G, G I F are algebraic, t is, according to Theorem 297, algebraic over F. We denote by f(x) the minimal polynomial of .. over F. Since this splits into linear factors over G, it follows that lies in G. Consequently Theorem 300 is proved. § 131. Normal Fields A field N I F is said to be a normal field if every irreducible element of F[x] has either no zero in N or splits into a product of linear factors over N.

NORMAL FIELDS

497

THEOREM 303. The concepts "normal field" and "splitting field" (over a field F) are identical. (According to this theorem it would be sufficient to use only one of the two terms; however, sometimes one is more suitable and sometimes the other.)

For the proof we consider an arbitrary (finite or infinite) number of non-constant principal polynomials

fi(x), fi(x), ...

(131.1)

over the field F and denote its splitting field by G. We show that G I F is normal. For this we take an irreducible polynomial g(x) over F which has a zero ) in G: g(%) = 0.

(131.2)

We have to show that g(x) is a product of linear factors over G. For every product f(x)'of a finite number of polynomials from (131.1) (131.3)

with elements

0 from G. We choose f(x) so that :A E F(01,

...,

(9; G),

(131.4)

as is obviously possible. By repeated application of Theorem 295 it follows that a = h ( 6 1 1 ,-.

., 8n),

(131.5)

where h is a suitable polynomial over F. Let us form the product

H(x) = H (x - h(01, ...,

(131.6)

extending over all the permutations of the indices 1, ..., n. According to the main theorem for symmetric polynomials (Theorem 268) and according to (131.3) the coefficients of H(x) are expressible in terms of those of f(x), thus H(x) E F[x].

But, by (131.5) and (131.6), H(A) = 0. Consequently, from (131.2) and the irreducibility of g(x), the divisibility 9(x) I H(x)

(131.7)

follows. Since, by (131.4) and (131.6), H(x) is a product of linear polynomials in G [x]. so, from (131.7), the same follows for g(x). Accordingly, G I F is in fact a normal field.

THEORY OF FIELDS

498

We then consider a normal field N F. This arises from F by the adjunction of certain algebraic elements al, x.,, ... Their minimal polynomials (over F) are denoted in order by f1(x), f2(x), ... As N is normal, these polynomials split into linear factors over it. All the zeros in N are designated by $I, $2, .... Then N. F(01. b.,, ...) (131.8)

But as all the al, x2, ... occur among the fir, fl....... so N = F(%, a.,,

.. .) S F(0, 0_,, ...).

Accordingly, (131.8) holds with "_" instead of and so N is the splitting field of the polynomials f1(x), f2(x), .... This proves Theorem 303. By a normal field of an algebraic field G I F we understand a normal field N I F such that N G, with the property that no field, different from it, is normal over F and lies between G and N. THEOREM 304. Every algebraic field G I F has (apart from equivalent extensions) only one normal field. For the proof we write G = F(ai, «2, ...),

where al, a2.... are algebraic elements over F. Let their minimal polynomials be A(x), f2(x)I - ..

(131.9)

.

By Theorem 303 it is evident that a field N I F is a normal field of G I F if. and only if, N is an overfield of G and is a splitting field of the polynomials (131.9) over F. But, the last-mentioned fields are equivalent to the splitting field of the same polynomials over G. Hence Theorem 304 follows from Theorem 301.

§ 132. Fields of Prime Characteristic In every field F of characteristic p (> 0) we have the very important rule

(x + f,)P = xP + fl"

((x, # E F).

This follows immediately from n

- it

(x +

p aP-L

and from P

pl

(O

p).

f

(132.1)

YIELDS OF PRIME CHARACTERISTIC

499

Furthermore (a - P)P

(XP - #P.

(132.2)

For if p > 2. thus 2 A. p, then we have (- I)P = - 1, therefore (132.2) now follows from (132.1) by applying it to -l3 instead of (l. If p = 2, (132.2) agrees essentially with (132.1), since now in general e = - e (e E F). From (132.2) and the zero-divisor-free nature of F it follows immediately that if a, j3 are different, then cP, NP are also different. This, together with (132.1), means that the mapping z --* a"

(132.3)

is a meromorphism of the module H. If F is finite, then (132.3), of course, must be an automorphism of F+. If, moreover, F is a prime field, then by Theorem 290 (132.3) must be the identical mapping of F+, as is obvious. Conversely, if (132.3) is the identical mapping of F+, then F is a prime field, since in F the equation xP - x = 0 cannot have more than p roots. Since, furthermore, because of the commutativity of F, in addition to (132.1), (cg9)P = aP/9P, i.e., the mapping (132.3) is also homomorphic with respect to multiplication, we have proved the following theorem: THEOREM 305. For every field F of characteristic p (> 0) the mapping

a - x"

(a E F)

is a meromorphism of it. This is an automorphism if F is finite; further it is the identical mapping if, and only if. F is a prime field. The rules (132.1), (132.2) also hold in every commutative ring of prime characteristic p. In particular, in the polynomial ring F[x] over the above field F n

Y- ai x` r=0

_ I aipxP' i -0

If, moreover. F is the prime field of characteristic p, then the important rule

(f(x))P =.f(x)

(f(x) E F[x], 0(F) = p)

follows. EXERCISE. If F is a field and n (> 1) is a natural number for which

a - a"

(aEF)

is an endomorphism of F, then the characteristic of F is necessarily a prime number p and n is a power of p. § 133. Finite Fields

THEOREM 306. Every finite field is of primepower order. Conversely, to every prime power p" (n >-- 1) belongs, apart from isomorphism, only one field F with 0 (F) = p". (133.1)

THEORY OF FIELDS

500

This is (absolute) algebraic of degree n and may be given over its prime field (by a simple adjunction) in the form

F = FP(),

(133.2)

where the primitive element 0 is the zero of an arbitrarily given irreducible polynomial f(x) (E Fp[x]) of degree n. The group F* is cyclic (of order p" - 1), therefore all the different elements of F may be given, by means of a primitive element p of F*, in the form 01110, . . ., op"-:'.

(133.3)

x"' - x = 11 (x - a),

(133.4)

Also aEF

according to which F is the splitting field of the left-hand side of (133.4) and so is normal. The full automorphism group of F is cyclic of order n and consists of the automorphisms

(k = 0, . . ., n - 1).

a --> a°k

(133.5)

Since every finite field must be of prime characteristic, so its module is a finite p-module, thus its order is a power of p. This proves the 'first assertion of the theorem. We want to show that to every prime-power p' (n 1) there is at least one field F which satisfies (133.1). Instead of this we shall now prove that the splitting field F of the polynomial (133.6)

XP" - x (E FP [XD

satisfies the requirement (133.1), where FP is the prime field of characteristic p.

The differential quotient of (133.6) is - 1( 0), so that (133.6) has exactly p" zeros in F. Therefore it suffices to show that these zeros constitute a subfield of F, for then this subfield is the splitting field of (133.6), and so necessarily equal to F. We now consider two zeros a,# (E F; l4 0) of (133.6):

aP"-a=0, pPl-(3=0.

Then we have

-

a#-I

= xj9 ` -

0Cq-1

= 0,

so that a - f, a#-' also belong to the set of zeros. This is, therefores a field. Consequently, the existence of an F with the property (133.1) i, proved.

FINITE FIELDS

501

In the following let F denote an arbitrary field which satisfies (133.1). Because O(F+) = p" the characteristic of F must be p. Denote the prime field of F by FP. Since F+ is an FP vector space, so, because O(FP) = p and by (133.1), this must be of rank n. This means that F is of degree n and algebraic. Since in the finite Abelian group F* (and even in F) every equation xR = 1

(k z 1) has at most k solutions, it follows from Theorem 217 that F* is cyclic.

A primitive element of F* is [according to (133.3)] also a primitive element of F. Accordingly, there exists a 29 which satisfies (133.2). From

[F : FP] = n and Theorem 295 it also follows that the minimal polynomial of t is of degree n; this is irreducible and has .0 as zero. Also the remaining assertion on #9, that to every f(x), in the theorem, there exists a O which satisfies (133.2) and f (z9) = 0, is true on account of Theorem 295, if we prove that all fields of order p" are isomorphic.

This assertion is a consequbnce of (133.4) and the uniqueness of the splitting field (Theorem 299). We prove (133.4) itself as follows. From O(F*) = p" - 1 it follows that al "-I = 1 for all elements a of F*. Thus (XP" - a = 0 for all elements a of F. Since, accordingly, every element of F is a zero of (133.6), (133.4) follows from this and from (133.1). The last assertion of the theorem still remains to be proved. An automorphism of F is determined by the image of its primitive element 0. But t

can only be transformed into a zero of its minimal polynomial, so that there can be at most n automorphisms of F. On the other hand, a -* aP, by Theorem 305, is an automorphism of F. Since (133.5) is the kth power of this automorphism, we have only to show that the n automorphisms given by (133.5) are different. Otherwise there are two integers i, k (0 < i <

< k < n - 1) such that for all the elements a of F we have aP'`

- aP'=0.

This is absurd since the degree of this equation is smaller than O(F). Consequently Theorem 306 is proved. THEOREM 307. Every subfield of a finite field F of order p" is of order pd,

where d is a divisor of n. Furthermore there is to every such d exactly one subfield of F of order pd. If d > 1 the primitive elements of this subfield are those a (E F, 0) for which

o(a) I pd - 1, o(a),f'pd' - 1

W1 d, < d)

(133.7)

(the second for all the d'). Since F is of degree n, by Theorem 292, the degree of every subfield of it is a divisor d of n, consequently this subfield has the order p".

502

THEORY OF FIELDS

Henceforth let d I n and F' be a subfield of F with O(F') = pd.

According to Theorem 306, F' is a splitting field of the polynomial xn"I

- X.

(133.8)

Since this has at most p`' zeros in F, there can be at most one F'. The existence of an F is proved if we show that (133.8) splits into linear factors over F. for then F contains a splitting field of (133.8). Since from d I n it follows that p,r - 1 I p" - 1,

x" -I - I

I xP"-.1

- 1.

XP - .Y xP, - X.

Hence, and from (133.4), the assertion follows. Only the last assertion of the theorem remains to be proved. Since the group F* is cyclic of order p" - 1, so F'* is the only subgroup of F* of order ir' - I and so consists of the elements a (96 0) of F such that o(a) i P`r - 1.

This implies (133.71). The condition that a is not in a subfield (0 F') of F' is similarly expressed. by (133.7..,). Hence follows (133.7) by which Theorem 307 is proved.

From Theorem 306 it follows as an additional result that over every prime field F, (p > 0) there are irreducible polynomials of every positive degree. More precisely, we have the following THEOREM 308. Orer the prime ,field FP (p ; 0) the .factor decomposition _`P

- _x - 11.1(x)

(133.9)

holds, where f(x) runs through those irreducible principal polynomials over FP whose degree is a divisor of it. The number of irreducible principal polynomials of degree it over FP is 1

nd

lrJ dp 1nl

dr.

(133.10)

For the proof let f(x) denote an irreducible principal polynomial of degree it over F, and let F be an extension field of FP such that O(F) = p".

FINITE FIELDS

503

First assume that 1(X) i x"" - X.

(133.11)

According to (133.4) there is an a (E F) with

f(x) = 0.

(133.12)

Since, according to this, Fp(x) is a subfield of degree d of F, from Theorem 292, d is a divisor of n. Secondly assume that d I n. Then by equation (133.12) a field F,(x) of order p° is defined. Because d I n and from Theorem 307, it may be assumed that this field is a subfield of F. Then, by (133.4), x is a zero of x" - x. Hence, and from (133.12), (133.11) follows.

Consequently we have shown that (133.11) holds if and only if d I n. Since now the right-hand side of (133.11) has no multiple divisors, this proves (133.9).

Let N,, denote the number of irreducible principal polynomials of degree n in FF[x]. From the equality of the degrees of both sides of (133.9) it follows that

p" = I dNd (I

(n = 1, 2, ...).

I,

Hence and from MOBtus's inversion formula (Theorem 222) we obtain (133.10). Consequently, Theorem 308 is proved. The interpolation may also be carried out, of course, in the finite fields according to LAGRANGE (Theorem 280) or according to NEWTON (Theorem

281). Moreover we have the following THEOREM 309. In a finite field F of order q (= p") every function

a -* a'

(x E_ F)

(133.13)

may be uniquely given as a polynomial of formal degree q - 1, by the formula

f(x) = 1 x'(1 - (x - 7.)Q-1) .

(133.14)

2EF

For every element x of F I (x 0), 0 (x = 0). Hence, and from (133.14),

.`(x) = x'. Accordingly, the function (133.13) is given by (133.14). We also see that

(133.14) is of formal degree q - 1. The assertion of uniqueness of the theorem is trivial. 17/a R.-A.

THEORY OF FIELDS

504

In the proof of the three following theorems we make use of the final inference given below, in Example 1.

TiEomm 310. If p is a prime number, F(x) an irreducible principal polynomial mod p of degree n from .7[x] and a(x) a further polynomial from 3' [x] with (d I n, < n),

(a(x))," * a(x) (modp, F(x))

(133.15)

then the congruence

f(a(x))

x (mod p, F(x))

(133.16)

has a solution f(x) in J [x].

Let F denote the finite field such that O(F) = p" and K a (primitive) element of F such that F(K) = 0. According to (1.33.15) (a(K))P° & a(K)

(d j n, < n).

This means, according to (133.5) and Theorem 307, that a(K) does not he in a proper subfield of F, i.e., it is a primitive element of F. Then for some polynomial f(x) from 7 [x] we have f(a(K)) = K. This means the same as (133.16), therefore Theorem 310 is true. THEOREM 311. If p is a prime number and

F(x)=x"+alx"-'+...+a"

(a,,...,a"E-7)

(133.17)

is an irreducible principal polynomial mod p of degree n from '7[x], then

a1+x+x"+.1rO+... +x°"-' =-0(mod p, F(x))

.

(133.18)

In the proof we apply the above F and K. Because of (133.5), KP' (i = 0,

., n - 1) are all the zeros of F(x) in the field F, from which, according to (133.17),

K+Kp+K#'+...+Ke-1= -al follows. This is identical in meaning to (133.18). THEOREM 312. If F(x), FI(x) mod a prime number p are irreducible principal polynomials of degree n from. .7 [x], then there is a polynomial a(x) from .7[x] with

Fl(a(x)) - 0 (mod p, F(x))

.

(133.19)

FINITE FIELDS

505

Moreover, we have

F, (a(x)) * 0 (mod p, F(x))

,

(k = 1, ..., n - 1).

(a(x))''t # a(x) (mod p, F(x))

(133.20) (133.21)

Let F again denote the finite field with O(F) = p". From this we take two primitive elements rc, A with

F(u) = 0 , F1().) = 0 . Then there is a polynomial a(x) (E 3[x]) with A = a(rc).

This implies that Fl(a(n)) = 0, which is identical with the congruence (133.19). From the irreducibility of F1(x) we have FF(A) a 0 and from (133.5) it follows that AI,k 0 A (k = 1, ..., n - 1). These inequalities are equal in meaning to (133.20) and (133.21), respectively. Finite fields were formerly also called "Galois fields" after their discoverer; this expression, however, is nowadays used in another sense (cf. p. 655). EXAMPLE 1. If R is a ring with unity element and a an ideal of it, then the isomorphism

(R/a)[xl - R[x]/ax

holds, where ax denotes the ideal of R[x] generated by the elements of a, which evidently consists of the polynomials a(x) with coefficients from a and is consequently

equal to A[x]. It is obvious that

,(ay+a)x'-

(la.

r=o

(a,E R)

f

is a suitable isomorphism. For instance (.7/(P))[x]

p[x]/(P)x

(p prime)

holds, so that the theory of polynomials with integer coefficients mod p is identical with that of polynomials over the prime field of characteristic p. In particular the finite field of order p" may be given as .7 [x] j(F(x), p),

where F(x) is an irreducible polynomial mod p with integer coefficients of degree n. EXAMPLE 2. Over the prime field F,, (p > 0) every polynomial

j(x)=x°-x+a

(aE F 96 0)

is irreducible. Because x°" = (x - M) 00-1

x°"_' - a

(mod }(x)) (n Z 1)

in general

xp'

- na (modRx)) (n = 0, 1 1,...).

THEORY OF FIELDS

506 Accordingly we have

f ( x ) I x"n - x. (fix), x"" - x) = I

(n = I , .... p-1 ) ,

whence, by Theorem 308 [equation (133.9)], the assertion follows. EXAMPLE 3. If

p, q are different prime numbers, then F"(x) =

prime field F" splits into q

m

1

X"

x - j over the I

(different) irreducible factors of degree in where in = o(p( mod q)).

This follows from Theorem 308, since, on the one hand (Fq(x), x", - x) = 1 (I < r < m), and on the other hand Fo(x) ! x'°' - x, furthermore, because (x° - 1)' _ = qx"-' even x° - I has no multiple factors over F,,. (For a generalization see Theorem 317.)

EXAMPLE 4. In order to give the finite field F of order 53 (= 125) take the irreducible polynomial x3 + x + 1 mod 5. The elements of F may thus be given by the help of an element 0 having the property 03 + & + 1 = 0 in the form

a02+bo+c

(a,b,r = 0....,4).

We compute with them as with polynomials in 0 with integer coefficients mod (5, 63 + 't) + 1). EXAMPLE 5. The conditions (133.7) in Theorem 307 are also representable in the form o (p(mod o(x))) d. (133.22) EXAMPLE 6. All the different primitive elements of a finite field F of order p" (n > I)

may be given, according to (133.7), in terms of a primitive element 0 of the group F* by Q', where i runs through those integers 1, .... p" - 2 which are not divisible

p" - 1

(d; n.
pd - 1

by any. Their number, according to (133.10), is equal to (133.23)

(i.) P".

din

EXAMPLE 7. Over every finite field F of order q (= p") there are irreducible principal polynomials f(x) of degree n for every natural number n. Take a finite field G

of order q" (= p) which we can suppose contains F, then G I F, according to Theorem 292, is of degree n. Consequently, the minimal polynomials of the primitive elements of G over F are exactly the polynomials f(x) of the required kind. Determine their number. EXAMPLE 8. Two polynomials f(x), g(x) over a finite field of order q generate the same

function in this field if, and only if, they are congruent mod (x° - x). EXAMPLE 9. If x is a generator of the group F* of a finite field F of order q, then, for its minimal polynomial f(x),

1(x)Ixk-Iraq-1Ik. The left-hand side is identical to assertion follows.

hk

= 1. whence, because o(k)

q - I,

the

FINITE FIELDS

507

EXERCISE 1. Over a finite field of odd order q the polynomial ay-i a+

+ (1- x) 2l

1

2

I1+X

is the square of a polynomial. Cf. REDEI (1946-48). EXERCISE 2. Besides finite fields there are no commutative rings R whatever in

which all the functions can be given by polynomials f(x) (( R[x]). Cf. REDEISZELE (1947, 1950a).

§ 134. Konig-Radon Theorem

We call the square matrix %0

a1 ... xn-2 an-I

at

a2 ... an-1. ao (134.1)

an-I 70 ... an_g an-2

of type n2 the cyclic matrix of the elements ao...., a1 which can belong to an arbitrary ring. [The law for the construction of (134.1) is that every row after the first is obtained from the preceding row by the cyclic permutation' (0 I ... n - 1) of the indices.] THEOREM 313 (KONIG-RADOS theorem). The number of roots different from 0 and from one another of an equation MO +a1C + ... + a4_.2X'

=0

(a; E F)

(134.2)

over a finite field F of order q is

q - 1 - r,

(134.3)

where r denotes the rank of the cyclic matrix of the coefficients of the equation.

For the proof let f(x) denote the polynomial on the left-hand side of (134.2) and (134.4)

P1, .. . N4-1

all the elements of F other than 0. Further, let A denote the cyclic matrix of the elements ICQ, ...,'Q__., [case n = q - 1 of (134.1)] and B the matrix R1.

R9-2 R1

(134.5) fR{

q

z

"4-1 ... F'q-1

508

THEORY OF FIELDS

This latter is called the Vandermonde matrix of the elements I, ..., By _l. Both

matrices A, B are square matrices of type (q - 1)2. Since every Pi-' is equal to 1, we obtain JWI)

9If(fl) ...

NI

(y-2'N1) (134.6)

9

17JlNy-1] ... 9
('y-1)

We now denote the number of roots of (134.2) by N, i.e., the number of vanishing terms of the sequence

.fi), ...,f(fy-1) For convenience we may suppose that these are the last N terms of this sequence. Then in the last N rows of the matrix (134.6) there are only zeros, so that the rank of this matrix is at most q - 1 - N. But it is exactly

q- 1 -N,

since a principal minor of this order is the Vandermonde determinant ., 9q !,_N, multiplied by f (#I) ... f (Ny_1-N) and so different from 0.

of the distinct elements #i 1,

Since, on the other hand, because of the diversity of the elements (134.4), the matrix (134.5) is regular, the rank of the matrix (134.6) just determined

is equal to that of A, i.e., equal to r. Since by this r = q - 1 - N, so N = q - I - r. Consequently the theorem is proved. Till now only that part of Theorem 313 was known which relates to prime fields. Cf. RADOS [1886]. For a similar theorem cf. REDEI-TURdx (1959). For the determination of the number of irreducible factors of degree k (k > 1) of polynomials f(x) over finite fields cf. SCHWARZ (1956).

§ 135. Cyclotomic Polynomials

The splitting field of the polynomial

x"-1

(n_>_1)

(135.1)

over the prime field F. is called the nth cyclotonic field (of characteristic p > 0). Of course the cyclotomic field for p > 0 is a finite field, so that the case p = 0 is the most important. When we speak briefly of a cyclotomic field we mean this case. For the determination of cyclotomic fields it is necessary to decompose the polynomial (135.1) into irreducible factors (over F,).

CYCLOTOMIC POLYNOMIALS

509

Here we wish to deal only with this problem, and return to cyclotomic fields themselves more explicitly in § 171. In an arbitrary field we call every zero of the polynomial (135.1) an nth root of unity in this field. It is said to be a primitive nth root of unity when it does not satisfy any equation

xk-1=0

(15k
i.e., it is not a kt' root of unity for any k (< n). The roots of unity in a field F are equal to the elements of finite order of the group F*. In particular, a primitive nth root of unity in F means the same as an element of order n of F*. As preparation for the solution of the above problem, we need the following

THEOREM 314. For every natural number n, all the nth roots of unity in a field F constitute a cyclic group whose order is a divisor of n. First of all it is trivial that the nth roots of unity in F constitute an Abelian group. Since (135.1) in F has at most n zeros, this group is finite. Since in F every equation (q prime) xq = 1

has at most q solutions, it holds afortiori for the given group, whence, by Theorem 217, it follows that it is cyclic. If it is of order k, then it contains an element a of order k. Since ion = I must hold for this, so k I n. The theorem is therefore proved. The following theorem brings us near to the solution of the given problem. THEOREM 315. Over the prime field F. of characteristic p (>_ 0) the factor decomposition

xn- 1= 11FAX) . din

(135.2)

holds for every natural number n not divisible by p, where

Fn(X) = II (xd - 1)p(nd-1) din

135.3)

is a polynomial of degree p(n) over F. whose zeros in a splitting field of (135.2) are exactly the primitive nth roots of unity. The polynomial FF(x) is called the nth cyclotomic polynomial (over FP). Usually Fn(x) is tacitly considered only for the case p = 0, and then Fn(x) may be interpreted as a polynomial over 9. Because of (135.3) and GAUSS'S

theorem (Theorem 207) we must then have Fn(x) E 7[x]. It should be noted that from the definition (135.3) the polynomial property of Fn(x) is not immediately obvious, since in general exponents - I also occur. The assertion referring to this forms the principal part of Theorem 315.

THEORY OF FIELDS

510

In order to prove the theorem we notice first of all that, according to Mosius's inversion formula (Theorem 222), applied in the group FF(x) of the field FF(x), (135.3) constitutes the only solution of the system of equations (135.2). We show that (135.3) lies in F ,[x], whence according to the formula (94.16) it follows that its degree is cp(n). For the proof of our assertion we make use of the splitting field F of the polynomial (135.2). The differential quotient nxn-1 of (135.2) is prime to (135.2) because p .4' n , thus it follows from Theorem 267 that (135.2) has no multiple zeros in F. Since, accordingly, the number of the zeros is n. these constitute, by Theorem 314, a cyclic group of order n. Consequently, over F, the factor decomposition

x" - I = (x - I) (x - o) ... (x - nn')

(135.4)

holds, where o is a primitive n' root of unity in F:

o(o)=n.

(135.5)

F = F(o).

(135.6)

From (135.4) it follows that

Since the equation xd = 1 (d I n) has the d distinct roots

(k=O,...,d- 1),

Ond -' k

so

Xd- 1

d-I =fl(x-and-'k)

k=0

After substitution in (135.3) we obtain 11

FF(x) =

fl (X -

Ond I k)p(nd -') .

d.n k=0

Since nd-1, like d, runs through all the different divisors of n, the former equation can be written as follows : nd-'-1 Fn(x) = 1din1 k=0 11 (x - Odk)p(d)

(135.7)

In order to continue the transformation of the right-hand side we consider for a fixed number c (= 0, ..., n - 1), some d, k which satisfy the conditions

dk= c; din; k=0,...,nd ' - 1.

CYCLOTOMIC POLYNOMIALS

511

These may also be characterized by

k= d .

dI(c,n),

Consequently, on the right-hand side of (135.7), the exponent of x - o" is equal to

E p(d)

d, (r.n)

By Theorem 222, this sum is equal to 0 or 1 according as (c, n) > I or (c, n) = 1. So from (135.7) we obtain n-1

/

(x - oc)

F (x) _

(135.8)

(c, n)=1

Accordingly, we have FF(x) E F[x]. On the other hand, by (135.3), we have F,(x) E FF(x). From both of these it follows that F.,(x) E F,,[x]. Because of (135.8) the last assertion of Theorem 315 is true, so completing its proof. THEOREM 316. Over the prime field of characteristic 0 all the cyclotomic polynomials are irreducible.

Let F denote the n' cyclotomic field, i.e., the splitting field of x" - 1. over ."t,. Then F = .70(P)

,

F,(Q)

= 0

where F,(x) denotes the nth cyclotomic polynomial and g a primitive n`h root of unity. Furthermore, we denote byf(x) the minimal polynomial of g

over.. We have to prove that f(x) = FF(x). Let p denote a prime number such that p,f' n. Then g° is a primitive nth root of unity. Its minimal polynomial over .Yo is denoted by g(x). First of all we show that g(x) = f(x). Let us assume that g(x) 96 f(x). Since e, g° are zeros of xn - 1, it follows by hypothesis that f(x), g(x) I xn-1 , and

xn - i = f(x) g(x) h(x)

(135.9)

for some h(x) in .70[x]. Since furthermore a is a common zero of f(x) and g(x°),

g(x") = f(x) k(x)

(135.10)

for some k(x) from .7 [x]. From GAUSS'S theorem (Theorem 207) it follows that the factors of the right-hand side of (135.9) lie in .7[x], and then the same follows from (135.10) for k(x).

THEORY OF FIELDS

512

For every polynomial 1(x) (E --Y [x]) its image corresponding to the homomorphism 3 - 3p is denoted by 1(x). Since then

3 [x] -

[x] (1(x) __> 1(x))

the equations

x" - 1 = f(x)g(x) h(x) ,

(135.11)

g(x") = f(x) k (x)

(135.12)

follow from (135.9) and (135.10), where we now have to think of the lefthand side of (135.11) as in 3p[x]. Because g(x1) = (9(x))"

it follows from (135.12) that f(x), g(x) have a non-constant common divisor. Accordingly, the existence of a multiple factor of x" - 1 (E [x]) follows

from (135.11). But this contradicts the result proved in the paragraph containing (135.4); so we have established the truth of the assertion g (x) =f(x).

We have thus seen that together with e, Np is also a zero of f (x). On the other hand, because p ' n, ep is also a primitive nth root of unity in F. By repeated application it follows that all the

,a

(a > 0; (a, n) = 1)

are zeros of f(x). But since these are all the primitive nta' roots of unity, it follows that F"(x) I f(x). Since the right-hand side is irreducible, f(x) _ = F"(x) follows. Thus Theorem 316 is proved. But, for the case of finite characteristic: THEOREM 317. Over the prime field of prime characteristic p such that p f' n the nth cyclotomic polynomial Fn(x) splits into e 97(n)

(e = o (p (mod n)))

(135.13)

different irreducible factors of degree e and is irreducible if, and only if, n is one of the numbers

n = qk, 2q", 4

(k >_ 0; q an oddprime number)

(135.14)

and the prime number p is a primitive number mod n.

For the proof we have recourse to a splitting field of x" - 1 over J",In this finite field, every zero of F"(x) is a primitive nt root of unity. We have to prove that the degree of o is equal to e. Because of the last

CYCLOTOMIC POLYNOMIALS

513

proposition in Theorem 306 it is sufficient to show that e is the minimum of the natural numbers d with the property

This equation is identical in meaning to the divisibility n I pd - 1, so that

e = o (p (mod n)) is in fact the minimum of these d. This proves the first part of the theorem. In order to prove the last assertion of the theorem we have to examine when the number (135.13) is equal to 1. This is the case if, and only if, e =

= p(n), i.e., o(p (mod n)) = p(n) , i.e., p is a primitive root mod n. In conformity with this, because of Theorem 226, the last assertion of Theorem 317 is also true. EXAMPLE 1. From (135.3) we obtain the special cases

FQ(x)=

xq X

-1

-

x"°- 1

Fq.(x) = x0 -

_

1 =

-X*_1+...+ x(q- 1)q +

+ x4 + i ,

(x°' - 1) (x - 1)

(x°-1)(x'-1)' where q, r are different prime numbers. We have:

Fi(x) = x - 1,

F4(x) = x2 + 1,

F$(x) = x + 1,

Fa(x) = x4 + x8 + x2 + x + 1,

F3(x)=x2+x+l

FB(x)=x2-x+1.

EXAMPLE 2. Since over the prime field of prime characteristic p xmat

- 1 = (xm - 1)'

,

Theorems 315, 316, 317 make the decomposition of x" - I into irreducible factors over every prime field possible. EXAMPLE 3. According to the above theorems the n(" cyclotomic field of characteristic p is of degree p(n) if p = 0, and of degree o(p (mod n)) if p > 0, p ,}' n. EXERCISE. The nth cyclotomic polynomial F"(x) is irreducible over the mth cyc-

lotomic field of characteristic 0 if, and only if, (m, n) 12.

§ 136. Wedderburn's Theorem THEOREM 318 (WEDDERBURN's theorem). Every finite skew field is afield.

In order to prove this, let F denote a finite skew field and Z its centre which is then a field. We write

O(Z) = q,

514

THEORY OF FIELDS

where q is a prime power. Since F+ is a finite-dimensional Z-vector space,

O(F) = q where it = [F : Z]. We have to prove that n = 1. With this end in view we assume that n > 2. It is evident that the group Z* is the centre of the group F*. The class equation (42.32) for F* consequently results in

q" -1=q- 1 +

0q-

(N*)I

O(N

(136.1)

where a runs through a system of representatives of those classes which are formed from the elements of F* - Z* and N* always denotes the centralizer of a in F*. If we add the element 0 to the group N*, then the set N so obtained evi-

dently constitutes a skew field, namely the centralizer of a in F. Hence Z c N c K, and so, by Theorem 292, for the degree r = [N : Z]

rIn, r
O(N*) = qr - 1

.

We now consider the nth cyclotomic polynomial F"(x) (E 7 [x]). Since. according to Theorem 316 this is irreducible and according to Theorem 315

is a divisor of x" - 1, but not a divisor of xr - 19 F,, (x)

x"- 1 xr- 1

This divisibility is retained after the substitution x = q, consequently from (136.1)

F"(q) I q - 1

.

Hence a contradiction will arise. By Theorem 315, F"(q) _ fl (q`t - 1)k(nd-q .

(136.2)

(136.3)

d n

We denote by k (>_ 1) the number of the different prime factors of n and notice at once that 9,(n) ? 92(2k) = 2k- `.

(136.4)

WEDDERBURN'S THEOREM

515

In (136.3) there are exactly 2' exponents µ(nd-1) other than 0. One half of them are 1, the other half -1. So it follows from (136.3) that 1 0. Consequently (136.2) implies that (mod q) and also that I

(136.5)

.

Since now, because q > 2, the inequalities 1

e, (qd - 1)-t > q-d ? 2 hold and F (x) is of degree qp(n), it follows from (1 36.3), (136.5), that qd-1

1

2k _ I

q,(n)<1. q(n)<2k'

2

This contradiction to (136.4) proves Theorem 318. The above simple proof originates essentially from WITT (1931). For a generalization

of the theorem cf. HERSTEIN (1954). Another proof of the impossibility of (136.2) for n > 2 is given by PICIERT (1951). EXAMPLE. Over the field of complex numbers

fl (x - a) x

where a runs through the primitive ni° roots of unity. Since ; at j = 1 and for n > 2 no a is equal to 1,

Iq - aI > q- 1,

>q - 1

and so the impossibility of (134.2) follows. This simple conclusion likewise originates from WrrT (1931).

§ 137. Pure Transcendental Field Extensions

Of all the field extensions - apart from the case of a simple transcendental adjunction - so far we have considered only the theory of algebraic extensions, which we have completed to a great extent, where the most important result was STEndTTZ's first main theorem (Theorem 300). Now transcendental field extensions will be studied generally, but it should be noted in advance that their theory is partly based on that of algebraic field extensions.

Take an arbitrary field G/F. For transcendental field extensions the following definition is fundamental. The element 9 (E G)

(137.1)

is algebraically dependent (over F) on a set (or more generally on a system) S (C G)

(137.2)

516

THEORY OF FIELDS

if a is algebraic over the extension field F(S)

(137.3)

.

In the other case e is said to be algebraically independent of S. Above all, the elements algebraically dependent on the empty set are the elements of G algebraic over F. These, according to the corollary of Theorem 296, constitute an extension field of F which we call the algebraic hull of F in G and here denote briefly by F. In the special case F = F the field F is called algebraically closed in the field G . F = G if, and only if, G I F is algebraic. Evidently a depends algebraically on S if, and only if, (137.4)

e E F(S) .

The condition (137.4) means that a satisfies an equation

aoe"+... +a"=0

(a0A0),

(137.5)

where orc, ..., a" are expressions in certain elements of the union set F U S. Since only a finite number of elements of S enter into such an expression, we have obtained the following important criterion: The element a (E G) is algebraically dependent on S (S G) if, and only if, it is algebraically dependent on a finite subset of S. Further we can ensure by multiplication of the equation (137.5) by an element of F(S) other than 0 that the elements a0, . . ., a" lie in the ring {F, S}. According to this the following criterion holds: The element a (E G) is algebraically dependent on S (C G), if and only if, there is a polynomial f(x, x1, ..., xover F so that with suitable elements col, ..., co," of S the conditions f1X, wi, ..., (0m)

0 , .f(e, col, ...,

0

(137.6)

are satisfied.

We call S (c G) an algebraically independent set over F if no element o (E S) is algebraically dependent on the difference set S - Q. From the above considerations it follows that the algebraic independence

of a set is a property of finite character. This means that a set S (S G) is algebraically independent (over F) if, and only if, all the finite subsets of S are algebraically independent. We now prove the following THEOREM 319. In a field G I F a set S (9 G) is algebraically independent over F if, and only if, for every polynomial f(xl, ..., xk) (96 0) over F and for arbitrary different elements ool, ..., wk of S ft(011 ..., wk) 0 0

(137.7)

PURE TRANSCENDENTAL FIELD EXTENSIONS

517

Because of the above remark it will suffice to prove the theorem for First we assume that S is algebraically a finite set S =
(137.8)

0. If n = 1 it is obvious. For n ? 2 we assume the assertion for n - 1 instead of n. Since co. is algebraically independent it follows from (137.8) and the criterion (137.6) that of
0.

.f(w1, ...,

(137.9)

Iyf we write more explicitly }(xl,

..., x ) =f0(xl, ..

+f,(xi, ..., xn-1) ,

,

(137.10)

then (137.9) implies that

(i=0,...,r).

Awl,

Hence because of the induction hypothesis: f (xI, .

.

0

.,

(i = 0, .

.

.,

r).

Because of (137.10) this implies that f(xl, . . ., 0, as stated. We then assume that S is not algebraically independent. Let, e.g., w1 Hence according be algebraically dependent on S- wl = <w2, ..., to the criterion (137.6) there exists a polynomial f(xt, ..., x,,) such that

0, Since then f(xl, . . ., x,,) 0 0, the proof of Theorem 319 is completed. A trivial consequence of the definition is that the elements of an algebraic-

ally independent set are transcendental over the fundamental field. On account of this the elements of an algebraically independent set are called independent transcendental elements over the same fundamental field. The field G I F is said to be pure transcendental if G arises from F by adjunction of independent transcendental elements, i.e., if G = F(S), where S is an algebraically independent set over F. For convenience we sometimes include F itself in pure transcendental extensions of F.

TmroRn.M 320. The pure transcendental extension fields of a field F are equivalent to the rational function fields over F.

THEORY OF FIELDS

518

It follows from the definition that the function fields mentioned are pure transcendental extensions of F. We now consider, conversely, a pure transcendental extension field G = F(wl, co., ...) of F. Assign to each of the elements (t)l, w2, ... an indeterminate x1, x.,, .. . and consider the polynomial ring F [x1, x,, ... ]. By J (w1, a).2....)

f(xi, x2, ...) -

this is mapped isomorphically onto a subring of G whose quotient field is exactly G, and the elements of F remain fixed. Then this isomorphism may be extended to a relative isomorphism of F(xi, x.2, ...) with G. Consequently the theorem has been proved. THEOREM 321. A field is algebraically closed in all its pure transcendental extension fields. According to the previous theorem it will suffice to consider a rational function field G = F(xl, x2, . . .). Wehave to show that the algebraic elements of G over F necessarily belong to F. It obviously suffices to consider the case G = F(x),

...,

where only a finite number of indeterminates are adjoined. If n = 1, by Theorem 293, the assertion is true. If n >_ 2 we assume the truth of the assertion for n - 1. Every algebraic element a of G over F is, a fortiori, algebraic over

is pure transcendental over G', Since, on the other hand, G = the truth of a E G' follows. Because of the induction hypothesis it follows that a E F. Consequently Theorem 321 is proved.

§ 138. Steinitz's Second Main Theorem

As an important preparation we show that the fundamental properties I, II, 111 formulated at the beginning of § 100 referring to linear dependence also hold for algebraic dependence in a field G I F.

For the fundamental property I it is a consequence of the definition of algebraic dependence and of the fact that we always have

w,EF(wl,

(w,,..

i= 1.....n).

STEINITZ S SECOND MAIN THEOREM

519

In order to prove the corresponding facts with respect to the fundamental

property II, we assume that for certain o, to,, ..., con (E G) the element and transcendental over F(w,, ..., 2 is algebraic over F(w,...., Since 0 is algeWe have to show that w,, is algebraic over F(Q..... braic over F(wl, ..., coj, there is a polynomial f(x.......,,) over F such that .f(x, Col...... wn) 0 0 ,

.f(P. w,...., w,,) = 0 .

(138.1)

Here we may assume that x. x,...... ,; are indeterminates even over. G. We arrange our polynomial in powers of x,,: .f(x,.

....x"))

91(x,.....

(138.2)

Now, because of (138.1,).

AT. w,.....

I, x,)

0.

Hence, by (138.2), it follows that at least one g1(x, (o,, ..., to. -I) is not equal the inequality to 0. Here, since p is transcendental over F(wj...., w,...., 0 must then hold. Thus, by (138.2), f(n. WI.. ... (')It- 1, x,,) 0 0 .

Because of (138.1.,) this means that w,, is. in fact, algebraic over the fi-.,I i F(t,,

..,

As regards the fundamental property III we have to show that if the elements

a. o,.....

co,..... (1),, (E G)

have the property that a is algebraic over

F, = F(u,..... o,,,) and every of is algebraic over F., = F(W1.....

then a is algebraic over F.,. Because of the assumption we have made, a is a , fortiori algebraic over

F,(w,......

= F(y,.....

w,.....

On the other hand, again by the assumption, and on account of Theorem 296. this field is algebraic over F.,. From both, because of Theorem 297, it

THEORY OF FIELDS

520

follows that oa is, in fact, algebraic over F2. Consequently we have proved the above assertions.

Because of this, all the statements of § 100 remain true when "algebraic dependence" is understood instead of "linear dependence". Some of the concepts, which arise from the definitions after this alteration, were introduced in the previous section. Here we make the definition: two subsets S, T of a field G I F are called algebraically equivalent over F if for all elements v, v of Sand T, respectively, a depends (over F) algebraic-

ally or,Tand ron S. We also have the following definitions: if a subfield T of a field G I F has the property that T I F is pure transcendental and G I T algebraic, and if S (9 T) is a set of independent transcendental elements (over F) such

that T = F (S), then we call this (in general not uniquely defined) set S and its cardinal number the transcendence basis and the degree of transcendence of the field G I F. The justification for these definitions lies in the following THEOREM 322 (STEINITZ'S second main theorem). Every field G I F has at

least one transcendental basis, furthermore its degree of transcendence is invariant. The transcendence bases of G I F are identical with the maximal algebraically independent subsets of G over F. First of all we prove the last assertion of the theorem. We consider a maxi-

mal algebraically independent subset S of G. In order to show that S is a transcendence basis of G I F we have only to prove that every element e of G - S is algebraically dependent on S. Because of the maximal property

of S the union set S U e is not algebraically independent. If a does not depend algebraically on S, then there is an element co of S which is algebraically dependent on the difference set

(SU0)-w. Then S contains a finite subset S' such that co I S' and co is also algebraically dependent on S' U e. Since, on the other hand, the set S' U co is algebraic-

ally independent, e is, because of Theorem 238, algebraically dependent on this set and so on S. Accordingly S is in fact a transcendence basis of G I

F.

Now, conversely, assume the latter. Then S is algebraically independent, furthermore every element of G is algebraic over F(S), i.e., algebraically dependent on S. Accordingly, S is a maximal algebraically independent set. Consequently the last assertion of the theorem is proved. Since, furthermore, algebraic independence is a property of finite character, it follows from the TEICHMULLER-TUKEY lemma (Theorem 17) that there

exists a maximal algebraically independent subset of G, and therefore a

STEINITZ'S SECOND MAIN THEOREM

521

transcendence basis of G I F. This means that the first assertion of the theorem is true. The second assertion still remains to be proved, i.e. the invariance of the degree of transcendence. For that purpose we consider two maximal algebraically independent subsets S, T of G. It is to be proved that these are equi-

potent. By hypothesis, the elements of the sets S, T are algebraically dependent on T and S, respectively, therefore these two sets are algebraically equivalent (over F). So if S, T are finite, then they are, according to STEINITZ'S exchange theorem (Theorem 240) equipotent. Therefore the assertion is true for this case. This fact is the basis of the following proof for the general case.

We briefly denote by (S', T')

(138.3)

a one-to-one mapping of a subset S' of S onto a subset T' of T algebraically equivalent to it. (Note that there are in general several mappings for a given pair S', T'.) We denote by c the set of all the mappings (138.3). We admit the case S' = T'= 0, i.e., where (138.3) is the "empty" mapping and thus obtain the result that S # 0. We make C a semiordered set by writing T,) (S", T") (S',

if, and only if, the mapping on the right-hand side is an extension of that on the left-hand side. We show that every ordered subset

(S', T) < (S", T") < ...

(138.4)

of S has an upper bound in 25. Because of (138.4), we first of all have

S'cS"c..., T'cT' c... We consider the union sets

So=S'US"U... (CS), T0=T'UT"U... (ST). Since every finite subset of So is contained in once of the sets S', S",..., and is therefore algebraically independent, this also holds for So and for similar

reasons also for To. Moreover it is evident that in S there is a uniquely determined mapping of the form (So, To) which is a common extension of all the mappings (138.4). This is the required upper bound. Accordingly from the KURATOWSKI-ZORN lemma (Theorem 15) the existence of a maximal element (S*, T*)

(138.5)

THEORY OF FIELDS

522

follows. We show that S* = S and T* = T by which we shall prove the theorem. if the assertion is false, then we may assume, e.g., that

of t

S*CS. Let al denote an element of S - S*. This is algebraically dependent on a finite subset of T, and so, in particular, on a set

T*UT1U... Ur,, where T1, ..., r; are suitable elements of T - T* and i is assumed to be minimal. The case i = 0 means that at is algebraically dependent on the set T* and so on the set S* algebraically equivalent to it. But this is a contradiction since the set S is algebraically independent. According to this only the case i >- 1 is possible. We take a minimal number of elements a2, ....

a; (E S) such that every r1...... r, is algebraically dependent on the set

S*Ua,U...Ua; (j>1). If we have j > 1, then we take a minimal number of elements r;+,.. . such that every al, ..., a; is algebraically dependent on the set

T*Url'U...Urk

., TA

(k>i).

Either we may continue this process infinitely obtaining a countably infinite number of at, a.2 . . . and an equal number of r1. re...., or the process is broken off after a finite number of steps. In the first case the union sets

S** =S* U at U a_ ... (9 S),

T**=T*UT1UT_...(CT) are algebraically equivalent. We complete the mapping (138.5) by the correspondences (r -= 1, 2, .. .) .

ar --->- Tr

Thus an element (S**, T**) of is obtained which is an extension of the mapping (138.5). By this we find ourselves contradicting the maximal property of (138.5). In the second case we have to deal with two algebraically equivalent sets

of the form

S**=S*Ua,U... Ua,,,( S), T**

T*UT] U

...

U r,, (S T) (rn,n>_ I).

STETNITZ'S SECOND MAIN THEOREM

523

This means that the fields F(.S**), F(T**) have the same algebraic hull

H = F(S**) = F(T**)

(138.6)

in G. Since now S** is an algebraically independent set (over F), the elements

crl, .., a,,, constitute a maximal algebraically independent subset of S over the algebraic hull F( S*). Likewise the elements T1,...,T,,

constitute a maximal algebraically independent subset of T over F(T*). But since, as in (138.6). F(S*) = F(T*)

.

because of (138.6), the remark on the finite case before (138.3) now holds, according to which we must have m = n. Completing the mapping (138.5) by the correspondences Cr -" T,

(r = 1.

.

.

.. 171)

we arrive at a similar contradiction. Consequently the theorem is proved. For a very short proof, supposing more set-theoretical knowledge, cf. K ERTESZ (1959).

§ 139. Simple Transcendental Field Extension

By Theorem 293, the simple transcendental field extensions of a field F are essentially identical to the rational function field F(x). Here this will be investigated more explicitly. (In other words, we are dealing with the pure transcendental extensions of F of degree of transcendence 1.) THEOREM 323. For every element y of degree n (>_ 1) of a rational function

field F(x), the field F(x) I F(y) is algebraic and of degree n. In particular,

ax+b cx±ci

(a. b, c. d

F; ad - be -A 0)

(139.1)

are all the primitive elements of F(x).

For the proof we put y =

J(j)

g(x)

(139.2)

THEORY OF FIELDS

324

with two relatively prime polynomials

f(x)=aox'+...+a, g(x) = bor + ... + b.,

(aorA 0; a.,....arEF) (bo : 0; bo, ..., b' E F)

over F such that n = max (r, s) . We take a further indeterminate z and form the polynomial

F(z) = y9(z) - f(z) (E F(y) [z])

.

(139.3)

Since, according to (139.2), this has the zero z = x, it suffices to show for the proof of the first assertion of the theorem that F(z) is irreducible and of degree n. It is evident that the degree of F(z) is at most n. Since furthermore, in F(z), the coefficient of z" corresponding to the cases r < s, r > s, r = s, is respectively

boy, - ao, boy - ao

and y does not belong to F, this coefficient never vanishes. Consequently the degree of F(z) is exactly n. Since, according to Theorem 293, y is transcendental over F, in the proof of the irreducibility of F(z) it can be regarded as an indeterminate. Then, according to (139.3), F(z) is a linear polynomial in y over F [z], moreover it is primitive, since f(z), y(z) are relatively prime. Accordingly, F(z) is irreducible in F[y, z], and also in F(y) [z] because of GAUSS's theorem (Theorem 207). Consequently the first assertion of Theorem 323 is proved. Further F(y) = F(x) if, and only if, n = [F(x) : F(y)] is equal to I, i.e., y is of degree 1. These y are exactly given by (139.1), which proves the theorem. THEOREM 324. All the F-automorphisms of the rational function field F(x) are given by the substitutions

+b x-- ax cx+d

(ad-bc=0; a,b,c,dE F)

(139.4)

and the group is isomorphic with the linear fractional group of the second degree over F. If y is an arbitrary non-constant element of F(x), then by the substitution

x -> y a mapping

f(x)

f(y)

9(x)

9(v)

(139.5).

SIMPLE TRANSCENDENTAL FIELD EXTENSIONS

525

of F (x) into itself is given, where f(x), g(x) are polynomials over F, such

that g(x) does not vanish, whence, because of the transcendence of y, it follows that g(y) is also non-vanishing. For the same reason (139.5) is one-to-one and so a meromorphism of F(x). It is also evident, conversely, that all the meromorphisms of F(x) are furnished by (139.5). Since F(x) is mapped onto F(y), this meromorphism is an automorphism if, and only if, F(y) = F(x), i.e., if y is a primitive element of F(x). Because of Theorem 323, the first assertion of Theorem 324 is proved. For the proof of the second assertion we consider two automorphisms of F(x) which we give according to (139.4) in the form

x

ax+b

cx+d'

x

a'x+b' c'x+d'

(139.6)

If, of the two linear functions given here the second is substituted in the first, then we obtain

(aa' + bc')x + (ab' + bd') (ca' + dc')x + (cb' + dd')

The comparison with

(aa'+bc' ab'+bd' c d I` c' d',

t ca' + dc' cb' + dd'

shows that a homomorphic mapping of the full linear group of degree two over F onto the automorphism group of F(x) arises if to each regular matrix

ab c dI

(over F) the automorphism furnished by (139.4)

This automorphism is the identical one if, and only if, a i.e., if is a scalar matrix dl

k0 0 k

(k E F,

0),

is assigned.

ax+b = x, cx+a

(139.7)

so that these matrices constitute the kernel of the homomorphism. Consequently, the factor group of the linear group mentioned with respect to the group of matrices (139.7), i.e., the linear-fractional group of the second degree over F, is isomorphic with the automorphism group of F (x). Consequently Theorem 324 is proved. THEOREM 325 (LUROTH'S theorem). Every subfield of the rational function

field F(x) I F has a primitive element. We consider a subfield G of F(x) I F where we may restrict ourselves to the case G F. Since, according to Theorem 323, F(x) I G is algebraic,

THEORY OF FIELDS

526

there is an irreducible polynomial with the zero = = x in the polynomial ring G [z]. After multiplication by an element of F(x) we can turn this polynomial into

F(x, z) = ao(x)z" + ... + so that ao(x), ...,

(ao(x)

(139.8)

0)

are relatively prime polynomials in F[x]. Moreover

[F(x) : G] = n.

(139.9)

Let in denote the degree of F(x, .s) with respect to N.

Because of (139.8) and the fact that ao(x)-I F(x. z) E G[z] all the a,(x)

(i

0.

lie in G, but not all in F, since x is transcendental over F. Of these quotients we take one which does not lie in F and denote it by 0. Then we can write 11 =

g(x)

G),

(139.10)

where numerator and denominator are relatively prime polynomials over F of degree at most m and not both constant. By Theorem 323 we then have [F(x) : F(O)] < m.

(139.11)

The polynomial g(x) f(z) - f(x) g(z) is different from 0 and has the zero _ = x. Since this also holds for F(x, z) and this is a primitive and irreducible polynomial in = over F[x], it follows from GAUSS'S theorem (Theorem 207) that (139.12) g(x)f(z) - f(x) g(=) = G(x, z) F(x, z)

for some polynomial G(x, z) from F[.x, _]. The degree of the left and righthand sides with respect to x is at most m and at least m, respectively, thus

both are of degree m with respect to x. Since, according to this, G(x. _) does not contain the indeterminate x we may write G(x, z) = G(z)

(E F[z] ).

We can even show that G(z) E F. We assume that this is false. According to (139.12) G(z) I g(x)f(z) - f(x) g(z)

The residues arising from the Euclidean division of f(z) and g(z) by G(z) are denoted by.f,(z) and g,(z), respectively. At least one of these

SIMPLE TRANSCENDENTAL FIELD EXTENSIONS

527

polynomials is different from 0, sincef(z), g(z) are relatively prime. Further G(z) I g(x).f1(z) - f(x) g1(z)

But since the left-hand side is of greater degree than the right-hand side, this must vanish. Hence it follows that neither of f1(z), gl(z) is equal to 0 and f1(z)

Ax)

g1(z)

g(x)

lies in F. This contradiction proves that G(z) E F. Since the left-hand side of (139.12) is of the same degree with respect to x

and z, the same follows for F(x, z). But since the latter has degree m and n, respectively, with respect to x and z, we have m = n. By (139.9) and (139.11) [F(x) : G] > [F(x) : F(#)].

Since, on the other hand, we have G Q F ('), it follows that G = F (0). Hence Theorem 325 is proved. § 140. Isomorphisms of an Algebraic Field Since we have practically an unlimited possibility for obtaining isomorphic

structures from a given structure, we never ask for all the isomorphisms of a structure. On the other hand, it is in general of great importance to study isomorphisms which map a structure into a given overstructure. (This is essentially identical with the problem of substructures we have formulated earlier, according to which we look for the determination of all the

substructures of a structure non-isomorphic with one another.) Here we wish to deal with such questions with regard to fields and particularly for relative isomorphisms. At first we admit arbitrary relative fields and later restrict ourselves to algebraic fields and even to those of finite degree. Those isomorphisms of a field G I F which map G into a fixed extension field H of G are called the isomorphisms of G I F into H. First of all we prove the following, almost trivial

THEOREM 326. Over a field F, let a field G and two equivalent overfields H, H' of G over F together with an isomorphism H of H I F with H' I F be given. If S1, S2, ... are then all the different isomorphisms of G I F into H, then HS1, HS2.... are all the different isomorphisms of G I F into H'.

It is evident that HS1, HS2, ... are isomorphisms of G I F into H' and different from one another. So it remains to be shown that all the iso18 R.-A.

528

THEORY OF FIELDS

morphisms of G I F into H' occur among them. For eacn such isomorphism S, H-1 S is an isomorphism of G I F into H, i.e., equal to an Si. Since then

S = HS,, the theorem is proved. THEOREM 327. To every extension field H of an algebraic field G I F there is a normal field N of G I F such that all the isomorphisms of G I F into H occur among the isomorphisms of G I F into N. We denote by y' an image of an element y of G which has been generated

by any isomorphism of G I F into H. Together with y also y' is algebraic over F. We adjoin to F all the images y' of all the y and denote by G' the subfield of H obtained. Since, among the isomorphisms considered, the identical isomorphism of G occurs, we have

G9G'. We denote by N an arbitrary normal field of G' I F and show that N is a normal field of G I F. Since N I F is normal and contains G, it follows from the definition of the normal fields of a field that, in any case, N contains a normal field No of G I F. It suffices to show that necessarily No = N.

We retain the above notations y, y' and consider the minimal polynomial f(x) of y over F. Since No is normal over F, an equation such as

J(x)=(x-yI)...(x-yn)

(y1=y; y1,...,ynE N0)

must hold. Because f(y) = 0, we have f(y') = 0. Because N ? No it follows that y' is equal to one of the yl, ..., y,,, thus belonging to No. Since this holds for all the y', we have G' S No. Therefore we obtain the result that the normal field N of G' I F contains the normal field No

(? G') over F. By the minimal property of N it follows that No = N. Accordingly, N is in fact a normal field of G I F. Since, finally, all the y' belong to G', and so to N, every isomorphism of G I F into H is also an isomorphism into N. Consequently Theorem 327 is proved.

NOTE. On account of the last theorem, of all the isomorphisms of an algebraic field G I F only those into a normal field need be considered further. By Theorems 304, 326 it is mostly unimportant which normal field of G I F

is taken. 1=lowever, we shall often consider isomorphisms of G I F into arbitrary extension fields of G. We shall now consider an algebraic field G I F and its isomorphisms into

an arbitrary extension field H of G. These map G onto subfields of H equivalent to it, which we call the conjugate fields of G over F in H. We similarly denote the images, furnished by the same isomorphisms, of an element y of G as the conjugate elements of y over F in H. It is evident that

ISOMORPHISMS

529

these conjugacies are equivalence concepts. Therefore if G1 (= G), G2, ..., or yI (= y), Y2.... are the conjugates of G and y, respectively, then we can also say that G1, G2, ... are conjugate fields and that yl, y2, ... are conjugate elements, respectively, in H (over F). If we do not put "in H" we always mean the conjugates in an (arbitrary) normal field of G I F. The conjugate fields and elements over F are also called relative conjugate fields and elements, respectively. If F is also a prime field, then we can also talk of absolute conjugate fields and elements. To avoid any misunderstanding it should be noted that if S1, S2.... are all the different isomorphisms of G I F into H and G1, G2, ..., and yl, y2, .. . the corresponding conjugates of G and an element y of G, respectively, then neither the G1, G2, ... nor the y1, y2, ... need be different, for, on the one hand, among the isomorphisms considered, automorphisms (different from 1) of G I F may very likely occur. On the other hand yl, y2, ... are all equal if y E F. The different fields and elements among the G1, G2, ... and V1, Y2.... may be called "all the different conjugates" of G and y in H, respectively. Without specifying "in H" we mean all the different conjugates in a normal field of G I F. As regards the conjugates of an element y of an algebraic field G I F we observe that if f(x) is the minimal polynomial of y over F, then f(y') = 0 for all the conjugates of y in an overfield H of G. Therefore y' is algebraic over F as already stated. But hence it also follows that the number of conjugates of an element is always finite and does not exceed the degree of this element. (For further details see below.) Considering briefly an arbitrary (not necessarily algebraic) extension field G = F(01, 02, ...)

of the field F which arises from this by the adjunction of the elements 491, 02, ..., we see that since the elements of G are expressions in t9I, e 2, .. .

(with coefficients from F), each isomorphism y ->. y' of G F is already

....

On account uniquely determined by the images of the generators 0, t 2, of this, we may also represent these isomorphisms in the form 01 -

;

,

02 __> 02' 9 ...

(140.1)

where i9 denotes the image of 0I. We have applied this type of notation in Theorem 324 already. Of course, homomorphic mappings of an arbitrary structure may also be similarly denoted by giving the images of generators. If G I F is algebraic and (140.1) denotes an isomorphism of it into an overfield H of G, then every t9; must be a conjugate of O; in H. Now, by the above, the number of conjugates of an algebraic field element is finite, and hence, by virtue of Theorem 296 and from the observation

530

THEORY OF FIELDS

above, the same follows for the number of conjugates of a field of finite degree, thus justifying the following two important definitions. By the reduced degree of an element a of an algebraic field G I F we under-

stand the number of distinct conjugates of a. Further, we understand by the reduced degree of afield G I F of finite degree the number of isomorphisms

of G I F (both in an arbitrary normal field of this field). The term "reduced degree" of an element is explained by the fact that by the above

it is at most equal to the degree of this element. A similar statement also holds for fields.

THEOREM 328. If F(i) I F is an algebraic field, N its normal field andf(x) the minimal polynomial of the primitive element 13, then the distinct isomorphislns of F($) I F into N are given by

0-*01

(i= 1,...,n'),

(140.2)

where 01, ..., 13,,, are the different zeros of f(x), i.e., the different conjugates of $ in N. Consequently the reduced degree n' of 0 is equal to the reduced degree of F(13) I F.

If 0 -> 0' is an isomorphism of F(8) I F into N, then, because f(t9) = 0, we also have f(Y) = 0. But since f(x) splits over N into linear factors, it follows that 0' is equal to one of the 01 mentioned in the theorem. Therefore

it only remains to be shown that an isomorphism of the required type is given by (140.2) for every fixed i. Again this is trivial, and, because of Theorem 295 this isomorphism may be given by g(1') -- g(0)

(g(x) E F [x))

Consequently the theorem is proved. A part of this theorem is generalized in the following THEOREM 329. If G I F is an algebraic field, a an element and N a normal field of it, then for every zero a' (E N) of the minimal polynomial f(x) of a there is an isomorphism of G I F into N which carries a into a'. COROLLARY. The reduced degree of a is equal to the number of distinct zeros of f(x) in one of its splitting fields and so depends only on a and F. Since f(x) splits into linear factors over N, N contains a splitting field of f(x). Thus, by Theorem 328, we have the isomorphism F(a) I F

F((x') I F(a -> a')

(140.3)

into N. Now, in conformity with Theorem 303, N is a splitting field over F,

furthermore it contains both the subfields F(a), F(a ). If we then apply Theorem 302 (with G = G' = N instead of G, G'), it then follows that

ISOMORPHISMS

531

N I F has an automorphism which is an extension of (140.3). This automorphism induces an isomorphism of the subfield G I F and maps a onto a'. Accordingly Theorem 329 is true. In order to prove the corollary we have to consider that the reduced degree of a is equal to the number of its conjugates in N. On account of Theorem 329 this number is the same as that of the distinct zeros of f(x) in N. But since N contains a splitting field of f(x), f(x) has in this case the same zeros

as in N. This completes the proof. THEOREM 330. If F, F1, F2 are three fields such that F c F1 c F2 and F2 I F is offinite degree, then the reduced degree of F2 I F is equal to the product of the reduced degrees of F1 I F and F2 I Fl. We denote the relative degrees of F2 I F, F, I F and F2 I F1 by f2, f1 and f12,

respectively. We have to prove that f2 =.fi.fi2 Let N denote a normal field of F2 I F. Then f2 is the number of the isomorphisms of F2 I F into N.

Each of them is of the form

F21Fk F2' IF.

(140.4)

This induces a certain isomorphism,

F'IF

(140.5)

(into N). Conversely, those isomorphisms (140.4) to which a fixed isomorphism (140.5) belongs are so obtained from them that all the isomorphisms of F2'

I

F1'

(into N) are obtained. The number of these isomorphisms [because of (140.4), (140.5)] is the same as that of the isomorphisms of F2 1 F1

(140.6)

into N. Since N contains a normal field of (140.6), the number of these isomorphisms is equal to f12. Since, furthermore, N also contains a normal field of the left-hand side of (140.5), the number of isomorphisms (140.5) is equal to fl. From all this it follows that f2 = J i f12, so that Theorem 330 is proved. THEOREM 331. An algebraic field G I F is normal if, and only if, every iso-

morphism of it into an arbitrary extension field is an automorphism, i.e., G I F has no conjugates except itself. For, if G I F is normal, thus is its own normal field, then it follows from Theorem 327 that all the isomorphisms of G I F into an arbitrary extension field are isomorphisms into G, and so are automorphisms. If, on the other hand,

G I F is not normal, then there is an element a of G such that its minimal

THEORY OF FIELDS

532

polynomial f(x) does not split into linear factors over G. So, if we take a. normal field N of G I F, then N - G contains an element a' such that f(a') = 0. Accordingly, by Theorem 329, G I F has an isomorphism into N which maps a into a' and G onto a field different from it. Consequently it is not an automorphism of G. Thus Theorem 331 is proved.

.,

EXAMPLE. We consider the field 2 = . o(t9) with 04 - 2 = 0. Since x' - 2 is irreducible over 2 is of degree four. The element a = 0E of 2 is of degree two, be-

cause a2 - 2 = 0. The polynomial x2 - 2 also has the zero a' = - a in _T. Now x4 - 2 has only the two zeros 0, -0 in 2, consequently 2 has only the two automorphisms 0 - t9 and 0 - -tA. Both have a as fixed element. Hence we see that Theorem 329 would be false with G in place of N.

§ 141. Separable and Inseparable Field Extensions It is also an intrinsically interesting problem whether a polynomial f(x), irreducible over a given field, can have multiple zeros over a suitable extension field; if this is the case, then its splitting field certainly suffices. This

question is closely linked with the last paragraph. A polynomial f(x) irreducible over a field F is called a separable or inse-

parable polynomial, according as f(x) has only simple or also multiple zeros in an (arbitrary) splitting field. An algebraic element a of a field G I F

is called a separable or inseparable element (over F) according as the minimal polynomial of a (over F) is separable or inseparable. Finally, an algebraic field G I F is called a separable or inseparable field, according as all its elements are separable or inseparable elements also occur among them. The terminology "of first kind" and "of second kind", originally introduced by STEINITZ, instead of "separable" and "inseparable", is rarely used in current literature.

THEOREM 332. Let F denote a field of characteristic p (>_ 0). If p = 0 there are no inseparable polynomials over F. If p > 0 all the inseparable principal polynomials over F are those irreducible principal polynomials which have the form f(x) = g(x")

(g(x) E F[x]) ;

(141.1)

for the irreducibility of a principal polynomialf(x) of the form (141.1) it is necessary and sufficient that g(x) is irreducible and not all its coefficients are pth powers in F. Let us consider an arbitrary principal polynomial f(x) over F. This is, according to Theorem 267, inseparable, if and only if, the condition (1(x), f'(x)) 4z 1

(141.2)

SEPARABLE AND INSEPARABLE FIELD EXTENSIONS

533

is satisfied and f(x) is irreducible. But now condition (141.2) for an irredu-

cible f(x) is equivalent to f(x) I f'(x), thus to f'(x) = 0. This last condition for an irreducible, thus non-constant, f(x) is equivalent, according to Theorem 262, to the fact that p > 0 and (141.1) holds. Therefore only the case in which the last two conditions are fulfilled need be further considered. Only the criterion of irreducibility formulated at the end of the theorem remains to be proved. If g(x) is reducible then, because of (141.1), so isf(x). If

g(x)=x"+aix' +...+a."

(a1.... ,a"EF),

then by (141.1)

f(x)=(x"+alx"-1+...+a,,)P, thus f(x) is again reducible. The case where, in (141.1), g(x) is an irreducible principal polynomial and not all of its coefficients are pth powers in F still remains to be proved. We assume that f(x) is reducible in order to obtain a contradiction by which the theorem will be proved. In the first place we consider the case where f(x) has only one (but then necessary multiple) irreducible divisor in F[x]. So now an equation such as

f(x) = (x" +

atx"-1

+ ... + a")'

(a,, ..., a" E F)

(141.3)

holds for some natural number q different from 1. We may assume that q is a prime. Then, of course, the expression in brackets on the righthand side of (141.3) need no longer be irreducible. The case q = p is absurd, since then, because of (141.1) and (141.3) we obtain g(x)=x"+alx"-I+... +a,,

which contradicts the supposition. Consequently q 0 p. Again because of (141.1), f'(x) = 0, so that the differential'quotient of the expression in brackets on the right-hand side of (141.3) must vanish. Hence, and from Theo-

rem 262, it follows that f(x) _ (h(xP))q

f or a polynomial h(x) over F. This, and (141.1), give the result g(x) _ _ (h(x))q, which is again absurd, since g(x) was assumed to be irreducible.

In the second place we consider the case where f(x) has at least two different irreducible factors over F. Then an equation such as f(x) = f1(x) A W

(141.4)

THEORY OF FIELDS

534

holds with non-constant relatively prime factors in F[x]. Since, according to (141.1), f'(x) = 0, so

fi(x).fi(x) + fi(x).fz(x) = 0 . Since (f1(x), f2(x)) = 1, we have

l(x) I .f; (x)

(i = 1, 2).

This gives f(x) = 0. Thus, by Theorem 262, f(x) = ga(x")

(i = 1, 2),

where gl(x), g2(x) are non-constant polynomials over F. Hence, and from (141.1), (141.4), we have g(x) = g1(x) g2(x), although g(x) was assumed to be irreducible. This contradiction proves Theorem 332. THEOREM 333. An algebraic element at over afield F of characteristic p > 0

is separable if, and only if, F(a") = F(a). Let f(x) denote the minimal polynomial of a over F. According to the definition a is separable if, and only if, f(x) is separable. First, letf(x) be separable. By f1(x) we denote the polynomial determined by

f,(x") = (f(x))" and show that fi(x)

is

(141.5)

irreducible. Let h(x) be an irreducible principal

polynomial and divisor of fl(x) over F. Then h(xP) I f1(xP), thus h(xP) I (f(x))P.

Because of the irreducibility off(x) an equation such as h(xP) = (.f (x))k

(1 < k S p)

follows. By differentiation we get kf'(x) = 0. On the other hand, because of Theorem 332, condition (141.1) is not satisfied, therefore f'(x) # 0. Thus

k = p, h(x") = (f(x))P =fi(xP),

h(x) = f1(x).

Accordingly fl(x) is in fact irreducible. Furthermore because of (141.5)

and the fact that f(a) = 0, f1(x) and f (x) are of the same degree and fl(ap) = 0. It follows that F(a), F(aP) are of the same degree, i.e., that we have F(aP) = F(a).

Next, let f(x) be inseparable. Then (141.1) holds whence g(ap) = 0. Accordingly aP is of smaller degree than a, therefore F((XP) C F(M). Consequently Theorem 333 is proved. By the first part of Theorem 332 inseparability in connection with fields of characteristic 0 is impossible. This is why we have only considered the case p > 0 in Theorem 333. For the sake of generality, however, we some-

SEPARABLE AND INSEPARABLE FIELD EXTENSIONS

535

times admit the case p = 0 in connection with inseparability problems, although this will then be trivial. We shall see that for every algebraic element a of a field G I F of characteristic p (> 0) a separable term occurs in the sequence a, aP, cP°, .... (Of course, all the subsequent terms are then separable.) Let f(x) be the minimal polynomial of a over F and e (>_ 0) the greatest integer for which there is a polynomial g(x) with

Ax)=g(xn.

(141.6)

If p = 0, then e = 0, pe = 1 and g(x) = f(x) trivially. This number e is called the exponent of the element a (over F) and is, as we shall now show, the least integer e (> 0) for which the element cP` is separable. Since, because of (141.6), g(x) is irreducible and, because of Theorem 332, is separable, and also because, according to (141.6), g(cc') = 0, cc' is separable. On the other hand, if e Z 1 the element cc" 1 is inseparable, since it is a

zero of the irreducible and consequently inseparable polynomial g(xP). Hence, the assertion is proved. If G I F is algebraic and the exponents of the elements cc (E G) are bounded above, as they are if G I F is of finite degree, then we call their maximum the

exponent of the field G I F. (This will be considered in the next section.) THEOREM 334. For an irreducible principal polynomial f(x) of degree n over afield F of characteristic p (>_ 0) the factor decomposition

f(x) = ((x - aI) ... (x - an,))" ,

(141.7)

holds over its splitting fields G where aI, ..., an, (E G) are all its distinct zeros and e and n' denote the common exponents and the reduced degree of the latter over F. Hence (141.8) n = pen' , and n' I n; furthermore it follows that f(x) is separable if, and only if, e = 0,

(i.e., n' = n). Let us take f(x) in the form (141.6) with maximal e (? 0), then g(x) is irreducible and evidently not of the form h(xP). Accordingly, because of Theorem 332, g(x) is separable and

g(x)=(x-(3i) ...(x-Nn) over one of its splitting fields with distinct PI, ..., fin, (where n' has, for

the moment, nothing to do with that of (141.7)). Then by (141.6) f(x) = (xPe _ NI) ... (x"e 18/a R.-A.

fl.') .

THEORY OF FIELDS

536

The factors of the right-hand side are each divisible over a proper splitting field of f(x) by a linear polynomial x - al, ..., x - (x.,. Hence it follows that

fi,=a?'

(i= 1,...,n').

(141.9)

and

f(x)=(xre-ar)...(xr`-a.n, for which we can also write (141.7). Since the fi, are different, so from (141.9) the same fellows for the a,. The remaining assertions of Theorem 334 need no further proof. THEOREM 335. Consider an element a and two subfields F c F1 of a given

field of characteristic p (>_ 0) and suppose that a is algebraic over F (thus also over F1). Let e, n, n' and e1, n1, nl, respectively, denote the exponent, degree and reduced degree of a over F and F1, respectively, then e1 < e,

n1 < n, n' < n'. Then if a is separable over F, it is (because el S e) also separable over Fl.

Denote by f(x) and f1(x) the minimal polynomial of a over F and F,, respectively. Because F c F1, we have fl(x) I f(x). Hence, and from Theorem 334, follows Theorem 335. THEOREM 336. A field G I F of finite degree is separable if, and only if, its degree and reduced degree are equal. On account of Theorem 296, take elements a,, . . ., a,, such that

G = F(al, ..., a) , write

F,=F(a1,...,a)

(i=0,...,s; F0= F,F3=G)

and denote by n, n' the degree and the reduced degree, respectively, of G I F and similarly by n,, ni the degree and the reduced degree of F, I F,_1(i = 1, ..., s). By Theorems 292, 330

n=n,...n,, n'=nl...ns.

(141.10)

Furthermore, by Theorems 295, 328. n,, n{ are equal to the degree and reduced degree, respectively, of a, over F7_1. Because of Theorem 334 n; I n,

(i = 1, ..., s).

(141.11)

First, assume that n' = n. From (141.10), (141.11) it follows that nl = n1. This means, according to Theorem 334, that al is separable over (F0 =) F. But since at, can be an arbitrary element of G, we have obtained the proposition that G I F is separable. Next, assume that G I F is separable. Then al, ..., a, are separable over F, whence, by Theorem 335, it follows that a, is separable over F1_1, i.e.,

SEPARABLE AND INSEPARABLE FIELD EXTENSIONS

537

according to Theorem 334, n; = n; (i = 1, . . ., s). Hence, and from (141.10) it follows that n' = n. Consequently Theorem 336 is proved. A theorem similar to Theorem 297 is the following THEOREM 337. If G I F is a separable field, then the separable elements of a

field H I G are also separable over F. We consider an element a of H separable over G. It is to be proved that a is separable over F, too. We denote by f(x) the minimal polynomial of a over G and by al, . . ., as_I the coefficients of f(x). Since f(x) is separable over G, it is separable over the subfield F(at,

.

. ., ac,-j)

of G, and then the same holds for the element a. Furthermore, since al ..., a, _ 1 are separable over F, so ai is, according to Theorem 335, separable over F(al,

.

(i = 1, . . ., s - 1) .

. ., ai -1)

For the sake of uniformity we write

ex =a, and denote by n, n' the degree and the reduced degree, respectively, of the field

Ho= F(aj,...,oc) over F. Again, the equations (141.10) then hold, if nI, . . ., n, and

n'1,

.

. ., n;

are defined as before. On the other hand, according to the above, ni = ni (i = 1, ..., s). Hence, by (141.10), n' = n. This means, by Theorem 336. that Ho I F is separable. In particular, the element a (= a,) of HU is then separable over F. Thus Theorem 337 is proved. THEOREM 338. A field F(a1, a2, ...) I F is separable if a1, a2, ... are separable over F. 0) are separable elements of afield G I F, then so COROLLARY. If a, P (/ are

a + #, a - /3, a.fl, a/-1 .

(141.12)

First of all we prove the special case of the theorem, namely that F(a) I F

is separable if a is. According to Theorem 334 the degree and reduced degree of a over F must be equal. According to Theorems 295, 328, this means that the degree and reduced degree of F(a) I F are equal. This means, according to Theorem 336, that F(a) I F is separable.

538

THEORY OF FIELDS

For the general case it will suffice to prove that F(al, ..., as) I F is separable if al, ..., as are. According to the above special case, in the sequence of fields F, F(a), . . ., F (al, ..., as) every term (from the second on) is separable over the preceding term. By repeated application of Theorem 337 the assertion follows. Consequently Theorem 338 is proved. The corollary is trivial. If G I F is now an arbitrary field, then according to this corollary its separable (thus algebraic) elements over F constitute a field F which we call the separable hull of F in G. An algebraic field G I F is called a pure inseparable field if F = F, i.e., if all the elements of G - F are inseparable over F. An (algebraic) element a of an arbitrary field G I F is said to be a pure inseparable element (over F) if F(a) I F is pure inseparable. An irreducible polynomial f(x) over an arbitrary field F is called a pure inseparable polynomial (over F) if the field defined by the equation f(x) = 0 over F is a pure inseparable field (over F). Notice that a pure inseparable field G I F is in general inseparable. The only exceptions are the fields F I F which are at the same time separable and pure inseparable. Likewise in an arbitrary field G I F only the elements of F are simultaneously separable and pure inseparable. Among the (irreducible) polynomials from F[x] only the linear ones are at the same time separable and pure inseparable. THEOREM 339. In an algebraic field G I F the separable hull F of F is marked by the fact that F I F is separable and G I F pure inseparable. Certainly F I F is separable. Furthermore G I F is pure inseparable; for if

there were in G - F a separable element over F, then this, according to Theorem 337, would be separable over F, and belong to F. Conversely, if Fl is a field between F and G such that Ft I F is separable and G I Fl pure inseparable, then, as above, Fl 9 F. Fl must equal F, since otherwise G - Fl would have a separable element over F, which is impossible. Consequently Theorem 339 is proved.

THEOREM 340. For a normal field N I F the separable hull F of F in N is likewise normal over F. For, since under every isomDrphism separable elements become elements. of the same kind, Theorem 340 follows from Theorem 331.

THEOREM 341. An element a of a field G I F of characteristic p (> 0) is pure inseparable if, and only if, there is an integer e (>_ 0) such that cc" E F,

(141.13)

and the least e with this property is the exponent of a. The pure inseparable elements of G I F constitute a field.

539

SEPARABLE AND INSEPARABLE FIELD EXTENSIONS

We begin the proof by noting that the elements a of G, for which there is an e (>--_ 0) satisfying (141.13), are obviously algebraic, moreover they constitute a field. We consider two such elements e, or (a # 0) of G for which with two integers e, f (> 0) NP`, a' E F.

We may assume that e =f Then (e - a)°` = e" - (IrP` E F ,

(ea-1)' = epe a-' E F ,

whence the assertion follows. Besides, we have established that if

Of`, ...,1 r E F

01,

.,13 E G)

,

then for every element co of G (01, ..., t9s) the same condition c)" E F is satisfied.

We now suppose that (141.13) is true. In order to prove that a is pure inseparable we consider an arbitrary element co of F(a). The polynomial

x"-CDP`(=(x-MY) has co for a zero and lies in F[x]. Hence it follows that the minimai polynomial of co is a power (x - co)k (k z 1). Thus co is separable if, and only if, k = 1, i.e., co E F. Since in conformity with this F is its own separable hull in F(a) I F, this field, and so also a, is pure inseparable. Conversely, we suppose that a is pure inseparable. We denote by e the exponent of a. Then c' is separable. Since, on the other hand, according to the supposition F(a) I F is pure inseparable, (141.13) follows. If (141.13) (for e >_ 1) also held for e - I instead of e, then the polynomial Xpe-1 - av-'

would lie in F[x] and have a for a zero, consequently the minimal polynomial of a would have a degree which is smaller than p This contradicts (141.6) which implies that e is the least number with the property (141.13).

From what has already been proved, and from the above, the only remaining assertion of Theorem 341 follows. THEOREM 342. Over a field F of characteristic p (> 0) all the non-linear pure inseparable principal polynomials are those polynomials xPe - a

in which a is not a p`b power in F.

(e>0;aEF),

(141.14)

THEORY OF FIELDS

540

The principal polynomials concerned are the minimal polynomials of those pure inseparable elements from an extension field of F which lie outside F. First let a be such an element and e its exponent. By Theorem 341

e is at the same time the least non-negative integer with the property (141.13), so that now e >_ 1. Furthermore it follows from (141.13) that

ae=a for an element a from F. Thus the minimal polynomial of a is a divisor of (141.14), and even equal to (141.14), since it must be of the form (141.6). There a cannot be a pth power in F, for otherwise (141.14) would not be irreducible. Conversely, if a is not a pth power in F, then, according to Theorem 332, the polynomial (141.14) is irreducible, and for every zero of (141.14), lying

in a suitable extension field, the condition (141.13) holds. Therefore, according to Theorem 341, a is pure inseparable. Since (141.14) is its minimal

polynomial, this is also pure inseparable. Consequently Theorem 342 is proved. THEOREM 343. A field G I F is pure inseparable if, and only if, all its elements

are pure inseparable over F. Assume at first that G I F is pure inseparable. For every element a of G its exponent e has the property that a' is separable, and so by hypothesis

lies in F. Consequently a is pure inseparable by Theorem 341. Conversely, if every element a of G is pure inseparable, i.e., if the fields F(a) I F belonging to it are pure inseparable, then it follows that F is its own

separable hull in every F(oc), and so also in G. This means that G I F is pure inseparable, so proving the theorem. THEOREM 344. Every pure inseparable field G I F is normal and its only automorphism is the identical one.

From Theorem 342 it follows that a pure inseparable element has no conjugate different from itself. Hence, and from Theorem 343, it follows that every isomorphism of G I F into an arbitrary extension field of G can only be the identical one. Because of Theorem 331, this proves Theorem 344. THEOREM 345. The reduced degree of a field G I F of finite degree and of

characteristic p (> 0) is equal to the degree [F : F] of the separable hull F of F in G. If N is a normal field of G I F, then every isomorphism of G I F into N induces an isomorphism of r F into N. Consequently, if we prove that conversely every isomorphism

FIF;-- F'IF

(p - to)

(141.15)

SEPARABLE AND INSEPARkBLE FIELD EXTENSIONS

541

of F I F into N may be extended in only one way to an isomorphism of G I F into N, then, by Theorem 336, the assertion follows. By Theorems 339, 343 all the elements of G I F are pure inseparable. Since this field is of finite degree, we may write

G = WI, . . ., o,) ,

(141.16)

where the elements

(i= 1,...,s)

(141.17)

for suitable el, ..., e., (>_ 0) belong to F. Now if A -> A* is an isomorphism

of G I F into N which continues the isomorphism (141.15), then from (141.17) it follows that

(i = 1 ,

t*r`' = ai

..., s) .

(141.18)

As at most only one system 6; , ..., 8 (E N) is possible with this property, so, because of (141.16), there can exist, in fact, at most one isomorphism

such as 7, -**.

On the other hand, from the normality of N I F the existence of

elements 0i, ..., tS (E N) which satisfy (141.18) follows, and it is obvious that an isomorphism of G I F into N which is a continuation of (141.15) is defined by s)

E F) .

Consequently, Theorem 345 is proved. Because of this theorem the reduced degree of a field of finite degree is always a divisor of the degree. This also follows from Theorems 328, 330, 334. By the degree of inseparability of a field G I F of finite degree we mean the

quotient of the degree and the reduced degree of this field. This is always equal to the degree [G : F], where F is the separable hull of F in G. It is trivial if the characteristic is 0, and a consequence of Theorem 345 if the characteristic is a prime number p. In this case the degree of inseparability is, because of Theorems 339, 342, a power of p. The degree of inseparability of a field of finite degree is equal to 1 if, and only if, it is separable. EXAMPLE 1. Since in a finite field of characteristic p every element is a pt' power,

it follows from Theorem 332 that there are no inseparable polynomials over finite fields.

EXAMPLE 2. Over the transcendental extension field F,($) of the prime field F, with characteristic p (> 0) the polynomial x° - aA is irreducible and inseparable.

§ 142. Complete and Incomplete Fields

A field F is called complete if only separable field extensions over it are possible; otherwise the field F is called incomplete. In other words, a field is complete if, and only if, every irreducible polynomial f(x) is sepa-

THEORY OF FIELDS

542

rable over it. In particular, all fields of characteristic 0 are complete, therefore we shall consider in this section only fields of prime characteristic. THEOREM 346. A field F of prime characteristic p is complete if, and only if, every element in it is a pth power. If in F every element is a pth power, then, from Theorem 332, it follows that there are no inseparable polynomials over F. If, on the other hand, a is an element of F which is not a pth power in F, then again according to this theorem, the polynomial xP - a is (irreducible and) inseparable. Consequently, Theorem 346 is proved. From this theorem it follows, e.g., that finite fields are complete, but, rational function fields of prime characteristic are incomplete. (Cf. § 141, Examples 1, 2.) All algebraically closed fields are complete.

For an arbitrary field F of characteristic p (> 0) we shall henceforth denote

the set of the pth powers of the elements of F by F". We know that F" is always a field, and that Theorem 305 implies that the meromorphism F

FP (a -).. aP)

(142.1)

holds. The content of Theorem 346 may then be expressed by saying that F is complete if, and only if, FP = F, i.e., if the meromorphism (142.1) is an automorphism.

We now define

(n=1,2,...).

F1=F, FF"=(FP"-')P

(142.2)

Then FP" is that subfield of F which consists of the elements aP" (a E F).

If F D F' then FP" z)

FP'+'

follows by applying the meromorphisrn a -> a"" for every n = 0, 1, .... Accordingly either

F=FP=FP'=

...

(142.3)

or

F:D FP:DP's...,

(142.4)

according as F is complete or incomplete. The intersection

D=FfFPfFP'fl...

(142.5)

is always complete, since obviously D° = D. If, in addition, G is an arbitrary complete subfield of F, then

G= G""cF"", thus G 9 D. This results in the following

COMPLETE AND INCOMPLETE FIELDS

543

THEOREM 347. For every field F of characteristic p (> 0) the subfield F n FP fl Fp' fl ... is complete and contains all complete subfields of F. It should be noted that the field F I FP is pure inseparable, since the p`h powers of the elements of F lie in F". This field F I F" will be important in further discussions, and already in the following fundamentally important .

definition. An element a of a field F of characteristic p (> 0) is said to be p-dependent

on a set (or a system) S 9 F, if (142.6)

e E FP(S).

Otherwise, Q is said to be p-independent of S. We shall show that the fundamental properties I, II, III, formulated at the beginning of § 100 with respect to linear dependence, also hold for p-dependence in the field F. First of all I holds, since o0i E F" (co,, . . ., co.) (i = 1, . . ., n). In order to prove II, we suppose that for certain elements Q, ooi, ... a> of F g E Fp(co1, ..., co.),

F"(col,

..., con-i)

.

We must infer that co,, E F"(o, w1, ..., con-0 .

(142.7)

By hypothesis an equation such as e =

p-1

aiwin i=0

holds for some elements ao, ..., ap_1 from R(col, ..., co,,), _1such that not all a1, . . ., ap_i vanish. Hence it follows that the degree of w on the right-hand side of (142.7) is smaller than p. Since co is pure inseparable over F", and so a fortiori over the right-hand side of (142.7), it follows from Theorem 342 that this degree is a power of p, thus necessarily 1. This implies that (142.7) holds. Finally, III also holds, since for any elements o, el, . ., en:, wi, ... W,

of F, from the assumption

a E F"(ei, ..., em), Lei E F"(w1, . . ., co.)

(i= 1'...m)

it evidently follows that a E F°(co1, ...,

From what has now been proved it follows that the conclusions in § 100 remain valid when p-dependence is understood instead of linear dependence and the definitions are suitably adapted. In conformity with this, we call a subset S of a field F of characteristic

p (> 0) a p-independent set when every element a (E S) is p-independent

544

THEORY OF FIELDS

of S - e. Two subsets S, T of F are called p-equivalent sets when every element of each of these sets is p-dependent on the other set; this simply means that FP(S) = FP(T).

(This definition can also be extended to the more general case, where S, T are systems of elements of F.) Finally, by a p-basis of F we mean a p-independent subset S of F such that F = FP(S). The following theorem is similar to STEINITZ'S second main theorem (Theorem 322). THEOREM 348. Every field F of characteristic p (> 0) has at least one p-basis, and the p-bases are the maximal p-independent subsets of F. Furthermore all the p-bases of F are equipotent. We can omit the proof since it is similar to that of Theorem 322 in that

we have only to replace the notion of algebraic dependence by that of p-dependence. The proof involves no difficulties. For a combined proof of Theorems 322 and 348 see MAcLANE (1938). Cf. KERTtsz (1960).

On account of Theorem 348, we call the cardinal number of an arbitrary p-basis of a field F of characteristic p (> 0) the degree of incom-

pletability of this field. This is an invariant of the field and is 0 if, and only if, the field is complete. If the field F is of finite degree of incompletability k, then F I FP is obviously of finite degree, and then [F

:

FP] = pk.

THEOREM 349. If G I F is a pure transcendental field of characteristic p (> 0) and degree of transcendence n, and if F is of degree of incompletability k and both of these degrees are finite, then G has degree of incompletability

n + k. The union of a transcendence basis of G I F and a p-basis of F then yields a p-basis of G.

It is sufficient to prove the theorem for the case n = 1, for then the general case is proved by induction. Let G = F (0), and let S be a p-basis of F. Then we have G = FP(S)(b) = F1(t91)(S, e) = GP(S, 0).

Since 0 does not belong to F, still less to S, it is sufficient to prove, for the first assertion, that S U 0 is a p-independent set in G. Instead of this, because of Theorem 238, it will suffice to show that S constitutes a p-independent set in G and 0 is p-independent of this set.

Instead of the first of these two assertions we prove more generally: if, for an element a and a subset A of F, a is p-dependent on A in G, then

545

COMPLETE AND INCOMPLETE FIELDS

this also holds in F as well as G. By hypothesis, a is an element of GP(A) = FP(8P) (A) = FP(A) (&")

and is algebraic over FP(A). But since & is transcendental over FP(A) (S F), by Theorem 293 a E FP (A). This means that at is p-dependent on A in F, which was to be proved. If, further, is p-dependent on Sin G, then 0 E GP(S) = FP(OP)(S) = FP(S)(9P).

This means that an equation such as P

$

AM

(f (x), g(x)

E

FP(S)[x])

holds. This contradiction completes the proof of the first assertion of the theorem.

The truth of the second assertion of the theorem follows from G = F(s) = FP(S)(0) = FP(SP)(S)(0) = GP(S, 0). THEOREM 350. If G I F is a separable field of characteristic p (> 0), then every p-independent subset of F is also p-independent in G. Further every p-basis of F is a p-basis of G. COROLLARY. The algebraic extension fields of a complete field are complete.

For the first assertion of the theorem it suffices, as in the proof of the last theorem, to prove that if an element a (E F) is p-dependent in G on a set A (c F), then the same also holds in F instead of G. We put G = F (S). Then a E GP(A) = FP(SP) (A) = FP(A)(SP).

(142.8)

On the other hand, the application of the meromorphism P -->. PP of G shows that GP I F° is separable. Since, then, the elements of SP are separable over FP(A), it follows from (142.8) that GP(A) I FP(A) is separable. But now a is pure inseparable over FP, and so also over FP(A), consequently a E FP(A). This establishes the assertion. In order to prove the second assertion of the theorem we denote a p-basis of F by B. Then we have F = FP(B) and so F C GP(B).

Consequently G I GP(B) is separable. Since this field is also pure inseparable,

it follows that G = GP(B). Hence, and from the first assertion already proved, it follows that the second assertion of the theorem is true.

THEORY OF FIELDS

546

If F is a complete field and G its algebraic extension field, then G I F is separable according to the definition of complete fields. Since further the only p-basis of F is now the empty set, the same follows for G from Theorem 350. This proves the corollary. THEOREM 351. If G I F is a field of characteristic p (> 0) and of finite degree, and F is of finite degree of incompletability, then F and G are of the same degree of incompletability. Since G I F, F FP are of finite degree, [G : F] [F : FP] = [G : FP] = [G : GP] [GP : FP]

from Theorem 292. The application of the meromorphism e

eP of G

results in [G : F] = [GP : FP].

Hence [F : FP] = [G : GP], i.e., the theorem follows. THEOREM 352. Every field F of characteristic p (> 0) has an extension field G with

GP = F.

(142.9)

If F is complete, this implies that G = F. We take a field F such that F'

and F, F are disjoint. Because F F'P

F

F", F.

Accordingly F may be embedded in F' so that the subfield FP is replaced by F. After embedding, the field obtained from F is denoted by G. Then equation (142.9) holds as required. If F is complete, then, because GP' = FP = F = GP, G is also complete, from which G = GP = F follows. This proves the theorem. By (142.9) the field G is, to within isomorphism, uniquely determined and, since G - GP = F must hold, we denote it by FP-'. FP-' is uniquely determined even up to equivalent extensions, for if G, G' are two fields such that GP = G'P = F and we assign to each A (E G) that A' (E G') for which AP = A'P, then we have the isomorphism

GIFsrG'IF(A--*A'). Moreover we define FP-° =

(FP-A+')P

(n = 2, 3, ...)

COMPLETE AND INCOMPLETE FIELDS

547

by recurrence. Then the terms of the ascending chain of fields

F9FP--'cFp_2c...

(142.10)

are uniquely determined up to equivalent extensions. Moreover each term is the pth power of the term following it. Of course, everywhere in (142.10)

either "=" or "e" holds [cf. (142.3), (142.4)]. THEOREM 353. To every field F there is a complete extension field F which

has the property that the between fields G such that F c G c F are incomplete. This field F is, apart from equivalent extensions, uniquely determined and algebraic over F.

Let p denote the characteristic of F. We may suppose that p > 0, otherwise. the theorem is trivial. As in Theorem 47 we form [as a counterpart to (142.5)] the union field of the chain (142.10) which we denote by F and prove that for this the requirements of the theorem are fulfilled. Evidently F° = F, therefore F is complete. Since each term of the chain is algebraic over F, the same holds for F. If G is a complete field between F and F, because of Theorem 346 it must contain all the terms of the chain (142.10), so that G = F. It still remains to prove the uniqueness part of the theorem. If F is a complete extension field of F, we can form a chain of fields in it as in (142.10).

If the union of (142.10) is again denoted by F, this is a complete subfield of V. Consequently, if we postulate that the between-field G such that

F c G c F is incomplete, then, necessarily, F = F. This completes the proof of the theorem. EXAMPLE 1.

The degree of incompletability of the rational function

field

.-9:",(xt, ..., x") is equal to n. EXAMPLE 2. In every complete field F of characteristic p (> 0) define a°-` (a E F;

n = 1, 2, ...) as the (only) root of the equation x" = a. For every k (= 0, ±1, ...) a - a°" is then an automorphism of F. EXAMPLE 3. If, for a subset S of a field F of characteristic p (> 0), F = F'(S),

(142.11)

then

F=P"(S)

(n=1,2,...)

(142.12)

also holds. This assertion is true if n = 1. We assume its truth for an arbitrary n. If we denote by S'" the set of elements a' (a E S), then, by applying the meromorphism

e - e°", the equation

F°" =

F

= F""+`(S"") (S) = F""+`(S°" , S) = FP' +`(S),

follows by induction. This proves (142.12) in the general case.

THEORY OF FIELDS

548

§ 143. Simplicity of Field Extensions THEOREM 354. The set of fields between a field F and its extension field G is finite if; and only if, G is a simple algebraic extension of F. COROLLARY. If an overfield G of a field F is a simple algebraic extension of F, then the same holds for all fields between F and G.

To prove the theorem we first suppose that G is a simple algebraic extension of F. Let 19 denote a primitive element of G and f (x) its minimal polynomial over F : G = F(t9), f(19) = 0.

If H is a field between F and G, then the minimal polynomial g(x) of 0 over H is a divisor of f(x). Since, according to this, there are only a finite number of possibilities for g(x), the finiteness of the number of fields H will follow if we show that different g(x) belong to different H. For this purpose we suppose that the same polynomial g(x) belongs to two such fields H1, H2. We put

H12=H,f1H.,. Because g(x) E H1[x], H2[x], it follows that g(x) E H12[x]. Since g(x) is irreducible over H1, it is also irreducible over H12. Consequently we have

[G:H12]-[G:HI]=[G:H2], that is, H 1= H 2.

Now suppose, conversely, that between F and G there are only a finite

number of fields. If G I F had a transcendental element e, then F(e), ... would be an infinite number of different fields between F and G, consequently G I F must be algebraic. Consider a chain F(e2),

F c F(O) c F(#1, 02) c

... c G

of fields between F and G. Since this chain is finite, by hypothesis, there must be an n with

G=

r9n)

It is sufficient to show that, if n = 2, G I F is a simple extension, then the same will follow by induction for all n. Therefore we assume that G = F(+91, '02),

where we may restrict ourselves to the case 01, 192 o 0. If F is finite, then also

G is finite. Consequently the assertion is now true, according to Theorem

SIMPLICITY OF FIELD EXTENSIONS

549

306. If F is infinite, there are an infinite number of different elements 01 + c62 (c E F). Thus, there are two different c, d (E F) such that F((91 + (1#2)= F(01 + (1192).

This field contains the elements 01 + co, #1 + d092, and so also their difference (c - d)-92. Consequently it also contains 192, i91 and so the whole

field G, to which it must be equal. Accordingly G is a simple extension of F. This proves Theorem 354 and the corollary follows. For the proof of Theorem 354 cf. WILKER (1951 - 52). THEOREM 355. An algebraic field extension F(a1i . . ., ak) I F is simple if the adjoined elements al, ..., ak are, with at most one exception, separable. Since the theorem is trivial for k = 1 we have only to consider the case k Z 2. It will suffice to prove it for k = 2, since if for k >_ 3 we suppose

it to be true for smaller k, there is a

such that

F(al,

..., ak--I) = F(P);

F(al,

..., ak) =

since then F(e, ak)

and, according to Theorem 338, of the elements e, ak at least one is separable,

the general validity of the'theorem follows. Now let F(oc, fi) I F be the given field, where a, fi are algebraic and, e.g., is separable. We denote the minimal polynomials of a and P over F by

f(x), g(x), respectively. We take a splitting field G of the polynomial f(x) g(x) over F(a, ,9) and denote by al

QQ

QQ

//

QQ

a), a2, ..., am; Y1 (= N)+ #22 ..., Nn

all the different zeros of f(x) and g(x), in G. If F is finite, then F(oc, 9) is

also finite. The assertion is, therefore, true. Then we suppose that F is infinite. Since every equation such as

(X(+ fijx = al + N1x

(j # 1)

has at most one solution x (E F), there exists an element c (E F) such that

a(+C9f0al+A

(i= l,...,m; j=2,...,n).

We show that for the element

0 =0cl+ c#1=a+cP the equation F(a, j9) = F (0)

holds.

THEORY OF FIELDS

550

The left-hand side contains the right-hand side, therefore we have only to prove that a, f3 E F (s). Because a = 0 - ci, it suffices if we show that E F(s).

(143.1)

Now

g(fl) = 0,

f(1' - e fl) = 0.

According to this, f is a common zero of the polynomials

g(x), f(( - cx).

(143.2)

No further common zero exists (in G), since for every other zero f1, (j 9& 1) of g(x) the inequalities 0 - c fl t

a;

(i = l , ..., in)

all hold, and so f(O - c f) 0 0. Furthermore, since g(x) is separable, f is a simple zero of it. We have now shown that the polynomials (143.2) have greatest common divisor x - fl. But since the coefficients of these polynomials lie in the field F(t), the same follows from Theorem 203 for x - f3. This means that (143.1) holds, and completes the proof of Theorem 355.

We now formulate an important special case with a slight improvement. THEOREM 356. Every separable field extension G I F of finite degree is simple, and an element 0 is primitive if, and only if, the number of its conjugates in a normal field of G I F is equal to the degree of G I F. The first assertion follows from Theorem 355. In order to prove the second assertion we denote the degree of G I F by n. This is now, according to Theorem 336, equal to the reduced degree of G I F, therefore this field has, in its normal field, exactly n isomorphisms. Hence, and from Theorem 328, the second assertion of Theorem 356 follows. THEOREM 357. A field extension G I F of finite degree and of characteristic

p (> 0) is simple if, and only if, for the exponent e and the degree of inseparability pf of G I F the equation e = f holds. We denote by F the separable hull of F in G. Then we have [G

If G I F is simple, i.e., G =

F]'= pf for a (primitive) element r9, then G = F(g)

and pf is the degree of 0 over F. But, since G I F is pure inseparable, this degree, by Theorem 341, is equal to pe, whence e =f.

SIMPLICITY OF FIELD EXTENSIONS

551

If, conversely, e = f, then we take an element a from G with exponent e. Then we have [F(a) : F] = pe = pf = [G : F], and so G = F (a). Since, on the other hand, F I F is separable, it follows from

Theorem 356 that F = F(p) for a separable element j9. From

G=F((x)=F(a,fi) and Theorem 355, it follows that G F is simple, whence Theorem 357 is proved. THEOREM 358. If a pure inseparable field extension F(aI, ..., ak) is simple,

then there is a primitive element among the adjoined elements al, ..., a,,.

We put G = F(al, ..., ak) and suppose that, e.g., the exponent of al, which we denote by e, is the greatest possible. Then e is at the same time the exponent of G I F, whence, according to Theorem 357, it follows that pe is the degree of inseparability of G I F. Since this field is pure inseparable, this means that [G : F] = pe. On the other hand, by Theorem 341, [F(a1) : F] = pe, so that G = F(a1). Theorem 358 is now proved. EXAMPLE. Theorem 355 is, in general, false if among the al, . . ., ak two inseparable

elements occur. Indeed, if G = Sr-, (x, y) is the rational function field of the inde-

terminates x, y over P-,, p > 0, and F = .g', (x°, y°) then G = F(x, y) with x, y° E F, whence we see, because of Theorem 358, that G I F cannot have a primitive element. This is also obvious, since [G : F] = p2 and every element of G I F has degree < p. According to Theorem 354, there are infinitely many fields between F and G. These are as follows: F(x + ay) (a ( F), F(y).

§ 144. Norms and Traces in Fields of Finite Degree We consider a field G I F of finite degree. Since this is a finite-dimensional

F-algebra, so for every element a its characteristic polynomial f(x), its norm N(a) and its trace T(a) (over F) are defined (see §§ 74, 75). These are called the characteristic polynomial, the norm and the trace of a with respect to G I F and denoted if necessary, more precisely by fGIF(x; a),

NGIF(a),

TGjF(a).

(144.1)

These last two terms are called the relative norm and relative trace of the element a (E G). First the following simple theorem holds:

THEORY OF FIELDS

552

THEOREM 359. If H I G, G I F are two fields of finite degree with

r = [H:G], then

(144.2) (a E G), fHIF(x; a) = (GjF(x; a))r (a))r THIF(a) = rTGIF(a) (a E G). (144.3) NHIF(a) = (NG,F

For the proof, assume an F-basis cot, ..., co, of G and a G-basis P1, of H. Then, according to Theorem 292,

..., er

(i = 1, ...,1; j = 1, ..., r)

w1ej

is an F-basis of H. Thus I

acol = E a,kwk k=1

(i = 1, ...,1)

(144.4)

with uniquely determined elements aik of F. Hence I

a(oleej =

k=1

alkCokQj

(i = 1, .

.

.,

1; j = 1, ..., r) .

(144.5)

Temporarily we arrange the basis elements co1LOj as follows:

wleh

, C'01,01, ...; co wr,

, Wlt-Or

The matrices connected with the linear mappings defined by (144.4), (144.5)

are denoted by A and A', respectively. Then A = (a11) is a square matrix of type la, and

A' =

is a square matrix of type (Ir)a in which r matrices A are placed diagonally and in the empty places are only zero elements. The corresponding characteristic matrices are

xU - A, xU' - A',

(144.6)

where U, U' denote the unity matrix of type 12 or (1r)a, respectively. Between

the determinants of the matrices (144.6) we have the equation

IxU'-A'I=IxU-Ajr. Hence, and from (74.4), (75.4), (75.5) equations (144.2), (144.3) follow. Consequently the theorem is proved.

NORMS AND TRACES

553

For a more detailed study of the concepts introduced in (144.1) we use the minimal polynomial (144.7) + ... + c,. 9(x) = xn + of the above element a of G I F, where n = [F(a) : F]. We prove that c,_1xn-1

(144.8)

.fF() F(x) = 9(x).

To do this we use in F(a) the F-basis 1 , a, ..., an-1. Because of (144.7) we have

(i = 0, ..., n - 2),

acct = (x`+1

... -

aan-1 = __co - Cla -

Cn_lan_'-

By comparing these equations with (144.4), we see that the above matrix is now specialized to 0

1

0

0

0

1

0

...

A=

(144.9) i

0

0

0

- CO - Cl

1

- Cn_2 - Cn_1

ii

1

This is called the concomitant matrix of the polynomial (144.7). Its characteristic matrix is x

-1

0

0

x

-1

0

xU - A=

(144.10) 0

0

x

CO

Cl

Cn_2

-1 X + Cn_1

If we add, in order, for j = n,.. ., 2, x times the jth column to the (j - 1)`h column, and place the first last, then we obtain a matrix with diagonal

-1, ..., -1, g(x) and only zero elements above the diagonal. By suitable elementary row transformations this becomes the diagonal matrix 1

-1 (144.11)

-1 9(x) 1

THEORY OF FIELDS

554

(of type n2). The determinant of (144.11) is (- 1)n-1 g(x). On the other hand, the determinants of the matrices (144.10), (144.11) differ from one another by the factor (- 1)n-1 because of the column permutation, so that the determinant of (144.10) is equal to g(x). (144.8) is now proved. This result, and Theorem 334, yield the following THEOREM 360. If a is an algebraic field element over a field F, then its characteristic polynomial is, with respect to F(a) I F, equal to its minimal polynomial over F and may be represented as a)In n ' (x fFtOF (x; a) = ( (ai = a), (144.12) i ('I

where n, n' denote the degree and reduced degree, respectively, of F(a) I F, and II, . . ., an the different conjugates of a in a normal field of F(a) I F. The norm and trace of a with respect to F(a) I F are: NF(-)IF(a) _ (al ... an.)nn'-"

(144.13)

TF(a);F(I) = n n'-I (II + ... + an). We prove the following THEOREM 361. A field G I F of finite degree is inseparable if, and only ij; always TGIF(a) = 0

(a E G).

If G I F is inseparable, then its degree of inseparability is a power of the characteristic p (> 0) of F. Hence, and from (144.32), (144.132) the "only if" of the theorem follows. In order to prove the "if" part we suppose that G I F is separable. According to Theorem 356, we can write G = F(a), where m ;A 0 may be assumed.

Let g(x) be the minimal polynomial of I. By hypothesis

g(x)=fl(x-ai) i=1

(a1=a)

holds [as a special case of (144.12)], where 21, ..., an are the distinct conjugates of a in a normal field of G I F. Now g'(x)

0.

According to Leibniz's rule we have g'(x) _

g(x)

(144.14)

Consider the identities n

xn - ain = (x - a,)

a;'-kxk-1

k=1

(i = 1, . . ., n).

NORMS AND TRACES

555

. ., a are different from 0, so by cancelling the first term of the left-hand side and dividing by - a/' the congruences

Since at, .

n 1

( x - ai)

(i = 1, ... , n)

a; kxk-1 (mod xn) k=1

are obtained in the polynomial ring F(a1, .

.,

.

an) [x]. After multiplication

by g(x) and addition, on account of (144.14) we obtain

x - at

g'(x)

g(x) > TF(-)IF (a/ k)xk`(mod x"). k=]

Since the left-hand side is different from 0, the same holds for at least one TF(,,)IF (ai k). Consequently Theorem 361 is proved. EXAMPLE. Since (144.11) is obtained from (144.10) by elementary transformations,

(144.11) is the normal form of the characteristic matrix (144.10). The elementary divisors are - 1,.. , - 1, g(x).

§ 145. Differents and Discriminants in Separable Fields of Finite Degree We take a separable field G I F of finite degree. The concept of the conjugates of the elements of G can often be easily modified as follows: we take a given fixed normal field N of G I F and denote by s1 (= 1), s2i ..., sn all the different isomorphisms of G I F into N, where n is the degree of G I F. By the conjugates of an element a of G I F (in N) we mean the elements a; = s,a

(i = 1,

.

. ., n),

(145.1)

which, unlike the "customary" conjugates of a, are not necessarily distinct. Usually we assume the order of the isomorphisms s2, ..., s,, as arbitrarily fixed, and then we call al the i' conjugate of a. (In particular, al = a.)

If we denote by f(x) and g(x) the characteristic polynomial of a with respect to G I F and the minimal polynomial of a over F, respectively, then by Theorems 359, 360

f(x) = g(x)n-- "

(145.2)

where m is the degree of a (over F). The isomorphisms sl, . . ., s,, are continuations of isomorphisms of F(a) I F and every isomorphism of F(a) I F into N has exactly nm-1 continuations among the s1, . . ., s,,. This means that in (145.1) all the different conjugates of a occur with the same multi-

THEORY OF FIELDS

556

plicity nm'1. Hence, and from (145.2), it follows that

f(x) = (x - (xl)

... (x - an).

(145.3)

Notice that the elements (145.1) are all the different conjugates of cc (in N) if, and only if, m = n, i.e., a is a primitive element of G I F. Now we call 6GIF((X) = f'(a) = (a - a2) ... (a - an) (145.4) the different and DGIF (a) = k-

1)(E) NGIF

IbGIF (a))

(145.5)

the discriminant of the element a (E G). The more precise terms relative

different and relative discriminant or - if F is the prime field of G absolute different and absolute discriminant, are also used. Clearly SGIF(a) E G,

DGIF(a) E F;

moreover we see that the different and the discriminant are invariant with respect to the choice of the normal field N of G I F. Both the different and the discriminant of a are unequal to 0 if, and only if, a is a primitive element of G I

F.

Henceforth we shall denote different and discriminant by b(a) and D(a). Because of (145.1) and (145.4), we have b(a) = (s1a - se(X)

... (sla - S.00-

Hence, and from (145.5), it follows, from Theorem 360, that

D(a) _ (- 1)(1)

i=1

s,b(a) _ (- 1)")

(sac - ska) . t. k=1

t#k

Consequently, by (145.1), we have

D(a) = fl

1si
(ai - ak)2.

(145.6)

According to this and Theorem 277, the discriminant D(a) of the element a

agrees with the discriminant of the characteristic polynomial of a with respect to G I F. It should be noted that (145.6) (cf. § 115, Example 2) may also be written in the form 2

D(a) =

(145.7)

DIFFERENTS AND DISCRIMINANTS

557

Further we consider n arbitrary elements

cv1,...,co"(E G)

(145.8)

(which need not now be conjugates) and denote by

(i, j = 1, ..., n)

wii = sew,

(145.9)

the jtb conjugate of aai. We put DGIF(wl,

..., co,,) _ I cot 12,

(145.10)

where I w;u I is the determinant of the elements (145.9) and call (145.10)

the relative discriminant or discriminant of the elements co , ..., Co". Henceforth let (145.10) be denoted by D(wl, . . ., co,). A permutation

of the elements (145.8) causes a permutation of the columns of the determinant I co, 1, so that the discriminant (145.10) is independent of the order of the w1, . . ., co, By comparing (145.7) with (145.10) we get

D(l, a,

. .

.,

an-1) = D(a)

(145.11)

so that (145.10) is a generalization of (145.7). We show that (145.10) may easily be traced back to (145.7). To do this, we now take for a a primitive element of G I F. This means that D(a) A 0. Then 1, a, . . ., a."-1 is a basis of G I F, and it follows that

wi = a,1 + ai2a + ... + a;"a"-1 (i = 1,

.

.

., n)

(145.12)

with uniquely determined elements a,1 of F. These constitute a matrix

A = (a,,).

(145.13)

Since from (145.12)

wui=ail +a:2aj+...+a,"(x;"-1

(j=1,...,n),

on account of the theorem of multiplication for determinants we have

Iw;;l=Iai;lII Hence, and from (145.7), (145.10), (145.13) follows that D(wl, ..., uw") = I A 12D(a),

(145.14)

THEORY OF FIELDS

558

where I A I denotes the determinant I aid 1. From this formula (145.14) we also see that D(w1, . . ., co,,) is always an element of the fundamental field F; of course this also follows immediately from the definition (145.10). Because of (145.12), wI, . . ., co,, are linearly independent over F if, and

only if, 1-4 1 96 0. Hence, and from (145.14), the following theorem is obtained :

THEOREM 362. The elements wl, ..., co,, of a separable field G I F of degree n constitute a basis if, and only if, their discriminant is not equal to 0.

We now express (145.10) in another form. The right-hand side may be written, according to (145.9), as I sjwi 12. This, by the theorem of multiplication for determinants, is equal to the determinant s1(wi w;) t=1

Consequently D(a)j, . . ., co.) = I T(w oj) I

,

(145.15)

where T denotes the relative trace TGIF(...) and we have a determinant of order n on the right. § 146. Ore Polynomial Rings Let a commutative ring J9 with unity element of prime characteristic p be given, which we shall later specialize to a field. Further we take a fixed power

q=p'(> 1)

(146.1)

of p. For every polynomial k

f(x)=Laixi=ao+alx+...+akxk i=o

(146.2)

from fi[x] we put k

fq(x) = > a,xq' = aox + alxq + i=o

... + akx0

(146.3)

and call this a q-polynomial over .1P or the q-polynomial corresponding to the polynomial f(x). We denote the set of all such polynomials by a'[x]q. It is evident that both the sum and the difference of two q-polynomials is again a q-polynomial, but in general, this is not true for their product. However, JJZ [x]q may be made to a ring - as will be shown - by defining the multiplication in it by .fq(x)0 9q(x) =fq(9q(x)).

(146.4)

ORE POLYNOWAL RINGS

559

(Cf. § 67, Exercise 1.) We call this ring 12[Xlq the Ore polynomial ring over J9. Since, for arbitrary elements a, f of R[x], (a + j9)° = 0 ° + S°, and so

also (a + fly = 04 + 1Q, the definition (146.4) may be written more explicitly as follows : atx4j 1

(F,

j=0

bixq, I =

0(Y-

1=0

E a.bi'x'

(146.5)

.

i=oj=o

Since the right-hand side belongs to 2[x]Q, a composition has in fact been defined by (146.4). Moreover, it is easy to see from (146.5) that this multiplication is associative and distributive on both sides with respect to addition. Accordingly R[x]Q is, in fact, a ring. In particular ax O x4 = ax4, x4 O ax = aQx4.

Hence, and from (146.5), we see that .5l [x]Q is commutative if, and only if, always

a4 = a

(a E R).

(146.6)

We now take for R a field .iu of characteristic p (>0). We prove the following

Tm oREM 363. If . is a field of characteristic p (> 0) and q (> 1) is a power of p, then the Ore polynomial ring Y[x]Q is Euclidean on the righthand side and, for a complete .7, completely Euclidean. For the proof we consider two elements

f4(x) = avgk+ ..., g,(x) = bx4` +

...

(k _> 1; b

0)

Of Y'[x]Q of degrees qk and q', respectively. Then f4(x) - ab-Q" -'xQk-` O g4(x) = f4(x) -

ab-q*-lg4(x)

is of degree < qk. If .7 is complete, then there is in it an element c with

a-bc4`=0. For this fq(x) - gq(x) 0 cx4k

1 = = '(x)

- gq(cxQk 1)

is again of degree < qk. Hence Theorem 364 follows. 19 R.- A.

I

THEORY O(' FIELDS

560

THEOREM 364. If 7 is a finite field of order q, then between the polynomial

ring 7[.1c] and the polynomial ring .r-[x], we have the isomorphism

15'[x] .:.f [x]q (f(x) -fq(x)).

(146.7)

For any two polynomials f(x) (9& 0), g(x) over .7 and the corresponding q -polynomials we hare the rule Ax) I g(x) -A(x) I gq(x).

(146.8)

Every ideal of Y[x] generated by q -polynomials may be generated by a single q -polynomial.

(On both sides of (146.8) the divisibilities are to be understood in the

ring 7[x].) Since the condition (146.6) is satisfied for ..g' (instead of. ), the first assertion of the theorem is partly trivial, partly a consequence of (146.5). In order to prove the remaining assertions, we carry out the Euclidean division of g(x) by f(x) :

g(x) = k(x)f(x) + r(x). Because of (146.7) it follows that gq(x) = kq(4(x)) + rq(x).

Since the q-polynomials contain no constant terms, the first term of the right-hand side is divisible by fq(x), while the second term is of smaller degree than fq(x). Accordingly the following four propositions are equivalent :

f(x) I g(x), r(x) = 0, rq(x) = 0, .fq(x) I gq(x) Hence (146.8) is proved. We have seen that for q-polynomials the residue occurring in the Euclidean division is again a q-polynomial. Hence the last assertion of Theorem 364 follows. An application of Ore polynomial rings will follow in the next section. For other interesting applications and generalizations cf. ORE (1933, 1934).

§ 147. * Normal Bases of Finite Fields By a normal basis of a separable normal field N I F of finite degree we mean an F-basis of this field which consists of conjugate elements, i.e.. the normal bases of N I F are the bases of the form s¢.), .

.

.,

(147.1)

NORMAL BASES OF FINITE FIELDS

561

where co is an element of N, and sl, . . ., s denote all the automorphisms of N I F. Here we prove the existence of normal bases for finite fields and in § 182 for infinite fields. Before doing so, some preliminary work is necessary.

The determinant of a cyclic matrix is called a cyclic determinant. Apart from sign X0 XI ... Xn_2 Xn_ , Xo ... Xn_2

D-

(147.2)

X1

Xn-I Xo

X2

may belong to an is such a determinant, where the elements x0, ..., arbitrary commutative ring. For the case where the elements of D belong to a field . " we prove that n

D = fl (x0 + 2ix1 + ... +

off,

l=I

where splitting field :

(147.3)

denote all the roots of the equation x" - I = 0 in a

X" - 1 = (x - 01) ... (x - o ).

(147.4)

It will suffice to prove the assertion for the rational function fields

.T=.

(147.5)

of arbitrary characteristic p (>_ 0). In the first place we consider the case p = 0. Then ol, ..., gn are different. To the first column of D we add Qi -1 times the kch column (k = 2, ..., n). Then the first column consists of the elements

Jiwhere

n-1

Hence f, I D (i = 1, ..., n). Since, furthermore, f1, .,f,, are linear, and so irreducible, and not associated in pairs, fl ... f I D. Mere both sides are polynomials of degree n. Thus D = cf1

.

.fn

THEORY OF FIELDS

562

with a constant c. The coefficient of xo on the left and right is equal to I and c, respectively, whence c = I follows. This proves (147.3) for this case.

In order to prove the remaining case we express the result obtained in another form. It should be noted that the X01, . . ., e,, may be given in the form (147.6) (I = 1, . . ., n), er = e,-'

where a denotes a primitive n`h root of unity. If (147.6) is substituted in (147.3), then we obtain an equation which is equivalent to the congruence n

D = fi (xo + z'-'xl + ... + z('-1)(n-')xn-1) (mod Fn(z)), t=1

where z is a further indeterminate and F,(z) denotes the nU' cyclotomic polynomial. Hence it follows that 1

D=

- 17 ,

1

i=1

(xo + z' 1x1 + ... + Z(i-1)tn-!) xn-1) + F,(z)Jy(xo, ..., cn-], z) (147.7)

with a polynomial f(xo, . . ., x,,_1, z) from 3 [x0, . . ., xn_1i z].

We now consider the case p > 0. We write n uniquely in the form n = p e m (e > _ 0, p, m)

with two integers e, m. Then, over J"Tp,

X. - 1 = (xm - 1)p`.

(147.8)

If, for the time being, the left-hand side is taken as a polynomial over 3, it is, according to Theorem 315, equal to

fJ Fd(x)

dIn

Similarly, the right-hand side is equal to (Gd(x))'e,

where Gd(x) denotes the dth cyclotomic polynomial over p. If the Fd(x) are likewise taken as polynomials over .7p, then from (147.8) it follows that

fl Fd(x) _ JIM(Gd(x))" d.n Since this is valid for all n = 1, 2, ..., the F,(x) from this infinite system

NORMAL BASES OF FINITE FIELDS

563

of equations may be uniquely expressed by the Gm(x). This could be obtained from Mdbius's inversion formula, but here we immediately find the solution F,(x) = (Gm(x))9'(vq

(147.9)

by substitution in (147.8). Finally we have to note that equation (147.7) remains valid if we take the polynomials occurring in it as being over -9p (instead of,7). If in (147.7) we replace z by a primitive mth root of unity e, from the mth cyclotomic field over .5rp, for which Gm(e) = 0, then, according to (147.9), FF(e) =0, and In

f D = i=1

+ e`-1 x1 + ... + e (i-1) 0-1)

(147.10)

If we define el, ..., e,, by (147.6), then these are now exactly the mth

roots of unity (over 9,), each occurring exactly pe times. Equation (147.4), because of (147.8), then holds for them. Consequently (147.10) also implies the formula (147.3) for the case under consideration. Timoim& 365. Every finite field G I F has a normal basis. For the proof we denote the characteristic of F by p and put

q = O(F), qn = O(G),

(147.11)

where q is a power of p and n = [G : F]. Evidently all the conjugates of an element a of G I F are a, aq

(147.12)

That these should also constitute a normal basis of G I F it is necessary and sufficient that they be linearly independent over F, i.e., that for all the polynomials f(x) (E F[x], 0 0) of formal degree n - I (147.13)

R(a) 0 0,

where f (x) denotes the q-polynomial corresponding to the polynomial f (x). In the proof corresponding Roman and German letters will always denote

a pair of corresponding polynomials from F[x] and the Ore polynomial ring F [x]q. We first deal with the case p,f' n, where we wish to construct such an a. For that purpose we put

_e - I = .fi(x) ...A(Y)

(147.14)

with principal polynomials f1(x),..., fk(x) irreducible over F. By supposition these are distinct. We define the polynomials F1(x), ..., Fk(x) by x" - I = f ( x ) F , ( x )

(i = 1, ..., k).

(147.15)

THEORY OF FIELDS

564

From Theorem 364 it follows that

(i = 1, .... k).

f,(x) I xq"- x

Since G is a splitting field of the right-hand side and every f,(x) has at least two terms, there exist elements al, ..., ak (E G, # 0) such that

(i = 1, ..., k).

f;(a;) = 0

(147.16)

Since, according to (147.14), (147.15) for i j, always fj(x) I Fi(x), by Theorem 364, f j(x) 1 `;(x). Because of (147.16) this means that

i(a) = 0

(i 0j).

(147.17)

We now prove that for

a =ai+...+xk

(147.18)

(147.12) is a normal basis. To do so we assume that f(x) is a polynomial of degree < n over F with f(a) = 0.

(147.19)

If we show that then necessarily x" - 1 I f(x), from which f(x) = 0 follows. we shall have completed the proof. We write 9,(x) = F;(x).f(x) (i = 1...., k). (147.20) According to Theorem 364, f(x) ; g,(x). Therefore, by (147.19).

g;(a) = 0. After substituting (147.18) this gives:

3,(ai)+...+g(ak)-0. Further, fi(x) I g,(x) follows from (147.20) by Theorem 364. and because of (147.17)

t1,(aj) = 0

(i # j)

Consequently g,(ad = 0

(i = 1, ..., k).

(147.21)

We consider, for an arbitrary i, the ideal (fi(x), cl;(x)) of Fix]. By Theorem 364 there is an h;(x) (E F[x]) with (fi(x), g,(x)) = (f), (x)).

NORMAL BASES OF FINITE FIELDS

565

From (147.16), (147.21) it follows that 4i(a1) = 0. Since a; # 0, it follows that %(x), and also h;(x), contains at least two terms. Since Vx) ! f1(x) g1(C),

then, by Theorem 364, h1(x) I fi(x), gi(x)

Thus, because of the irreducibility off (x), it follows that fi(x), h,(x) ai e associated and so we have. i(x) I g1(x). This means that

.,(x) I F,(x) f(x) because of (147.20). But since, because of (147.14), (147.15), (1(x), F,(x)) _

= 1, so

(i = 1, ..., k).

l (x) I f(x)

Hence, and from (147.14), x" - I I f(x), which proves the theorem for the case p,I'n.

We now consider the general case. We have to prove that for a suitable x the elements (147.12) are linearly independent. This property is, according to Theorem 362, equivalent to the fact that the discriminant D(a, xy, ..., aq'-') is different from 0. This discriminant is evidently the square of the determinant a?i '- 2

1

of order n. Consequently, the theorem to be proved is equivalent to x`I" - X 'f' I X""'-2 I

.

Instead of this we may write

.r`'" - xf' I

x`7(;+i-2)

(147.22)

where (i + j - 2) denotes the least non-negative residue of i + j - 2 mod 11.

(Of course, both sides of (147.22) are to be interpreted as polynomials over F.) Thus we see that the theorem to be proved is equivalent to (147.22), therefore it suffices to prove (147.22). The preceding result may be formulated by saying that (147.22) is true

for p,I' n. For p 1n we put

M

-

(147.23)

p

and suppose that, contrary to the assertion, .X,711 _ CI IX4(i}I-_')I.

((47.24)

THEORY OF FIELDS

566

We shall show that (147.24) then holds for m instead of n. We shall then have proved the theorem since by repeated application we shall obtain the contradiction that (147.24) also holds for p, n. Because of (147.23) we have

.e - 1 = (xm - 1)p.

(147.25)

Over a suitable overfield of F

"' - 1 = (x - el) ... (x - em)

(147.26)

The right-hand side of (147.24) agrees apart from sign with the cyclic determinant whose first row is x, xQ, ..., xq'-'. This determinant, according to formula (147.3). and according to (147.25), (147.26), is the pth power of the polynomial

F(x) _

r=I

(,r + e1x4 +

... + e; -

(147.27)

Further, since the left-hand side of (147.24) has no multiple divisor, (147.24) is equivalent to x4"

- x I F(x).

We put

1(x)=x+xqm+x42m+

...

(147.28)

+x4"-m.

(147.29)

(The right-hand side consists of p terms.) Then q(x))ql

l + _ q;+M + X' +21N =.lq

Furthermore, according to (147.26), T = 1 (i = 1,

/

by (147.27) m

F(x) = In other words

(j = 0, 1, ...).

+ ... + xqJ+a-m

(1(x) + Q,#(x))4 + ... +

. .

.,

m). Consequently, er-'0(x))4°'-').

F(x) = G(1(x)),

(147.30)

where nt

G(x) _ fl (x + e1 x4 + ... + oj't-I

xqm- )

(147.31)

We see from (147.3) and (147.26) that G (x) is, apart from sign, the right-

hand side of (147.24) for n: instead of n. In order to reach the required contradiction, we have thus to show that [similarly to (147.28)] x4m - x I G(x).

(147.32)

NORMAL BASES OF FINITE FIELDS

567

We obtain, by Euclidean division,

G(x) = (xg"' - x)Q(x) + R(x),

(147.33)

where the degree of R(x) is smaller than q". On the other hand, according to (147.29)

(f(x))gM - 1 (x) = xg" - x, then, because of (147.33),

G(I(x)) as R(t(x)) (mod x"' - x). Since, according to (147.30), the left-hand side is equal to F(x), it follows, by (147.28), that tq"

- x R(I(x))

The degree of the i;ght-hand side is the product of the degrees of 1(x) and R(x), and so, by (147.29), is smaller than q". From this, R(x) = 0. Because of (147.33), we see that (147.32) holds. The proof of Theorem 365 is now complete. For a shorter proof, assuming more preliminary knowledge, see BOURBAKI (1939). Besides, it would have been sufficient to prove Theorem 365 for the case where F is the

prime field of G (i.e. q = p), but this would not have shortened the above proof.

CHAPTER IX

ORDERED STRUCTURES In the field fa the well-known ordering relation < may also be defined purely algebraically, that is, with the aid of the compositions valid in it. The same idea, by far-reaching generalizations, leads to the concept ofordered

structures. The investigation of ordered fields by ARTIN and SCHREIER was a pioneering work. Later SZELE and JOHNSON extended the examination to skew fields and rings, respectively.

§ 148. Ordered Structures

Let S denote a module or a ring, in particular a skew field or field. Correspondingly we understand by a positivity domain of S a semimodule or a semiring P of S, respectively, such that for every element a of S, of the three possibilities

x=0, xEP, -aEP

(148.1)

exactly one is true. For a given positivity domain P we call an element a(

0) of S either positive or negative with respect to P according as a E P

or -a E P. Further we introduce a relation < in S by the definition

a
(z,9ES).

(148.2)

We shall see that this relation is an ordering relation in S. We say that S is an ordered module or ring (skew field, field), respectively, with respect to the positivity domain P. We prove more precisely the following THEOREM 366. If S is an ordered module or ring (skew field, field) with respect to the positivity domain P, then an ordering relation is defined in it by (148.2), for which

x
(a,/9,yES),

(148.3)

and in the second case also

a.ya<;'/9, ay
(148.4)

ORDERED STRUCTURES

569

IJ; conversely, an ordering relation < is defined in S with the properties )148.3), (148.4) of which only (148.3) comes into consideration in the case of a module, then S is ordered with respect to the positivity domain consisting of the elements a (E S, > 0).

NoTE. If, in particular, S = .5 then the positive elements of ., in the usual sense, evidently constitute a semiring P with the property (148.1), so that 9ro is an ordered field. Accordingly, ordered structures are to be regarded as a generalization of the rational number field. The properties (148.3), (148.4) are called the monotonicity of the addition and multiplication, respectively, in ordered structures. From (148.3) the stronger property 7<

+<+

(148.5)

follows at once; furthermore +

x<j

< ;' a a <

<-a.

2 <0b -a;-0,

(148.6)

(148.7) (148.8) (148.9)

as is obvious (a, f3, y, b E S). The second part of the theorem implies that S

may be turned into an ordered structure by directly defining an ordering relation with the properties (148.3), (148.4) in it. It should be noted that the positivity domain P belonging to it is uniquely determined, since this must, obviously, consist of the elements a (E S) such that a > 0. Since S may in general contain several positivity domains, we denote the ordering rela-

tion belonging to a positivity domain more precisely by a < A(P) if necessary.

In order to prove the theorem we first suppose that S is ordered according to a positivity domain P. We have to show that an ordering relation is, in fact, determined by (148.2). This assertion consists of two parts, one that the rule

x<(1, f
(148.10)

holds for a, P, Y (E S) and the other that always exactly one of the three possibilities

holds for a, 0 (E S).

01=i4, a
(148.11)

ORDERED STRUCTURES

570

Since P is a semimodule, the rule:

fl -a.y-P EP=; -xr P holds because (/3 - a) + (y

x. Because of (148.2) this means

that (148.10) is satisfied. Since, by (148.2), (148.11) may be represented in the form

/3-x=0, f - aEP, - (i3-x)EP, we see from (148.1) that the assertion concerning (148.11) is true. From (148.2) the validity of (148.3) follows at once. If S is a ring, then from the left-hand side of (148.4)

/3-x,1'EP follows from (148.2). As P is a semiring, so

(8-a);'EP, i.e..

Yi3 - ya, /By - x;' E P. Because of (148.2) this implies the validity of the right-hand side of (148.4). This proves the first part of the theorem.

We now suppose that S is ordered and. that (148.3), (148.4) hold. [If S is a module then only (148.3) holds.] We denote by P the set of elements a of S with the property a > 0. From the supposition a, /3 > 0, by (148.3), (148.4), we deduce that a + /B, aj9 > 0. Hence it follows that P is a semimodule or semiring, according as S is a module or a ring. Further, because of the supposition, for every element a of S exactly one of the possibilities

a=0, 0
is true. We can write for this

a=0, x>0, -a>O. Hence we see that for P the condition required in (148.1) is satisfied. Hence P is a positivity domain. From the equivalence of the three propositions

a0, f - xEP we see that the supposed ordering relation (in S) corresponds to this positivity domain P. Consequently Theorem 366 is proved.

ORDERED STRUCTURES

571

For two elements a, P (a < f) from an ordered structure S, by the interval (a , f) we mean the set of those elements of S which satisfy the condition

A module or a ring is called orderable if it can be ordered in at least one way.

THEOREM 367 (LEVI'S theorem). A module M (0 0) is orderable if, and

only if, its characteristic is 0. If this is so and N is an arbitrary subsemimodule of M which does not contain the element 0, then M can be ordered in at least one way so that all the elements of N become positive. If M is ordered in some manner and a is an element different from 0 in M, then either c > 0 or a < 0. Hence, and from (148.9), not > 0 or

na < 0, respectively, for all n = 1, 2, .... Since then we have in both cases no, 0, M must, in fact, be of characteristic 0. Now suppose that M has characteristic 0. Then every element of M different

from 0 generates a subsemimodule which does not contain the element 0. i.e., which has the property required in the theorem. Therefore it suffices to prove only the last assertion of the theorem. With this end in view we consider the set of those subsemimodules of M

which contain N but do not contain 0. This set contains, according to Theorem 48, a maximal element P. Since this P is a subsemimodule of M which contains N, we have only to prove that P is a positivity domain. Since, further, P does not contain the element 0, it suffices to show that for every element a (# 0) of M either a or -a, but not both, belongs to P.

Since a + (-a) = 0 and P is a semimodule which does not contain 0, so of a, -a at most one can belong to P. We suppose that neither of these elements belongs to P; the contradiction arising from this will prove the theorem. On account of the supposition the semimodules {P, a}, {P, -a} are proper extensions of P, therefore both must contain the element 0. Accordingly, the equations

It+ma=0,

r-na=0

hold with appropriate 11, r (E P), m, n (E . 1 ) From these we have

nla+In r=0. But since the left-hand side is an element of P and the right-hand side not, we have a contradiction. Consequently Theorem 367 is proved. Because of this theorem we need only further examine a ring (skew field, field) R for orderability. Since, if R is ordered, the module R+ is also always

ordered (according to the same positivity domain), we may restrict ourselves, by Theorem 367, to the case where R is of characteristic 0.

ORDERED STRUCTURES

572

First, we observe the following: if R is ordered and a, j3 are arbitrary elements of R, other than 0, for which the refore one of the four case 1)

a>0, P>0.

2)

1a>0,

3)

(a<0,

4)

I J3>0.

1f3<0.

Jx<0, J3<0

must hold, then correspondingly we have

aJ3>0, af<0. a43<0, a#>0.

(148.12)

In case 1) the assertion follows from (148.4). In case 2) we have a > 0,

-# > 0, whence it follows first that -al4 > 0. so that a# < 0. Case 3) is the dual of 2). In case 4) we have -a > 0, -9 > 0, whence afl > 0 follows. Rule (148.12) is now proved. Since in all four cases a(3 # 0, this implies that every orderable ring is necessarily zero-divisor-free. Further, the next more general rule follows from (148.12): in an ordered ring R every product

al ... an (21, ..., an # 0)

(148.13)

is positive or negative according as the number of the negative factors is even or odd.

Now we introduce the following fundamental concept in an arbitrary ring R. We consider all the prod ucts of the form as /9i9 ... vv with factors from R different from 0 and the further products which are obtained from them by permutation of the factors (e.g., a2, afl2a, aflai9). The semiring P0 generated by these products is called the positive kernel of the ring R.

This terminology is justified by the circumstance that according to the rule formulated in (148.13) the generators mentioned, and so, generally, all the elements of P0, must be positive, no matter how R was ordered. We call a ring (skew field, field) formally real when its positive kernel does not contain the element 0. (The reason for the term "formally real" will be given in § 156.) THEOREM 368 (JOHNSON'S theorem). A ring R (# 0) is orderable if, and only if, it is formally real. According to what has been stated above the assertion "only if" is true.

In order to prove -if" we suppose that R is formally real, i.e., its positive kernel P0 does not contain the element 0. To every subsemiring H of R with the property H

Po

(148.14)

we construct a certain subsemiring H' of R such that H

(148.15)

ORDERED STRUCTURES

573

as follows: we consider all the products of factors (s 0) of R with the property that a number (> 0) of the factors belong to h' and the remaining factors may be collected in pairs of equal elements. The semiring generated

by these products is to be taken for H'. It is evident that this satisfies (148.15). We then denote by> the set of those subsemirings H of R with the property (148.14) for which the subsemiring H' does not contain the element 0. Since, in particular, for H = P, evidently H' = P,,, the set k) is not empty. For every chain

Hl cH., c....

which is formed of the elements of 5i, the semiring

H = H1UH.,U... exists, by Theorem 47. Evidently for the corresponding semirings H'. H H;, ... the similar relation

H' does not contain the element 0, therefore H E .. This implies that the semirings H1, Hz, ... have the upper bound H (in s)). From the Kux&TOwsKI-ZORN lemma (Theorem 16) it follows that 5')

contains at least one maximal element. We denote by P such an element and show that P is a positivity domain of R, which proves the theorem. (In addition, P' = P, but this may be disregarded.)

Since P is a subsemiring of R, we have only to prove that the condition (148.1) is satisfied. Because P E 9), 0 is not an element of P. Therefore

it will suffice to show that for an arbitrary a (E R, # 0) exactly one of a, -a belongs to P. We cannot have a, -a E P, for this implies that x + (-a) = 0 belongs to P, which is false.

We suppose that neither of a, -a belongs to P. The following contradiction will prove the theorem. Since a lies outside P, the subsemiring Pt = {P, a}

of R is larger than P, whence it follows that

0EPi.

(148.16)

We denote in general by rro a product of factors (o 0) of R for which a number (> 0) of factors belong to P and for which the remaining factors

ORDERED STRUCTURES

574

may be collected in pairs of equal elements. Further, we denote by nl every such element of R for which and is a no. It is evident that the semimodule generated by all the no, nt is equal to P;+. Because of (148.16) we therefore have an equation of the form

Yno +Ynl=0,

(148.17)

where the sums are finite and at least one is not empty. Moreover, since every no belongs to P' and 0 does not belong to P', it follows that, in (148.17),

the second sum is not empty.

But now, on account of the above supposition, the elements a., -a play the same role. Thus, there is also an equation

E n;, - Y nl = 0,

(148.18)

similar to (148.17), where again the second sum is not empty and the no, ni are defined exactly as the n and n,, respectively. From (148.17), (148.18) we get

(>no)(Eno)+(Y_ no (z a,') = 0. Since every product no no is a no and every product n, ni is likewise a pro,

it follows that the left-hand side belongs to P', thus 0 E P'. This contradiction proves Theorem 368.

NOTE. According to this theorem, "orderable" and `formally real" define the same property for a ring. Of these notations, both equally valid, we prefer the latter. It is evident that the subrings of a formally real ring are also formally real.

Theorem 368 is essentially simplified when a skew field or even a field is under consideration. Before discussing this we first note that the positive

kernel of a skew field F consists of all the non-empty sums of the form

(c,EF, 00; k>0).

a ...ak

(148.19)

This follows by complete induction from the definition of the positive kernel of rings and from m floc = Q2aV

(a, Y E F; Y o 0; e = aN ; U = 1' -').

... ak = (al ... ak)2, it also follows that the positive kernel of afield consists of all the sums of the form Further, since in the commutative case a` 72

+ ... + an

(ac, E F, 0 0; n > 0).

(148.20)

ORDERED STRUCTURES

575

THEOREM 369 (PICKERT-SZELE theorem). A skew field F is formally real

and only if, -1 may not be written as a sum of the form

1... aK

(ai E F).

(148.21)

Let P0 denote the positive kernel of F. Because of JOHNSON'S theorem (Theorem 368) it suffices to show that P0 contains the element 0 if, and only if, -1 is representable in the form (148.21). If this is so, then

+Ea...ak=0,

from which it follows that because 1 = 12, 0 E Po. Conversely, if 0 E P0, i.e., we have an equation of the form w1 .

.

. wr + E 4 ... Mk = 0

lwi, ai E F; wi # 0),

we obtain

1 = I (w; 1)2 ... (a) 1)2 a ... aR Consequently, Theorem 369 is proved. THEOREM 370 (ARTiN-SCHREIER theorem). A field F is formally real if,

and only if, -1 may not be written in the form a1 +

... + a

(ai E F).

(148.22)

This follows immediately from Theorem 369. Cf. ARTIN-SCHREIER (1926), SZELE (1952), JOHNSON (1952). Theorem 369 (for

the countable case) was first published without proof by PICKERT (1951) (p. 248, Exercise 13). Lately PODDERJUGIN (1954) has extended Theorem 368 to non-associative

rings (also with zero divisors), where one has to modify the ordering slightly.

THEOREM 371. If G I F is a field of (finite) odd degree and if F is formally

real, then G is also formally real.

For the proof we put n = [G : F] _- 1 (mod 2). Because of the supposition and LEVI'S theorem (Theorem 367), F is of characteristic 0, thus we may put G - F(ad). Let f(x) be the minimal polynomial of s over F. If n = 1 the theorem is true. If n >-- 3 we assume its truth for all odd integers less than n instead of n. If G is not formally real, then an equation of the form

-1=

k

i=I

(f(x))2 +.f(x)9(x)

(f1 x ......fk(x), 9(x) E F[x]), (148.23)

where all the fi(x) are different from 0 and of degree < n - 1, follows from the fact that G ki F[x]/(f(x)) and from Theorem 370. Let r denote the maximum of the degrees of f1(x)...., fk(x). Because of (148.23), the

ORDERED STRUCTURES

576

degree of f(x)g(x) is at most 2r. It must be exactly 2r, since otherwise from the comparison of the coefficients of x2r it would follow that in F the sum of the squares of certain elements other than 0 vanishes, which is impossible, since F is formally real. Consequently, the degree of g(x) is equal to 2r - n, and so an odd number which is smaller than n. Therefore g(x) has an irreducible factor gl(x) also of odd degree, smaller than n. We now take an extension field G, = F(t91) of F such that g1(O1) = 0. According to the induction hypothesis G1 is formally real. On the other hand, it follows from (148.23) that

This contradicts Theorem 370 thus proving Theorem 371. Of course we must modify the concept of isomorphism and substructure when applied to ordered structures. We call an isomorphism between two ordered structures S (148.24) S' (x -k x') an order preserving isomorphism of S with S', if the condition

x<#=* a'<X

(148.25)

is also satisfied. (Of course this also holds for "C+" instead of "r*".) If S = S' we then speak of an order preserving automorphism. It is evident that an isomorphism such as (148.24) is order preserving if, and only if, the image of the positivity domain of S is exactly the positivity domain

of S'. Two ordered structures are considered as being not essentially different when they are obtained from each other by an order preserving isomorphism.

Of two ordered structures S, T we say that T is an order preserving sub-

structure of S when T is a substructure of S and every relation a < (3, valid in T, retains its validity in S. We also say that the ordering of S is an xtension of the ordering of T.

If instead of a substructure T an ordered structure T' is embedded in an

ered structure S so that every relation a < fl, valid in S, retains its ty after the embedding, then we speak of an order preserving embedding.

tly this is possible if, and only if, T is order preserving isomorphic

a ollowing assertion is almost trivial: if S is a structure ordered sa to the positivity domain P, then every substructure S' of the as S is ordered according to the positivity domain

P,=S'nP ring of S is an extension of that of S'.

(148.26)

ORDERED STRUCTURES

577

It will suffice to prove the assertion, e.g., for the case where S, S' are rings. Then P, P' are semirings. Since, for an element x of S, of the three possibilities (148.1) exactly one is true, it follows from (148.26) that for an element x of S', of the three possibilities

a=0, aE_ P', - aE P' exactly one is true. According to this S' is ordered according to the positivity

domain P'. Finally, if for two elements a, (3 of S' the relation a < (3 (P') holds, then fi - a E P', and therefore (3 - a E P, i.e., x < (3 (P). THEOREM 372 (GRATZER - S CHMIDT theorem). If S is a zero-divisor-free

Everett extension ring of an ordered ring R (0 0), then the ordering of R may be extended in one, and only one, manner to an ordering of S.

Let P be that positivity domain of R according to which R is ordered. It is to be proved that P may be extended to a positivity domain of S in only one way. We denote by P' the set of additive inverses of P, i.e., that of negative elements of R and show first of all that for every a (E S, 0) either

Pa, aP S P or Pa, aP S P'.

(148.27)

Since R is an ideal of S and P c R, so Pa, aP S R. Moreover, since S is zero-divisor-free, Pa, aP S PUP'.

Thus in order to prove (148.27), it suffices to show that for any two

elements , i of P ya E P

ar E P.

(148.28)

part According to the principle of duality it will suffice to prove the of this. For this we suppose that the left-hand side of (148.28) holds. Then Earl E P, thus the right-hand side of (148.28) holds. Hence (148.28), therefore also (148.27), is proved. Now in order to prove the theorem we first suppose that Q is a positivity domain of S for which (148.29) Q ? P. We have to prove the uniqueness of Q. We even prove that Q consists of the a such that Pa, aP S P

(a E S)

(148.30)

holds.

For a E Q, because of (148.29), Pa, aP c Q. But Pa, aP S R. Therefore, since according to (148.26), P = R fl Q, (148.30) follows. Conversely, if (148.30) holds for a a, then a E Q follows because of (148.29).

ORDERED STRUCTURES

578

Hence we have proved that Q is the set of elements a having the property (148.30).

Conversely, let Q denote the set being considered. We shall show that Q is then a positivity domain of S and (148.29) holds, which will prove the theorem. From the definition of Q it follows that it is a semiring and that (148.29) holds. If we take (148.27) into consideration then it also follows that for every a (E S) exactly one of the three possibilities

a=0, aEQ, -aEQ holds. Therefore Q is a positivity domain of S. Consequently Theorem 372 is proved. Cf. GRATZER-SCHMIDT (1957).

THEOREM 373. The ordering of a ring R with at least one central element

(;4 0) may be extended in one, and only one, way to an ordering of the quotient ring R' of R. Let P denote the positivity domain of R belonging to the given ordering. Because of the supposition, R must be zero-divisor-free. Therefore [according to (47. 1)] the elements of R' are given by

x9-1

(a E R; fi E Z, 0 0),

(148.31)

where Z denotes the centre of R. (Cf. also Theorem 94.) First, we assume an extension of the ordering of R to an ordering of R'. Let P' be the positivity domain of R'. Then P = R n P'

(148.32)

holds. We shall show that the element (148.31) belongs to P' if, and only if, (148.33),

a.(3 E P,

whence the assertion of uniqueness of the theorem follows. If the element (148.31) belongs to P', then, because (32 > 0, so also does 2 x(3 = 00-1 F(148.32).

(148.34)

belong to P', thus (148.33) follows from If, conversely, (148.33) holds, then by (148.32) ocfl E P', consequently the element (148.31) belongs to P', because of (148.34) and f-2 E P'. The assertion is now proved. In order to prove the existence proposition in the theorem we denote by P' the set of those elements (148.31) for which (148.33) holds. We shall show that P' is a semiring. For that purpose we consider two elements. y(3

', ;)b-I

(x, y E R; /3, 6 E Z,

0)

ORDERED STRUCTURES

579

of P' for which therefore the conditions af3, yb E P

are satisfied. There

of -1 +

yb-'

= ((xb + yj9)

xii '

yb

xy(f b)-'.

(148.35)

Since the elements

xy 3b = a,B yb

(ab + fly) (f1b) = xf9b2 +

belong to P on account of the assumption, the left-hand sides of equation (148.35) belong to P'. Consequently P' is a semiring. Moreover, since of the three possibilities

xf1=0, af1EP. -x(3EP

(xER;fEZ,

O)

exactly one is true, for an arbitrary element (148.31) of R', of the three possibilities a. f3-' = 0,

x,B-' E

P',

- xft __' E P'

likewise exactly one is true. Accordingly P' is a positivity domain of R'. Since P' P, the ordering of R' according to P' is the extension of that of R according to P. Consequently Theorem 373 is proved. As GRATZER-SCHMIDT (1957) have observed, Theorem 373 includes the

commutative case of Theorem 372, since all the ideals (A 0) of a zerodivisor-free commutative ring have a common quotient field. In every ordered ring with unity element e, this is positive because e = s2. Consequently s, 2e, ..., must all be positive, too. Because of Theorem 373, the following then holds: THEOREM 374. The ring 7 and the field .,-.-0 hare each only one ordering, and their positivity domains consist of the numbers m and mn-'. respectively, (m, n E <.Y ).

NoTE. This simple theorem is of great importance as it furnishes the abstract algebraic definition of the positive integers and the positive rational

numbers. Namely, the positive rational numbers are essentially the elements of the positivity domain P of an ordered prime field Fo of characteristic 0. Further the positive integers are essentially the common elements of the module generated by the unity element of Fo and of P. Hence-

forth, if an ordered ring with unity element is being considered we tacitly identify the unity element with the number 1; this means the order preserving embedding of 7 or, for an ordered skew field or field, the order preserving embedding of.7o.

ORDERED STRUCTURES

580

EXAMPLE 1. The positive kernels of 3 and .moo consist of the positive integers and the positive rational numbers, respectively. EXAMPLE 2. Contrary to Theorem 374, the module .7+ has exactly two orderings

in which the positivity domain consists, in the one case, of 1, 2, . . ., and in the other, of -1, -2..... Yet, these two ordered modules are not essentially different since n -- -n is an order preserving isomorphism between them.

EXAMPLE 3. Let V be the 3-vector space with the basis a),, ..., co.. The elements of the form n) (CA > 0; ek, ..., C. E .7; 1 < k CkWA + . . . + of V constitute a subsemimodule which is obviously a positivity domain P of V. In V. ordered according to this P, a1W, + .

.

. + anw < b1w, + ... + bn("

(a1 b, E 9)

if. and only if, for some k (1 < k < it) ak< bk. EXAMPLE 4. If a ring R is formally real and P denotes its positive kernel, then, for arbitrary elements n, i4 (# 0) of R, efl2, nx1 E P0

(n == 1, 2, ....),

whence it follows that a2f2, nay ; 0, and then that 0, nac 96 0. This is the direct proof that a formally real ring must be zero-divisor-free and of characteristic 0 which, of course, follows from JOHNSON'S theorem (Theorem 368), too. EXAMPLE 5. The complex field .F-, (i) is of degree two and because i2 + 1 = 0

is not formally real. Accordingly, Theorem 371 loses its validity for fields G I F of even degree. EXAMPLE 6. The minimal extension ring with unity element of a formally real ring

is formally real as a consequence Of SZENDREI'S theorem (Theorem 201) and of Theorem 372. (Cf. the original proof of JOHNSON (1952) for his Theorem 368.) EXERCISE. From Theorem 97 it follows that: if a semi-skew-field is a positivity domain of its difference ring then this is a skew field.

§ 149. Archimedean and Non-Archimedean Orderings

An ordering of a structure S is called an Archimedean ordering, if for every

pair of elements x, t3 (> 0) of S there is a natural number n such that

not > .

(149.1)

Otherwise we speak of a non-Archimedean ordering. The above terms are derived from the axiom of Archimedes according to which.

for two intervals AB, CD, there is always a natural number n such that nAB > > CD. THEOREM 375. An Archimedean ordered ring is necessarily commutative.

and if it has a unity element then it has no non-identical order preserving automorphism.

ARCHIMEDEAN AND NON-ARCHIMEDEAN ORDERINGS

581

Let R denote an Archimedean-ordered ring. We consider two positive ) there is, according to (149.1), an elements a, /3 of R. To every m (E

n (E 4'') such that (n - 1)o c::!9 m(3 < na.

(149.2)

Hence

m(af - floc) = a m# - mil a < a na - (n - I )o:

of. = a2 .

Since this holds for all in (E -t'), it follows, again from (149.1), that

a/3-floc <0. By applying this to /3, a instead of a, /3 we obtain j9a - of <_ 0, so that a/B= floc. Accordingly, R is in fact commutative. Moreover we suppose that R contains the unity element e. Fore instead of a, (149.2) reads as follows:

(n - 1)85mfl
(n - 1)e_<m#'
-e<m(/l-/3')<8

for all in, whence We then have 0' = 0 so proving the theorem.. If an element a (> 0) of an ordered ring R with unity element has the

property that for all n = 1, 2.... either a > n or na < 1 holds, then we call a an infinitely large or infinitely small element, respectively. We show that the ordering of R is Archimedean if and only if, R has neither infinitely large nor infinitely small elements.

The assertion -only if" needs no proof. In order to prove "if" we con-

sider two positive elements a, /3 of R. If there are elements in, n (= 1, 2, ..

such that

a l ,

then a < mnf follows. This proves the assertion. The propositions a > n, na-1 < 1 are identical for an arbitrary element

a (> 0) in a field. Thus if a is infinitely large, then x-1 is infinitely small and conversely. Hence it follows that the ordering of a field is Archimedean if it has no infinitely small elements.

The extension of a non-Archimedean ordering can, of course, be only non-Archimedean. On the other hand, the following holds: THEOREM 376. In the case of an algebraic field G I F the extension of an

Archimedean ordering of the field F to an ordering of G is necessarily Archimedean.

ORDERED STRUCTURES

582

COROLLARY. An absolute-algebraic field can only be Archimedean-ordered.

in order to prove the theorem it suffices to show that to every positive element a of G there is a k (= 1, 2, ...) with a < k. Let n-t 1=0

be the minimal polynomial of a.. By hypothesis there is a k (= 1, 2....) such that

k> 1 -a, if a >_ k, then a

(1 =0,....n- 1).

1-a;, i.e.

a,> 1 -a

(i=0,...,it - 1),

from which it follows that n-t f(a) ;> an + (I - a)

57,

a' = 1.

i=0

Since this contradicts f(a) = 0, we have a < k. Consequently, we have proved the theorem and, on account of Theorem 374, the corollary also follows. EXAMPLE 1. Let V be the.3-vector space with basis w1, ..., w,. The ordering of V given in § 148, Example 3, is evidently non-Archimedean. We obtain an Archimedean

ordering of V if we take a real transcendental number 0 and define the positivity domain as the semimodule of those elements C1w1 + ... + Cn(un

(Cl, ..., c,, E .7)

of V, for which we have c1+c211

(This example relies, of course, on the usual ordering of the field of real numbers with which we shall only deal later on.) EXAMPLE 2. Let R = .fo[x]. The semiring consisting of the polynomials f(x)(r R)

with positive leading coefficients is a positivity domain of R. The corresponding ordering of R is non-Archimedean (although it is the extension of the Archimedean ordering of .? ). On the other hand, we obtain an Archimedean ordering of R if we take a real transcendental number 0 and define as positivity domain the semiring of those f(x) for which f($) > 0. EXAMPLE 3. The last two orderings of R = .$-0[x] obviously furnish, by virtue of Theorem 373, a non-Archimedean and an Archimedean ordering of the quotient field .7,,(x). In the first case an element a0x" + ... + any

b of T (x) is positive if ao b > 0. Moreover, all the order preserving automorphisms of .7-.(x) are then given by

x + cx + d, where c, d (e > 0) are elements of .?-,,. (For the second case cf. Theorem 375.)

583

ARCHIMEDEAN AND NON-ARCHIMEDEAN ORDERINGS

EXAMPLE 4. We construct a non-Archimedean-ordered skew field as follows: let

(r,(x)) = (..., r0(x), r1(x), ...) be an infinite sequence in both directions with terms from P-,(x) which, however, vanish to the left from a certain place on. in the set of these sequences we define addition and multiplication by (r,(x)) + (s,(x)) = (r;(x) + s,(x)), (ri(x)) (si(x))

rk(x) s,-k(x°k)), k

where we have to sum over k = 0, ± 1, ..., and a denotes a non-identical automorphism of , 0(x). It is easy to show that is a non-commutative skew field. Now order ."o(x) in a non-Archimedean way as in Example 3, and take an order preserving automorphism of .F'0(x) for ,s. Let an element of be positive if its first nonvanishing term is positive. A positivity domain is now defined in i. As for the rest, we can take the elements of it in terms of a new indeterminate y as a formal power series (cf. § 167)

The construction (without ordering) also holds if we take a field of prime characteristic p instead of .7-0, and leads in this case to a non-commutative skew field of the same characteristic p. EXERCISE 1. A homomorphism S - S' (fin -- o') of ordered structures is said to be order preserving if from e < a (o, a E S) o' < a' always follows. Show that the kernel of such a homomorphism, especially in the case of rings S, S' (0 0), consists only of 0 and infinitely small elements. (POLLAK.)

EXERCISE 2. Every ordered ring without infinitely large elements admits of an order preserving homomorphism, where the image of the ring is different from 0 and Archimedean-ordered. (POLLAK.)

§ 150. Absolute Value in Ordered Structures If S is a module, ring, skew field or field, then we obtain a very simple

classification of S by taking any two elements a, - a (a # 0) to form a class and taking 0 alone as a class. If, moreover, S is ordered, then every class contains exactly one non-negative element by which the classes are then represented. This happens in the following important definition: by the absolute value I a I of an element a of an ordered structure S we understand the non-negative member of the pair a, - a. The absolute value has the following properties:

10I=0,IaI>0(a0 0), ja.+

-a,

(150.1) (150.2)

(150.3)

(In modules only (150.1), (150.2) are to be considered.)

584

ORDERED STRUCTURES

Of these only (150.2) needs proof. Since

it follows that

x+ji.-a-rg
This establishes (150.2). We shall prove that from (150.1) and (150.2)

>_ilal-Iii

follows.

(150.4)

We apply (150.2) to a - f7 instead of x: i.e.

Similarly

Ia:_!9a-h'"+

Ia-i

flIlaI-

These may, because of (150.13), be summarized in

(lal - IfD

la! -

From this and (150.14), (150.4) follows. As a ccnsequence of (150.Q and (150.3) should be noted :

-'=x'".

(150.5)

NOTE. If S = .70 the absolute value I a I (a E .. T) is, because of Theorem 374, uniquely determined. Accordingly, the absolute value in an arbitrary S is

to be regarded as a generalization of the absolute value defined in S. An extensive further generalization in fields will be the subject of the next chapter. EXAMPLE. An ordered ring S is homomorphically mapped by at -- I a I, with respect to multiplication, onto the semigroup of non-negative elements of S.

CHAPTER X

FIELDS WITH VALUATION Soon after STEINITZ created the general theory of fields, KURSCHAK (1913) guided by the p-adic fields of HENSEL (1908) made a far-reaching discovery that the concept of absolute value in fields is capable of generalization. From this he developed the theory of valuation for fields, one of

the most important chapters of algebra with the widest possibilities of application. In this theory, among many others, the pure algebraic foundation of the notion of the field of real numbers was developed, where it came to light that the p-adic fields are analogous concepts. OsTxowsiu (1918, 1935)

has contributed largely to the further development of the theory of valuation. Concerning the treatment of the theory of numbers on a valuationtheoretical foundation cf. HASSE (1949). § 151. Valuations Let a field _9" and an ordered field F be given. The ordering relation in F is denoted by "<", the absolute value in it by "I I". Unless otherwise indicated, small Roman and Greek letters (also marked with indices) denote elements of .7 and F, respectively. The unity element is denoted in both fields by 1 without risk of ambiguity. Since, according to LEVI'S theorem (Theorem 367), F must be of characteristic 0, we may identify the prime field of F .70. These conventions also apply to the following paragraphs. We call .5T a field with valuation when 'a mapping a -+ T(a) of .5' into F is

given with the properties

qu(o)-0, q(a)>0(av

o).

(151.1)

qq(a + b) 5 q,(a) + q-(b)

(151.2)

T(ab) = q(a)p(b)

(151.3)

.

Then, q7(a) is called the value of the element a and F the value field and T itself is called a valuation, or, more precisely, the 99-valuation. 585

586

FIELDS WITH VALUATION

We have q{a - b) >_ I p(a) - p(b) I

(151.4)

,

q,(-a) = p(a) P(1) = I q.(a ' I) = p(a) I

(151.5)

(151.6)

,

(a # 0)

.

(151.7)

For b = 1, 99(a) = 99(oc)99(1) follows from (151.3). This together with (151.1) proves (151.6). (151.7) follows from (151.6) and (151.3). (99(- 1))' = 1 follows from(151.7), therefore 99 (-1) 1. This, together with (151.1), implies that q' (-1) = 1. Consequently, (151.5) follows from (151.3). 99 (a - b) >_ T (a) - T (b) follows from (151.2), with a-b instead of a. Similarly, 9,(b - a) ? 9'(b) - 9%(a)

also holds. Because of (151.5) the left-hand sides are equal and one of the right-hand sides agrees with the right-hand side of (151.4), therefore (151.4) is also true. Now we have to consider an important special case where -7itself is an ordered field and F = _9'. Let T(a) = I a I

This is possible since then (151.1), (151.2), (151.3) are satisfied according to (150.1), (150.2), (150.3). Hence in ordered fields the absolute value is a valuation which we call the absolute valuation and often denote by co:

co(a) = j a i

.

(151.8)

We have now shown that valuation is a generalization of absolute value. Evidently

q(0) = 0 , qu(a) = I (a # 0) ,

(151.9)

is always a valuation and is called the trivial valuation. THEOREM 377. In every field with valuation all the roots of unity have the value 1.

COROLLARY. The finite fields and, more generally, the absolute-algebraic fields of prime characteristic can have only the trivial valuation.

If 99 (a) > 1, then 99 (a) = 1 + S for some b > 0. Hence, and from the binomial theorem, 9' (a") = (1 + b)" > I f o r all n = 1, 2, ..., therefore a is not now a root of unity. If, furthermore, 92 (a) < 1, then q, (a-') > 1, thus a just as a-1 is again not a root of unity. This proves the theorem.

VALUATIONS

587

Since, in a finite field, the elements other than 0 are roots of unity, the corollary follows from the theorem. NOTE. We shall see that, besides the fields mentioned in the corollary, all other fields have non-trivial valuations. It is this generality which gives

valuations such a wide range of applications. According to (151.3), a valuation is always a homomorphic mapping of . into F with respect to multiplication. More precisely, according to (151.1), it maps .' into the semifield consisting of the zero element and the positivity domain of F. According to (151.2), homomorphism with respect to addition is not required. That, of course, would be impossible, too, since it would involve an isomorphism of F which is incompatible with (151.1).

§ 152. Convergent Sequences and Limits

We consider a valuation (p of the field .7 with the value field F and retain the conventions made in § 151 with respect to notation which we extend, however, by using here and in the following paragraphs i, j, k, n, r, s, N, N' to denote elements of 41'. The ordering relation is denoted by < both in 41" and in F. A countably infinite sequence al, a2, . . . consisting of elements of Y is denoted by [a,]. In particular, [a] will denote the sequence a, a,. . . We denote positive elements of F by s, S, p, v. We call [ai] a (p-convergent sequence if for every s (E F, > 0) there is a natural number N such that

4p(ai - ak) < a

(1, k > N).

(152.1)

We say that a (E.9') is the cc-limit of a sequence [ai] if, for every s (E F, > 0),

there is a natural number N such that

(i > N).

,p(ai - a) < a

(152.2)

We denote this by 92-lim ai = a.

(152.3)

(If necessary, the more precise notation' lim instead of lim is used. If 92-lim a; = 0

,

(152.4)

then we call [ail a p-zero sequence. This means, in other words, that for every

s (E F, > 0) there is a natural number N such that

rp(ai)<s

(i > N).

(152.4)

Further (152.3) is equivalent to the fact that [ai - a] is a q,-zero sequence. For a fixed valuation, we omit the attribute "q2" in the notation now introduced (and also in those to be introduced later on) when there is no

588

FIELDS WITH VALUATION

possibility of ambiguity. If Yis an ordered field and unless otherwise indicat-

ed, then in place of q' we have to understand the absolute valuation co. This will always be related to the ordered field F (instead of -7). The above concepts are well known in analysis for the case when 'P- (= F) is the field of real numbers, or of complex numbers, and ' means the "absolute valuation" in those fields. We shall not take this into consideration, however, but we shall obtain these fields as special cases of the present investigations.

We can write every element of .7as a limit, for a = lim [a]. We shall show that the limit of a sequence, if it exists, is always uniquely determined. To prove this we suppose that the sequence [a;] has two limits, a and b. Then, not only (152.2) holds, but, with a suitable natural number N'. also

q(a,-b)N').

For i > N, N' we have

9,(b-a)=q#a;-a)-(a;-b))<_q,(a;-a)+92(a,-b)<2e. If a

b fore=-2
THEOREM 378. Every sequence with a limit is convergent. Assume the validity of (152.3); then for every e (E F, > 0) there is a natural

number N such that (p(ai -- a),

q(ak - a) < 2

(i, k > N).

Since this gives

T(a,-ak)= p((a,-a)-(ak-a))
THEOREM 379. A sequence [a;] converges if, and only if, for arbitrary nl, n2, ... (E [a; - a;+,,,] is a zero sequence. COROLLARY. In afield, the set of convergent sequences is uniquely determined

by that of the zero sequences.

The assertion "only if" is trivial. In order to prove "if" we suppose that [a;] does not converge. Then there is an s (E F, > 0) such that we can find

for every i (> 0) suitable r, s (> i) with gq(a,, - a,) > 2e. Since we have 2e < q(ar

- a) < q(af - a,) +

cp(ai

- a,)

589

CONVERGENT SEQUENCES AND LIMITS

the first or the second term of the right-hand side is at least equal to e; we put ni = r - i or ni = s - i, respectively. Since, accordingly, we always have 99(ai - ai+R,) z e, it follows that [ai - ai+,,,] is not a zero sequence. Consequently the theorem is true, from which the corollary follows immediately. THEOREM 380. To every convergent sequence [ai] from Jr there is a p (E F, > 0) such that 92(ai) < i

(i = 1, 2, ...);

(152.5)

moreover, if [ai] is not a zero sequence, then there is also a I' (E F, for which, with a suitable natural number N, we always have cp(a) > v

(i > N) ;

0)

(152.6)

further, in the special case." = F, q' = co, this N may be chosen so that

either ai > r (i > N) or ai < - v (i > N).

(152.7)

From (152.1), it follows for k = N + I that q2(a,) < e + p(aN+i)

(i > N).

Accordingly none of the T (al), 97(a9).... is greater than ,u = max ((9'(a>.). .... q.(aN), e + (p(aN+I)) ,

so that (152.5) is proved.

If the assertion in (152.6) were false, then to every pair r, N there would be a k (> N) such that 92(ak) < r .

According to (152.1). N may be so chosen that for all i (>N) we have Ip(ai - ak) < I'. Hence 9,(a,) < 92(a, - ak) + 97(ak) < 2v .

If we apply this with v = - , then we see that [ai] is a zero sequence. This contradiction proves the assertion in (152.6). For the special caseY= F, 99 = w, (152.6) implies that for every i (> N) either a, > r or -a, > I'. Consequently, if (152.7) is false, for every N there

is a pair i, k (> N) such that a, > v, -ak > r.

Since

I

ai - ak I > 2r

follows from this, [ai] is not convergent. This proves Theorem 380. THEOREM 381. If [ai] converges so does [q,(a)]. If [ai] has a limit, then [92(a)] has too, and then lim 92(ai) = cp(lim a).

(152.8)

590

FIELDS WITH VALUATION

Since, according to (151.4), we always have I 9v(u) - 90) I < 9w(u - v) , from the definitions (152.1) and (152.2) it follows that the theorem is true. We define the sum and the product of two sequences [ai], [bi] by

[ai] + [bi] = [ai + bi], [ai] [bi] = [aib,] In this way we obtain a commutative ring which is isomorphic with the complete direct sum of a countably infinite number of fields.7. From here on we consider the sequences [ai] as elements of this ring. It is obvious that here [0] and [1 ] are the zero element and the unity element, respectively. The additive inverse of [bi] is [-b,] so for subtraction we have

[ai] - [bi] = [a, - bi] The multiplicative inverse of [bi] is [bt 1] provided that all the bl, b.2, .. . are distinct from 0; so for division we have [a,] [bi]-1 = [a, b, 1] THEOREM 382. LetO denote each of the four fundamental compositions. If the

sequences [a,], [bi] converge, then [a,Obi] converges; if they have a lim-

it, then [a,Obi] also has a limit, and lim (a,(Dbi) = lim a,Olim b, ,

provided that for division b1, b2, . . . are different ,from 0 zero sequence.

(152.9)

and [bi] is not a

We shall prove the theorem first for addition and multiplication. Let [ai], [bi] be two arbitrary sequences. Because of the identities

(u+v)-(u'+v')=(u-u')+(v-v') and uv - u'v' = v(u - u') + u'(v - v')

we have

9q((ai + b) - (at + bk)) < 92(ai - ak) + 9p(bi - bk) ,

(152.10)

92(aibi - akbk) < 9p(br)9v(a, - ak) + V(ak)99(b, - bk).

(152.11)

If [ai], [bi] are convergent, then for every s (E F, > 0) there is an N (> 0) such that E

92(ai - ak), 49(bi - bk) <

2

(i, k > N) .

From this, and from (152.10), it follows that [ai + bi] is convergent. Further, by (152.5), there is a p (E F, > 0) with 99(bi), 99(ak) < p p

(i, k = 1, 2, ...)

CONVERGENT SEQUENCES AND LIMITS

591

and to every s (> 0) an N (>0) such that 4v(a, - ak) , 92(b, - bk) < 2µ

(i, k > N).

Then the right-hand side of (152.11) is less than e, thus [a, b,] is convergent.

We now suppose that lim ai = a, lim bi = b. The above reasoning may be repeated if ak is everywhere replaced by a and

bk by b. We obtain as a result lim (a, + b,) = a + b, lim a,b, = ab. This proves that part of the theorem which relates to addition and multiplication.

Since if [b,] converges then so also does [-b,], and if lim b, = b then lim (-b,) = -b, the assertions relating to subtraction are reduced to the case of addition. Finally, in order to prove the assertions referring to division we suppose that the conditions stipulated at the end of the theorem are satisfied. We shall show that if [b,] converges then so also does [b, 1] and that if Jim b, = b then lim b, 1= b - '. This reduces that part of the theorem which refers to division to the case of multiplication.

If [b,] converges, then according to (152.6) there are certain v and N such that always (i > N). p(b) > v For every e (E F, > 0) N may be chosen so large that 99(bk - b) < v2e

(i, k > N).

From these it follows that 4'(bi 1 - bk 1) = 99(b,)-19P(bk)-199(bk - b) < e ,

therefore [bi 1] converges.

If, in addition, lim b, = b (# 0), then we may similarly conclude that lim bj 1 = b-1. For this, we have only to write everywhere b instead of bk and to subject v to the further condition 92(b) > v. Consequently Theorem 382 is proved. THEOREM 383. The convergent sequences [a,] constitute a ring in which the set it of zero sequences constitutes a maximal ideal. This it is called the ideal of zero sequences. The first assertion and the ideal property of it are a consequence of Theo-

rems 380, 382. It remains to be proved that the ideal it is maximal. For this it is sufficient to show the following: if [a,], [b,] are convergent se_ 20 R.-A.

592

FIELDS WITH VALUATION

quences, but [ai] is not a zero sequence, then [bi] is an element of the ideal

(n, [ai]). This assertion is equivalent to the following one: There is a convergent sequence [xi] such that [ai] [xi] _ [b,] (mod n)

.

(152.12)

Now, because of (152.6), the sequence [ai] contains only a finite number of vanishing terms, consequently, after the addition of a suitable zero sequence, a sequence [ai ] is obtained from [ai] with only terms different from 0. Then [a'] = [ail (mod n) . Since furthermore, according to Theorem 382, [a; ] is convergent and is not

a zero sequence, it follows, again from Theorem 382, that the equation

[ail [xil = [bil determines a convergent sequence [xi]. Thus (152.12) holds and Theorem 383 is true. We understand by a perfect field a field with valuation in which every convergent sequence has a limit. Sometimes the valuation of a perfect field is called a perfect valuation.

It is evident that a field with the trivial valuation is always perfect. We shall become acquainted with other perfect fields later. In perfect fields we shall introduce the concepts of infinite series and infinite products as follows: let. rbe a perfect field with respect to the valuation 99. To every countably infinite sequence [ai] formed of the elements of.Twe can assign, temporarily but formally, i.e., without any more closely defined sense, an infinite sum (i.e., consisting of an infinite number of terms) 00

i=1

ai=a1+a2+...

(152.13)

and an infinite product (i.e., consisting of an infinite number of factors) 00

ai = a1a2...;

(152.14)

q=

these are in general called an infinite series and an infinite product, respectively. The finite sums and products

si=a1+... +ai, pi=a1...a,, formed from them, are called the partial sums of (152.13) and the partial products of (152.14). More precisely, si is called the i`s partial sum and pi the ith partial product. We now call (152.13) a WW-convergent infinite series,

CONVERGENT SEQUENCES AND LIMITS

593

if the sequence [s,] formed from the partial sums is 97-convergent, i.e., because of the perfectness ofSf it has a 99-limit; then we say that this infinite series "exists" and is equal to this limit:

at = a1 + a2 + ... = 92-lim sl = 92-lim (a, + ...a)

.

1=1

Otherwise we call the infinite series 97-divergent. For this, it is usual to say that the infinite series does not exist. Furthermore, we call (152.14) a 92-convergent infinite product, if [pi] has a 92-limit different from 0; for this we say that the infinite product "exists" and is equal to this limit: OD

fl of = a1 a2 ... _ 92-lim pt = 92-lim a1 ... a; .

t=1

Otherwise we say that the infinite product is 92-divergent or "does not exist". From the definition it immediately follows that every existing limit may

be turned by the formula 92-I'M c, = c1 + (c2 - c1) + (c3 - c2) + ...

(152.15)

into an infinite series. Similarly we have the formula 92-lim c, = c1(c2cl 1)(c3e2 1) ...,

(152.16)

provided that the left-hand side does not vanish and no ci is equal to 0. Obviously, for the convergence of a1 + a2 + ... and a1a2..., the conditions lim a, = 0 and lim a, = 1, respectively, are necessary but, in general, not sufficient conditions. If necessary (152.13) and (152.14) are written more precisely as and

92-s al

99-11 a; , i=1

i=1

respectively.

Following the corollary of Theorem 379 we define two valuations of a field with the same ideals of zero sequences as equivalent.

NOTE. Let F be an ordered extension field of F whose ordering is an extension of the ordering of F. If 99 is a valuation of a field Y with the value

field F, then by the statement

(a) = 99 (a) (a E Y) a valuation 0 of. F is

defined with the value field F. Although these valuations at first sight appear

to have the same status, it may very easily happen that they are not equivalent. For example, if [al] is a 92-zero sequence whose terms are different from 0, and if F has a positive element a such that there is no element

sequence. (Cf. the e with 0 < e < E, then [a,] is evidently not a Example at the end of § 153.) However, we shall deal with the equivalence of valuations later in § 157.

594

FIELDS WITH VALUATION

§ 153. Perfect Hull It is of great importance that every field .7 with valuation - as will be seen - may be extended to a perfect field Y in which all the sequences, convergent in3, have a limit. Moreover we can subject .Y to certain further

conditions by which we attain (in a certain sense) uniqueness, but Y will depend essentially on the valuation of.''. This last fact affords many possibilities for the construction of overfields of a given field. By way of introduction we give some definitions. We say that a valuation q/ of an extension field 3' of the field 3, with valuation q', is an extension of the valuation 9), if the value field of 3' contains that of Jr in an order preserving way and, for the elements a of 3, q' (a) = 99'(a). If a fixed extension of 99 is considered, we often likewise denote it by 99.

A perfect extension field .3' of a field with valuation _9' is called a perfect hull of 3, if the valuation, of ' is an extension of the valuation 9) of .7 and consists of the qi-limits of the 92-convergent sequences formed from

the elements of 3. By a (bicontinuous or) topological isomorphism of a field .7 with a field c.P,

both with valuation, we understand an isomorphism of .7 with c.P which transfers the zero sequences, formed from .5r, just into the zero sequences, formed from cP. If there is such an isomorphism we call the fields Jr, cO (continuous or) topologically isomorphic. If between two fields 3, cP an isomorphism a --> a° holds and 99 is a valuation of .i , then a valuation 99° (with unaltered value field) is evidently defined in c.P by 99°(a°) = 97(a), such that a is now a topological isomorphism between _" and 4t.

Two extension fields c0, 4g' of a field .7 (all with valuation are called topologically equivalent (over ..3"), if their valuations are extensions of those of 3 and there is a topological isomorphism c.P 1.7 -- dt' 13. From now on let a field.7 with the valuation 99 and the value field F be given. (The notations are governed by the conventions made at the beginning of § 151, 152.) We shall construct a perfect hull of -7 and show afterwards that it is, in the sense of topological equivalence, uniquely defined.

To do this, we denote the ring of 97-convergent sequences [a;] by R and the ideal of zero sequences of ll by n. Since according to Theorem 383, n is maximal,

3 = T/n

(153.1)

is, by Theorem 130, a field. We denote its elements by [a1 ]' = residue class [a1] (mod n).

(153.2)

The special elements [a]' evidently constitute a subfield .7" of .7 such that

.7

.7' (a -# [a]') .

(153.3)

PERFECT HULL

595

We suppose that in .}' the corresponding embedding of Y instead of .g' is carried out, while we retain this notation for Y after the embedding, so that Y is now an overfield of Y. We define a certain valuation - (with a suitable value field) in Y, which will be an extension of q.'

We first deal with the case J r = F, q' = a). For this case we write F instead of .}'. In conformity with (153.2) we can take an arbitrary ele-

ment of F in the form [a;]' = residue class [a;] (mod n) , where it now denotes the ideal of zero sequences of F and [a;] is an w-convergent sequence. We call this element positive and write it as [aj]' > 0,

if there is a S (E F, > 0) and an N (E J) such that (i > N). a; > b (153.4) We have to show that the positive elements of F are uniquely determined by this. For this purpose we consider an element [fi]' of F equal to [a;]'. It has to be proved that the condition corresponding to (153.4) is also satisfied for fl;. The assumption implies the congruence [a;] _ [fli] (mod n). Since accordingly [(3; - aj] is a zero sequence, we can choose N above to be so great that besides (153.4) we also have

< Hence

a;

- 2s

,

(i>N).

then, because of (153.4),

(i>N), by which we have shown that in F the positive elements are uniquely defined.

Evidently these constitute a subsemiring of F which we denote by P. We show that P is even a positivity domain of F. For this purpose we consider an element [a;]' other than 0, of F, for which the sequence [a;] is therefore convergent, but not a zero sequence. It has to be shown that of the elements [a;]', - [a;]' exactly one belongs to P. According to Theorem 380, (152.7), there are a v (E F, > 0) and N (E 4") such that we have

either aj > v (i> N) or aj < - v (i > N). For the latter we may write -a; > v, so that condition (153.4) is satisfied,

for exactly one of the a,, - aj, i.e. of the elements [a;]', - [a;]' (= [-a,]') exactly one is positive. Accordingly P is in fact a positivity domain of F.

FIELDS WITH VALUATION

596

We put

P=Pf1F.

(153.5)

Since F is embedded in F, P consists of those a (= [a]') for which a > 0 (in F). This implies that P is the positivity domain of F. Since, because of

(153.5), P e P, the ordering of F according to the positivity domain P is an extension of the ordering of F. According to this the corresponding absolute valuation of F, which we denote by 11 11 or by w, is likewise an

extension of the absolute valuation of F. This will be taken as the required valuation of F, which we shall retain in what follows. Before passing to the general case we wish to show, as a preparation for it, that every co-convergent sequence [at] in F has an 6-limit and for it w-lim at = loci], .

(153.6)

We consider an arbitrary positive element [et]' of F. According to the definition in (153.4) there are then a 6 (E F, > 0) and N (E such that we always have et > 6 (i> e"). On account of the assumption, we may choose N so that

ai - akI <

2

(i,k>J )

also holds. Accordingly 6

6

et+ai - ak> 2 , et- at +ak> T. Again by (153.4), we have from this [et + at]' > ak, [ei - at]' > - Mk i.e.,

Vi]' + [ail' > ak,

[et]' - [ai]' > - ak ,

so that finally 11 Mk - [at]' 11 < [et]' for all k (> N). This implies (153.6).

We call attention to the consequence of (153.6) that every w-zero sequence is an c-o-zero sequence, which we shall often apply without reference.

We now consider the general case. We wish to define in .3r a valuation (P which is an extension of qq. (The preceding special case is not excluded. The

valuation to be given for this special case - as will be seen - turns out to be 161.)

Let [at]' be an arbitrary element of Y. We shall show that by 0([ai]') = w-lim q'(ai) (=[m(a,)]')

(153.7)

PERFECT HULL

597

a valuation 0 of . 1 ' (with the value field F) is defined with the desired properties.

For this, we must first of all show that the right-hand side of (153.7) exists and is uniquely determined by [ai]'. Since [ai] is qr-convergent, [ip(ai)] is, according to Theorem 381, an ca-convergent sequence. Hence, and from

what was proved above, the existence of the right-hand side of (153.7) follows.

It still remains to be shown that the equation cw-lim 9' (a) = w-lim 9' (b)

(153.8)

follows from [ai]' = [bi]' . By hypothesis [ai - bi] is a 99-zero sequence, consequently [q9 (ai - bi)] is, according to Theorem 381, an co-zero sequence. Because 99 (a) - q' (b,) I < 9, (ai - bi), 1

[9, (a) - 99 (b;)] is then an co-zero sequence. Hence, and from Theorem 382, (152.9) follows the assertion (153.8). What we have proved so far may

also be expressed by saying that a mapping 99 of Yinto F is defined by (153.7).

Further we have to show that ? is a valuation of Y. If [a,]' = 0, i.e., [ai] is a 9,-zero sequence, then [q:(ai)] is, according to Theorem 381, an co-zero sequence, thus, according to (153.7), ([a,]') = 0. If, on the other hand [a,]' 0 0, i.e., the p-convergent sequence [ai] is not a 9,-zero sequence, then (153.7) and (152.6) show that 99([a,]') > 0. Accordingly, 0 has the property given in (151.1). If 0 had not the property given in (151.2),

then there would be two elements [a,]', [bi]' of .i ' such that

([a,]' + [bi]')

- ([a,]') - 9'([b,]') > 0.

Hence, by (153.7), the existence of an s (E F, > 0) and N (E d) such that

,(ai + b) - 9,(a) - 9p(bi) ? s

(i > N)

would have followed. Since this is absurd, according to (151.2), it follows from this contradiction that Q' has the property given in (151.2). From (153.7) and Theorem 382, it follows that, together with 99, also has the property

given in (151.3). Accordingly, qi is in fact a valuation of Y. It is also an extension of q', for since Yis embedded in Y, it follows from (153.7), that 9'(a) = m([a]) = co-lim q,(a) = qu(a).

FIELDS WITH VALUATION

598

It still remains to be shown that in the special case Y = F, q = w we have 0 = w. For this case the definition (153.7) becomes: ik[ar]') = w-lim w(a;) . The right-hand side is, according to (153.6), equal to [w(a1)]'. This, according to the definition of tv, is equal to Hence the assertion has been proved. For the following we fix for the valuation 9 defined by (153.7), which in the case Y = F, 99 = w becomes the absolute valuation w. We now prove the following theorem: THEOREM 384. Every field J r with valuation has, to within topologically equivalent extensions, one, and only one, perfect hull, and that is the field Y defined above by (153.1) and (153.7).

First of all we prove that 3' is a perfect hull of Jr. (We continue to use the previous notation.) We begin with the proof that for every 99-convergent sequence [at], (153.9) -lim a, = [a,]'. (In the special case." = F, 99 = w, (153.9) becomes (153.6), which has already

been proved, but the following proof is also valid for it.) Take an element

is (> 0) of.}'. From (153.4) there exists a 6 (E F, > 0) such that

b<E. For every term a of

it follows from (153.7) that

([a,]' -

w-lim 99(a; - a.).

But there is an N (E,4) for which we always have

4'(ai-

a 2

<

E

2

(i,n>N).

Because of the preceding work we may choose N so that also

From these we get

<2' ([a,]' -

E

(n > N),

by which (153.9) is proved. This means that Yconsists of the -limits of the q-convergent sequences [a;].

In order to complete the proof that.3is a perfect hull ofYwe still have to verify only that

' is perfect. For this purpose we consider a qi-convergent

PERFECT HULL

599

sequence [ai] (formed of elements ai of 7), in order to prove that it has a -limit, which will verify the assertion. (We deal with a generalization of one part of the former statement involved in (153.9).) Now, first of all, we consider the case where we may form an (5-zero sequence [St] of the elements of F consisting only of positive terms. (There

is no need that this should always happen;

cf. HAUSCHILD-POLLAK

(1965).) Since according to (153.9) every element of.}' is the -limit of a 99-convergent sequence, there is for every i an ai (E 5) such that

,&i - ai) < bi

(i = 1, 2, ...) .

Further there is for every s (E F, > 0) an N(E- ) such that E

0(a-i - ak) , bi <

3

(1,k>N).

Since it follows from this that

9'(ai - ak) = 0(ai - ak) = m(Cai - ak) - (a, - ai) + (ak - ak)) < E (i, k > N), [ai] is a 99-convergent sequence. We write

a = t-lim ai . Then, for every s (> 0) from F there is an.N' such that

T(ai-a)<2

>

We can choose N' so great that also 1

a;<2 Hence

9'(ai -a-) = 0((ai - a'', + (ai - a.)) < e ,

so that O-lim ai = a. This implies that the assertion is true for the case considered.

In the other case there is no w-zero sequence in Y consisting only of positive terms. As [g1(ai - ai+i)] is an w-zero sequence, it has only finitely

many terms different from 0. According to this, there is an N (E "40") such that a, = dN(i > N) whence ip-lim ai = dN. This proves that.' is a per-

fect hull of Y. The proposition of uniqueness of Theorem 384 still remains to be proved. For this, we need the following preparatory 20/a R. - A.

600

FIELDS WITH VALUATION

THEOREM 385. Let.r "be a non perfect field with valuation with the value field F ands" a perfect hull of Y with value field F. To every positive element of F there is then in F a smaller positive element. (In this theorem ,7, F need not mean the fields so denoted, prior to Theorem 384.) For the proof let 99 and q denote the valuations of .7 ands', respectively.

We also consider-7 as being valued by 9-9 and denote this valuation of .' by 991. (The two valuations 99, q' of .7 differ from each other only in that F and F are their respective value fields.) We denote by 91 and 91, the ring of the q,- and q'1-convergent sequences, respectively, formed from.7. Further we denote by n and n1 the corresponding ideals of zero sequences of R and 911, respectively. We shall show that then 91 = ll and n = n1. As .9" is a perfect hull of .7, every 9,-convergent sequence from .7 has a -limit, thus it is a fortiori -convergent, i.e., qq,-convergent. According to this we have 919; B11. On the other hand, every 9,1-convergent sequence from .7 is also 9-9-convergent, thus, because F 9 F, it is even 9,-con-

vergent. Since now 911 c 91, it follows that 91= M1. Furthermore, since every 9,1-zero sequence from,7 is a 9-9-zero sequence, it is, again because F c F, a 9,-zero sequence. Therefore n1 c U. Since nl is, by Theorem 383, a maximal ideal of (91, =) 91, we have 111 = it. Now lets be a positive element of F. We have to show that there is an a in F such that 0 < s < s. First we consider the case where we can form a qq-zero sequence [ai] from .7 with infinitely many terms different from 0. We have already

proved that [a;] is a --zero sequence, so there is a term a, such that 0 < 99(a,) < e. Then, q (a;) = 99 (a,) belongs to F, so the theorem is true in this case. In the other case every 9,-zero sequence from 3 contains only finitely many terms different from 0. In every 9-convergent sequence from.' the

terms are then equal except for finitely many. This implies that every q-limit belongs to-9', i.e., Y=,7. Since this is, however, a contradiction, Theorem 385 is proved.

Finally the proposition of uniqueness of Theorem 384 is contained in the following as the special case .7 = cPt, a = 1: THEoREM 386. A bicontinuous isomorphism

' -- co (a - aa)

(153.10)

between two fields .7, a with valuation may always be extended to a bicontinuous isomorphism between two arbitrary given perfect hulls of .7T' and cQt.

We denote perfect hulls of _57 and &t by .t and

the valuation and

value fields of .7, &t, -7, c by q, V, , and F, G, F, G, respectively. If f is perfect, there is for every 9,-convergent sequence [ac] from . an element a of,7 such that a = T-lim a,. Since then a = q-lim a; must also hold,

PERFECT HULL

601

3 = .7. Also from the supposition, it follows that c0 is perfect, therefore i4 = c4. According to this the theorem is true in this case, so that we may henceforth assume that neither .7 nor cet is perfect. We denote an arbitrary 92-convergent sequence from.' by [ai] and shall show that (153.11) -lim a, --> i-lim aa, is a bicontinuous isomorphism of ..'onto c0. Since (153.11) is a continuation of (153.10), this will prove the theorem. The left- and right-hand sides of (153.11) run through all the elements of .f and 4 respectively, therefore 3 is mapped by (153.11) onto c4. We shall

show that this mapping is one-to-one. The proof will rely on the direct consequence of Theorem 385 that a sequence in Y is a i-zero sequence if, and only if, it is a p-zero sequence. ('If"' follows from Theorem 385, "only if" is trivial.) For two 42-convergent sequences [ai], [bi] from .7

-lim a, = -lim bi if, and only if, [a, - b,] is a -zero sequence, i.e., a 99-zero sequence, i.e., if [oai - vbi] is a y,-zero sequence, i.e., a yi-zero sequence, i.e., if -lim aai = +p-lim orb,.

This proves that the mapping (153.11) is one-to-one. Its homomorphy property follows from Theorem 382. If we show that every i-zero sequence becomes a yrzero sequence under (153.11), then for the same reason the remaining requirement will hold, thus completing the theorem. We take a sequence [a,] from Y. Every term ai of this sequence may be considered as a limit a, = 99-lim aik k

(i = 1, 2, ...) ,

where the aik are suitable elements of.'. We prove that [a-i] is a i-zero sequence if, and only if, to arbitrary natural numbers N1, N2, ... may be assigned further natural numbers n1(> N1), n2 (> NO ...

(153.12)

92-lim a,,, = 0 .

(153.13)

with the property r

First, we suppose that ri] is a -zero sequence and N1, N2.... are any natural numbers. To every positive element a of F there is then an N (E.4*) such that always

j(ai)<2

(i>N).

FIELDS WITH VALUATION

602

Also we may chose n1,, n2, ... satisfying (153.12) and such that IG (a,,,, - ai) < Hence 99 (a,n)

(t = 1, 2, ...) .

2

(ai,,,) < e, and we have proved (153.13).

Conversely, we suppose that for arbitrary N1, N2, ... the n1, n2, ... with the property (153.12) may be chosen in such a way that (153.13) is valid. Let is be an arbitrary positive element from F. According to Theorem

385 there is a positive element E of F such that 2e < E. We choose the N1, N2, ... in such a way that we always have &en, - ai) < e (ni > Ni, i 1, 2, ...) . Because of the supposition the nI, n.2,

. .

. may then be chosen in such a way

that for a suitable N (E J") we always have faint) _

faint) < e

(i > N).

Hence 45 (a) < 2 e < 9 (i > N), i.e., 15-lim a, = 0. Consequently we have shown that the assertion concerning the f)-zero sequences formulated in (153.12) and (153.13), formed from ", is true.

For the same reason a similar statement holds also for the

TP-zero

sequences formed from 4. From both these, and (153.10), it immediately follows that by (153.11) every q-zero sequence is in fact mapped into a w-zero sequence. Thus Theorem 386 is proved. EXAMPLE. In Theorem 385 it is necessary to suppose that F" is not perfect. Suppose

that the field.. has the trivial valuation, so that it is perfect and consequently equal to its perfect hull S r, and let F = i and F = .moo (x) be the value fields of .7" and X. respectively, where F, as in § 149, Example 3, shall be ordered in a non-Archimedean way. (See Theorem 374.) Now x-1 is a positive element of F, and there is no element

eofFsuch that 0<e<x-'.

§ 154. The Field of Real Numbers

As one of the most important applications of valuation theory first of all arises the field of real numbers which we define as the perfect hull of the field 5 with the absolute valuation and denote by o). The elements and subfields of 9( 'o) are called real numbers and real number fields, respectively. By Theorems 384, 385, o; is an Archimedean-ordered field provided with the absolute valuation. We always denote the ordering, the absolute value and the limit by <, I I and lim, respectively. THEOREM 387. Every Archimedean-ordered field F, perfect with respect to

the absolute valuation, is order preserving isomorphic with .f(o). Further every Archimedean-ordered module M or ring (field) R is order preserving isomorphic with a submodule or subring (subfield) of ol.

THE FIELD OF REAL NUMBERS

603

It may be supposed that is a subfield of F. We denote by co or I I the absolute value in F. By Theorem 386, for the first assertion of Theorem 387, it will suffice to prove that every element a of F is the limit of a sequence from Y0. By the supposition, to every natural number n there is one and only one integer kn with kn 5 not < k,, + 1. We put an = n-'I,,. Then we have an 5 a < an + + n-1, so that

(n= 1,2,...).

Ia - ocI < n-I

On the other hand, for every element a (> 0) of F there is a natural num-

ber N (> s-1). Then for n > N

Ian-al 0) and a further element a from M. Then, for every natural number n there is an integer kn with k,,y 5 na < (k,, + 1)y. Evidently lim n-1kn = a

exists and depends only on a. Further a -* a is an order preserving isomorphism of M into 9'( ). The assertion about R may be reduced to the case where R is a field. For, according to Theorem 375, R is commutative and zero-divisor-free, so that, by Theorem 373, the ordering of R may be extended to an ordering of its quotient field F, which is obviously Archimedean. The ordering of F may be extended further to an ordering of its perfect hull F. Since, according to the first assertion, already proved, of the theorem, F is order preserving isomorphic with 7(o), the proof is now complete. A function f (x) defined in o) is called continuous at the point x = c

if, to every element a (> 0) of

0?,

there is a further element b (> 0)

from 30) such that the condition

I1(c+h)-f(c)I <e

(IhI <6)

is satisfied. Iff(x) is continuous at every point of the interval (a, b), then we say that f(x) is continuous on this interval. Finally, a function f(x) is said to be continuous if it is continuous at every point of its domain of definition. THEOREM 388 (BOLZANO'S theorem). If a function f(x) defined in

O) is

continuous on the interval (a, b) and f(a) < 0 and f(b) > 0, then it has at least one zero in this interval.

For the proof, we suppose that the theorem is false. We write a0 = a, b0 = b and determine the real numbers a,,, b,, (n = 1, 2, ...) so that an+I = an, bn+I = 2 (an + b,,) or an+I = 2 (an + bn), bn+l = bn ,

FIELDS WITH VALUATION

604

respectively, according as f12 (a

+

I

is positive or negative. Then

111

ao < al S ... and bo > bl. >_ ..., and

f(a,J<0, f(b,J>0, b"-a"=2-"(b-a)

(n=0,1,...).

It is evident that the sequences [a"], [b"] have a common limit. We put

c=lima"=limb". Then a S c < b and

a"
(n=0,1,...).

Furthermore, by hypothesis, J(c) 96 0.

Therefore, and because of the continuity of f(x), we can choose a number S (> 0) so that f(x) is either positive everywhere or negative everywhere on the interval (c - 8, c + b). But since, for sufficiently large n, a", b" lie in this interval, we reach a contradiction, so proving Theorem 388. so

If two functions f(x), g(x) are continuous at a point x = c, then also are f(x) + g(x), f(x) - g(x), f(x)g(x) and (for g(c) # 0)

f(x)g(x)-1, too. We need not prove this since the proof is similar to that of Theorem 382. It follows that every polynomial f(x) (E 9( 'O)[x]) is continuous. Tm;oREM 389. Every polynomial fl x) over .5o) of odd degree has at least one zero in o). We may suppose that f(x) is a principal polynomial:

fix) = x" + alx"-' + ... + a"

(2,f' n)

.

Take a real number c such that

c> l,c> Ia,I +... + Then f (c) > 0, f (- c) < 0. This, by Theorem 388, proves Theorem 389. Tm?OREM 390. Every equation

x"-a=0

(n>-1; aE.5o),>0)

(154.1)

has exactly one positive root in -7(o. Since the left-hand side of (154.1) is negative for x = 0 and positive for x = 1 + a, so by Theorem 388, (154.1) has at least one positive root. There

cannot be two positive roots since b" < c" follows from 0 < b < c. The theorem is now proved.

THE FIELD OF REAL NUMBERS

605

Because of this theorem we introduce for the positive root of the equation (154.1) the notation

a(154.2)

We define more generally the power a' for a positive real basis a and for a rational exponent ,bmy putting amn-1

= (a n-I

(a E _7(o), > 0; m E 7, n E -44'

.

(154.3)

It is now necessary to show that the right-hand side depends only on a and mn-1. In other words, we have to show that we always have (am)"-' = (ak)'-' (m, k E .7; n, I E -t'; mn-1 = kl-1). The left-hand side and the right-hand side are positive roots of the equations

x" = am and x = ak , respectively. Because of Theorem 390, we may replace these equations, without altering their positive roots, by xn' = ank.

xn1 = am' ,

Now, because ml = nk, these equations are identical, whereby the uniqueness of definition (154.3) is proved. We shall prove that

(ab)' = arbr

,

arts = eras,

(a')s = ars

(a, b E (o), > 0; r, s E Yo) .

(154.4)

First of all, it follows from Theorem 390 that (154.43) is true ifs is a natural

number. Let us therefore take two natural numbers m, n such that rm, sn are integers, then ((ab)r)m = (ab)rm = a'mb'm

= (ar)m

(b')m

= (arb')m,

(ar+s)mn = a(r+s)mn = armn asmn = (ar)mn (as)mn = (ar a3)mn ,

((a)s)mn = (ar)smn = arsmn = (ars)mn .

From these equations, and again from Theorem \\390, it follows that (154.4) is true. Further, we shall prove the two properties of monotonicity

0
(a,bE_9(o); rEYo,>0), (a E 3(o), > 1; r, s E .7o)

.

(154.5) (154.6)

606

FIELDS WITH VALUATION

For the proof of (154.5) let m denote a natural number such that rm is an

integer. Then a'' < b'm, (a'), < (b')', from which (154.5) follows. To prove (154.6) we take two natural numbers m, n such that rm, sn are integers. Because rmn < smn we have 1 < asm ,", and arm" < asm", whence (154.6) follows. We define the power a` for every positive real number a and for every real number c by

d'= lim a(n

(c=limes),

(154.7)

where [c"] is an arbitrary sequence from 70 with limit c. To justify this definition we must show that the right-hand side of (154.7) always exists and is independent of the special choice of [c,, ], i.e., it is uniquely determined by a, c. From this, and because c = lim c, it follows that (154.7) becomes a

valid equation for a rational c, thus giving a generalization of the concept of power.

We shall shorten the proof somewhat. We shall show, first, that to arbitrary elements a (> 1), e (> 0) of .5 o) there is an N (E 41') such that

an-' < 1 + e

(n > /V).

(154.8)

Since (154.8) is equivalent to a < ([ + E)" and this is valid for every n such that a < I + n e, the assertion is true. Now we shall show that the right-hand side of ([54.7) exists for every convergent sequence [c"] from .70. If a = 1 it is trivial. Furthermore, since (a-')`" = (a`")-', it suffices to consider the case a > 1. We take an integer r such that r > C1,C2,... Let a be an arbitrary positive real number. On account of (154.8), we take an n with

a"-'<1+a'e.

Then we take an N' (E a-) for which we always have

Ck - c,I
(k,1>N').

Then

a`k - ac' 15 a'(d`k`h' - 1) < a'(a"-I - 1) < g. Since accordingly [a`"] is convergent, the right-hand side of (154.7) exists.

In order to prove the uniqueness of the definition (154.7), we take a further sequence [d"] from .ro such that c = lim d,,. We denote the "mixed" sequence c1, d1, c2, d,., ... by [e"]. According to the preceding work, all three limits lim a'R , lim a" d , lIm a`"

THE FIELD OF REAL NUMBERS

607

exist. Since the first two limits separately must be equal to the third, they are equal to each other. Accordingly, a` is uniquely defined by (154.7). We leave to the reader the simple proof that (154.4), (154.5), (154.6) also remain valid for real numbers r, s and then a` is a continuous function of both variables a, c. The definition of an ordered module will henceforth also be related to the multiplicative notation. The meaning of an ordered Abelian group is now obvious. We have to interprete correspondingly order preserving automorphisms and isomorphisms, in connection with Abelian groups, and we may speak, in the corresponding sense, of the order preserving isomorphisms between two structures, one of which is an Abelian group, the other a module. We denote the positivity domain of .7jo) by P. This is the subsemifield of .7"(o) consisting of the positive elements. Correspondingly, the group of positive real numbers is denoted by P". This is ordered and has for its positivity domain the semigroup of those real numbers which are greater than 1. THEOREM 391. All the order preserving automorphisms of the group of positive real numbers are given by

a-->a`,

(154.9)

where c can be any positive real number. (This theorem may be regarded as a definition of the power a` for a, c > 0. Since a° = 1 , a = (a`)-1, we can now arrive at the complete definition.) For the proof of the theorem we denote the group of positive real numbers, as above, by P'<.

First of all we shall show that an order preserving automorphism of P"

is given by (154.9). If b is an arbitrary element of P", then (b`-')` = b. Hence we see that P" is mapped onto itself by (154.9). Furthermore since, as seen above, (154.41), (154.5) hold for c instead of r, (154.9) is in fact an order preserving automorphism of P'. Conversely, we consider an order preserving automorphism a -->f(a)

(154.10)

f(ab) = f(a)f(b) ,

(154.11)

a < b = f(a) < f(b) .

(154.12)

of P". Then for a, b E P"

We first show that

f(d) = (f(a))`

(a E P", I E J) .

(154.13)

608

FIELDS WITH VALUATION

For t = 0 it is trivial. Further, according to (154.11), f(1) = 1, f(a-t) = = (f(at))-1, while (f(a))-t = ((f(a))t)-1, so that it is sufficient to prove (154.13) for t > 0. We put t = kl-' for two natural numbers k, 1. Because of (154.11) we then have ((d ))h' = f(a) = f(ak) = (J (a))k = ((f(a))t )f .

Hence, and from Theorem 390, (154.13) follows. We shall also show that every equation

a"=b

(a,bE. (0,; a>0,b>0; a

1)

(154.14)

is solvable with a real number x. For, it is obvious that there are integers k, l such that ak < b < ar. Hence, and from the continuity of ax, according to BoLZANO's theorem (Theorem 388) the assertion follows. We now take a fixed element b (# 1) and an arbitrary element a of P" . According to what has been proved for (154.14),

f(b) = b` , a = bd

(154.15

for two suitable real numbers c, d. We take two sequences [r"], [s"] from .moo such that

r1, r2, ... < d < sI, s2, ... ,

lim r" = lim s" = d.

(154.16)

(154.17)

From (154.152), (154.16) we get

b'"
(n= 1,2,...).

Hence, and from (154.12),

.f(b'")
(f(b))'" < f(a) < (f (b))s"

,

so that, by (154.17) and (154.15), .f(a) = (f(b))d = (bc)d = (bd)` = a` .

Here, because of (154.12), c must be positive. Consequently Theorem 391 is proved. By this proof we have also shown that (154.14) always has a solution x in Yob. Of course there is only one solution. We call it the logarithm of b

THE FIELD OF REAL NUMBERS

609

(to the base a) and denote it by logo b. This definition is representable by the formula (154.18) (a, b E .3lo). > 0; a # 1) . b aIO$ab For log. b we write log b when the base a is fixed. Without any difficulty we can obtain the rules

log. be = log. b + log. c loga b` = c log. b"

(b, c > 0) , (b > 0) .

(154.19) (154.20)

THEOREM 392. All the order preserving isomorphisms of the group of positive real numbers onto the module of real numbers are given by a -+ log, a,

(154.21)

where c can be any real number > 1. (This theorem may also be regarded as a definition of the logarithm log, a for c > 1. From here, since log,_ I a = - log, a, we can obtain the complete definition.) The assertion of the theorem that (154.21) yields only the desired isomorphisms is easy to see on account of (154.19) and (154.6). We choose one of the isomorphisms (154.21) and denote it by at,. All the required isomorphisms are then given by

a=ace, where e runs through the order preserving automorphisms of the group of positive real numbers. According to Theorem 391 these a are given by

a--Boa=a', where r is a positive number. For a (E F'(o), > 0) we have, because of (154.20)

as = a. ea = aca' = loge a' = r loge a. But since

(Cr-I)rbca = clogca = a

we have as = logs a with d = c'-'. This proves Theorem 392. As regards Theorem 391, it should be noted that the group of positive real numbers also has automorphisms which are not order preserving. The same holds with respect to Theorem 392. Cf. HAMEL (1905). For the theory of real numbers see KALMAR (1952).

610

FIELDS WITH VALUATION

EXERCISE 1. A set S of real numbers bounded above has one and only one upper bound c with the property that every interval (c - h, c) (h > 0) contains at least one element of S. We call c the Weierstrass upper bound of S. EXERCISE 2. By a partial sequence of a sequence [a1] let us understand every sequence [a,,] with 1 < n, < n2 < ... Every sequence of real numbers in an interval has convergent partial sequences (BOLZANO-WEIERSTRASS theorem).

§ 155. The Field of Complex Numbers

For every real number a, a2 + 1 > 0, so that the polynomial x2 + 1 is irreducible over .7(O). Thus according to Theorem 295 the field .o)(i) with i2 + I = 0 exists and we call it the field of complex numbers. Its elements and subfields are called complex numbers and complex number fields, respectively. The real number fields also occur among these but are usually excluded.

Since -7to) (i) is algebraically of the second degree over 20), the complex

numbers may be taken uniquely in the form a + bi, where a, b denote real numbers. These are sometimes called the real part and the imaginary part of the complex number. THEOREM 393. When, in an ordered field F, every positive element is a square

and every polynomial from F[x] of odd degree has a zero in F, then the complex field F(i) is algebraically closed. COROLLARY (the so-called Fundamental Theorem of Algebra). The field of complex numbers is algebraically closed.

In order to prove the theorem, first of all let us refer to the fact that, according to the JOHNSON'S theorem (Theorem 368), F is formally real.

Since, according to this, the equation a2 + b2 = 0 has only the trivial solution a = b = 0 in F, by Theorem 177, the complex field F(i) does in fact exist. We consider an arbitrary non-constant principal polynomial f(x) over F(i). We have to prove that this has a zero in F(i). First of all we shall prove this assertion for the special case where f(x)

lies in F[x]. Let n = 2`no

(2` 11 n)

(155.1)

be the degree of f(x). If t = 0 then 2 ,j' n, therefore f(x) has a zero in F. If t > 0 we assume the assertion for t - 1 instead of t. We put

f(x)_(x-00...(x- 00, where al, ..., a are elements of a splitting field

G = F(i, al, ..., cc ) of f(x) over F(i). It will suffice to prove that at least one of the al, ..., an necessarily lies in F (i).

THE FIELD OF COMPLEX NUMBERS

611

For this purpose we put /I* _

I2 "I

= 2'-'no(n - 1)

(155.2)

and form the polynomials

Fk(x)_

fl

(x-a,as-k(a,+x,,))

(k=1,...,u*+1) (155.3)

of degree n*. Temporarily we consider the a.I, ...,x as indeterminates. Fk (x) is then a symmetric polynomial in al, ..., x,,. Hence by the main theorem for symmetric polynomials (Theorem 268) it follows that Fk(x), together with f(x), lies in F[x]. Further, because2,f'no(n - 1) (since t > 0 and (155.1), holds), and for the degree n* of Fk(x) the factor decomposition

(155.2) holds, it follows from the induction hypothesis that every Fk (x) has at least one zero in F(i). Now the number of pairs r, s occurring in (155.3) is equal to n*, whereas the number of polynomials Fk(x) equals n* + 1. Consequently there is a pair r, s (1 < r < s <- n) and two distinct k, 1(= 1, .... n* + 1) such that

a,a, + k(a, + a,) , x,as + 1(a, + x.,) both he in the field F(i). This is of characteristic 0, so that aS E F(i).

(155.4)

Hence

(ar - a,)2 E F(i) .

(155.5)

But the formula

(a + bi) (2a + 2 , /a2 + b''=) _ (a + Jag + b2 + bi)'2

(a, b E F) (155.6)

holds in F(i) where the radical Ja` + b2, according to the supposition, is an element of F. We may suppose that this is non-negative. The second factor of the left-hand side of (155.6) is then a non-negative element of F and

vanishes only for a = b = 0. Again according to the supposition, it also follows from (155.6) that every element of F(i) is a square in it. Therefore (155.5) may be replaced by or, - a,, E F(i). Hence, and from (155.42), we obtain a a, E F(i) by which the assertion for the special case f(x) E F[x] is proved.

From here the step leading to the general case f(x) E F(i)[x] is not difficult. We write f(x) = g(x) + ih(x)

FIELDS WITH VALUATION

612

with two polynomials g(x), h(x) from F[x]. The product

(g(x) + ih(x)) (g(x) - ih(x)) = (g (x))2 + (h (x))2 lies in F[x], and so has a zero a + bi in F(i) (a, b E F). If it is a zero of the first factor, then the proof is complete. Otherwise

g(a + bi) - ih(a + bi) = 0. Applying the isomorphism i --> -i of F(i) I F we get

g(a - bi) + ih(a - bi) = 0, i.e., f(a - bi) = 0. Consequently Theorem 393 is proved. As the suppositions of this theorem, according to Theorems 389, 390, are valid for F = Y(o), the corollary is likewise true. The above proof goes back to LAGRANGE whose explanations were, however, incomplete in one point. The first complete proof of the Fundamental Theorem of Algebra was due to GAUSS. According to STEINITZ'S theorem (Theorem 300) every field has an algebraically

closed extension field but only the existence of such a field is proved so that the importance of the Fundamental Theorem of Algebra remains undiminished. Although

it was from the very first called the Fundamental Theorem of Algebra, it achieved the character of a pure algebraic theorem only since the researches of ARTIN - SCHREIER

(1926). Furthermore it should be pointed out that the Fundamental Theorem of Algebra, in spite of its importance, in the present stage of development of algebra is not correctly named, but this name is retained for historical reasons.

By Theorem 328, . 7(0) (i) I.9r o) has only one non-identical automorphism

i ->- - i. Since here a + bi becomes a - bi (a, b E Y(o)), we call these numbers conjugate complex numbers (more precisely "conjugate complex numbers over .Fto,"). Here two conjugate complex numbers are denoted by a, &, and we say that a is the (complex) conjugate of a.

The field Y(O)(i) is not orderable since i2 = - 1, so that we can not define any absolute value in it in the usual sense of the word. But we call the non-negative real number I

a I = (acac)1 = (a2 + b2)1 ,

(155.7)

by long-established custom but very incorrectly, however, the absolute value of a for a complex number a = a + bi (a, b E to)) We shall show that this defines a valuation which we (again incorrectly) call the absolute valuation of Jr(o)(i). [Since we are dealing with an extension of the absolute valuation of Y(o), the absolute value in _7 (o) and Y(o)(i) can be similarly denoted.]

It is evident that I a = 0 only for oc = 0. For two arbitrary complex numbers a, i we have aP6i4 = oca(3B, whence, because of (155.7), the homo-

THE FIELD OF COMPLEX NUMBERS

613

morphy property I ap I = I a I I P I follows. It still remains to be shown that

I«+flI
This inequality is, because of (155.7), equivalent to any of the following:

«+# I2

(Ial+I

I)2,

I, («P+«p)2<4«aRP,

(aPizfl)250.

The last inequality is trivially true, since the real part of MA - a# vanishes. This proves the assertion. The absolute algebraic complex numbers are called algebraic numbers. These constitute a subfield of .7to) (i) which we denote by vl?o and call the field of algebraic numbers. According to Theorem 297, ,,eo is algebraically closed and at the same time the algebraic hull of .70 in.9r(o)(i). It follows also that o is the algebraic closure of Yo. (This explains the notation we have introduced.) The algebraic number fields shall simply mean the subfields of -eo. The real elements of,,-go are called the real algebraic numbers. These constitute the field of real algebraic numbers which is obviously equal to

o n .(o). THEOREM 394. Every ordered absolute algebraic field is order preserving isomorphic with a real number field. This follows from the corollary of Theorem 376 and from Theorem 387. EXAMPLE 1. For the square root of complex numbers we have the formula

Ja + bi =

ja+Vas+b$ + 2

i

-a+ Jas+b2 2

(a, b E Jr(,))

(155.8)

where we have to take for the radical aE -+b2 its non-negative value and for the remaining two radicals on the right-hand side such values that their product results in b. [This formula can also be easily obtained from (155.6).] EXAMPLE 2. Although K,0)(t) is not orderable, there are orderable (non-real) complex

number fields. In particular it follows from Theorem 394 that a complex number field of finite degree is orderable if, and only if, it is isomorphic with a real number field. For instance, let us take the three roots al, aE, as of x3 - 2 = 0 from X (0)(t). Among the three conjugate fields Kto) (ak) (k = 1, 2, 3) there is one real and two complex ones. Thus all three are orderable. EXAMPLE 3. From Theorem 394 we obtain the following more general consider-

ation: Let F be an absolute algebraic field of characteristic 0. In order to determine all the orderings of F, take all its isomorphic mappings into ddo fl .(o). To each such isomorphism there belongs an ordering of F, namely that for which the considered isomorphism is order preserving. All the orderings of F arise in this way.

614

FIELDS WITH VALUATION

EXAMPLE 4. In particular it follows from Example 3 that a number field of finite degree has as many orderings as isomorphisms in c74 n .F_(0). For instance, ,F'0( r2-) has two orderings. Which? EXAMPLE 5. We wish to denote, analogously to dlo, the algebraic closure of the prime field ., (p > 0) by c 1l, This field may also be obtained as the union of a suitable

chain of its finite subfields. For this purpose take a sequence n1, n2, ... of natural numbers with n, 1 n2 1 ... such that every natural number is a divisor of a suitable nk. Then take for every nk the uniquely determined finite field Fk of order pk contained in d4,. It is also evident that F1 c F2 s .. , and c4, = F, U F2 U ... EXAMPLE 6. The field (p > 0) has noteworthy properties. All its automorphisms are the mappings a -- a°k, where k may be any integer (cf. § 142, Example 2). They form an infinite cyclic group. They all map every subfield of d4, onto itself. For every natural number n, cl, has exactly one subfield of degree n. Also at,, has an infinite number of infinite subfields. For example, for every prime number q the union field of the subfields of degrees q, q2, . . . is an infinite subfield. The group of the roots of unity of order prime to p is isomorphic with c4;. EXAMPLE 7. If R is a countable ring, then, because of Theorem 13, there are only

countably many polynomials f(x) of a fixed degree n over it. Hence, and from Theorem

12, it follows that R[x] is countable. It is now obvious that all the fields at,, (p > 0) are countable. It follows more generally that the algebraic closure of a countable field is countable. We similarly arrive at the conclusion that, for a countable field F. every polynomial ring F[x,, ..., x.], furthermore the polynomial ring F[x1, x2, ...] of countably many indeterminates and the quotient field F(x1, x2, ...) are countable. Since.((,) is not countable (cf. the Example on Theorem 11), it follows from the above and STEINITZ'S second main theorem (Theorem 322) that the degree of transcendence of 9-(,,) is not countable. EXAMPLE 8. From the Fundamental Theorem of Algebra it follows that every irreducible polynomial over is linear or of degree two. EXAMPLE 9. In (93.1), (93.2), (93.3) we defined the character group G of a finite Abelian group Al. We now wish to specialize this definition so that we take for the

group, denoted there by 3, the group

(i)* of complex numbers other than 0.

This is possible, since this group, by the Fundamental Theorem of Algebra (corollary of Theorem 393) and by Theorem 314, contains exactly one cyclic subgroup of order m for every natural number m. For the specialization under consideration, the characters of the group 4 are called its complex characters. As it is usual to consider only these

characters (for finite Abelian groups), they are briefly called the characters of 4. The definition therefore runs as follows: let G be the set of homomorphic mappings of 4 into P',0, (i)*. Denote by as (E . 10, (i), # 0) the image of a (E 4) under an a (E G), for which (c(ab) = )

aab = as a b

(a, b E 4; a E G)

(155.9)

and define the product ai3 of two elements a, ig (E G) by ((ai3)a =) (a E 4; a, i3 E G). ai9a = as fla

(155.10)

G now becomes a group, namely the group of (complex) characters, or the (complex) character group of 4. All the assertions in § 93 hold for this case, as well as the fol low-

ing important character relations: _ In for a = e,

as

a C4 -

10 for a ll

e,

aE as

_

n for a = c 10 for a

e,

(155.11)

REALLY CLOSED FIELDS

615

where n = 0(4) (= O(G)), while e and e denote the unity elements of 4 and G, respectively. It is usual to call the sums in (155.11) character sums. Because of the principle of duality for finite Abelian groups it is sufficient to prove, e.g., (155.11,). Let S(a) denote the left-hand side of (155.111). For every b (E 4), it follows from b4 = 4 and (155.9) that S(a) = Y aab = Y, ((xa ab) = ab Y as = ab S(a), a

a

a

so that ((xb - 1)S(a) = 0.

If at 96 e, then there is a b such that ab 96 1, whence S(a) = 0. On the other hand ea = 1 for all a, thus S(e) = n. Consequently (155.11) is proved. For the applications of complex characters to number theory cf. HASSE (1950). EXAMPLE 10. The algebraic numbers with minimal polynomials in .R70[x]are called integral algebraic numbers. These constitute a ring (proof by Theorems 207, 268) with the quotient field c-4o; this ring is not a field and has no irreducible elements.

§ 156. Really Closed Fields

A formally real field without proper formally real algebraic extension fields is said to be really closed. (It would be more correct to say "really algebraically closed".) THEOREM 395. In a really closed field F every algebraic equation of odd degree is solvable, and F may only be ordered so that the positive elements are squares.

For the proof we take a polynomial f(x) of odd degree over F. We have to show that f(x) has at least one zero in F. Since f(x) has an irreducible factor of odd degree we may assume that f(x) itself is irreducible. Then since, according to Theorem 372, the extension field F(a) with f(a) = 0 is formally real, we must have F(a) = F, a E F. The first assertion of the theorem is now proved. In order to prove the second assertion, we denote the set of squares other than 0 from F by P, and suppose that F is ordered in some manner. It suffices to prove that P is necessarily the positivity domain of F. For that purpose consider an element y ( 0) of F - P. Then since x2 - y is irreducible, there exists an extension field G = F( .,/y). Because G D F, G is not formally real, so that there exists an equation

- 1 = k=1 , (k + Yk V r)2

lake Pk E F; k = 1, ..., n).

In this the coefficient of .Jy must be equal to 0, consequently

-1 =k (ak +Yky). QQ

(156.1)

616

FIELDS WITH VALUATION

Since F is formally real, it follows that y is not a sum of elements from P. Since such a sum cannot be 0 either, we see that P is closed with respect to addition. Since a similar statement holds for multiplication, P is a semiring. Since, moreover, the elements n

n

k=1

k=1

lie in P and (156.1) agrees with a + fly = 0, -y = q9(#-1)2 belongs to P. It follows that for every element S of F one of the propositions

6=0, 6EP, -bEP is valid. But since two of them evidently cannot hold, P must be the positivity domain of F. Consequently the theorem is proved. THEOREM 396. A formally really field F is really closed if, and only if, the complex field F(i) is algebraically closed. If F is really closed, then, according to Theorems 393, 395, F (i) is algebrai-

cally closed. Conversely, let us assume the latter. Since [F(i) : F] = 2 is a prime number, there is no field G such that F c G c F (1). Since, on the other hand, according to the assumption, F(i) is the algebraic closure of F, it follows that every proper algebraic extension field of F is isomorphic with F(i). Since this field is not formally real, because i2 = -1, F is really closed. This proves Theorem 396. THEOREM 397. The field ,eo (1 'tol of the real algebraic numbers is, apart from isomorphism, the only really closed absolute algebraic field, and the field - 'o of the algebraic numbers has degree 2 over it. We write

For every element a = a + bi of vgo (a, b E Y(o) the conjugate a = a - bi t?o. As i E -o, so

lies in

a=.(a+a), b=4(a-a) o and so also in .7. Hence a E 9'(i), and consequently ufo = .}'(i). This means, by Theorem 396, that Y is really closed, proving that the last assertion of Theorem 397 is true. Conversely, let us consider a really closed absolute algebraic field LP. By Theorem 394, we may assume that 4g is a subfield of .7. The subfield 4 (i) of o is algebraically closed, because of Theorem 396, so that 4 (1) = moo.

lie in

Since [vfo :.f] = [,,go : 4t], and 4

Y, it follows that dot = Jr. This

completes the proof of Theorem 397. For a proof of the existence and uniqueness of the really closed algebraic field which makes no use of d,, see VAN DER WAERDEN (1955).

ARCHIMEDEAN AND NON-ARCHIMEDEAN VALUATIONS

617

§ 157. Archimedean and Non-Archimedean Valuations

Those valuations in which the value field is Archimedean-ordered are the ones most frequently used. In this case, by Theorem 387, we can assume without loss of generality that the value field is equal to.7(o), and then we speak of a real valuation. Henceforth we shall consider only these valuations. First, the following theorem holds : THEOREM 398. Two real valuations 99, ip of a field . are equivalent if, and

only if, there exists a positive real number K such that V(a) = (gp(a))K

.

(157.1)

This is evidently true when cp or ' is the trivial valuation. In the rest of the proof we shall disregard this case. The assertion "if" is true, since in (157.1) the q'-zero sequences and p-zero sequences formed from -9ragree with one another. In order to prove the "only if" part we assume that q, tp are equivalent. First we shall show that

q7 (a) < p (b) a ip (a) < ip (b) .

(157.2)

From the left-hand side it follows that 99(ab-1) < 1, therefore [(ab-I)"] is a 99-zero sequence, consequently a p-zero sequence. This means that +o(ab-1) < 1, i.e., the right-hand side of (157.2) holds. By symmetry this proves (157.2). We now take a fixed element c such that q'(c) > 1. Such a c always exists,

since q' is not the trivial valuation and 9(c)9(c-1) = I for c 0. Because of (157.2), +p(c) > 1. We consider an arbitrary element a (0 0) of.9rand put 93(a) = (q'(c))" ,

V(a) _ (+Y (c))"'

for two suitable real numbers a, a'. For these we show that a = a'. We take two integers in, n (n 0 0) with

Then (92(c))mn -' < (92(c))"

= 92(a) ,

and so 99(c'")<99(an

From this and from (157.2), we get o(c'") <

and then (Y'(c))n n

tV(a"),

' < V(a) = (w(c))"

(157.3)

FIELDS WITH VALUATION

618

As this gives

m

n

<_ cc', so a <_ a'. Likewise a' < a, consequently a = a'.

Finally we put ip(c) =

(99(c))K

for some positive real number K. Hence, and from (157.3), the validity of (157.1) follows, since a is equal to a' for all a 0. Even a = 0 is not an exception, therefore the theorem is true. A real valuation 99 of a field Y is said to be an Archimedean or nonArchimedean valuation, according as there are elements n (= 2, 3, ...) in .s' such that 9'(n) > 1, or qq(n) < 1 for all n. Since the trivial valuation is non-Archimedean, it follows from the corollary of Theorem 377 that the fields of prime number characteristic have only non-Archimedean real valuations. The absolute valuations of Y (0) and of .7(0)(i) are Archimedean-. THEOREM 399. A real valuation T of a field.Y is non-Archimedean if, and only if, we always have p(a + b) < max (qu(a), qq(b)) . (157.4)

If (157.4) holds, it follows by induction that gq(al +

... +

a") < max (9q(a1), ..., 99(a")) .

Therefore 99(n) 99(1) = 1 for all n = 2, 3, ..., and so 99 is nonArchimedean. We now suppose that 99 is non-Archimedean. It follows that

((p(a + b))" = 99 ((a + b)") <

ko

On account of the supposition 99

((;)an_kbk)

11

kI

1

(n = 2, 3, ...) .

<_ 1. Therefore from the homo-

morphy property of 99, (q9 (a + b))" 5 (n + 1)µ" (,u = Max (qu(a), qq(b))). If now a and b are not both 0, then

(j&-'99(a + b))" < n + 1

.

(157.5)

On the other hand, for every positive 6

(l+by >1+(;)a2 and so it follows from (157.5) that p -'99(a + b) < 1, i.e., (157.4) holds. Since (157.4) holds also in the case a = b = 0, Theorem 399 is proved.

619

ARCHIMEDEAN AND NON-ARCHIMEDEAN VALUATIONS

NoTE. Since from (157.4) it even follows that qq(a + b) < 99(a) + p(b), we can also characterize the non-Archimedean valuations as follows: a non-

Archimedean valuation 99 of a field Y means a homomorphic mapping a -> qu(a) of .7+ into . 'tot such that (157.4) and 99(0) = 0, q(a) > 0 (a hold. Since, from (157.4), for every positive real number x (99(a + b))" < max (9,(a))K

0)

(9' (b))K) ,

,

this implies that both qu(a) and V(a) = (qu(a))" are non-Archimedean valuations. By Theorem 398, we see that an Archimedean and a non-Archimedean valuation can never be equivalent. EXAMPLE. In connection with Theorem 398 it should be noted that (157.1), for a given Archimedean valuation 99, does not yield for all x (> 0) a valuation w, since for this to be the case we must have

(P(a + b))" 5 (q,(a))" + (99(b))". The K such that 0 < it < 1 are always suitable, since then from 9'(a + b) < qu(a) + 9'(b) according to a well-known theorem we get (99(a + b))' S (qu(a) + 9p(b))" < (9p(a))" + (9v(b))'

In the special case F =

.

(.), qu(a) = ; a I no x > I is suitable.

§ 158. Exponent Valuations By contrast to what we have done up to this point, from now on we shall denote a field with valuation by F, while.{(,) will always represent a value field.

We consider a non-Archimedean valuation 99 of a field F. By the note at the end of § 157, this implies a mapping a --> qu(a) of F into .7(o) with the properties c(o) = 0, 99(m) > 0 (a

0) ,

(158.2)

97(aJ) = 9'(a)99(J9) ,

cp(a + (9) < max (qu(a), T(f))

(158.1)

.

(158.3)

Instead of this function q, it is convenient to introduce the function w(a) = -I og" 99 (a)

(a 0 0) ,

(158.4)

where we take an arbitrary fixed real number g (> 1) as base of the logarithm. (Here we have excluded a = 0 although this is immaterial.) We call

620

FIELDS WITH VALUATION

w an exponent valuation of F and shall prove that it has the following properties :

w(a) E 9o;

(cc E F, * 0) ,

(158.5)

w(aq) = w(a) + w((3) (aft # 0) ; (158.6) (158.7) w(a + P) > min (w(a), w(f)) (a, 6, a + f # 0) . Before giving the proof we note that Z min (w((xl), ..., w(Q) (al, . . ., a,,, al + ... + (Xn#0) w(al + ... + (158.8) follows more generally from (158.7) If, moreover, a is a root of unity in F, q9(a) = 1, and so w(a) = 0. Besides, from g (-a) = qu(a), q9(a-) = 99 (a)-1 the properties w(- cc) = w(a)

w(a 1) = - w(a)

(0100), (0,00)

(158.9) (158.10)

follow.

The equivalence of the two definitions of w above becomes obvious if we write (158.4) in the form 99(a) = 9-W(a)

(a 96 0) .

(158.11)

(The appearance of w (158.11) in the exponent is the reason for the term `exponent valuation".) NoTE. We also use the exponent form w, defined by (158.4), of a real valuation 99 if we can prove that this valuation is non-Archimedean. We see from (158.5), (158.6), (158.7) that an exponent valuation w of a field F means a homomorphic mapping a --> w(a) of the group F* into the module . '(,,,) which also satisfies condition (158.7).

If necessary we suppose that the definition of w (a) is extended to include

the case at = 0 by putting w(0) = oo, which means that w(0) is "greater" than all the other w(a). Here (158.7) remains true. (158.6) remains also true if we put oo + w(a) = Co. We emphasize once more that non-Archimedean valuations and exponent valuations differ from each other only immaterially, so that the previous definitions and statements, so far as they concern non-Archimedean valuations, may also be applied directly to exponent valuations. Similarly, future statements about one of these valuations will be taken as valid for the others. We say that the (non-Archimedean) valuation 99 and the expo-

nent valuation w correspond to each other, if (158.4) [or (158.11)] holds for a fixed g (> 1). For instance, a sequence [a;] from F is a w-zero sequence if, and only if, to every real number C there is an N (E -/") such that w(a;) > C

(i > N) .

EXPONENT VALUATIONS

621

On account of Theorem 398, the exponent valuations equivalent to an exponent valuation w(a) are given by ctii'(a) ,

where c can be any positive real number. THEOREM 400. Instead of (158.7) for w(a) # w(fl) we even have

w(a + p) = min (w(a) , w(p))

(158.12)

.

Let us assume, e.g., that w(a) > w(8). If the assertion is false, then, because

of (158.7), w(a + fl) > w(8). On the other hand, from (158.7) and (158.9), w(fl) =

a + a + p) > min (w(oc), w(a + 6))

.

From both results w(fl) Z w(a). This contradiction proves the theorem. Exponent valuations are, in several respects, simpler than Archimedean valuations, as we shall see later. First of all, in addition to Theorem 379, the following also holds : THEOREM 401. A sequence [a;] from afield F with the exponent valuation

w is convergent, if [a; - ai+i] is a zero sequence. For this, it is necessary that the terms of the sequence [w(a;)], except for finitely many, are equal, unless [a;] is a zero sequence.

If [a; - a,+,] is a zero sequence, then, for every real number C there is a natural number N such that

w(a;-(zi+I)>C

(i>N).

Because 1-1

ak - al = X (ai - a! +I) !=

(0
and because of (158.8), it follows that

W(ak - a;) > C

(k, 1 > N).

The first assertion of the theorem is now proved. In order to prove the second assertion, we suppose [a;] to be convergent and not a zero sequence. To every real number C there exists a natural number N such that we always have w(a; - (xk) > C (1, k > N). (158.13) If the assertion is false, then for every i (> N) there is also a k (> N) with w (a;) # w (ak) .

FIELDS WITH VALUATION

622

For such a pair a;, ak, because of (158.9) and Theorem 400, (158.13) becomes min (w(00, w((Xk)) > C C.

From this we obtain w(a;) > C so that [a;] is a zero sequence nevertheless. This contradiction proves Theorem 401. THEOREM 402. If [a;] is a convergent sequence, but not a zero sequence, from a field F, which is perfect with respect to an exponent valuation w, then

w(lim a) = w(aN) = w'(aN+1) =

...

(158.14)

for a suitable N.

In order to prove this, we put a = lim a,. By Theorem 400 we have: w(M) 0 w(a)

w(a, - a) = min (w(at), w(a)) .

But since [a; - a] is a zero sequence, Theorem 402 follows. We now consider an arbitrary field F with an exponent valuation w, in order to define the following important notions. By the value module of F we understand the submodule of Y o) consisting of all the values w(a). When it is denoted by %, then according to (158.6) we have the homomorphism

F* -

(a - w(a))

.

(158.15)

Further we take into consideration those elements a of F for which w((X) z 0.

(158.16)

[Thus, these elements are 0 and the elements of F which are mapped onto the non-negative elements of fin the homomorphism (158.15)]. We shall show that these elements a constitute a subring of F, which we denote by R = R,4, and call the valuation ring of F (with respect to the exponent valuation w). We notice at the same time that, because w(l) = 0, R contains the unity element of F. In order to verify the ring property of R, let us take two elements a, fi from R. Because w(a), w(f) >- 0, it follows from (158.6), (158.7), (158.9) that w(a - A), w(afl) >_ 0 ,

and then a - fl, afi E R. Accordingly, R is, in fact, a ring. We shall show that the elements a with w(a) = 0 are all the units of R.

(158.17)

EXPONENT VALUATIONS

623

For, the units of R are those elements a of F for which both w(a) Z 0 and w(a-') >_ 0 are satisfied. Because of (158.10), the assertion follows. We shall show that the elements a of F with the property w (a) > 0

(158.18)

i.e., the elements of R other than the units constitute a maximal ideal of R, which we denote by p = pw and call the valuation ideal with respect to the exponent valuation w. We shall also show that p is a prime ideal. The ideal property of p follows at once from (158.6) and (158.7). The maximal property of p follows from the fact that every ideal of R containing

a unit must be equal to R. Since, by (158.6), if w(a/4) > 0, then either w(a) > 0 or w(fl) > 0, so p is a prime ideal. (This also follows from Theorem 186 as well as from the above.) is a field, From Theorem 130 it follows that the factor ring R/.p = which we denote by 2a = tSW and call the valuation factor field with respect to the exponent valuation w. THEOREM 403. Two exponent valuations w, w' of afield F are equivalent if, and only if, the valuation rings R, R, are equal. It is clear that Rw = F if, and only if, w is the trivial valuation. Hence the theorem follows for the case where one of w, w' is the trivial valuation. Therefore we exclude this case in future. If now w, w' are equivalent, then according to Theorem 398, w'(a) = cw(a)

(a E F)

(158.19)

for a positive real constant c. Since the propositions w(a) z 0, w'(a) > 0 are then equivalent, R,, = R,,.. Conversely, let us assume the latter. From this it follows that ,pW = ipW therefore

w(a) > 0 . w'(a) > 0. We consider two elements at, # such that w(a), w(f) > 0, whence w'((x), w'(fl) > 0. For arbitrary integers m, n, according to (158.6), w(a'"(3") = mw(a) + + nw(fl). The same holds for w' instead of w, thus

mw(a) + nw(f) > 0 a mw'(a) + nw'(f) > 0. Hence it evidently follows that w(a)(w(fl))-' = w'(x)(w'(f))-'. This implies the existence of a positive real number c which has the property that (158.19) holds for all a such that w(a) > 0. Because of (158.10), the same is true for

w(a) < 0. Since, finally, the units a of Rw = Rw are characterized both by w(a) = 0 and by w'(a) = 0, (158.19) is true generally. Thus Theorem 403 is proved. 21 IL-A.

FIELDS WITH VALUATION

624

According to this theorem the determination of the possible (non-equivalent) exponent valuations of a field F is reduced to the examination of all its valuation rings. THEOREM 404. All the valuation rings of a field F are F itself and those maximal subrings of F with unity elements which are not fields. First of all we suppose that R (A F) is a valuation ring of F. Let w denote

the exponent valuation of F belonging to it. Because w(1) = 0, we have 1 E R. From the definition in (158.16) it follows that for every element a of F other than 0 at least one of (x, ac-' belongs to R. Since then F is the quotient field of R, R itself cannot be a field. We still have to show that for every element a of F - R the ring {R, a) is equal to F. By hypothesis we have w(a) < 0. Consequently for every element # of F -nw(a). Hence there is a natural number n such that w(/3a - ") ? 0, fa " E R, # E { R, a},

therefore {R, a} = F. We now consider, conversely, a maximal subring R of F containing the unity element which is not a field. We have to prove that R is a valuation ring of F. First of all we show that, for every element a of F other than 0, one of a, a-1 lies in R. With this aim we suppose that a is not an element of R, so that it follows that the ring {R, a) is equal to F, and hence a

I =f(a)

for a polynomial f(x) over R. Consequently there is a principal polynomial F(x) over R with

F(a-I) = 0.

(158.20)

Let n denote the degree of F(x). Since, by hypothesis, F is the quotient field of R, there is an element P (A 0) of R with e,

ea_I....

Pa-("-1) E R

.

(158.21)

Here we may postulate that e is not a unit in R. Hence it follows, because

PP-' = 1, that e-' does not lie in R. If e-' lay in the ring {R, a-1}, this would imply that 9 e- = g((X-' )

for a polynomial g(x) over R, whose degree may be taken as n - 1 at most, because of (158.20). But, because of (158.21), it follows from this that

P-I = PP 2 = Pg(a-')

EXPONENT VALUATIONS

625

belongs to R, which is impossible. Since, according to this, lie in the ring {R, a-1} then

e_2

does no

R c {R,a-1} c F. Now by hypothesis, " = " must hold on the left-hand side, so that it follows that a-1 E R, as was asserted. We now denote by U the group of units of R and by that module which arises from the factor group F*/U after passing over to the additive notation. Then, f consists of the cosets ocU (a E F, 0); further the rule aU -I- 19U = ajU

(158.22)

holds inffor the addition of elements. a relation < by We define in ocU < flu cs a-'# E R - U .

(158.23)

This is well defined, since if ocU = a1U, 19U = fl1U then the propositions " fl E R - U, al-'til E R - U are equivalent. Since, for a, fl, y (E F, 0) the rule

a

f-'y ER-U=a-')'ER-U

holds, this relation < is transitive. For any two elements ocU, flu of4', of the three propositions

aU = flu, aU < flu, flu < aU exactly one holds, for if a laFof does not belong to U, then according to what has been stated exactly one «_ Ill,

,q_la ( =

(a-1N)-1)

belongs to R - U. Hence, we have defined an ordering relation < in 44f by (158.23).

Addition is monotone since, if

aU
0) it follows from (158.23) that ocyU < flyU,

i.e., by (158.22),

aU -j- yU < flu -- yU

.

According to the second part of Theorem 366 the modulecAf is ordered; further its positivity domain is the semimodule consisting of the elements

626

FIELDS WITH VALUATION

aU (> U), where we have to take into consideration that U is the zero

element of.

We show that the ordering of

is Archimedean. For this purpose

let us consider two elements aU, 9U (> U) of Lam. Since a'1 does not belong to R, the ring {R, a-1} is equal to F, and so 14-1 E {R, a-1}. Accordingly

14-1 = h (a-') for a polynomial h(x) over R. Therefore it follows from a E R that fl- Ian E R for a suitable natural number n. This implies that 14U < naU, so that f is in fact Archimedean-ordered. For any two elements a, fl of F with a, fl, a + j rA 0 the relation (a + fl)U >_ min (oU, 14U)

(158.24)

also holds. Let aU <_ 14U. Then according to (158.23) a-116 E R, thus a -'(a + 14) E R, i.e., aU < (a + #)U. Hereby (158.24) is proved. By Theorem 387, /K is order preserving isomorphic with a submodule of tat. We carry through the corresponding embedding, but keep the previous notation for the elements of L f. By (158.22), (158.24), an exponent

valuation of F is then defined by w(a) = aU (cc E F, # 0). The valuation ring of F belonging to it consists of those elements a (E F) for which either

a = 0 or a U >_ U. Now we see from (158.23) that these a are exactly the elements of R. Consequently Theorem 404 is proved. A supplement of this theorem is the following THEOREM 405. The valuation ideal of a valuation ring R (# F) of a field F is the only proper prime ideal of R. Since, by Theorem 404, R is different from 0 and not a field, there is an element a in it which is neither 0 nor a unit. Since a-1 does not belong to R, the ring {R, a-' I is equal to F. Therefore for every element fi (& 0) of a proper

prime ideal p of R an equation #-' = f(a-) holds for some non-constant polynomial f(x) over R. If its degree is denoted by n, it follows that a"f(a -') E R,

a" = 9 . a"J (a_ 1) E ) .

Since p is a prime ideal, it follows that a E p. Accordingly, we have found that the elements other than the units of R all belong to ,p. Since, on the

other hand, p can contain no units, it is necessarily the valuation ideal. Since this is evidently a proper prime ideal of R, the proof is complete. We see that the definition of an exponent valuation of the field.F (,) relies only on its module .""( , and also remains meaningful if instead of this module we have recourse to an arbitrary ordered module. In this way, according to KRIJLL (1932), the so-called general valuation w of a field F is defined as a homomorphic map-

ping a - w((x) of the group F* onto an ordered module.,it with the property w(a + j9) > min (w(a), w(8))

(a, fi E F; a, /3, at + f 96 0).

DISCRETE VALUATIONS

627

Again of is called the value module, which occurs instead of the value field. If f is Archimedean-ordered, then we return to the exponent valuations Accordingly, general valuations essentially include all the exponent valuations, but not the Archimedean valuations. Cf. also PICKERT (1951) and FucfLS (1951).

ExAMPLE. In a field which is perfect with respect to an exponent valuation, an infinite series a, + a$ + . . . is convergent, by Theorem 401, if, and only if, lim a, = 0.

§ 159. Discrete Valuations A non-trivial exponent valuation w of a field F with cyclic value module

is called a discrete valuation. So we may now take the value module in the form {c}, where c is a positive real number. Because of the order preserving isomorphism .Y* ti { c} (n -+ nc) it is possible in the case of a discrete

valuation, by passing over to an equivalent valuation, to take,7+ as a value module. In this case we call the discrete valuation normed. THEOREM 406. A non-trivial exponent valuation is discrete if, and only if,

among the positive elements of the value module there is a minimal one. We have to prove only the assertion "if". We consider a non-trivial expo-

nent valuation w such that among all the positive values w(a) there is a minimal value w(b). To every w(a) with a # 0 we may assign an integer n such that nw(b) 5 w(a) < (n + 1)w(b) . Then

0 5 w(a) - nw(b) = w(ab-") < w(b). Now w(a) = nw(b) follows from this because of the assumption. Hence the theorem is proved. THEOREM 407. The valuation ring belonging to a discrete valuation is a

principal ideal ring with (prime decomposition and to within associates) only one prime element which is a generator of the valuation ideal. Conversely, if a principal ideal other than 0 belongs to an exponent valuation as valuation ideal, then the valuation is discrete. In order to prove this, let us consider an exponent valuation w of a field

F with the valuation ring R and the valuation ideal p. First of all we suppose w to be discrete. Let a be a proper ideal of R. Since the elements of a have non-negative values, a contains an element

a (# 0) such that for all the elements P (# 0) of a, w(f) > w(a) > 0. Hence w(fla-1) Z 0, i.e., floc-' E R. This means that P lies in the principal ideal (a), i.e., a = (a). Thus R is a principal ideal ring. In the following, for the sake of convenience, we suppose that w is normed. Let n denote an element of p such that w(n) = 1. For every element a (00) of

R, w(a) = k is a non-negative integer. From w(aaa-k) = 0 it follows that

FIELDS WITH VALUATION

628

an-k = e is a unit of R. Because a = nnk, R has the single prime element n. Asp consists of 0 and the a such that w(a) = k > 0, sop = (n). Conversely, let us suppose that the valuation ideal p is a principal ideal (n) other than 0. For every element a of p, an-1 E R, thus w(an-1) Z 0, w(a) Z w(n). According to this w(n) is the least positive element of the value module. This implies, by Theorem 406, the validity of Theorem 407.

THEOREM 408. Let F be afield, perfect with respect to the nonmed discrete

valuation w, with the valuation ring R and the valuation ideal p. Take fixed elements ..., n_1i no, nI,... from F with

(i = 0, ± 1, ...)

w(n;) = i

(159.1)

and a system of representatives 91 of R mod p, containing the element 0. Then the infinite series a = amnm + am+lnm+1 + ...

(am,

am+1,

.. E 91; am }o 0)

(159.2)

are all the different elements (# 0) of F, where m can be any integer, and w(a) = m .

(159.3)

(If, on account of Theorem 407, we choose an element n such that p = (n), we may then write n; = n'.) We already know (cf. the Example at the end of the preceding paragraph) that the infinite series (159.2) is convergent. In order to prove (159.3) we denote the i`h partial sum of the right-hand side of (159.2) by a;. Because of (159.1) and Theorem 400, w(i) = w(amnm) = m

(i = 1, 2, ...) .

Since, on the other hand, a = lim at,

lim (aj - a) = 0. Hence, and from Theorem 400, it follows that w(a) m is impossible and so (159.3) is true. Then we show that every element a (# 0) of F may be written in the form (159.2). For if we put m = w(a), then w(an;,f) = 0 follows, thus anml is a unit in R. Accordingly am (mod p)

for some am (

0) from 91. Since w(mn

- xm) >_ 1, so

w(a - amnm) >_ m + 1.

629

DISCRETE VALUATIONS

We suppose that for an s (> m) with any an, ..., as (E R) whatever, W (a - an,n,n - ... -- a ac) >- S -1-

1.

(159.4)

This is, according to the above, in fact the case particularly for s = m. If we

now temporarily denote the expression in brackets on the left-hand side s + 1, and so w (fins+1) >_ 0. Accordingly of (159.4) by fl, then w

s+l = a:+l (mod p) for an as+1 from RJR, thus we have

w(f -as+1n.,+1)Zs+2. This means that (159.4) is satisfied for s + 1 instead of s. By induction it follows that elements am (s' 0), am+l, ... of ll exist such that (159.4) holds for all s = m, m + 1, .... This implies the validity of (159.2). Lastly, in order to prove the uniqueness of the representation of a in the form (159.2), it should be noted that, by (159.3), the number m on the righthand side of (159.2) is uniquely determined by a. Consequently every further representation, similar to (159.2), of a is of the type : a = I9mnm + 1m+1nm+1 +

where Ym, m+t,

are elements of M. After subtraction we obtain

0=(01m-Ym) am +((Xm+1-Nm+1)nm+1+ If a,, - fl, were the first non-vanishing coefficient on the right-hand side, then the right-hand side would have the value s which is, however, false. This completes the proof of the theorem. § 160. ,p-adic Valuations

So far we have become acquainted with only one application, though a very important one, of valuation theory which consists in constructing the field (o) as the perfect hull of the field .5 with the absolute valuation and, proceeding from here, the field,) (i) as the algebraic closure of Y(o). Now we shall explicitly give certain further valuations which similarly

lead to important applications. Let a field F and a subring R of F with unity element and with prime decomposition be given. Suppose also that F is the quotient field of R.

FIELDS WITH VALUATION

630

Let n be a prime element of R. The elements a (96 0) of F may then be represented in the form

a=

aw(")

(K, A E R; n X K, A)

T,

(160.1)

where the exponent w (a) indicates an integer uniquely defined by a. (Notice

that if a E R, 0 then Tcw(") I I a.) We shall show that w is a discrete and normed exponent valuation of F, which we call the z-adic valuation of F and denote by w,,. Further, we shall show conversely, that every discrete valuation is equivalent to a FI-adic valuation. It should be noted that w,, (for fixed F and R) depends only on the prime ideal ,p = (n), therefore w,, is also denoted by w, and called a p-adic valuation. We repeat that the normed discrete valuations and the (n-adic or) p-adic valuations are identical.

For the proof we consider a further element fi (s 0) of F and write this as in (160.1) in the form

vER;aXµ,v).

_IFIw(") V

(160.2)

Since

Av

w(aj9) = w(a) + w(fl). If further, e.g., w(a) < w(p), then from (160.1) and (160.2) we obtain

a+#=

(rcv + Ap.,k),w(a)

(k = w(fl) - w(a) Z 0) .

vv

This gives w(a + i) w(a). Thus, w(a + fi) Z min (w(a), w(,B)) always holds. According to this, w is an exponent valuation. It is also discrete and normed, since the value module is evidently equal to 7+. Let us consider, conversely, a discrete valuation w of F. Let the valuation ring and the valuation ideal be denoted by R and ,, respectively. By Theorem 404, F is the quotient field of R. Furthermore, R, according to Theorem 407 is a ring with prime decomposition and ac is the unique prime element for which 4 = (ac). Thus the ,p-adic valuation w,, of F certainly exists. We even show that wand w , are equivalent. For this purpose let us consider an arbitrary element a (A 0) of F. This may be represented in the form

x= TK nwv

,

where K, A are units of R. Thus w(a) = w(9c) wp(a), so that W. wy are, in fact, equivalent.

P- ADIC VALUATIONS

631

Now, in general, we understand by a p-adic field any field which is perfect with respect to a p-adic valuation. (In the literature, it is usual to call only certain special cases of ?these "p-adic fields". Cf. HASSE (1949). Further we call the perfect hull of a p-adic valuated field the p-adic hull of this field. The p-adic hulls of algebraic number fields are called p-adic number fields. The elements of a p-adic number field are said to be p-adic numbers. We know from Theorem 408 that the elements of a p-adic field F (since this is just a field perfect with respect to a discrete valuation) may be uniquely given as infinite series

a = an,'G"' + am+1 arm+I +

(am, am+l, ...E R; am # 0)

,

(160.3)

where n is a generating element of the valuation ideal p and 91 denotes a system of representatives mod p of the valuation ring R of F, containing the element 0. We call (160.3) a normed p-adic series. (Hence the term p-adic field. Of course (160.3) is convergent and represents an element of the field,

are arbitrary elements of the valuation ring. Then when am, am+l, (160.3) is simply called a p-adic series.) _ In particular, we have to consider the p-adic hull F of a field F. Let w denote the p-adic valuation, R the valuation ring of F belonging to it and p itself the valuation ideal of R. We denote by w the valuation of F (in exponent form). Every element (-A 0) of r arises as a = w-lim ai , where [a;] is a w-convergent sequence from F. According to Theorem 402,

w(x;)=c for a suitable N (E 4") and an integer c for all i > N, whence w(a) = lim w(a,) = c .

Consider a further element P (# 0) of F : # = w-lim 9, , where [i4;] is a suitable w-convergent sequence from F. If, moreover,

a+#

0, then w(ac + 0) = lim w(a; + 9,) .

Hence, it immediately follows that the valuation w (together with iv) is non-Archimedean. We have also found that the value module of w like 21/a R.-A.

632

FIELDS WITH VALUATION

that of w is .7+, thus w is a discrete valuation of F. Let the valuation ring R and the valuation ideal belong to it. Then w itself is the -adic valuation of F and consequently is a P-adic field. Let n be a generating element of ,p, i.e., an element from F such that w(n) = 1. Since w is a continuation of w, w (7c) = w (n) = 1, thus 7L is also a generating element of . Furthermore, since the elements ( : A 0) of F are the w-limits of w-convergent sequences from F, this means, in other words, that the infinite series (160.3)

formed from F furnish all the elements a (0 0) of F. Since an, elements (

0, the

0) of R and P are given by m

0 and m > 0, respectively. We wish to apply this to the case F = . With this end in view we

proceed from the subring 7 of .9' ; this has . for quotient field and is a ring with prime decomposition. Consequently, a p-adic valuation of 3 belongs to every prime number p, which we call the p-adic valuation of 5 and denote by wp. wp (a) is, according to the general definition (160.1), that integer for which an equation of the form

a=

pw,(a)

(k, I E J"; p X k, 1)

(160.4)

holds, where a can indicate any element of .50 other than 0. The corresponding p-adic hull of 0 is denoted by p) and called the p-adic number field.

(Of course p1 has characteristic 0.) Its elements are the p-adic numbers. Since in J' the numbers 0, . . ., p - 1 constitute a system of representatives mod p, the p-adic numbers are given by the p-adic series a = am pm + am+l pm+l + .. .

(ann am+1, ... = 0, ..., p - 1) , (160.5)

where m can be any integer. The cases where am # 0 in (160.5) are the normedp-adic series, which furnish all the different p-adic numbers. Ap-adic number a is called a p-adic integer, if a may be represented in the form (160.5) with m Z 0. The p-adic integers constitute a ring, which is called the ring of p-adic integers. This is the valuation ring of pl with respect to wp. The units of this ring are called the p-adic units. These are the special cases of (160.5) given by m = 0, am 0. From (160.5) it follows that the elements (0 0) of 3 p) may be uniquely given in the form pop, where @ is a p-adic unit and m an integer. It is evident that the (infinitely many) p-adic number fields .21, Y(3), . . . are not pairwise isomorphic. For an easy application of p-adic number fields see HASSE (1935). EXAMPLE 1. The p-adic field contains non-rational algebraic elements, i.e., algebraic elements lying outside F.. For instance, let us take an equation such as

x"- a =0

(a E J; p,f'n, a),

(160.6)

633

P-ADIC VALUATIONS

which has no solution in .f`-,. This means that a is not an nth power in J. Furthermore, let us suppose that in 3 the congruence

x" - a = 0 (mod p)

(160.7)

has a solution at, (= 1, ..., p - 1). We shall show that (160.6) then has a solution a in .",,, for which moreover a = ao (mod p) . (160.8) Since ao - a = 0 (mod p), and because of the polynomial theorem, we may determine

a sequence of integers a,, a2, ... (= 0, . . ., p - 1) so that all the congruences

(a,+alp+...+a,p'?-a=0 (modp'+') (i=0,1,...)

...

hold. It is evident that the p-adic number a = ap + a1 p +

is then a solution

of (160.6), for which (160.8) also holds. EXAMPLE 2. The field .t',,, is not algebraically closed, since, e.g., the equation xE - p = 0 has no solution in it. This follows simply from the fact that p is a prime element in the valuation ring of . ,,,. We call the algebraic elements of . ",,, algebraic p-adic numbers. EXAMPLE 3. The field .3r",,, is not countable, as is obvious from (160.5). Hence, by the same inference as in § 155, Example 7, it follows that the degree of transcendence

of F(,) is not countable. EXAMPLE 4. The field .97,,, is not orderable. For the case p 96 2 we see this as follows: Since the congruence x2 + p - 1 = 0 (mod p) is solvable in .7, there exists, according to Example 1, a p-adic number a such that a2 + p - 1 = 0. Because -1 = = aQ + p - 2 the assertion follows from the ARTIN- SCHREIER theorem (Theorem 371). If p = 2 prove as in Example I that in ' (2) the equation as + 7 = 0 is solvable. Because - I = a2 + 6 the former inference again holds. EXAMPLE 5. The polynomial x°-1 = 0 splits into linear factors over the field

.-%,,) [i.e., -2-&) contains p - 1 (p - 1)`h roots of unity ]. This follows from Example 1,

since the congruence x" = 0 (mod p) has the solutions 1,

.

. .,p - 1.

§ 161. Ostrowski's First Theorem THEOREM 409 (OSTROWSKI's first theorem). All the real valuations of the fields F are (apart from equivalent valuations) the trivial valuation, the absolute valuation and the p-adic valuations. In the first place we know that the valuations listed above are not equi-

valent to one another. We now consider an arbitrary non-trivial real valuation q' of .5. At first we suppose 92 to be Archimedean. Since p(1) = 1, we have qq(n) 5 n (n = 0, 1,

. .

.). Because q7(- a) = qu(a)

,p(n)<=Ink

,

(n=0,±1,...).

(161.1)

We shall show that p(b) < (max (1, qu(a)))'°s"b

(a, b = 2, 3, ...) .

(161.2)

FIELDS WITH VALUATION

634

For every natural number n we may put

b' =co+ cia+... +Cmam (0
where co, ..., c., are integers. Since then a' _< b", it follows that

m<_nlog,b.

(161.4)

Further from (161.1) and (161.3)

(99(b))" < a (1 + q7(a) + ... + (9,(a))) . If, therefore, we put

It = max (1, 9' (a)),

(161.5)

then because of (161.4) (qq(b))" < a(m + 1),u"' < a(l + n log, b)µ" Iogab , thus, (µ-IoSab qq(b))"

(n = 1, 2, ...) .

< a (1 + n logo b)

(161.6)

Because of the inequality

(1+S)">1+in)

2

(6>0; n=2,3,...)

the left-hand side of (161.6) can therefore be at most 1, i.e., q,(b) < , Iosab

By (161.5) this proves (161.2). By hypothesis there is an integer b (> 1) such that qq(b) > 1. Hence; and from (161.2), we see the validity of q'(a) > I for all integers a (> 1). Thus, according to (161.2) p(b) 5 q,(a)loeab = bloga O(a)

i.e., 1ogb q ( ( b ) 5 log. p (a)

(a,b = 2, 3, ...) .

Since, a, b may be interchanged, "=" holds here. Consequently

log. p(a) = K

(a = 2,3,...)

OSTROWSKI'S FIRST THEOREM

635

for a fixed positive number K, because q,(a) > 1. Hence, because q,(-a) _ = q,(a) and 99(0) = 0, q7(1) = 1,

9,(a)=IaII` This then holds because of the homomorphy property of 99 for all a from .9. According to this, and Theorem 398, 99 is equivalent to the absolute valuation .70. Secondly we suppose q, to be non-Archimedean. Now (p (a) 5 1

(a E J) .

We shall show that especially the a such that p(a) < I constitute an ideal

of 7. This follows from the fact that for two elements a, b of ..7 in the case q, (a) < 1 we always have ,(ab) = q,(a)q,(b) < 1

and if p(a) < 1, p(b) < 1 q,(a - b) 5 max(q,(a), q,(-b)) < 1

.

This ideal is prime, since for q,(ab) < I at least one of p(a) < 1 and p(b) < 1 must hold. Let (p) be this ideal, where p is a prime number. Then

p(p) < 1 .

(161.7)

Furthermore q,(a) = 1 for all the integers a prime to p. For every element a it follows that ( 0) of . 99(a)

= (9, (p))wP(a) ,

where wp denotes, the p-adic valuation of .,. Thus if we pass from q, to an exponent valuation according to the rule 9'(a) = g-w(a)

where g denotes a fixed real number (> 1), then w(a) = Kw,(a) for a fixed real number K, which, because of (161.7), must be positive. Consequently, according to Theorem 398 the proof of Theorem 409 is completed. § 162. Hensel's Lemma

We prove the following theorem, important in itself, which at the same time is preparatory to the following paragraphs. THEOREM 410 (HENSEL'S lemma). Let F be afield, which is perfect with respect to a non-trivial exponent valuation w, with the valuation ring R and

FIELDS WITH VALUATION

636

the valuation ideal p. Then, if f (x), go(x), ho(x) are three polynomials over R such that the leading coefficients of go(x), ho(x) are prime to P and

f(x) * 0 (mod p),

(162.1)

fl x) - go(x)ho(x) (mod i,) ,

(162.2)

(4,, go(x), ho(x)) = 1

(162.3)

hold, then there exist two further polynomials g (x), h (x) over R such that g(x), go(x) are of the same degree,

f(x) = g(x)h(x) ,

(162.4)

and

g(x)

go(x) (mod ip)

,

h(x) _ ho(x) (mod ,p).

(162.5)

COROLLARY. For every irreducible polynomial

f(x)=aco+ar1Y+...+ax" over F

min (w(arn), ..., w(a)) = min (w(acJ, To prove this we denote the degrees of the polynomials f(x), go(x), ho(x)

by n, r, s, respectively, (n > r + s; r, s >_ 0). We may assume that r since otherwise the theorem is true. By (162.2),

f(x) - go(x) ho(x) - 0 (mod ,) .

(162.6)

On account of (162.3) there exist polynomials G(x), H(x) over R such that

go(x)G(x) + ho(x)H(x) - I - 0 (mod ,p) .

(162.7)

We take from p an element n (# 0) for which w(n) is not greater than the least value of the coefficients of the left-hand sides of (162.6) and (162.7). Then

J (x) - go(x)ho(x) (mod a) ,

go(x)G(x) + ho(x)H(x) - I (mod a)

(162.8) .

(162.9)

We prove that there are polynomials uj(x), vj(x) (i = 0, 1, ...) over R

of degrees 5 r - 1 and 5 n - r such that for the polynomials gj(x), hj(x) defined recursively by

gj+i(x) = gj(x) + ni+luj(x) ,

(162.10)

hj+1(x) = hj(x) + nj+IVj(x)

(162.11)

HENSEL'S LEMMA

637

the congruences

f(x) = g.(x)h,(x) (mod n'+1)

(i = 0, 1, ...)

(162.12)

hold.

According to (162.6), (162.12) holds for i = 0. We suppose that for an m (> 0) the ui(x), v;(x) (i = 0, . . ., m - 1) are determined as required so that (162.12) is satisfied for i = 0, . . ., m. Then we wish to determine a suitable pair of polynomials um(x), v,,,(x) such that (162.12) is also satisfied

for i = m + 1. This requirement is that f(x) = gm+t(x)hm+](x) (mod n"'+2)

for which we can write, because of (162.10), (162.11), f(x) - gm(x)hm(x) °- am+1(gm(x)vm(x) + hm(x)um(x)) (mod n'"+2). (162.13)

The left-hand side is, according to the case i = m of (162.12), equal to nm+'Fm(x), where the second factor is a polynomial of degree 5 n over R. Thus because of (162.10) and (162.11), (162.13) is identical with go(x)vm(x) + ho(x)um(x) - F,(x) (mod n) .

(162.14)

On the other hand, it follows from (162.9), on multiplication by Fm(x), that go(x) G(x) Fm(x) + ho(x) H(x) Fm(x) _- F,(x) (mod n).

In order to obtain from this a suitable solution of (162.14), we determine two polynomials qm(x), um(x) by Euclidean division so that H(x)Fm(x) = go(x)gm(x) + um(x)

and um(x) is of degree <- r - 1. Moreover we put vm(x) = G(x)Fm(x) + ho(x)gm(x)

.

Then (162.14) holds, because of (162.15), but the degree of vm(x) will, in general, not be <- n - r. However, this inadequacy is avoidable, if we omit from vm(x) the terms divisible by n. If we continue to denote the remaining polynomial by vm(x), then (162.14) remains unaltered, and the new vm(x) must be of degree < n - r. Otherwise, go(x)vm(x) would be a polynomial of degree > n with a leading coefficient not divisible by n, but this is an obvious contradiction of (162.14), since ho(x)um(x), Fm(x) are of degree 5 n. From this contradiction it follows by induction that the above assertion concerning (162.12) is true. Let us notice that by (162.10), (162.11) the degrees of the polynomials

g;(x), hi(x) are equal to r and, at most, n - r respectively (i = 0, 1, ...). If we now denote, for an arbitrary k (Z 0), the coefficients of xk in the

FIELDS WITH VALUATION

638

polynomials gi(x), h,(x) by ai and fli, respectively, (i = 0, 1,.. .), then by (162.10), (162.11)

n'+1 I ai+I - 'Xi, A+I - lgi , whence the existence of the limits a(k) = w-lim ai, 1(k) = w-lim Ni

follows. Thus if we write g(x) = a(,)x' + ... + a(o),

h(x) = N(n-,)xn' + ... 1g(o)

then it follows from (162.12) that (162.4) is satisfied. Furthermore because of (162.10) and (162.11), (162.5) is also satisfied. Hence the theorem is proved.

In order to prove the corollary, let us multiplyf(x) by a suitable element (

0) of F, by which we obtain min (w(orn),

..., w(a )) = 0 ,

so that now f(x) E R[x]. If the corollary were false, then we would have 0, so that w(an), f(x) __ (an +

... + a,x')

l (mod p)

(w(a,) = 0)

would hold for an r such that 0 < r < n. Hence according to the theorem the reducibility of f(x) follows. This contradiction proves the corollary. EXAMPLE 1. Let f(x) be a primitive polynomial over .I, and p be a prime number. Let us assume that f(x) = gjx)h::(x) (mod p) (go(x): h)(x) E J[xl), (p, go(x), h,(x)) = 1

where the leading coefficients of go(x), ho(x) are prime to p. Then a factor decomposition

f(x) = g(x)h(x),

holds in .F-(,)[xl where the coefficients of the factors are p-adic integers, the congruences

g(x) = gu(x), h(x) = h,(x) (mod p) hold and g(x), go(x) have the same degree. This follows from HENSEL'S lemma applied

to F = 5 7(,). Of course, the fact that h(x), ho(x) are of the same degree may be substituted for the condition about the equality of the degrees of g(x), go(x). A similar fact is valid for HENSEL'S lemma in the general case.

EXAMPLE 2. If f(x) is a primitive polynomial over 9, and p a prime number and c an integer such that

f(c) - 0 (mod p), f(c) * 0 (mod p),

HENSEL'S LEMMA

639

then there is an element a of .K&1 such that

f(a) = 0, a = c (mod p). This is the special case go(x) = x - c of Example 1. By further specialization to the case f(x) = x" - a (p,}'n, a) we again arrive at Example 1 of § 160.

§ 163. Extensions of Real Perfect Valuations for Field Extensions of Finite Degree THEOREM 411. If F is afield, which is perfect with respect to a real valuation

99 and G I F is a field of degree n (n >- 2), then G admits one, and only one, real valuation V, which is an extension of 97, and that is given by (163.1)

tV(a) = 9'(NGIF(a))e .

Furthermore G is perfect with respect to V. (If 99 is Archimedean, then the assumptions are fulfilled, according to the later Theorem 412, only for n = 2.) First of all we shall show that a valuation +p of G is defined by (163.1). It is evident that V(0) = 0 and V(a) > 0 (a 0 0). Since, according to Theorem

176, a - NGIF(a) is a homomorphic mapping of G" into F', the property yt(afl) = yr(a)tp(j) follows. We still have to show that' has the additive property of valuations. First, let us consider a non-Archimedean valuation 99 and show at the same time that tp is also non-Archimedean. Let a, j9 be two elements of G

with a, fi, a + # # 0. Without loss of generality we may assume that +p(a) 5 1p(f). We have to show that then

tp(a + 9) < t'O Since the left-hand side is equal to 4p(af -1 + 1) p(9) and moreover ep(a9-1) 5 1, it suffices to show the following: for every element y of G such that tp(y) 5 1. (163.2) V (Y + 1) 5 1. Let

f(x)=xm+alx'n-1 +... +a,,,

(163.3)

be the minimal polynomial of y over F. According to Theorem 360, (-1)nram.

NF(Y)IF(Y)=

Hence, and from Theorem 359, (Y) _

(-I)na.,:-1

.

FIELDS WITH VALUATION

640

Thus, from (163.1), V(y) = (9' (am))"' .

(163.4)

But f (x - 1) is the minimal polynomial of y + 1. Its constant term is equal to f (- 1), and so from (163.3) and (163.4),

tV(y+ 1)=(9(1 -al+... ±a,,,)),.

(163.5)

Let w denote the exponent valuation corresponding to the valuation P. Then w(1) = 0. Further, from ip(y) 5 1 and (163.4), first p(am) 5 1, then w(am) > 0. Because of the irreducibility of f(x), the validity of w(aI) >_ 0 (i = 1, ..., m) follows from the corollary to HENSEL'S lemma (Theorem 410). Accordingly,

w(1 -al+...±am)z0. This means that the right-hand side of (163.5) is at most 1, by which (163.2) is proved. Secondly, let us consider the case of an Archimedean valuation q'. According to Theorem 412 of the following section we may restrict our-

selves - as has been already noted - to the case n = 2. We again denote by y an element of G for which qq(y) 5 1. Similarly, as above, it is sufficient to prove that q,(y + 1) < y,(y) + 1.

(163.6)

We may suppose that y lies outside F. Because n = 2, the minimal polynomial of y over F may then be put in the form

f(x)=x2+ax+b. As before V(V) = (cv(b))'

+V(y + 1) = (,p(1 - a + b))*.

,

Accordingly, assertion (163.6) means that

T(1 - a + b) < ((p(b))* + 1)2.

Since qq(1 - a + b) <- 1 + p(a) + ga(b), it suffices to show that ,P(a) < 2(99(b))t

,

i.e., (99(a))2 < 499(b)

We assume the reverse: (99(a))2 > 4q((b).

(163.7)

REAL PERFECT VALUATIONS

641

Since f(x) is irreducible, VA 0. Hence, and from (163.7), m :A 0. We form

a sequence [c,] from the elements of F such that a

c1=2, c,+1=-a-

b c

(i=1,2,...).

(163.8)

This is possible, since the c1, c2i ... turn out to be different from 0, indeed, we shall show that

(i = 1, 2, ...).

99c. ? 2 99a)

(163.9)

Because (p(2) 5 2,

If (163.9) is then assumed for some i, then, because of (163.7) and (163.8), we obtain 99(c1+L) = 4'(a +

Al

> 9?(a)-

9%(b) > 4'(a) - 2

Vi(a) >

2 9'(a)

Hence (163.9) follows by induction. We now show that our sequence [c,] is convergent. We have

c,+s - c1+1= -

b c1+1

+

b c,

= b c1+1 - c, c, c,+I

Therefore from (163.9) it follows that m(c1+2 - c,+1) < (99(a))2

9'(c,+1- c)

If the first factor of the right-hand side is denoted by q, then according to (163.7) 0 < q < 1, and we obtain 99(c,+1 - c,) 5 q'-1 9'(c2 - c1)

(i = 1, 2,. ..).

This gives

,-1 99(c, - Ck) <

i=k

92(c,+1 - c) 5 92(c2- c1) Y. q1

qk-1

1

< 1-

whence we see that for 1 < k < 1, [c,] is ,p-convergent.

q

9' (c2 - c1)

FIELDS WITH VALUATION

642

We put c = 9'-lim c,. By (163.8),

c= -a - b, c2+ac+b=0. c But, because c E F, this contradicts the irreducibility of f(x). Consequently we have shown that (163.1) defines in both cases a valuation of G. From now on we deal with both cases together. For a E F, NGIF (a) = a", and then, from (163.1), V(ac) = 99(cc). Accordingly, y, is an extension of T. To continue the proof we need the following PROPOSITION. Let wl, . ., co,,

be an F-basis of afield G F of degree it.

Further let F be perfect with respect to the real valuation 99, and op be a real valuation of G, which is an extension of 99. A sequence [ai] from G with the basis representation

at = an) col + ... + a(,") co"

(a;i), ..., a(in) E

F; i = 1, 2,...)

of its terms is yp-convergent if and only if the sequences

[a;II ],

(163.10)

..., [a,">]

are (p-convergent, and then

o-lim a; = a(i'col + ... + a(")COn

(163.11)

where

a(k) = 97-lim a(k)

(k = 1, ..., n).

(163.12)

Since from (163.10) and (163.12) for every i (= 1, 2, ...) y!(act -a(' (j), -

... -

n

a(' W") :!9

k=]

p(a ") - a(k)) 'P(Wk)

both the "if" assertions of the proposition are true. In order to prove the "only if" assertion, we call the sequence [at] an m-sequence, if a;"'

a(") = 0

(i = 1, 2, ...).

For the 1-sequences the assertion is true, since for these

We denote by m one of the numbers 2, . . ., n, and assume that the assertion for the (m - 1)-sequence is true. We consider a p-convergent m-sequence [act]. It suffices to prove the assertion for this sequence, since from this the assertion will follow by induction for the case m = n, i.e., for the general case.

REAL PERFECT VALUATIONS

643

if the sequence [aJ')] is 99-convergent, then the sequence loci - aim) CJm] _ [all)wl + ... + a(m-1)wm_1J 77

is p-convergent and at the same time an (m - 1)-sequence. For this case it follows from the induction hypothesis that the assertion is true. If, on the other hand, the sequence [a(im)] is not 99-convergent, then we infer from this a contradiction and so prove the proposition.

By hypothesis there exists a positive number s and natural numbers n1, n2, ... such that g9(aim) - a;+Rr) > s

(i = 1, 2, ...).

(163.13)

We put (a; - ai+", )

f'i = (asm) -

(I= 1, 2,. ..).

(163.14)

Since now, according to Theorem 379, [a; - a;+";] is a 'N-zero sequence, so also, because of (163.13), (163.14), [,;] is a si-zero sequence. A glance

at (163.10) and (163.14) shows further that [f; - wm] is an (m - 1)sequence. This is, according to the above, p-convergent, and W-lim WI - wm) = -wm . i

By the induction hypothesis we obtain from the formula (163.11) an equation of the form -co. = b1w1 + ... + bra-lwm-1 with elements b1, . . ., bm_1 from F. Since, because of the basis property of w1, ..., w", this is impossible, the proposition is proved. Hence the assertion of Theorem 411 that G is perfect with respect to V follows immediately.

We still have to prove the uniqueness assertion of the theorem. For that purpose we anticipate a special inference from the proposition. If p-lim a; = 0, then, from (163.11), i

a(k) = q'-lim aik) = 0 i

(k = 1, ..., n).

On the other hand, by (163.10) and Theorem 360, NGIF(a,) is a homogeneous polynomial in the ail), ..., o(,) of degree n with coefficients from F, which

depend only on the wl, ..., w" and their conjugates. Hence there follows for the sequences [a;] considered in the proposition, the rule: y,-lim a; = 0

qi-lim Nc i F((X;) = 0.

(163.15)

644

FIELDS WITH VALUATION

We now suppose that vp is any real valuation of G and an extension of (p. We have to prove that (163.1) necessarily holds for this +p. We suppose

that for an element a of G (163.1) is false. Then we have either 1P((X") < < 99(N(a)) or V(an) > q)(N(a)), where we have put N = NGIF. Corresponding to these cases we write N(a)

Mn

which implies that +'(Q) < 1.

(163.16)

N(e) = 1.

(163.17)

Further evidently

From (163.16) +p-lim e'= 0. Then, according to (163.15), 4p-lim N(e) = 0, i.e., 99-lim (N(CO))' = 0, contradicting (163.17). Consequently we have

proved Theorem 411. EXAMPLE. On account of Theorem 359, definition (163.1) may be replaced by (163.18)

VW = T (NF(-) F(a))IF(a):F1-1.

Hence we see that rp((x) depends only on F, p, a. Thus Theorem 411, apart from the statement '°G is perfect with respect to +p", holds for all algebraic field extensions G I F of a field F which is perfect with respect to gyp.

§ 164. Ostrowski's Second Theorem THEOREM 412 (OSTROWSKI's second theorem). The Archimedean-valued perfect fields are topologically equivalent to one of the fields .5o), 5o)(i) valued by the absolute valuation.

Let F denote a field which is perfect with respect to an Archimedean valuation 92. Since the characteristic of F must be equal to 0, it may be supposed that .9 is a subfield of F. From OsTROwsKI'sfirst theorem (Theorem

409) it follows that the valuation of ..5 must be equivalent to the absolute valuation. Thus we may assume that F contains the perfect hull 5o) of Y0 and that (p is an extension of the absolute valuation of J 57(o). First, we distinguish two cases which we shall, however, soon consider together. If x2 + I is reducible over F, then it may be supposed that 5o)(i), too, is a subfield of F. Since furthermore [J9( '0,(!) : 5r(o)] = 2, it follows from Theorem 411 that for the elements a + bi of o)(1)

99(a + bi) _ 1(a + bi)(a - b 1)11 = (a2 + b2)'

a + bi

OSTROWSKI'S SECOND THEOREM

645

i.e., ois also valued by the absolute valuation. For this case we shall show that F = ob(i). If, on the other hand, x2 + I is irreducible over F, then the complex field

G =F(i) over F exists and we may suppose that .5o)(i) is a subfield of G. Because [G: F] = 2 it follows from Theorem 411 that G has one, and only one, real valuation, which is an extension of 99, and that it again agrees with the absolute valuation for the elements of.$o)(i). Here G is, according to Theo-

rem 411, perfect. It means that for G the former case holds, whence, anticipating the result, it follows for that case that

Jp(i) = G D F

.off,

[7o)(i) :. 0)] _ [G : F] = 2.

Since F = .Yjo) follows from this, it suffices to prove for both cases the following :

If F is an extension field of o}(1) and (p a real valuation of F, which is an extension of the absolute valuation of 90;(i), then F = .jo)(i).

For the purpose of the proof we suppose that F D .o)(i). Let a be an element of F - .7(0,(i). We denote by a the Weierstrass lower bound (cf. § 154, Exercise 1) of the set of all values (p(a - z) for z E .7()(i). (This means that for these z we always have q^(a - z) >_ a and that there is for every h (> 0) a z such that q(ac - z) < a + h.) In any case, a Z 0. First of all we show that there exists a zo (E Jo)(i)) such that q:(a - zo) = a.

(164.1)

According to the definition of a we may form from .9o)(i) a sequence such that lim 97(u -

a.

(164.2)

Since, on the other hand, we have

5 97(a - zj + qu(a), there is a real number C (> 0) such that I Z. l = 4i(zn) < C. According to the BOLZANO - WEIERSTRASS theorem (§ 154, Exercise 2) we may

therefore form a convergent partial sequence from [z.]. If we retain the for this, then (164.2) remains valid, and notation

limz=zo,

646

FIELDS WITH VALUATION

for a zo from -o) (i). It then follows that

a=lim p(oc by which we have proved (164.1).

Together with the preceding a, a - zo also lies in F(o)(i). Therefore, if we write a instead of a - zo then (164.1) goes over into qu(a) = a. Since a # 0 here, it follows that a > 0. Thereby we have the result that there is an element a of F and a positive real number a, for which the following always holds :

T(a - z) = qu(a)

(z E F(o)(1), I z I < a).

(164.3)

We shall show that for every a with the property (164.3) 'AM - z) = qq(a)

(z E 5o)(11, I z I < a).

(164.4)

For this purpose we take a natural number n and a primitive nth root of unity a from Y0)(i). Then, according to (164.3), Pn_IZ)

9p(a - z)9:(a - aaZ) ... 92(a -

=

p(an

- Zn) S

s q)(an) + p(z") = a"+ I z I".

If we apply (164.3) to the factors of the left-hand side from the second on, then we get 4'(a - z)an-1 < an + I Z I., i.e., n

rr

9'(a-z)Sa1+ [

(n=1,2, ...).

a 1

J

For I z I < a this gives (p(a - z) < a. Hence, and from (164.3), follows (164.4).

We generalize (164.4) to 99(a - nz) = qu((X)

(z E

z I < a; n = 1, 2, ...).

(164.5)

For n = 1, (164.5) is the same as (164.4). Let (164.5) be assumed for some n. If we then apply (164.3) to nz + z instead of z, (164.3) follows

from the assumption, thus also (164.4) for a - nz instead of a. Thus, again from the assumption,

tq(a - nz - z) = tp(a - nz) = p(a) (z E .(o)(i); I z I < a). Hence (164.5) is proved by induction. Now, from (164.5)

tP(a - z) = T(a) = a

(z

E'o)(')).

OSTROWSKI'S SECOND THEOREM

647

So, for any two complex numbers z1, z2,

Iz1-z2I=9q(z1-Z2)<4'(a-z1)+q'(a-z2)=2a. By this contradiction we have proved Theorem 412. EXAMPLE. Every Archimedean-valued field F is topologically equivalent to a sub-

field of the field . (,)(i) valued by absolute valuation. This follows immediately, when we apply Theorem 142 to the perfect hull of F.

§ 165. Extensions of Real Valuations for Algebraic Field Extensions The proposition of uniqueness in Theorem 411 (see also § 163, Examples does not remain valid, if continued valuations of algebraic field extensionof non-perfect fields (with valuation) are included. Concerning this the fol lowing theorem is valid: THEOREM 413. Let an algebraic field G I F and a real valuation 92 of F be given. Let r denote the perfect hull of F, and N an algebraic extension

field of F which continues a subfield, isomorphic with the normal field of G I F. Let the real valuation of N, which continues p, be denoted again by 97. Then all the real valuations w of G, which continue the valuation of F, are given by +V(a) = q7(a'),

(165.1)

where a -+ a' denotes every isomorphic mapping of G I F into N. Let a, fi be arbitrary elements of G. According to (165.1) tV(afl) = 97(((c)') =

= 92(a')990') = w(000'

'(a + fi) _ q((a + a)') = 92(a + fl) < 99(a') + q0') = o(a) + o(fl), thus , is a real valuation of G. When a E F, we have a' = a, tN (a) _

(a),

thus tN is an extension of q7.

Conversely, let p denote a valuation of G of the required type. Furthermore let G denote the perfect huff of G, where we may assume that F is a subfield of G. The valuation of G may likewise be denoted by v. The subfield F(G) of G has an extension field No isomorphic with N over Let a -b a' be a corresponding isomorphism:

No IG,& NIF(a-bal.

648

FIELDS WITH VALUATION

Similarly, as above, we see that a valuation 92t of N is defined by p(a) = g21(a') which is an extension of the valuation 99 of r, and thus identical with the valuation 92 of N. Because G c No, (165.1) is fulfilled as required. Of course formula (165.1) does not, in general, yield only different valuations V. Concerning this cf. PICKERT (1951).

EXAMPLE. We consider the special case of Theorem 413, for a simple algebraic extension G = F(8). Let f(x) be the minimal polynomial of ?? over F and

f(x) = fi(x) ...f.(x) its irreducible factor decomposition over F. Then it will suffice to admit only the isomorphisms 0 -- 0k (k = 1, ..., r), where each @l, . . ., @, (E N) is a zero of fl(x), .. , f,(x), respectively. We see from the trivial consequence of Theorem 411 that in the continued valuation +p, given there, two conjugate elements over F have the same +p-value.

§ 166. Real Valuations of Number Fields of Finite Degree

We apply Theorem 411 in order to determine all the real valuations of the fields of finite degree. For prime characteristic we have to deal with finite fields, which, however, can have only the trivial valuation, so that

we may restrict ourselves to the algebraic number fields of the form G = .7(o)(O). According to OSTROWSKI's first theorem (Theorem 409) there has are two cases to be distinguished according as the prime field

the absolute valuation or a p-adic valuation. So we are considering the Archimedean or non-Archimedean valuation of G, respectively. In both cases we obtain the required valuations on account of Theorems 411, 413

(where in both cases we have to take into consideration the Examples concerned). We denote by f(x) the minimal polynomial of 9 (over .5J, and by n its degree. We first determine the Archimedean valuations of G. Since Y0 has, with respect to the absolute valuation, the field . io> for perfect hull, so we must determine the irreducible factors of f(x) over 5 ol. We proceed most easily from the factor decomposition

f(X) = (X - 01) ... (X -

(&'1 = ) Let r1 and 2r2 be the number of real or non-real conjugates of 0, respectively (r1 + 2r2 = n). Then f(x) splits into the product of r1 linear and r2 quadratic irreducible factors over ,. If 19k is real, then a valuation of G is given by over

(166.1)

V)(9(6)) = 19('0k) I

where g(x) denotes the polynomials over F of degree < n. If '9k is not real, then we have to put _ V(9(0)) = 19(O,) 9(9k) 1

,

REAL VALUATIONS OF NUMBER FIELDS

649

where Ok denotes the complex conjugate of 11k. The right-hand side is again equal to I g(0k) 1, therefore all the Archimedean valuations of G are given by formula (166.1), where ?9k has to run through the conjugates 01, ..., of 0, however, so that of each pair of conjugate-complex ele-

ments only one is admitted. The r1 + r2 Archimedean valuations so obtained are evidently distinct. Secondly, we define the non-Archimedean valuations of G. We give these more easily as exponent valuations. Let cop denote the p-adic valuation

of.$, where p may be an arbitrary prime number. We also denote the extension of w, to ible factors:

0)

by wp. We decompose f(x) over 5p) into irreduc-

f(x) = f1(x) ... f,(x) All the exponent valuations w of G, which continue the valuation wp of .7o, are then given by

w(g(o)) =

WVp(N(9(0k)))

(166.2)

k

where Ok denotes a zero of fk(x) in a suitable extension field ofpl, nk the degree of f(k)(x), and N the norm with respect to f(p)(0k) 1 -7;pl, while g(x) means the same as in (166.1). In connection with the above see § 170, Example.

§ 167. Real Valuations of Simple Transcendental Field Extensions Let a simple transcendental extension field F(x) be given, i.e., the rational function field of an indeterminate x over a fundamental field F. We wish to determine those real valuations of F(x) which are extensions of the trivial valuation of F. Since the required valuations of F(x) are non-Archimedean, we may take them in the form of exponent valuations. Therefore, let w denote an exponent valuation of F(x) such that w(a) = 0 for a E F, 0 0. We distinguish two cases, one when we have always

w(f(x)) > 0

(167.1)

for the non-constant polynomials f(x) (E F[x]), the other when

w(f(x)) < 0

(167.2)

holds, too, but we note in advance that these two cases will not be essentially distinct. When wv(f(x)) = 0 always holds, all the polynomial quotients have

FIELDS WITH VALUATION

650

w-value 0, thus F(x) is then trivially valued as the quotient field of F[x]. As we wish to disregard this, we suppose in the first case that ">" occurs in (167.1).

In the first case letp(x) denote a polynomial over F such that w(p(x)) > 0. Among the irreducible factors of p(x) at least one must occur with positive value, therefore we may suppose that p(x) itself is an irreducible principal polynomial. Now, if we replace w, if necessary, by an equivalent valuation, we may also assume that w(p(x)) = 1. (167.3)

If f(x) is now a polynomial non-divisible by p(x) over F, then we shall show that w(f(x)) = 0. Let p1(x), f1(x) be such that 1 = P(x)Pj(x) + Ax) ft(x)

(P,(x),.fi(x) E F [x])

Hence, because of (167.1),

o = w(1) z min (w(p(x)) + w(pl(x)), w(f(x)) + w(fl(x))) Z min (w(p(x)), w(f(x))).

From this w(f(x)) < 0, and so w(f(x)) = 0 follows from (167.1).

We now consider an arbitrary polynomial g(x) over F. This can be written uniquely as

g(x) = (P(x))k4(x)

(9(x) E F[x],P(x) f' 9(x))

From what has been proved so far and from (167.3) it follows that

w(g(x)) = k . Because of the rule w

g(x) f h(x)

w(g(x)) - w(h(x))

(167.4)

the values of all the elements of F(x) are now known. Further, we see [cf. (160.1)] that w is that .p-adic valuation of F(x) which belongs to the prime polynomial p(x). Since, conversely, all the irreducible principal polynomials

p(x) (E F[x]) are admissible and yield non-equivalent .p-adic valuations, the present case has been entirely explored. In the second case we take a principal polynomial p(x) over F of minimal

degree with the property w(p(x)) < 0 .

(167.5)

We shall show that p(x) must be linear. For let us suppose that

p(x) = x° + PA(x)

(n>_2),

(167.6)

SIMPLE TRANSCENDENTAL FIELD EXTENSIONS

651

where p1(x) aenotes a polynomial of degree < n, then by hypothesis we have

0,

w(pt(x)) >_ 0, w(x)

w(x")

0.

w(p1(x)))

0.

Hence, from (167.6), w(p(x)) > min (w(x")

,

Since this contradicts (167.5), p(x) is in fact linear. Then

p(x)=x+a for some element a from F. But, because of w(f) = 0 and Theorem 400, it follows from (167.5) for every element 9 (# 0) of F that w(p(x) + j9) = w(p(x)).

This together with (167.5) means that all the linear polynomials of K [x] have a common negative value. When we thus replace w, if necessary, by an equivalent valuation, then we get

w(x) = -1. For every polynomial

9(x)=aoxm+... +am

(am00)

over F we have w(g(x)) _ -m as a result of Theorem 400. If we again apply rule (167.4) we obtain for the value of an arbitrary element of F(x) other than 0, the formula wl

x

h(x) j= n - m,

(167.7)

where m and n denote the degree of the numerator and denominator, respectively. (Moreover (167.7) also holds for g(x) = 0, since then, according

to our previous convention m = - oo, while on the other side w(0) = Co.) Conversely, we shall show that a (discrete) exponent valuation of F(x) is defined by (167.7), which we call the degree valuation of F(x).

If gf(x) is a polynomial (5& 0) over F of degree m, (i = 1, . . ., 4), then 91(x) w(---

93(x) I 92(x) 94(x) 1

_ m2+m4-ml-m3=w

91(x) 92(x)

+w I

and W

93(x)

91.(x) l

92(x)1 + 194(x) I =

91(x)94(x) + 92(x)93(x)

w 1

92(x)94(x)

93(x) 9a(x)

,

FIELDS WITH VALUATION

652

On the right-hand side of the last equation the degree of the numerator is at most max (m1 + m4, m2 + m3), but that of the denominator equals m2 + m4, thus the right-hand side of this equation is at least

m2+m4-max(ml+ m4, m2+m3)= = min (m2 - m1, m4 - m3) = min w 91(x)

,

x 9a(x)

92(x)1

19.I(x)1

According to this, (167.7) is in fact an exponent valuation. To sum up we have the following THEOREM 414. All the real valuations of the rational function field F(x), which extend the trivial valuation of a field F, are (apart from equivalent valuations) the ,p-adic valuations belonging to the irreducible principal polynomials p(x) (E F [x]) and the degree valuation of F(x). (Clearly the above valuations are not equivalent to one another.) It is important, as we have already suggested, that the exceptional position of the degree valuation of F(x) as opposed to the .p-adic valuations of F(x) (in a sense to be stated precisely below) is only apparent. It is well known, but is worth stressing, that the indeterminate x does not

have an invariant meaning, for the field F(x); it may in fact be replaced by any element Y

_ ax+ ;tx +

(z, "'

y, b r-- F; ab -

0) .

(167.8)

Then, according to Theorem 323, all the primitive elements y of F(x) for which F(y) = F(x) are given by (167.8). We choose, in particular,

y_x 1

(167.9)

and denote by wj, the .p-adic valuation of F(y) (= F(x)) corresponding to the prime polynomial p(y) = y. Let g(x) h(x)

be an arbitrary element (0 0) of F(x). In order to determine its wy value, we denote by m and n the degree of the numerator and denominator, respectively. We have g(x)

9(y-1)

y

y"h(y-1)

653

SIMPLE TRANSCENDENTAL FIELD EXTENSIONS

Both numerator and denominator of the second factor of the right-hand side are polynomials of y with a constant term other than 0, therefore they are prime to y. Consequently x

tip,,

h(x) = n - m.

We now see the correspondence with (167.7), so that the p-adic valuation xY of F(y) (= F(x)) is identical with the degree valuation of F(x). Thus we have seen that the degree valuation of F(x), after we have introduced the primitive element y =

Ix

,

is converted into the p-adic valuation corrrespond-

ing to the prime polynomial y of F [y].

The p-adic valuation of F(x) corresponding to the polynomial x - a is also called the valuation (of F(x)) belonging to the place a. Correspondingly,

the degree valuation is also called the valuation belonging to the place 00

(since, after we have introduced the primitive element

I,x this becomes

the valuation belonging to the place 0). The above valuations are called the point valuations of F(x). If, in particular, F is algebraically closed, i.e., only linear polynomials are irreducible over it, then point valuations are all the valuations of F(x). Let w,, denote the point valuation of F(x) belonging to the place a (a E F or a = oo). If, for an element r(x) of F(x), w.(r(x)) = n or

-n

(n = 1, 2, ...)

,

then we say, using the terminology of function theory, that x = a is an n -fold zero or an n -fold pole of r(x), respectively. As to every valuation in general, so also to every point valuation wa there belongs a perfect hull of F(x). Its elements, according to (160.5) may be uniquely given by the (p-adic) infinite series f(x) _= am(x - oc)m + an:+1(x - 00"' + ...

am+I, ... E F) ,

(167.10)

where m can indicate any integer. These are similar in form to the power series in the theory of functions, therefore they are called formal power series. For the degree valuation (x = co ), we have to replace x - a in (167.10) by

-.xI

In function theory one is generally interested only in the substitution values f(E) for 5 E F in connection with a power series f(x), i.e., we understand f(x) as a function of the variable x. Therefore the notion of power series in algebra and that in function

654

FIELDS WITH VALUATION

theory are related in the same way as the polynomial notions of both theories. (Cf. Exercise 2.) EXERCISE 1. Quite generally, if 99 is a valuation and a -- a' an automorphism of a field F, then tp(a) = p(al) defines a valuation ip of F. Prove that all the place valuations of the above-considered field F(x) can be obtained from one of them in this way by means of F-automorphisms. EXERCISE 2. The above notions introduced with respect to F(x) are meaningful even when F is arbitrarily valued (but now Theorem 414 is no longer valid). Let e.g.,

F = F-(,) be the p-adic number field. We take into consideration a formal power series (167.10) where now a, am, am+1, ... are elements of JF,," . With an element of F-(," we carry out the substitution x = :

!l') = am( - ex), + am+l(E - (z) '+1 + .. . However, this equation has a meaning only if the right-hand side is convergent, and then represents an element of .F'(,). The , for which this is the case, constitute the domain of convergence of the formal power series (167.10). Show that if p 96 2 the domain of convergence of ap

f(x) =A E

1l

(E) x"

n

which are not units, and that then

consists of those p-adic integers

(f($))2 = 1 + $ . For further investigations of this kind cf. HASSE (1963). EXERCISE 3. Take for basis the rational function field F = .70(x1, ..., x"), form the rational function field G = F(x) and denote by G the perfect hull of G belonging to the degree valuation. Put

AX) = (x - x1) ... (x - xn) = x" + g(x)

In G we have I

X - x,

_

I

a0

x

X= "

xi

1

x'r , wx)

_

1

x" i

gx) ''

00

(

X" )

If these are inserted into

"

1

f' (x)

x - x,

f(x)

we obtain a short proof of WARING'S formula (Theorem 279).

CHAPTER XI

GALOIS THEORY The main object of the theory of GALOIS is to obtain a detailed survey of the subfields of a separable normal field of finite degree. The applications

of this theory far surpass its objective and are deeply involved in many questions of algebra, number theory and geometry. It was one of the earlier

established branches of knowledge of modern algebra and is admirably neat. The twenty years old GALOIS, killed in a duel to the great sorrow of posterity, far exceeded with his life-work the stage of development of the mathematical sciences of that time. As to generalization cf. CARTAN (1947) and NAKAYAMA (1949-50).

§ 168. Fundamental Theorem of Galois Theory A separable normal field N I F of finite degree is said to be a Galois field and the group of its F-automorphisms is called the Galois group of N I F. If F is a prime field, then N is called an absolute Galois field. In this case the Galois group is the full automorphism group of N. We can express the definition of Galois fields in two other forms: Galois fields over an arbitrary field F agree with those separable algebraic fields F(ad) for which the minimal polynomial of 0 splits into a product of linear factors over F(O). The validity of this transformation of the above definition follows from Theorems 299, 303, 356. Furthermore, it follows from Theorems 331, 336 that a field N I F is a Galois field if, and only if, its degree [N : F]

is finite and equal to the number of automorphisms of N I F. Galois fields constitute the object of the theory of GALOis, or briefly Galois theory, and in this the Galois groups as equally important topics will also be considered. More precisely, we shall mostly deal with a fixed Galois field and its Galois group only. We shall show that two isomorphic Galois fields N I F, N' I F have isomorphic Galois groups. We shall even show that from every isomorphism

NIF;zt N' I

(a->. soc)

follows the isomorphism

d ^ 4'

(a - sas-1)

between the corresponding Galois groups c0, cdi'. 22 R. -A. 655

GALOIS THEORY

656

It is evident that a->sas-1 is an isomorphic mapping of N' onto itself for every a E 44, i.e., an automorphism of N'. The elements of the fundamental field F remain fixed here, thus sas-1 belongs to 4'. Accordingly s4ts-1 e_ C cp'. Likewise s-1c4 s S 4, i.e., 4' c s4s-1. Therefore sc4s-1 = 4t'.

This means that a -* sas' is a mapping of cP onto 4'. This mapping is obviously one-to-one and homomorphic, so proving the assertion.

Galois theory is concerned with certain other fundamental notions in connection with a Galois field N I F and its Galois group 4. In order to introduce these we next consider a quite arbitrary field N and its full automorphism group cP. If G is a subfield of N, we denote by G the set of those elements a of 44 for which the equation as = a

is satisfied by all a E G. If furthermore V is a subgroup of c0, then we denote similarly by - the set of those elements a of N for which the given equation

is satisfied for all a E A. Evidently G is a subgroup of 4, furthermore is a subfield of N. We call G and X the invariance group of G and the invariance field of, respectively. (In other words : G is the group of G-automorphisms of N, furthermore A' is the field of those elements of N which

are fixed under the automorphisms contained in X.) According to the above we have defined a unique correspondence

X->X where X runs through the subfields of N and the subgroups of 4. We call this the correspondence in the sense of Galois theory. Substantially it means a certain mapping by which the set of subfields of N is mapped into the set of subgroups of co and this into the former set. We say that G is the subgroup of c belonging to the subfield G (c N) and the subfield of N belonging to the subgroup ° (9 c.4). From the definitions we obtain the following rules immediately: G1 c G2

G,

G2

1719- Ir2 Ir1X2

(G1, G2 S N) .

G 1,'

2C(4)

(168.1)

(168.2)

Further we shall prove that GDG

(G S N),

(168.3)

Since, for every element a of G, the condition as = a is satisfied by all the elements a of G, we have a E G, whence (168.3) follows. (168.4) is proved similarly.

FUNDAMENTAL THEOREM OF GALOIS THEORY

657

We prove further that

(G c N),

G=G (

c c4)

(168.5) (168.6)

.

From (168.1) and (168.3) it follows that the left-hand side of (168.5) is contained in the right-hand side. Also the converse of this follows from (168.4) by application to = G. Hence, (168.5) is proved and (168.6) may be proved similarly.

TinoREM 415. For a group Re of automorphisms of a field N, the relative field Nla' is of finite degree if, and only if, W is finite, and then (168.7)

[N : Jr] = O(M).

(168.8)

is Thus, if N The automorphism group of N I is of finite degree, then l° is finite, consequently according to (168.4) X, too, is

finite. Conversely, we suppose that M° is finite. We next show that N I A' is then of finite degree and

[N:?]s0(

(168.9)

.

For this purpose we write n

(168.10)

It is sufficient to show that any n + I elements w1, ..., can+1 of N over Jr are necessarily linearly dependent. We consider the system

(awl) x1 + ... + (awn+l)xn+1 = 0

(a E -7)

of n homogeneous linear equations with n + 1 unknowns x1,. .., xn+1. This has, according to Theorem 248, a non-trivial solution. There is thus a least natural number m (_< n + 1), for which the equation system

(awm),n = 0

(a E M°)

(168.11)

is satisfied by any elements 1, ..., $n, of N, where m does not vanish. Of course, we may assume that (168.12) $m E X. We take an arbitrary element b of X, replace a in (168.11) by b-1a, and then

carry out the automorphism b: (acal) (b$1) + ... + (awm)

0

(a, b E a°)

CALOIS THEORY

658

Hence, and from (168.11), (168.12), we obtain by subtraction,

(awl) ($t - b$t) -I- ... + (awm-i) (em-i -

0.

Because of the minimal property of m

bpi = i

(i = 1, ..., m - 1; b E)

must then hold. Therefore fit,

- , m-i E .

. This and (168.12) imply,

for the special case a = 1 of (168.11), that oli.... , alm, and a fortiori are linearly dependent over X. (168.9) is now proved. oh, ..., Since, accordingly, the field NIA' is of finite order, so, from Theorem 335, for the order of its automorphism group the inequality

O(,YC) 5 [N : C]

(168.13)

follows. Since, moreover, from (168.4) 2C ', equality must hold here as well as in (168.9) and (168.13). This implies the validity of (168.7) and (168.8), and consequently Theorem 415. THEOREM 416. A field N I G is a Galois field if, and only if, there exists a finite group c° of automorphisms of N such that

G = X.

(168.14)

For the proof we suppose, first, that for afinitegroup l° of automorphisms of N the condition (168.14) is fulfilled. Hence, and from (168.8), [N : G] _

= O(. Moreover, it follows from (168.7) that G = T. Then [N : G] = O(G).

(168.15)

Since G is the automorphism group of N I G, so, according to the above, (168.15) means that N I G is a Galois field. Conversely, we suppose N I G to be a Galois field, i.e., that (168.15) is satisfied. We write A' = G. Since ° is finite, [N : G ] _ [N : G] follows G, i.e., = G. from (168.8) and (168.15). This and (168.3) give Consequently Theorem 416 is proved. THEOREM 417 (fundamental theorem of GALOIs theory). Let N I F be a Galois field and 4 its Galois group, and let X --> X be the correspondence in the sense of Galois theory. Then the set of subfields of N I F is mapped oneto-one onto the set of subgroups of c4 by

G->G

(F9G9N)

(168.16)

and

[N : G] = O(G)

(168.17)

FUNDAMENTAL THEOREM OF GALOIS THEORY

659

for elements assigned to each other. The inverse mapping of (168.16) is

r-. jr

( ° c Co.

(168.18)

SUPPLEMENT 1. For arbitrary subfields G1, G2 of N I F and for arbitrary

subgroups -71, -72 of cp

621

X1 c 4°2 a dr1 {Gl, (:;2} = Gl (1 G2, {DL

i'

2}_

1fl

2,

`2

{G1 n G2 = {Gl, G2},

1n

(168.19) (168.20) (168.21)

2= ['?'I, I2). (168.22)

SUPPLEMENT 2. For the conjugates aG (a E cot) of a subfield G of N I F

aG =aGa 1.

(168.23)

SUPPLEMENT 3. A subfield G of N I F is (normal i.e.) a Galois field over F if, and only if, G is normal in c.P, and then the Galois group of G I F is isomorphic

with thefactor group c4/G (= F/G). If the elements of a system of representatives mod G of cog act on the elements of G, all the different automorphisms of the field G I F are obtained, i. e. the elements of its Galois group. NOTE. Apart from (168.17) the contents of the fundamental theorem and of Supplement 1 may be so stated that the first or second lattice of the subfields of N I F is isomorphically mapped by (168.16), onto the second or first lattice of the subgroups of cot, respectively. In Galois theory, particularly

in Supplement 3, lies the justification for the similarity of the terms "normal field" and "normal divisor". The fundamental theorem may be applied to an arbitrary separable field G I F of finite degree as one has recourse to its normal field N I F which is evidently a Galois field and is called the Galois field of the field G I F; then the subfields of G I F are assigned, because of (168.21), to those subgroups of the Galois group of N I F which contain the invariance group G of G. In order to prove Theorem 417 we denote the full automorphism group of N by Since cP is the Galois group of N I F, F = d, .

(168.24)

On the other hand, there is, according to Theorem 416, a subgroup coo of vL' such that F=coo.

GALOIS THEORY

660

By Theorem 415, c o =o. and so F = 4io. Hence, and from (168.24), it follows that 40 = 4i, consequently

F=4.

(168.25)

Further, from Theorem 415, [N : cQ] = 0(c4). Hence, and from (168.24), (168.25),

[N : F] = O(F) .

(168.26)

We now consider an arbitrary subfield G of N I F. Since both N I F and N I G are Galois fields, we may apply (168.24), (168.25), (168.26) instead of F. Consequently (168.27) G=G and (168.17) holds. Since G ? F, according to (168.1), G c F, thus, according to (168.24), G c co. Accordingly, by (168.16), the set of subfields of N I F is mapped into the set of subgroups of cQ.

In order to show that this mapping is onto the latter set, we consider 9 4 it then an arbitrary subgroup 2l° of c.P. We put G = R. Because follows from (168.2) and (168.25) that G ? F, therefore G is a subfield

of N I F. Since V is finite, it follows from Theorem 415 that ' = 'V. Since then

AT, the mapping (168.16) is in fact onto the set of subgroups

of c.4.

In order to verify that this mapping is one-to-one, we suppose that for two subfields G, H of N I F the assigned groups are equal: G = R. Since, besides (168.27), VI = H also holds, it follows that G = H, i.e., that the mapping (168.16) is one-to-one. The mapping G

G, inverse to this, may be given, according to (168.16),

as G -* G. Since here G runs through all the subgroups A' of 4, this mapping agrees with (168.18). Consequently Theorem 417 is proved.

From the fact that (168.16) and (168.18) are one-to-one, and from (168.1), (168.2) we obtain the rules (168.19), (168.20).

In order to prove (168.21) we bear in mind that {G1, G2} Q G1, G2. According to this and (168.1), {G1, G2} c Gl, G2 ,

thus

{G2} S G1 fl G2 . On the other hand, the right-hand side is evidently a part of the left-hand side, whence (168.211) follows.

Similarly (168.221) is obtained from (168.2). (i = 1, 2). In order to prove (168.212) we apply (168.221) to By the subsequent application of the mapping X -k X, because X = X, we obtain (168.212).

FUNDAMENTAL THEOREM OF GALOIS THEORY

661

Similarly we obtain (168.222) from (168.211) by application to G. = Xt (i = 1, 2). Consequently Supplement I is proved.

In order to prove Supplement 2 we denote the elements of G by a. Then aG consists of the elements am. Consequently, the group aG consists of those elements s of 4 for which the condition

saa = as

(168.28)

is satisfied for all or E G. For (168.28) we may write a-1saa = or. Thus the required s are those elements of d for which a'sa E G, i.e., s E aGa-1. Hence Supplement 2 is proved. Lastly, in order to prove Supplement 3, we must notice that, because of Theorem 331, G I F is normal if, and only if, G has no conjugates in N over F except itself. This implies that

aG = G

(168.29)

for all a E c.0. Since (168.29) means the same as aG = G, for which we can

write, according to (168.23), aGa-1= G, the above-mentioned condition consists in the fact that G is normal in 4. Hence the first assertion of Supplement 3 is proved.

For the following statements we assume that G I F is normal, i.e. a Galois field. For every a (E c4) let a1 denote the mapping

a-* ax of G. This is, according to the above, an automorphism of G I F. It is again evident that all the different al form a group c01. For this the homomorphism 4 '" X11 (a-* a)

holds trivially. Its kernel consists of the elements a of cP for which the equation as = a is satisfied for all a E G. Since, accordingly, this kernel is equal to G, the isomorphism

4 /G

c.01 (a G -* a)

(168.30)

follows. This results in

[N : F] = 0(4) = O(G) O(4l) whence, according to (168.17) and Theorem 292,

O(Q = [N : F] [N : G]-1= [G : F].

(168.31)

Now cPl is a subgroup of the Galois group of G I F. Both are of the same order because of (168.31), and so equal. This, with (168.30), completes the proof of Supplement 3.

GALOIS THEORY

662

THEOREM 418. Let N I F1, N F2 be two Galois fields and (41, CP2 their Galois groups. With respect to F = FI fl F2 as fundamental field, N is a Galois field if, and only if, the group c44 _ {c4l, 4t2} is finite, and then this is exactly the Galois group of N I F.

If N I F is a Galois field, then F is its Galois group. On the other hand, according to (168.212),

F = F1 fl F2={FI,F2}=(4I,c02)=4. Hence it follows that co is finite and is the Galois group of N I F.

If, conversely, c4 is finite, then, according to Theorem 416, the field N 14 is a Galois field. Since, moreover, according to (168.221), at = {41, (^ti2) = 41

n42= F1 fl F2 = F ,

Theorem 418 is proved. THEOREM 419. Let N I F be a Galois field and P I F a further field, both with a common overfield. Then the field {N, P} I P is a Galois field and every

automorphism of it is the extension of one, and only one, automorphism of N I F. Furthermore the Galois group of {N, P} I P is isomorphic with a subgroup of the Galois group of N I F.

For the proof we put N = F(O)

.

Then {N, P} = P((9).

Letf(x) and g(x) denote the minimal polynomials of $ over F and P, respectively. Then g(x) I f(x).

(168.32)

Since f(x) is separable, the separability of g(x) follows from (168.32). Since furthermore f(x) splits into linear factors over N, it follows from (168.32) thatg(x) splits into linear factors over {N, P}. Accordingly {N, P} I P is, in fact, a Galois field.

If 0 -* 0' is an automorphism of the last mentioned field, so that g(X) = 0, then, because of (168.32), f(O') = 0; we are therefore concerned with the extension of an automorphism of N I F. Since both automorphisms are uniquely determined by 19', one is uniquely determined by the

other. If we assign the first to the second, then evidently we obtain an isomorphism of the Galois group of {N, P} I P with a subgroup of the Galois group of N I F. Theorem 419 is thus proved.

FUNDAMENTAL THEOREM OF GALOIS THEORY

663

TmOREM 420. Let N I F be a Galois field with the Galois group 4 and

(F =) Go c G1 c

... c Gr (= N)

(168.33)

a chain of subfzelds of it, and let

(C4=)

'oD'r1D...Z) -'r(=1)

(168.34)

be the chain of the related invariance groups. All the fields

Gi I Gi_1 (i = 1,

..., r)

(168.35)

are Galois fields if, and only if, (168.34) is a normal series of 4. Furthermore, the Galois groups of the fields (168.35) are then isomorphic with the factor groups

,:77i-1/ri

(i= 1,...,r).

(168.36)

COROLLARY. The Galois group c.P of N I F is solvable if, and only if, there is a

chain of fields (168.33) in which all the fields (168.35) are Galois fields of prime degree. The theorem follows immediately from the Supplement 3 of Theorem 417. From Theorem 138 it then follows that the corollary is also true.

In certain cases we name the Galois fields according to the properties of their Galois group. Thus a field N I F is called cyclic, Abelian or solvable, respectively, if it is a Galois field and its Galois group is cyclic, Abelian or

solvable, respectively. Of course a Galois field of prime degree is always cyclic.

§ 169. Stickelberger's Theorem on Finite Fields

According to Theorem 306, the number of automorphisms of a finite field is equal to its degree, thus finite fields are absolute Galois fields and, according to the same theorem, cyclic. Although the fundamental theorem of the Galois theory is almost trivial for them, nevertheless its application yields valuable results. An example of this is the following important theorem). If f(x) is a polynomial of degree THEOREM 421 n without multiple factors over a finite field F of characteristic p A 2 and m the number of irreducible factors of f(x), then its discriminant is a square if, and only if 2 1 n - m. For another proof cf. the Example at the end of § 174. For the proof we may suppose that f(x) is a principal polynomial. We first consider the case wheref(x) is irreducible, i.e., m= 1. Then we have recourse to the finite field G = F(a) where f (a) = 0. Since G is (absolute) cyclic, 22/a R.-A.

GALOIS THEORY

664

so G I F is cyclic of degree n. Let s be a primitive element of its Galois group. Then

f(x) = (x - a) (x - sa) ... (x - sn-1a) We write S=

11 (sta - ska) 051
.

.

(169.1)

Then, according to Theorem 277,

D=b2 is the discriminant of f(x). On the other hand, it clearly follows from (169.1) that sb=(-1)n-'6.

Since, according to this, (because p 0 2) b lies in F if, and only if, 2 n - 1, the theorem is proved for the case m = 1. For the general case an irreducible factor decomposition f(x) = f1(x) ... fm(x) holds in F[x]. Hence, and from the multiplication theorem for discriminants (§ 113, Example 2), it follows that the discriminant of f(x) is of the form D = DI ... DmL02 ,

(169.2)

where Di denotes the discriminant of f;(x) (i = 1, ... , m) and N an element of F. Now the group F*, according to Theorem 306, is cyclic, furthermore O(F*) is an even number because p # 2. Consequently it follows from (169.2) that D is a square in F if, and only if, the number of those Dl which are not

squares is even. If ni denotes the degree of f(x) (i = 1, . . ., m), the above condition, according to the case already dealt with, implies that

(n1- 1)+...+(nm- 1)=n-m is even. The theorem is now proved.

§ 170. The Quadratic Reciprocity Theorem Let a, b (b # 0) be relatively prime integers. If the congruence x2 =- a (mod b)

(x E -7)

is solvable, then we say that a is a quadratic residue mod b.

665

THE QUADRATIC RECIPROCITY THEOREM

For every odd prime number p and for every integer a prime top we define

a

the Legendre symbol pj so that this is equal to 1 or -1, according as a is a quadratic residue or not a quadratic residue mod p.

If a' denotes the residue class a (mod p), then j_!_j = 1 means that a'

P is a square in the group ..5 of the prime residue classes mod p. This group is cyclic of even order, according to Theorem 225. Hence the rule abo

p 1=

a (Aj t

(P X a, b)

(170.1)

I

immediately follows. THEOREM 422 (quadratic reciprocity theorem). For arbitrary different odd prime numbers p, q 1)p21 °21

P)fqJ

(170.2)

IqP = (-

THEOREM 422' (first complementary theorem to the quadratic reciprocity theorem). For every odd prime number p p-i

=c-1) 2

(170.2')

THEOREM 422" (second complementary theorem to the quadratic reciprocity theorem). For every odd prime number p p'1I2

tP _ (-

1)

8

(170.2")

We shall carry out the proof applying STICKELBERGER's theorem (Theorem

421) to the special case

F=Srp, f(x)=x"- 1 (Only the cases n = q, is = p 2

((n,2p)=1).

l will be taken into consideration.)

Since f'(x) = nx"-1, the Sylvester determinant of J (x), f'(x) is equal to n". Thus the discriminant of J (x) is D

=

(-

1)(Dn"

GALOIS THEORY

666

Because p .4' n, this is an element of J r, other than 0, thus f (x) has no multiple factors over .3p. Because 2 f' n we have

(it)

n 2

1

(mod 2).

Further the factor n"-1 of D is a square in .3,. Consequently we have, according to the theorem mentioned, P"-1

(- 1) z it )

( l

(170.3)

=

where m is the number of irreducible factors of f(x) over .3D. Next we put it = q. Then J (x) = (x - 1) Fq(x) ,

where FF(x) indicates the qa` cyclotomic polynomial over. Thus from 92(q) = q - 1 and Theorem 317 it follows that

m=1+ q-1

(e =.o (p(mod q))).

e

Accordingly, we now have

it - m =

q

e

1

Because of the meaning of e, p is the q e

(mod 2).

1

th power of a primitive element

of the group rq . Accordingly, p is a square in -7'* if, and only if, q e

1

is

even. Consequently, (170.3) now becomes q-1

( 1(-1) 2 q

Pj

P

q

q-1 (

1(

1

l

`q1 1P1

((

l P

2

(170.4)

THE QUADRATIC RECIPROCITY THEOREM

667

Since the left-hand side remains unaltered after interchanging p and q it also follows that p 1

(_1)2

((- l)z

P

q

1

(170.5)

Moreover

P11(-1)z P-1

3

,

(170.6)

3, it follows from (170.5) for q = 3.

since, if p = 3, this is trivial, and if p Because

()=_i.

1, (170.6) agrees with (170.2'), by which Theorem 422' is proved. 31

Because of (170.2') the equation (170.4) goes over into (170.2). This proves Theorem 422'. z We now write n = P

2+

1

.

[The condition (n, 2p) = 1 is fulfilled ! ]

Then we have n I p4 - 1, thus

f(x)Ix"-x. Hence it follows, according to Theorem 308, that the irreducible factors of f(x) over Yp can be only of degrees 1, 2, 4. According to the same theorem the product of those irreducible factors of f(x), whose degree is 1 or 2, is equal to the greatest common divisor

(AX), x1' - x) Because

x" ' - 1 - x(x" - x) = x2 - 1

(y

J lx) I x1'+1 - 1,

this greatest common divisor is equal to

(f(x), x1' - x, x2 - 1) , thus finally equal to x - 1, where we have taken into consideration the fact that because 2 I' n, x + I is not a divisor of f(x). This shows that the irreducible factors of f(x) over.$ , except the factor x - 1, are all of degree 4. Thus m=l+pz-1 8

Accordingly, we now have 2

n

-m-p

_ 8

1

(mod 2).

OALOIS THEORY

668

Since further, n

2

1

=

p2 - 1

is even, (170.3) becomes

4 p2 + 1

a

2

p Because of (170. 1) the left-hand side is equal to

12 . Consequently Theorem I

P1

422" is proved. The proof as carried out is essentially due to MIRIMANOFF-HENSEL (1905) and is one of the most elegant of almost a hundred proofs for the quadratic reciprocity theorem and the supplementary theorems. The quadratic reciprocity theorem may be formulated in field theoretical terminology

independently of the Legendre symbol as follows: Let p, denote the proposition that p is a square in .X",. If among the different odd prime numbers p, q at least one

is = 1 (mod 4), then the propositions p q, are equivalent. If on the contrary both are =- -1 (mod 4), then exactly one of the propositions p qp holds. (A really admirable correlation between .g', and X.!) ExAMPLE. As an application let us determine by the process derived in § 166 all the real valuations of the number field

G =.P-.(V-11 ) of degree two. The minimal polynomial of

11 is

f(x) = x2 - 11. We take an arbitrary element (96 0) of G in the form

a+b

11

(a, b E .X"o).

Since, over P'(0), the decomposition

f(x) = (x - J11) (x + J11), holds, we see, according to (166.1), that G has exactly the following two Archimedean valuations:

4gi(a+b 11)=Ia+b

11I,

.p2(a+b 11)=Ia-b

11`.

In order to determine the non-Archimedean valuations, we denote by w, that valuation of 9'(P) which extends the p-adic valuation of .9'o, where p can be any prime number. If f(x) is irreducible over X(,), then (166.2) yields the one and only one valuation w(a + b

11) = 2 w,,(a2 - l 1 b2)

of G ; if on the other hand f(x) is reducible over . ,), i.e., there is an element y of X(,) such that y2 = 11, then we likewise obtain the two valuations vi(a + b Vi -I) = w,(a + by),

v2(a + b

l 1) = w,(a - by)

THE QUADRATIC RECIPROCITY THEOREM

669

of G. The first case always occurs if x2 = 11 (mod p)

(170.7)

11

is insolvable, i.e., - _ -1. This condition, according to Theorem 422, may be p-i expressed in the form (-1) 2 111) = -1. The left-hand side depends only upon

the residue class p (mod 44), where we are concerned with the prime numbers p = 3, 13, 15, 17, 21, 23, 27, 29, 31, 41 (mod 44).

To this first case belong p = 2 and p = 11, too, since then xt - 11 (mod p2)

is insolvable. The rest of the p are given by 111 I = I and they may be characterized by p = 1, 5, 7, 9, 19, 25, 35, 37, 39, 43 (mod 44).

Since for these p congruences (170.7) has a solution co (0 < co < p), for which

f(x) _ (x - co) (x + co) (mod 11), the existence of a required element y follows for them from HENSEL'S lemma (Theorem

410), so that these p all belong to the second case. In order to determine in this case the n,-values (1 = 1, 2) of the elements of G it is still necessary, of course, to compute y explicitly. For that purpose we put

y = co + ct p + cz p2 + ...

(0 S cl, cz.... < p)

and compute recursively c1, c2, ... from the congruences (co + cl p + ... + ck pk)z m 11 (mod pk+')

(k = 1, 2, ...) .

Thus we obtain, e.g., if p = 5, co=1,

c1 = 1 ,

c2 = 2,

c3 = 0,

cs = 0,

c5=2,...,

thus y = 1 + 5 + 2.52 + 2.55 + ... Naturally for given p, a, b we need to calculate only a finite partial sequence c0, ... ck of co, c1, ..., for which already

a ± b (co + ... + ck p) r 0 (mod pk+1) holds, for it suffices to determine the above v,-values (1 = 1, 2).

§ 171. Cyclotomic Fields

Among finite fields the cyclotomic fields (§ 135) belong to the simplest kind of absolute Galois fields; of course, these are also very important. We shall now consider them more explicitly.

670

GALOIS THEORY

THEoREM 423. Let N denote the nth cyclotomic field of characteristic 0.

It is abelian of degree q(n), and its Galois group is isomorphic with the group ,71(n)* of prime residue classes mod n. If, in particular, n = q is a prime number, then the field N is cyclic of degree q - 1. N is the splitting field of the polynomial x" - 1 over Y0, therefore we may always suppose that N is a subfield of o)(i). As stated above N is of finite degree, separable and normal, i.e., a Galois field. Since all the nth roots of unity from 9 O)(i) are powers of a primitive nth root of unity,

N = Y(e) , where a denotes a zero of the nth cyclotomic polynomial F"(x), i.e., a primitive nth root of unity. Since F"(x), according to Theorem 315, is of degree qq(n) and, by Theorem 316, is irreducible, it follows that N has degree T(n).

Further F"(x) = fl (x - `) a

where a has to run through those numbers 1, ..., n which are prime to n. These give, in the form e-> ea

all the different automorphisms, - i.e., elements of the Galois group -

of N. The product of two such automorphisms a -> ea, a -* eb is, because (e")b = cab, equal toe - a"b. Since, further, in ° the exponent a is only taken mod n, because o(e) = n, we see that the Galois group of N has the property required in the theorem. Since this group is Abelian, N is also Abelian. Because of Theorem 225, it also follows that the last assertion of Theorem 423 is true. THEOREM 424. For a prime number q the splitting field G of the polynomial x9 - 1 over an arbitrary field F is cyclic, and its degree [G : F] is a divisor

ofq-1.

Let p be the characteristic and F,, (p >_ 0) the prime field of F. If q = p then G = F, so that we need only consider the case q 0 p. It follows, from the supposition, that G contains a splitting field Z of.x4 - 1 over Fp. Furthermore

G={F,Z}.

Now Z I F. is a cyclic field such that

[Z:F,,]Iq-I since, for p = 0, this follows from Theorem 423 and for p > 0 from Theorem 317. On account of Theorem 419,

{Z,F}IF=GIF

671

CYCLOTOMIC FIELDS

is likewise cyclic, moreover, the divisibility relations

[G:F]I[Z:Fp]Iq-] hold. Consequently Theorem 424 is proved. EXAMPLE 1. Let N denote, for a prime number q, the q1 cyclotomic field of characteristic 0. We determine all the subfields of N. Since N, according to the end of Theorem

423, is cyclic of degree q - 1, exactly one subfield of N of degree e belongs to every divisor e of q - 1, and no further subfields of it exist. Let

of=q-1.

(171.1)

Let a (E N) denote a primitive qt° complex root of unity and g a primitive number mod q. Then N= Jro(e). According to GAUSS the sums

n, = e`` + e``+` + ... + e°`+`('

,)

(i = 0, ..., e - 1)

(171.2)

are called f-termed periods of N. Since 1, . . ., e°-$ is a basis of N, the same holds

with respect to e, ..., e°-1. (The second is a normal basis.) After rearrangement of the elements, this basis may be written as follows: (171.3)

Consequently it follows from (171.2) that the periods no, ... ne_1 are different. Now let s denote the primitive automorphism e-

(171.4)

ea

of N, which is a primitive element of the Galois group of N. Then, according to (171.2),

snl = nt+1

(i = 0, ..., e - 1; ne = no).

(171.5)

Hence

skn,= r,

eIk

for every i = 0, ..., e - 1. This implies that all the subfields

G, = .

0(x,)

(i = 0, ..., e - 1)

of N have the same invariance group G, = {se} ,

and so by the fundamental theorem of GALOIS theory (Theorem 417), are equal. Moreover the relation [N : Gt] = O({se}) =

q-1 e

follows. Consequently we have obtained the result that Go (= G1 = ... = G,-,) is just the subfield of N of degree e. EXAMPLE 2. We see from (171.2) that the symmetric polynomials of no, ..., ne_1 are also symmetric polynomials of the basis elements (171.3). Consequently

h(x) _ (x - no) ... (x - ne_1)

(171.6)

GALOIS THEORY

672

is a polynomial over `.o. Since it is of degree e, because of Example I it is irreducible,

thus it is the (common) minimal polynomial of the f-termed periods no, ..., ne_1.

These are thus conjugate as we see also from (171.5). Hence it follows that the periods no, ..., napart from the order, are independent of the particular

choice of e and g. It also follows that h(x) = 0 is the defining equation of the subfield of degree e of the qth cyclotomic field and the periods no, ..., ne._l constitute a normal basis of it. EXAMPLE 3. We compute the polynomial (171.6) for the case q = 7, e = 3 (f = 2). As primitive number mod 7 we take g = 3. Because 3'= 1, 3, 2, 6, 4, 5 (mod 7)

(i =0,...,5),

according to (171.2),

no=e+eo, n1=28+e', na=22+eb. We compute

no=2+na, ni =2+no, ns=2+n,, noel = nl + n8 , n1n2 = no + X2 , Wont = no + al

If ak denotes the kth elementary symmetric polynomial of no, all nq, then it follows that

al = -1, QE=2al=-2,

as = none - na = (al. + na)nz = 2 + al = 1.

Consequently (171.6) now reads

h(x)=x3+x'-2x-1. § 172. Cyclic Fields Of the Galois fields, the simplest are the cyclic ones. We have dealt with examples of cyclic fields, and shown that all the finite fields and certain cyclotomic fields are absolutely cyclic. Here we wish to consider cyclic fields in general. In particular, we shall study those which are of prime degree, as these will be sufficient for our later purposes. We need the following preparation. If a is an element of a field F and n (>--_ 2) a natural number, then the (ntb) radical ./a means according to our

former definition a (in general not uniquely determined) root of the socalled binomial equation

x"-a=0.

(172.1)

We now extend this notion by admitting roots of(172.1) as values of . Ja which lie in any overfield of F. In this case we call /au more precisely a radical over the of F, which field F. Every such radical furnishes an extension field

CYCLIC FIELDS

673

we say arises from F by the adjunction of a radical. Of course, F(f) is, in general, not uniquely determined by F, cc and n. There is an extremely important special case, when the equation (172.1) corresponding to the radical '.Ja is irreducible. Then we call " a an irreducible radical. For this case, according to Theorem 295, the field F(Ja) is uniquely determined up to equivalent extensions (over F). We call " a a reducible radical, when (172.1) is reducible. (Cf. Examples

1, 2, 3.) When we say that "a field F contains the nth roots of unity", then it means that the polynomial x" - 1 over F splits into linear factors, i.e., all the radicals \/i over F already lie in F. THEOREM 425. A field F(Ja) I F is cyclic if the fundamental field F contains the nth roots of unity and its characteristic is not a divisor of n.

It will suffice to consider the case a 0. For the sake of brevity let us put 19 = Ja. Then we deal with the field G = F(ZJac) = F(8)

,

where 0 is a root of the equation (172.1). By hypothesis, F contains a primitive nth root of unity The factor decomposition

x" - a=x" - 0"= (x -0)(x - e8)... (x - e"-I8) holds over G. Since 0, . . ., a"-t0 are distinct, it follows that G I F is separable

and normal, therefore a Galois field, and that its automorphisms may be given in the form

0->eh9. The product of this with a further automorphism8-*et8 is 8-*e'ei8. Thus the Galois group of G I F is a homomorphic image of the cyclic group {e}, consequently it is itself cyclic. This proves Theorem 425.

One part of this theorem admits the following important converse. THEOREM 426. If G I F is a cyclic field of degree q of characteristic p (>_ 0),

where q is a prime number other than p and the fundamental field F contains the qth roots of unity, then there exists an element a of F such that G = F(.,/a). Let a denote a qth root of unity (in F), 0 a primitive element of G, for which G = F(d)

therefore holds. Also let s be a primitive automorphism of G I F. Then we call every expression (e, 8') = 0 + e(s8) + e2(s'o) + ... + eq-I(3q-I8,)

(172.2)

GALOIS THEORY

674

a Lagrange resolvent. It will be shown that among these Lagrange resolvents occurs a suitable element a of G. Because se = e we have

S(0, ,O) = s8 + e(s'0) + ... + eq-1(sgo) .

Since further sq = 1, comparison with (172.2) results in the equation

s(e, t) = e-1(e, ) . Then (172.3)

s ((e, o')q) _ (e, 0)q

On the other hand, it follows from (172.2) after summation for all the that

E (e, ) = q8 . e

The right-hand side is not 0, because p r q, therefore it lies outside F. Consequently, for at least one e, (e, t9) lies outside F and therefore is a primi-

tive element of G. We take such a e and put co = (e, 0) .

According to what has been said, G = F(co). Since s denotes a primitive element of the Galois group of G I F, it follows from (172.3) that coq belongs to F. Hence, Theorem 426 is proved. Theorem 426 even holds, according to FRIED (1956), for all natural numbers q such that p }' q. As regards the generalization of Theorem 426 for an arbitrary n (= 1, 2, ...) instead of q, see BoORBAKI (1939) and LUGOWSKI-WEINERT (1960).

THEOREM 427 (ABEL'S theorem). A polynomial (172.4)

of prime degree q over afield F is reducible if, and only if, a is a q`b power in F.

In this case (172.4) splits into linear factors over F if, and only if, F contains the q`s roots of unity. For the proof we suppose that (172.4) is reducible over F, i.e., there exists

a principal polynomial f(x) (E F[x]) of degree n (1 < n < q - 1) such that

f(x)Ixq-a.

(172.5)

Over a suitable extension field of F,

xq - a = (x - 13) (x - elI) ... (x -

eq-

)

(172.6)

675

CYCLIC FIELDS

with 994 = a,

e4 = 1

(172.7)

.

We take the constant term of f(x) in the form

(8 E F). From

(172.5) and (172.6),

P=

(172.8)

'0k0"

for some integer k, which we do not determine more precisely. According to (172.7) and (172.8),

N4=a"

Now the equation

qy + nz = I has a solution y, z (E J) because (q, n) = 1R.zIt follows that

a = qy+nz = (ayY)"

thus a is a qts power in F. From now on we suppose that a = 994 for some 99 (E F). Then

x4- (X =x4-994 has the factor x - 99. What has been stated so far proves the first part of the theorem. Since x4 - a = 994((99-Ix)4 - 1) ,

we see that x4 - a splits over F into linear factors if, and only if, the same holds for x4 - 1. This proves the second part of Theorem 427. The generalization of the first part of this theorem is the following THEOREM 428 (VAHLEN-CAPELLI theorem). Over afield F a polynomial

x" - a

(n ? 2; a E F, 0 0)

(172.9)

is reducible if, and only if, a = YRd

(d l n, > 1; j9 E F)

(172.10)

or

4 In,

a= -4y4

(y E F).

(172.11)

The proof is founded upon the following PROPOSITION. If f(x) is a separable principal polynomial and g(x) a nonconstant polynomial over a field F, and the irreducible factor decomposition

g(x) - 99 = fl gi(x) 1=1

(172.12)

676

GALOIS THEORY

holds over the extension field G of F, defined by

G = F(s)

(f(z9) = 0) ,

(172.13)

then the polynomial

h(x) = f(g(x))

(172.14)

has the irreducible factor decomposition over F:

h(x) _ [J N(g;(x))

(N denotes NG(X),F(X))

.

(172.15)

Thus the degrees of the irreducible. factors of h(x) are multiples of the degree

of f(x) and, in particular, h(x) is irreducible over F if, and only if, g(x) - 0 irreducible over G.

First of all we prove the Proposition. Because of (128.4), f(x) _ = N(x - 0). If x is substituted here for g(x), equation (172.15) follows from (172.12), (172.14). Since in this the factors of the right-hand side belong to F[x], we have only to show that they are irreducible over F. With this end in view we denote an irreducible divisor of h(x) over F by k(x). From (172.12), (172.15) it follows that for some i (= 1, ...,.r) gi(x) I k(x).

(172.16)

We can write

1(x) _ (x - iI) ... (x - 0.) so that N = F(t91,...,

(0 = 'b),

is the normal field of G I F. Because of the

separability of f(x), X91, ..., 0," are distinct. Now N(gi(x)) is the product of m factors, which result from g,(x) by applying the isomorphisms # --> 0, (s = 1, ..., m) of G I F (in N), and which, according to (172.12), are, in order,

divisors of g(x) - 01, ..., g(x) - 1,,, and consequently pairwise relatively prime. Since because of (172.16) they are all divisors of k(x), which follows

by applying the same isomorphisms, we find that N(gr(x)) I k(x).

Since the right-hand side is irreducible over F, it follows that every irreducible factor of h(x) over F is associated with one of the r factors of the righthand side of (172.15). This proves the Proposition. We shall now prove Theorem 428. For the assertion "if" this is simple, for if (172.10) holds, then

x"-a=x"-fld is divisible by x"" - f, but if (172.11) holds, then we have R

R

R

x" - a = x" + 4y4 = (x 2 - 2yx 4 + 2y2)( x 2 + 2yx 4 + 2y2).

CYCLIC FIELDS

677

Conversely, we suppose that (172.9) is reducible over F. We have to show that then (172.10) or (172.11) is true. If n is a prime number, then, according to the first part of Theorem 427, (172.10) is true. Now let n be composite. We suppose that the assertion is false for n but true for smaller n. Hence it follows that all the polynomials

xd - x

(d I n, < n)

(172.17)

are irreducible over F.

We denote by p (Z 0) the characteristic of F and first consider the case p,f'n. We begin with the case

n = q`

(q prime number, e >_ 2).

Since x4 - a, by (172.17), is irreducible over F and, because q # p, is even separable, a separable field G such that [G : F] = q is defined over F by G = F(O)

(VQ = 0C).

(172.18)

Since, moreover, we have

TV - a = (xV-')a - a, it follows from the reducibility of (172.9) and from the Proposition that

x°`-' is reducible over G.

-

19

(172.19)

If q # 2, it follows hence and from the induction hypothesis that there is an equation of the form

P_A4 On the other hand (since p

(AEG).

2), a = NGIF (0')

follows from (172.18). Since we now have a = (NGIF (A))4,

we have obtained the result that (172.10) is true for d = q.

If, however, q = 2, then it follows from the reducibility of (172.19) and from the induction hypothesis that

either 0 = (a# + v)2 or t = -4 (0 + v)4,

GALOIS THEORY

678

where µ, v denote suitable elements from F. In the first case 0 = µ2a + v2 + (21Av - 1)19,

thus µ2a + V2 = 0, 2µv - 1 = 0, and hence (because p

2)

a= -4v4. In the second case

0=0+ 4(µ2a+ v2+2µv+9)2= = 4(p2a + v2) + 16µ2 v2oc + (16p v(µ2a + v) +1)0,

thus (µ2a + v2)2 + 4µ2v2a = 0,

16µv(µ2a + v2) + 1 = 0,

and hence

j4.

1 8µv

So we see (in both cases) that (172.11) is t rue.

We now consider the case where n is not a prime power, i.e., it has a decomposition

n=uv

(u,vE7; (u,v)=1; u,vZ2).

Since, according to (172.17),

x" - a, x° - a are irreducible over F, and also

x"-a=(x°)"-a=(x')°-a, it follows from the Proposition that every irreducible factor of x" - a over F has a degree divisible by u and v, i.e., by n. This results in the contra-

diction that x" - a is irreducible over F. Lastly we consider the case p I n. Since, by hypothesis, a is not a pt power in F, the field G defined by

(P = a)

G = F ($)

(172.20)

is, according to Theorem 341, pure inseparable over F with [G : F] = p. Then

x"-a=(xp-," -#)°

(172.21)

An irreducible factor over F of this polynomial cannot be a power of .Cp-1n

- 0,

(172.22)

679

CYCLIC FIELDS

since 6, . . ., #°-' lie outside F. Consequently (172.22) is reducible over G. Therefore, because of the assumption, 4 1 n,

O= -4y4,

where fi, y are elements of G. (In the second case p must be odd, but we do not need this.) Since a = #P, it follows that a = (F'P)d

or

at

= -4 (yP)',

respectively. Since now G I F is pure inseparable and of degree p, it follows

from Theorem 343 that PP and yP lie in F. This contradiction completes the proof of Theorem 428. Cf. CAPELLI (1901), who considered merely a number field of finite degree.

Evidently the validity of Theorem 426 disappears if q = p. In this case, however, the following holds : TttEoi t 429. If G I F is a cyclic field of degree p and of characteristic p (p prime), then there is a A such that G = F(A) and AP - A E F.

Let s denote a primitive automorphism of G F and Y(s) the natural group ring of {s}. For an element of this and for an element of G we define the operator product n

w= E a,(s'w)

ais` 1

(ai E 7; w E G).

t=o

It is evident that G+ is transformed by this into an Y(s)-module (with the customary properties of an operator), furthermore F+ is an admissible submodule. In particular, (because sP = 1 and pw = 0)

(s- 1)P0.) =(SP- 1)w=0.

Now we take an co from G - F. For this (s - 1) w = sco - w 0 0, thus there is a k (1 <- k <- p - 1) such that

(s - I)kw

0,

(S - 1)k+1 CO = 0.

We put

e = (s - 1)k-1 Co. Then

(s- 1)e's0, (s- 1)Ze=0.

GALOIS THEORY

680

For the element e

7.=

(s - 1)e

so

sA

se

SQ

s(s-1)e = (s- 1)ae+(S- 1)e = (s- l)e =a + i,

then holds, as also does

s(AA-A)=(sA)"-sA=(A+ 1)°- (A + 1)=Ap-A. Since, according to this, A E G - F, AP - A E F, so A satisfies the required conditions, which proves Theorem 429. This simple proof is due to FRIED (1956). EXAMPLE 1. The field .g", (V2-5) is not uniquely defined as a subfield of .W(,) (10,

since .J25 has the four values 511"

(n=0,...,3).

Thus .F'0(.J25) can be either 3s"0(

) or .g-O( ,r--5 s ). These are not isomorphic. EXAMPLE 2. The n' roots of unity (n > 2) over a field may be written as Vi. This radical is always reducible.

EXAMPLE 3. If F is a field of prime characteristic p, then the field

F(°

m)

(a E F; n > 1) is, apart from equivalent extensions, always uniquely determined. It suffices

to prove this for the case where a is not a pth power in F. Then the assertion follows from the fact that, according to Theorem 428, is irreducible. (The same results from Theorem 427.) Since, moreover, V is uniquely determined within this radical is usually denoted by a°-". (Cf. § 142, Example 2.) EXERCISE. The following generalization of Theorem 426 holds. If G I F is a cyclic field of degree n and of characteristic p (>_ 0) where p,?n and F contains the nth roots of unity, then we can obtain G from F by the adjunction of an nth radical.

§ 173. Solvable Equations

Let us take a field F. We take an integer t (>_ 0) and give, over F, polynomials of the form

9; = gi(x1, ..., x,_1)

g = 9(x1, .. , xr)

(1

(173.1)

and integers q1, . . ., qt (>_ 2). (In particular, g1 is an element of F.) If t91, ..., 0t are elements of an extension field of F, where

..., $;_1)

(1= 1, ..., t),

(173.2)

then we call

0 =g(131,...,$e)

(173.3)

SOLVABLE EQUATIONS

R

681

a radical expression over F belonging to the polynomials g1, ..., g1, g and the root exponents q1, . . ., q, The 01, . . ., 0, are themselves called generating radicals. Allowing for the obvious ambiguity, if necessary we say more precisely that 0 is a value of this radical expression. We can (173.3) an irreducible radical expression, if the generating radicals occurring in it are irreducible (i.e., every 13; is an irreducible radical over F (01, . . ., 0I-1)).

Radical expressions can be easily characterized. In order to do this, we put Gi = F(01,

. .

(i = 0, ..., t).

., 0I)

(173.4)

Then G; = Gr-1(01),

MP E G,_I

(i = 1, ..., t),

(173.5)

(173.6)

0 E Lr L.

Hence we see that 0 is a radical expression over F if, and only if, there exists a chain of fields

F=Go9G1c...9G,

(173.7)

such that every term arises from the preceding by the adjunction of a radical and 0 is an element of Gt. Incidentally we sometimes call (173.7) a chain of fields belonging to the radical expression 0.

As an example of a radical expression (with t = 2) we have:

x+ A V a+ (µ + v (.,/ a)3), J e+ or

+

(Ja)z

(a, K, ..., r E F).

We call the irreducible equation

f(x) = 0

(f(x) E F[x])

(173.8)

solvable (by radicals) if it has a root which is a radical expression over F. By the above this means that there is a chain of fields (173.7) whose last term contains a root of (173.8) and that moreover (173.5) holds. On the other hand, suppose that f(x) is the product of separable polynomials over F. Let N I F be a splitting field of f (x), which is thus a Galois field. We call it and its Galois group the Galois field and the Galois group of the equation f(x) = 0, respectively (but cf. § 174). Since N I F is, apart from equivalent extensions, uniquely determined, the Galois group of an equation is uniquely determined to within isomorphism. The solution of equations by radicals was the main problem in early

algebra which had no other (algebraic) methods for the solution of an algebraic equation. Although Theorem 299 entirely solves this problem by the help of the splitting field, nevertheless solvable equations still form

an important branch of algebra, this being explained by the fact that

GALOIS THEORY

682

at least one solution of them is easily determined, i.e. by reducing them to

the solutions of binomial equations. One of the best results of Galois theory will turn out to be the close connection between the solvability of an equation and its Galois group. Before beginning our considerations we take a simple example: every cyclotomic equation F (x) = 0 is solvable (over . o), since every root of it may be given as the

radical expression V1. Still this is not a satisfactory solution of the problem of determining the solutions of this equation, since not all the values of this radical are solutions, but only the primitive nth roots of unity among them. This remark is reasonable,

in particular, already for the case n = 3, where the equation x2 + x + 1 = 0 is involved;

now, besides the previous expression, the radical expression 2 (-1 + V=3) is also a solution of the equation, and that, "fortunately", for all its possible values, therefore it is far preferable to the other.

For a solvable equation (173.8) an irreducible radical expression is called

a solution formula for this equation, if all its values are solutions of the equation and, conversely, all the solutions are furnished by its values. Next we prove the following THEOREM 430. Every irreducible radical expression over a field F, which furnishes a solution of an irreducible (solvable) equation (173.8), is a solution formula for this equation.

For this we suppose that the radical expression 0 given by (173.3) is irreducible and a solution of (173.8). Here 0, is a solution of the equation xq` - g,(01, ..., oi_1) = 0

U= 1, ..., t)

(173.9)

whose left-hand side is irreducible over F(81, . . ., O'i-D We consider a further value 0' of the same radical expression. For this

0' = 01, ..., 07,

(173.10)

where 0l, ..., 6' are elements of an extension field of F such that 0 is a solution of xqt

- g.(1 , ..., O'_-D = 0.

(173.11)

We have to prove that 0' also satisfies the equation (173.8).

To do this, we show that the isomorphisms F(01,

.,,&j) I F x F(a9l, ... 0r) I F (01 -* #1, ..., 0, -+ 0j'),

(i = 1, . . ., t) (173.12)

hold. For i = 1 it is true since, according to (173.9) and (173.11), 01, O' are solutions of the equation xq' - g1 = 0, irreducible over F. We then consider an i (= 2, . . ., t) and make the induction hypothesis that the

683

SOLVABLE EQUATIONS

isomorphism (173.12) holds for i - 1 instead of i. By this the left-hand side of (173.9) is converted into that of (173.11). Since, of these two polynomials, the first is irreducible over F(01, . . ., #t_1), and since t; and 0; are roots of (173.9) and (173.11), respectively, it follows from Theorem 298 that (173.12) is true for the i concerned and therefore in general.

We now take into consideration the isomorphism (173.12) for i = t. Because of (173.3) and (173.10) this maps 0 onto 0'. Hence it follows that, together with 0, 0' is also a root of (173.8) as asserted. We have still to prove that conversely every root of (173.8) is a value

of the radical expression (173.3). With this end in view we consider a normal field N of F(01, ..., i9) 1 F. From f($) = 0 and (173.3) it follows that N contains a splitting field of f (x). Therefore it is sufficient to prove that all the zeros of f(x) lying in N occur among the values of the radical expression (173.3).

Let 0' be such a zero :

f(0') = 0, 0' E N. By Theorem 329 there exists an isomorphism e -> sLo of F(01, ..., 0) 1 F such that 0' -1- s O. Now from (173.3)

so = 9(sh'1, ..., so), and it follows from (173.9) that sli, is a root of x41 = 9,(s#1, ..., s$,_1)

(i = 1, ..., t).

Accordingly, 0' = sO is, in fact, a value of the radical expression (173.3). Consequently Theorem 430 is proved. As a further preparation we prove the following THEOREM 431. Every normal subfield G I F of a solvable field N I F is solvable.

Since the assertion is trivial for G = F and N, we may assume that F c c G c N. Let co denote the Galcis group of N I F, which is, by hypothesis, solvable. Further let cY, denote the invariance group of G. According to Supplement 3 of the fundamental theorem of GALOis theory (Theorem 417),

Y is a normal subgroup of cPi, furthermore the Galois group of G I F is . isomorphic with the factor group d I Now, according to the JORDAN-HOLDER theorem (Theorem 136), the

normal series c0 X D 1 may be refined to a composition series

(`j _)Xp aYl

...

Xr (_ t) D ... D 1.

According to Theorem 138 in this all the factors Xi/

r+l are of prime

GALOIS THEORY

684

order. According to the second isomorphy theorem (Theorem 126')

`4/X D t1/

D ... D Xr1J7(= 1)

is likewise a composition series whose factors are again of prime order. Accordingly 4g/P, on account of Theorem 138, is solvable, so that the proof of Theorem 431 is complete. THEOREM 432 (main theorem for solvable equations). Let

f (X) = 0 (f(x) E F[x])

(173.13)

be a separable equation over a field F of characteristic p (Z 0), further let cP be its Galois group. If (173.13) is solvable, then C4 is likewise solvable. If, conversely, 4 is solvable and in the case p > 0 all the prime factors of 0 (4) are smaller than p, then (173.13) is likewise solvable and has a solution formula in which all the root exponents are prime numbers, which do not exceed the greatest prime factor of 0 (4). In order to prove the first part of the theorem we suppose (173.13) to be solvable. Then there is a radical expression t over F with f(t9) = 0.

(173.14)

On account of the rule

we may suppose that in 0 all the root exponents are prime numbers. These, and the corresponding radicals, are denoted by q1, . . ., q, and' 191, ..., so that i97' E F(191, ..., X9,_1)

(i = 1, ..., t),

0EF(61,.. hold. We show that q1i . . ., qt (if p > 0) may be supposed to be different from p.

First let us consider the case, where for some i (>_ 0)

g1, ,qr#p; q,+1=...=q,=p By Theorem 341, of the fields F(61,

..., 0,) 1 F,

F(191, ..., 't) I F(01, ..., 0t)

the first is separable, the second pure inseparable. Since 19 is separable, it follows that 19 E F(01, ..., 19,). Accordingly, the (superfluous) radicals a91+1, , 0t may be omitted, so that the assertion is true for this case.

SOLVABLE EQUATIONS

685

Secondly consider the case, where there exists an i with

qi = p, q;+1 9 p It suffices to prove that 0 may be transformed into a radical expression, where the root exponents

q1, . , q;- l, q;+1,p,p,q,+2,-- ,qi occur in order of succession, since hence, and from what has been said above, the assertion will follow by induction. For our purpose we insert in the sequence of radicals 791, . . ., 09, between

19; the term

u*_0+1' from which we obtain the sequence $1,

..., $;-1, (97 (_ X91+1), 991, O'i+1, ..., 19'.

Since

0*Q'+' = ( +i')° E F(19i , . . ., 9f) c F(191, . . ., 0r-1), = 07' E Fo91, . . ., 0,-,) c F(61,

..., 49;-v $*),

091P+1 = 6* E F(01, ..., 09;-1, #

,

01)

hold, the truth of the assertion follows. We now put G = F(01, ..., 0,). The splitting field of the polynomials

x41 - 1,...,x4'over G is denoted by G'. Then G' contains a splitting field F of the san. polynomials over F, furthermore evidently

G'={F',G}. By Theorem 424, F may be obtained from F by successive cyclic field extensions. Since finite cyclic groups are solvable, it follows from Theorem 420 that here extensions of prime degree suffice.

Further let us denote the normal field of G' F by N. Together with this, N I F is separable and thus a Galois field. Lets denote an automorphism of it.

Because of the supposition we can put (Oct-, E

Hence sO,

=

°j sx; --I.

GALOIS THEORY

686

Evidently we obtain N from F if we first successively adjoin all the conjugates sh1 of #,, by which we obtain a normal field over F, then we successively adjoin to this all the conjugates she of 02, by which we again obtain a normal field over F, and continue in the same way. But since F contains all the q;th roots of unity (i = 1, . . ., t), it follows from Theorem 427 that the adjunction of each s1, effects a cyclic extension whose degree is q, or 1. Together with the preceding statement this results in the existence of a chain of fields

F=No cN1 c... cN,=N, in which all the N,+1 1 N, are Galois fields of prime degree. This means, on account of Theorem 420, that the Galois group of N I F, i.e. N I F itself, is solvable.

Since, on the other hand, because of (173.14), N I F contains a splitting

field of f(x) over F, i.e., the Galois field of equation (173.13) as a normal subfield, it follows from Theorem 431 that this field, and thus also the Galois group c0 of (173.13), is solvable.

In order to prove the second part of the theorem, let us suppose that it is true for smaller O(o2). We denote by G I F the Galois field of the equation (173.13) and consider a composition series of its Galois group: (`xd =) -7p D'71 D ... Z) -7r (= 1). To this belongs, according to Theorem 420, a chain of fields

(F=)Go cGxc

... cG,(=G),

in which G; I G,_1 is cyclic of a prime degree q, (i = 1, ..., r). Hence

O(d)=q1...q,. A splitting field of the polynomials

xqI - 1,...,xgr- 1 over G is denoted by N. This contains a splitting field No of the same polynomials over F, which is obtained from F by the adjunction of all the q,th roots of unity (i = 1, . . ., r). Because of the induction hypothesis it follows from Theorem 424 that we can obtain No from F by successive adjunctions of irreducible radicals, where only prime numbers occur as root exponents which are smaller than the greatest prime factor of 0(c4). Finally, we consider the fields

N,={No,G,}(9 N)

(i= 1,...,r; N,=N).

Since we have

N, = {Ni-,, Gi},

687

SOLVABLE EQUATIONS

Ni I N;_1 is, according to Theorem 419, a Galois field, and [Ni Ni-11191 Since, furthermore, N,_ 1 contains all the qih roots of unity, either we have

Ni = Ni_1 or we obtain Ni from Ni_1 according to Theorem 426 by the adjunction of an irreducible qih radical (i = 1, . . ., r). If these radicals are adjoined to No, we obtain N, = N. By consideration of the above, because G c N. the second part of Theorem 432 follows. EXAMPLE 1. In the second part of Theorem 432 the condition "in the case p > 0

all the prime factors of 0(4) are smaller than p" may not generally be omitted. In order to give examples for this, we remark in advance that in Theorem 432 for a finite field F (since then by (173.13) a finite field is again defined) there are very simple conditions, as in this case equation (173.13) is always solvable and its Galois group c1/j is cyclic. For, the solvability of (173.13) follows from the fact that every element of a

finite field is a radical, and is either 0 or a root of unity. Nevertheless it can happen

that (173.13) does not have a solution formula. This is trivial in the special case F = .`, ffor non-linear equations (173.13)] since E has only the two elements 0, 1, therefore over it there exist no irreducible radicals at all. (There is no contradiction to Theorem 432, since now the above condition is not fulfilled.) EXAMPLE 2. Let f(x) be an irreducible polynomial of degree p over .2",. According to the above observation the equation f(x) = 0 is solvable, yet it has no solution formula. Otherwise it must also have (cf. the beginning of the proof of Theorem

432) a solution formula wherein only prime numbers other than p occur as root exponents, but this is impossible, since, by the equation f(x) = 0, a field of degree p is defined. EXAMPLE 3. Over the rational function field F = .z-$ (z) let us consider the separable equation

x2+x+z=0.

Its Galois group is of second order, thus solvable, although the equation itself is not solvable. (Of course, the condition quoted in Example I is not fulfilled.) Every root of unity over F is algebraic over 9 g. Thus if we adjoin to F arbitrary roots of unity, then the equation (because of the transcendence of z) remains irreducible. After adjoining suitable roots of unity we may always restrict ourselves to the adjunction

of irreducible radicals. When, for such an adjunction, the root exponent is odd, then the equation again remains irreducible. Finally, the adjunction of square roots leads to a pure inseparable extension, consequently it does not furnish a solution of the equation.

§ 174. The General Algebraic Equation First of all let us consider over a field F an equation (174.1) f(x) = 0, whose left-hand side is the product of separable principal polynomials

which we suppose without loss of generality to be different. Let ce, denote the Galois group of this equation. We put f (x) = (x - oc1) ... (x - xn) 23 R.- A.

(%I,. .., an

G),

(174.2)

GALOIS THEORY

688

where G = F(a1, . . ., an) denotes the Galois field of (174.1) over F, so that 4 is simultaneously the Galois group of G I F. If now t' -* n' is an arbitrary automorphism of G I F, then, because of (174.2),

J (X) = (x - aI) ... (x - an). Hence it follows that al ... and

a1...an

(174.3)

is a permutation of the roots of (174.1). Since furthermore al, ..., an are generators of G (over F), so the above-considered automorphism N -> t)' is already determined by the permutation (174.3); consequently the permutations

(174.3) must constitute a group isomorphic with c4. Generally we tacitly identify the Galois group of the equation (174.1) with this permutation group of the roots al, . . ., an. The index of this group, in the full permutation

group of the set
that we arrive in this way at different permutation groups, which moreover are of different degree, but necessarily are always isomorphic with one another.

According to the above preliminary observation we define the general algebraic equation of degree n as an equation

xn - xlxn-1 + ... + (-1)'-x, = 0

(174.4)

over the rational function field F(x1, ..., xn), where F can be an arbitrarily given field. This notation is justified by the circumstance that every algebraic equation of degree n over F arises from (174.4), when the indeterminates x1, ..., xn are replaced by elements of F. THEOREM 433. The general algebraic equation of degree n is separable and affectless. COROLLARY (RuFFirn-ABEL theorem). The general equations of degree

at least five are insolvable. Let a1, . . ., a,, be the roots of the equation (174.4) in an extension field of F(x1, ..., x,,). Then x1, . . ., x,, are the elementary symmetric polynomials of a1, . . ., an:

x1=act +...+an,

xn=a1...an.

THE GENERAL ALGEBRAIC EQUATION

689

Hence we see that F(al, . . ., (zn) is the splitting field of the equation (174.4) over the given field F(xi, . . ., x,,). We have recourse to the equation

(x-ti)...(x-tn)=x"-six"-1+...+(-1)"s,,=0,

(174.5)

whose roots t1i . . ., t" are new indeterminates over F. The s1, ..., s" are the elementary symmetric polynomials of t1,. .., t":

S1ti+...+to Sn = ti .. to .

Let us consider (174.5) as an equation over the field F(s1, ..., sn). Since its roots t1i ..., t" are different and F(t1, ..., tn) is its splitting field, this is, at the same time, its Galois field over F(s1i ..., sn). Its Galois group consists of all the permutations of
.

. ., XnJ ^' F [51, . . ., Sn] 77

f(X1, . . ., xn) -*J(Sl,

holds. In every case (174.6) holds with "-." instead of

.

. ., Sn))

(174.6)

", i.e., as a homo-

morphism. The kernel of this homomorphism is formed by the polynomials f(x1, ..., xn) for which f(si, ..., s,) = 0. But this equation is valid because of the proposition of uniqueness of the main theorem for symmetric polynomials (Theorem 268) only if f(t1i . . ., tn) = 0, i.e., f(x1, ..., x,,) = 0. Since, according to this, the kernel of the abovementioned homomorphism is 0, we have proved (174.6). The isomorphism (174.6) may be uniquely continued to an isomorphism F(x1,

.

. ., x,,) :.,

F(si.... , sn)

of the quotient field. Since in this the polynomials occurring in (174.4) and

(174.5) correspond to each other, it may be further extended to an isomorphism F(ai,

..., an)

F(ti, ..., tn)

of the splitting fields. This isomorphism maps the roots ai, ..., an of (174.4) in any order into the roots ti, . . ., to of (174.5). Hence it follows

GALOIS THEORY

690

that al, . . ., a are different, i.e., equation (174.4) is separable. Simultaneously it also follows that the Galois groups of the fields F(al, ..., an) I F(x1, . . ., xn),

F(t1,

.

. ., tn) I F(s1,

. .

., sn)

are isomorphic. Hence, according to what has been stated above, Theorem 433 is proved. Since the full permutation group of degree n is not solvable

if n > 5 according to Theorems 84 and 85, it follows that the corollary is also true. in the above proof the main theorem for symmetric polynomials might easily be dispensed with. In this way we could obtain a second proof of this theorem. Cf. VAN DER WAERDEN (1955).

According to the above corollary the splitting field of an equation of degree at least 5 generally cannot be obtained by adjunction of radicals from

the fundamental field. The most we can establish in this respect is the following

THEomm 434. Let f(x) denote a separable principal polynomial of degree n

with discriminant D over a field F of characteristic p 0 2, further let N I F be the Galois field off (x) and c0 its Galois group. D is a square in F if, and only if, 4t is a subgroup of the alternating group vgn. If this is not the case, then N contains a subfield F(.JD) of second degree over F, and the Galois group of N I F(J) is equal to the intersection c.P For the proof we put f ( x ) = (x

- a) ... (x - a,.)

(a1, ..., a. E N).

Then 4 and vgn may each be interpreted as a subgroup of the full permutation group of the set . According to Theorem 277 (also for n = 1)

Let s denote an element of co. sS = S if, and only if, s belongs to Accordingly, 6 lies in F if, and only if, 4 is contained in vgn. Hence, the first assertion of the theorem is proved, the remaining assertions follow trivially. EXAMPLE. We can obtain from Theorem 434 a second proof for STICKELEERGER's

theorem (Theorem 421). With this end in view, we apply Theorem 434 to a finite field F of characteristic p (> 3). Now, by Theorem 306, 4 is cyclic of order n, thus 4 is a subgroup of or,, if, and only if, n is odd. Hence we see that from the first assertion of Theorem 434 it follows that Theorem 421 is true for the special case m = 1. We then

obtain the general case of this theorem as in the second part of the first proof.

TSCHIRNHAUS TRANSFORMATION OF POLYNOMIALS

691

§ 175. Tschirnhaus Transformation of Polynomials

Let a non-constant principal polynomial f(x) over a field F be given. Take a further polynomial 99(x) over F. Let the factor decomposition

f(x) =x"+alx"-1+...+a,;=(x-aI)...(x-a") (175.1) hold over a splitting field of f(x) where the al, ..., a" are zeros of f(x). We form the principal polynomial

g(x) _ (x - Oai)) ... (x - 99(a,.))

(175.2)

with the zeros ..., 4p(a"). From the principal theorem for symmetric polynomials (Theorem 268) it follows that g(x) lies in F[x]. We introduce the notation g(x) = f (x) "

(175.3)

and say that this polynomial arises from the polynomial f(x) by the Tschirnhaus transformation with the polynomial 99(x).

THEOREM 435. If the result g(x) = f(x)) of the above Tschirnhaus transformation has only different zeros P1,

..., P", then we compute the (necessarily

distinct) zeros al, . . ., a", off(x) according to the rule

x - a; = (f(x), (p(x) - #,)

(i = 1; . . ., n).

(175.4)

By (175.4) we also select an order of succession of the al, . . . , a", moreover F(a,) = F((3,), as the proof shows, whence F(al, . . ., a") = F(f1, ..., #") for the splitting fields of f(x) and g(x).

For the proof of the theorem we choose the order of the al, ..., a" according to (175.2) by

9'W _ f,

(i = 1, ..., n).

Since then cv(ak) - f, = 0 only if k = i, x - a, must be the greatest

common divisor of f(x) and 99(x) - fl,. Hence the theorem is proved. Accordingly, with regard to the determination of all the solutions of an

equation f(x) = 0 (with distinct roots) the transformed polynomials g(x) =Ax)", quoted in the theorem, are equivalent to the given polynomial f(x). In the application we often proceed so that we seek a q((x) for which as many coefficients as possible of

g(x)=x"+blx"-I+...+b"

GALOIS THEORY

692

vanish. As the most simple example we take a linear polynomial

gp(x)=x+c

(cE F).

Because of (175.1) and (175.2)

g(x)=f(x-c)=(x-c)"+al(x-c)"-1+...+a" now holds. Here the coefficient of x"-1 is equal to -nc + a1. This vanishes for

provided that n is not divisible by the characteristic of F. So, by this we have settled that the problem of solving all equations of degree n over a field of characteristic p (p,r n) may be restricted to equations of the form

x"+b2x"-2+ ...+b"=0.

(175.5)

(In this special application it is not necessary to presuppose the distinctness

of the roots.) Thus if

instead of the general equations of degree n we need to consider only equations (175.5) with indeterminates b2,. . ., b". Of course, we could also have obtained the result (175.5) without the general notion of Tschirnhaus transformations. For further applications cf. BIEBERBACHBAUER (1933) and GARETT (1956).

EXAMPLE. If we restrict ourselves, as in Theorem 435, to the case in (175.3), where

f(x) and g(x) have no multiple zeros, then the similar equation

(g(x)) _ (1(x))' ,

(175.6)

follows from (175.3), where on the right-hand side we have to understand the Tschirnhaus transformation [with op(x)] of the ideal (Ax)) (cf. §122). Namely, the elements of

the ideal on the right-hand side of (175.6) are those polynomials h(x) over F, for which the condition h(p(x)) E (1(x))

is satisfied. This condition is identical to f(x) I h(q0)), i.e., according to (175.1), it is equivalent to the validity of all the equations h(cv(a,)) = 0

(t = 1, ..., n).

Since, according to (175.2), this amounts to the same as g(x) I h(x), it follows that (175.6) does, in fact, exist.

§ 176. Equations of Second, Third and Fourth Degree We consider for n = 2, 3 or 4 an equation of degree n

x"+aix"-I+...+a"=0

(176.1)

over a field F of characteristic p (>_ 0). Here we always suppose that p # 2

EQUATIONS OF SECOND, THIRD AND FOURTH DEGREE

693

and when n = 3, 4 we also assume that p 3. In both the last cases we also assume for the sake of convenience that al = 0, by which, according to the inference of the preceding paragraph, we lose no generality. Since the full permutation group of at most, fourth degree is solvable, by Theorem 432, equation (176.1) is solvable under the convention adopted,

and it also has a solution formula. We now intend to deduce this formula.

The procedure consists in assuming from the outset a splitting field ..., x,J, where xI, . . ., x denote all the roots of (176.1), then to

F(x1,

construct this splitting field by the help of adjunctions of irreducible radicals

and to express the roots xI, . . ., xn by the same radicals. In this way we have not only set up a solution formula, but also simultaneously effected the actual decomposition of the left-hand side of (176.1) into the form

x"+alx"-I+...+a,, =(x-xl)...(x-xn), where the x1, . . ., x,, are given by radical expressions over F. In connection with this we shall show that this decomposition also remains valid in the

reducible case, by which we obtain the complete solution of (176.1). We denote by D the discriminant of (176.1). We may put

JD = II

(xi - xk),

(176.2)

since, according to Theorem 277, the square of the right-hand side of (176.2) is equal to D. It should be noted that, by Theorem 434, the Galois

group of (176.1) over F(JD) is equal to a subgroup of the alternating group vf,, . CASE n = 2: (176.1) now reads

x2+alx+a_-=0.

(176.3)

On account of (113.4) we have

D = ai - 4a2,

(176.4)

JD = xl - x2 .

(176.5)

further because of (176.2)

2 = 1, so, according to the above statements, F(xl, x2) = F(-,/Y)). Accordingly, x1, x2 may be expressed as elements of F(.,/5). This can be done using equation (176.5) and the equation As t

xl+x2=--a1

GALOIS THEORY

694

arising from (176.3). Then we obtain

x1=

2

(-al+ / ), x2= 2 (-a1-.J),

(176.6)

where we have taken into consideration that p # 2. Since, conversely,

x2+a1x+a2=(x-x) (x-x2) always follows from (176.4) and (176.6), the formula (176.6) gives all the solutions in the present case. CASE n = 3: We now write (176.1) (with a1 = 0) in a simpler notation in the form

x-3+ax+b=0.

(176.7)

In order to avoid the necessity to distinguish between different cases, we assume that the coefficients a, b are indeterminates over F. This means that (176.7) is regarded as being over the rational function field F (a, b) as a fundamental field, which we still denote, for the sake of simplicity, by F. According to the final observation of the preceding paragraph, (176.7) is then like the general equation of third degree, by which we mean that Theorem 433 can be applied to equation (176.7), i.e., this is affectless. The result for this case can be applied to every other case, if we replace the indeterminates a, b by elements of the fundamental field originally given. Since equation (176.7) is affectless, it follows, as above, that its Galois group over F(,FD) is equal to f3. On account of (113.5) we now have

D = - 4a3 - 27b2.

(176.8)

Further, according to (176.2),

VD = (x] - x2) (x1 - x3) (x2 - x3) =

xix., - Y- x32Lx3 , 3

(176.9)

3

where the symbol > is defined for arbitrary functions f(xl, x2, x3) by 3

E J (XI, x2, x3) = J

x2, x3) + fl X21 x3, x1) + f(x3, x1, x3]

a

As we have O(vE3) = 3, the Galois field F(x1, x2, x3) I F(N46) is cyclic of degree three. We adjoin both third roots of unity

_=

2 (- 1 + J- 3), 02= 2

(176.10)

EQUATIONS OF SECOND, THIRD AND FOURTH DEGREE

which is equivalent to the adjunction of the radical ,

695

. Of course,

also the field F(x1, x2, x3, J-3) 1 F(JD ,) is cyclic of degree three. We take the Lagrange resolvent [(172.2)]: y = x1 + Cx3 + e2x3 .

(176.11)

(The following will be a similar construction to that in the proof of Theorem 426.) From (176.11) and p3 = 1 we have

y3 = E xi + 3e E xix2 + 3e2 E xix3 + 6x1x2x3 . 3

3

3

After the substitution of (176.10) and then (176.9) we obtain y3 =

xi 3

2

6

xix2 + 6x1x2x3 + 2 V - 3 JD ,

(176.12)

where E denotes the summation over all the permutations of <x1, x2, x3>. 6

From the formula p3 = - ai + 3a1a2 - 3a3 (§ 115, Example 1) we get Z Al = - 3b. 3

Further,

Y, xl 3 After substituting

E x1x2 = E xix2 + 3x1x2 x3. 3

6

Y_ x1 = x1 + x2 + x3 = 0,

x1x2x3 = - b

3

we obtain

E xix2 = 3b. 6

Thus, from (176.12),

y= 3I- 2 b+ 2 y-3D , where

- 3D =

(176.13)

JD (therefore the occurrence of J - 3D does

not imply that any new radical has been adjoined). Since y is obviously an irreducible radical of degree three over the field F(%D , /----3), it follows that

F(x1, x2, x3, J- 3) = F(/D, J- 3 , y).

(176.14)

Hence, it is possible to compute x1, x2, x3 as elements of the right-hand side of (176.14). 23/a R.-A.

696

GALOIS THEORY

For this purpose we have at our disposal the equations

x1+x2+x3=0

(176.15)

and (176.11). Similarly to (176.11) we put (176.16) y' = x1 + e2x2 + ex3. For this a formula similar to (176.13) holds (see below), but it can also be computed as follows: because e3 = I and (176.10),

yy'

_ 3

+ (e + e2) E xlx2 =

3

3

xi -

3

x1x2 = (Z x1)2 3

- 31 x1x2 3

follows from (176.11) and (176.16). According to (176.15) and

E x1x2 = x1x2 + xlx3 + x2x3 = a 0

we have

(176.17)

YY' = -3a, therefore y' is determined in terms of y.

We now compute the roots x1, x2, x3 from (176.15), (176.11), (176.16),

taking into consideration e2 + e + 1 = 0 and p # 3,

(Y+y'),

x1=

x2 = 3 (e2Y + ey') ,

(176.18)

1

X3 = 3 W + e2Y')

The equations (176.8), (176.10), (176.13), (176.17), (176.18) together furnish the required result. Since with the radical expressions x1, x2, x3 so computed, the decomposition

x3+ax+b=(x-x1)(x-x2)(x-x3) holds, this also remains true in the case where the indeterminates a, b are replaced by arbitrary elements of the original fundamental field, so that our result, apart from the case when a = 0, is generally valid.

We can express the result in another form too. If /- 3 is replaced by

- J- 3, e becomes e2, thus (176.11) becomes (176.16). Accordingly, from (176.13) we obtain

y' = 3 -

22

b-3

- 3D

(176.19)

697

EQUATIONS OF SECOND, THIRD AND FOURTH DEGREE

with respect to which it should be noted, however, that the radicals (176.13), (176.19) are not arbitrary inasmuch as (176.17) must bold for them. After substituting (176.8), (176.10), (176.13), (176.19) in (176.18) we now obtain all three roots of the equation (176.7) in the form

x=

a

-

b

+ JI 212 + 13 I3

+ s - 2-

X2)2

+ Tr (176.20)

where the radicals occurring here are always to be chosen so that the

product of the two terms of the right-hand side should be -

. This

formula (176.20) is called the Cardan formula (for the equation of3 degree of course, the case a = 0 is no longer an exception. Finally, denote the two radicals in (176.20) by u and v, so that (176.20)

then assumes the form x = u+ v uv = six pairs of values are possible: u, v; 0u, 02v;

02U,

I. Then for u, v the following 3

Ova v, u; 0v,

e2U;

e2v, eu ,

of which the first three already furnish the three solutions of (176.7).

CASE n = 4: We can now take the given equation (176.1) (because a1= 0) in the form

x4+ax2+bx+c=0,

(176.21)

Again it will suffice to consider the case where a, b, c are indeterminates, i.e., that a rational function field F (a, b, c) is the fundamental field, which we denote, for the sake of simplicity, by F. The solution of (176.21) will be obtained by its reduction to the previous case n = 3. Because of the supposition, the Galois group of (176.21) is the full permutation group '94. All its composition series are,, according to Theorems 84, 85, the

.94D f4JG4z) HD 1,

(176.22)

where G4 denotes the four-group consisting of the permutations 1, (1 2) (3 4), (1 3) (2 4), (1 4) (2 3) ,

(176.23)

and H is an arbitrary subgroup of order two of G4. The invariance fields belonging to the subgroups (176.22) form a chain

F c F(,JD) c F. c F12 c F(x1, . . ., x4)

(176.24)

OAI.OIS THEORY

698

with [Fe : F] = 6, [F12 : F] = 12. Our next intention is to construct the field F6.

With this end in view we consider the elements

Y1=(x1+x2)(x3+x4), (176.25)

Y2 = (X1 + x3) (x2 + x4) ,

Y3 = (x1 + x4) (x2 + x3)

and assert that

F6

(176.26)

= F(Y1, y2, y3).

On the one hand (176.25) furnishes fixed elements of the permutations (176.23), whence F(Y1, Y2, Y3) c Fa

(176.27)

.

On the other hand, we determine the equation of degree three with the roots yl, y2, y3. For this we must determine the elementary symmetric polynomials of yl, y2, y3. These are in shortened notation [Yll = 2[x1x2] = 2a [YIY21 = [xix221 + 3 [xix2x3] + 6 x1x2x3x4

YLY2y3 = [x3jx2x3] + 2 WX2x3x4] + 2 [x2jx2x3] + 4 [xlx2xsx4l

where, e.g., [x2jx2] denotes that symmteric polynomial whose terms all have the coefficient 1 and result from xix22 by permutations of <x1, ..., x4). Since [xix2]2 _ [x2jx22] + 2 [xlx2x3] + 6 x1x2x3X4 = a2,

[x1] [XIX2x31 =

[x2jx.2x3] + 4 xlx2x3x4 = 0

,

X1x.2x3x4 = C

[xlx2x3] = - 4c,

[xix2]=a2+2c, [YLY2l =a2-4c. Furthermore [x1] [x1x21 [xlx2x3l = [xix2x3] + 3 [xlx2x3x4] + 3 [x1 2xg] + 8 [x2jx2x3x4] = 0 ,

[JGIX2X3X4] + 2 [ 'IX2x3X4] = 0 , [X1X2x3]2 =

[X1X21X1X2x3X1 =

[x2jx2x321 + 2 [x2Ix2X3x41 = b2,

[xix2x3x4] = ac

EQUATIONS OF SECOND, THIRD AND FOURTH DEGREE

699

hold, so that we have

[xixaxs] _ - 2ac + bz , 2ac, [-41x2x3x4] [xix22x3] = 4ac - 3b2 ,

Y1Y2Ys = - Y.

Consequently yl, y2, y3 are the three roots of the third degree equation

y3-2ay2+(a2-4c)y+b2=0,

(176.28)

which (or, at least its left-hand side) is called the cubic resolvent of equation

(176.21). [Often y3 + 2ay2 + (a2 - 4c) y - b2 = 0 is also called so.] We shall show that the discriminant of (176.28) agrees with that of (176.21). We obtain from (176.25)

Y1 - Y2 = (xl - x4) (x3 - x2) ,

Y1 -/Y3 = (x1 - x3) (x4 - x2) ,

Y2 - Y3 = (X1 - x2) (x4 - x3)

whence the assertion follows. Since D is not now a square in F, it follows from what has been proved,

and from Theorem 434, that the equation (176.28) is affectiess, i.e., [F (3'1, )'21 Y3) : F] = 6. Hence, and from (176.27), (176.26) follows. From (176.28), y1, y2, y3 may be computed by the Cardan formula. Accordingly, it still remains to construct the extension field F(xl, ..., x4) of F(y1, y2, y3) by the adjunction of radicals and to express XI, ..., x4 in terms of these radicals. This is easy since this field, according to (176.22), (176.24), has the four-group G 4 for Galois group, thus the adjunction of two suitable square roots will be sufficient. In order to carry through this construction, we take into consideration XI + x2 + x3 + X4 = O ,

(176.29)

therefore (176.25) may be written as y1 = - (X1 + x0)2,

y2 = - (XI + x3)2,

y3 = - (X1 + x4)2.

Accordingly,

x1 + x2 =I1 X1 + X3 =

-Y21

X1 -+4

Y3,

(176.30)

whence evidently F(xI,

. .

., x4) =

(176.31)

GALOIS THEORY

700

According to what was said in (176.31) a square root is dispensable. We also prove this directly by showing that Yi -,I-Y2 %-Y3

= -b .

(176.32)

Since, for the product of the left-hand sides of (176.30),

(x1 + x2) (xi + x3) (xl + x4) = xi [x11 + [xlx2x3l = -b, so

(176.32) is true. Lastly we obtain from (176.29), (176.30) 1

x1= 2 (,/-Y1+

2+

3),

/

G,

X2

(176.33)

1

x4 = 2 (- N

/

Yl -

/

-Y2 + V -Y3)

These formulae thus furnish all the roots of the biquadratic equation (176.21), where yl, y2, y3 denote the three roots of the cubic resolvent (176.28) and the three square roots in (176.33) satisfy condition (176.32). The solution formula (176.20) is wrongly attributed to CARDAN as he obtained it from TARTAGLIA, but it is not known whether the latter was the discoverer of this formula. Formulae (176.33) are essentially due to FERRARI. EXAMPLE 1. We obtain the Cardano formula by a simple but clever method as follows: put x = u + v, with two new unknowns u, v, in (176.7). We then have

(u + v)3 + a(u + v) + b = 0, i.e.,

u3+v3+(3uv+a)(u+v)+b=0.

if we subject u, v subsequently to the condition

3uv+a=0, then the former equation reads

u3 + v3 + b = 0. The solution of the system consisting of the last two equations entails no difficulties

and leads at once to (176.20). Of course, this elementary process gives no real insight into the cubic equation. (See also the following paragraph.)

EQUATIONS OF SECOND, THIRD AND FOURTH DEGREE

701

EXAMPLE 2. Also the biquadratic equation (176.21) admits an elementary solution, if we seek a factor decomposition

x°+axs+bx+c=(x2+rx+s)(x2-rx+t) of its left-hand side. For this purpose one must determine a solution r, s, t of the system of equations

s+t-r$=a, r(-s+t)=b, st=c,

which can be easily done. EXAMPLE 3. Over a field with characteristic other than 2 the polynomial

y2 + bx +

JOY -

2 ay - 4 (a$ - 4c)

4

n the two indeterminates x, y has the property that its discriminants, with respect to y and x, are the biquadratic polynomial x4 + ax2 + bx - c and its cubic resolvent y3 - 2ay2 + (a2 - 4c)y + b2, respectively. [Cf. R DEI (1959c)].

§ 177. The Irreducible Case

First of all let us consider an arbitrary polynomial over Col without multiple factors which we decompose into the product of irreducible factors : ,f (x) = 11(x) ...1,(x) q1(x) ... qs(x)

The lk shall be assumed to be linear and the qk quadratic. Since the discriminant of an 1k is equal to 1 and that of a qk is negative, it follows from the multiplication theorem for discriminants (§ 113, Example 2) that the discriminant of J (x) is positive if, and only if, s is even, i.e., the number of

the non-real zeros of f(x) [in 501(i)] is divisible by 4. We now examine a cubic equation over 9'tO) without multiple roots, which we assume without restriction of generality to be in the form

x3+ax+b=0,

(177.1)

further we denote the discriminant of (177.1) by D. According to the above, the number of the real roots of (177.1) is equal to 3 or 1 according as D is positive or negative.

On the other hand, we know that for computing the roots of (177.1) according to the Cardan formula it suffices to adjoin to the fundamental field for a fixed value of _-3D the three values of 22

b+2

(177.2)

702

GALOIS THEORY

[cf. (176.13)]. If there is only one real root, i.e., if D < 0, then among the values of the radical (177.2) a real one always occurs, so that this single real

root of (177.1), on account of the Cardan formula, may be computed within .7(o. It is different if all three roots are real, i.e., if D > 0, since then the radical (177.2) has no real value at all. This case of the cubic equation, in which it is has three real roots, is called the irreducible case. The terminology refers to

the fact that in this case equation (177.1), applying the Cardan formula, is reducible only after the adjunction of non-real radicals. [From (176.18) it

is clearly seen how the three real roots then arise as a sum of pairwise conjugate complex numbers.] One might think that in the irreducible case the difficulty could be avoided; namely, the Cardan formula might be replaced by another, which

in the case of three real roots would only require the adjunction of real radicals. The centuries old efforts regarding this problem were brought to an end with the Galois theory in the following theorem. THEOREM 436. If f(x) is an irreducible polynomial of third degree with three real zeros over a subfield F of then none of these is expressible by real radicals (over F). For, let D (> 0) be the discriminant of f(x). After the adjunction of .JD

we obtain a field F(.JD) of at most second degree over F, so that f(x) remains irreducible. Hence and from Theorem 434 it follows that the Galois

is the alternating group 'e, group of f(x) over We suppose that f(x), after the adjunction of certain further real radicals 01, ...,14',,, will be reducible. We may assume that all the root exponents therein are prime numbers. In the considered sequence of radicals there is a term #k= /w (q a prime number) such that, of the fields G = F(.[, t91, ..., Ok-1),

G1

= G (;/w)

f(x) is irreducible over the first and reducible over the second. If the radical Jco is reducible, then co is, according to ABEL'S theorem (Theorem 427), a qt' power in G, whence, because .Jw E 9o), we obviously have .,/w E G, G1= G. But this is absurd, thus ,A.) must be irreducible. Hence [Gl : G ] = q. Since, on the other hand, G I contains a zero of f(x), we have 3 1 q, i.e., q = 3. According to what has been said above, the Galois field of J (x) over G is of degree 3. Since G1 contains a zero of f(x) and is likewise of degree 3 over G, so G1 I G itself must be a Galois field. Thus the irreducible polynomial

f(x) over G decomposes into linear factors over G.I. Likewise, according to Theorem 427, G1 must contain all third roots of unity which is impossible,

because G1 9

o). Consequently the above theorem has been proved.

EQUATIONS OF THIRD AND FOURTH DEGREE

703

§ 178. Equations of Third and Fourth Degree over Finite Fields

If f(x) = 0 is an equation over a field F, then the question arises as to how many roots lie in the fundamental field F itself. If, e.g., F = .off, and f(x) is a polynomial of third degree (over Y(0)) without multiple zeros, then the number of these roots of f(x) = 0 is 1 or 3. On the other hand, for other fundamental fields this number can have the three values 0, 1, 3, e.g., if F is a finite field. Here, and from now on, we shall deal only with this case and we shall examine the question raised with regard to equations of third and fourth degree. The KONIG-RADOS theorem (Theorem 313) answers this question generally for finite fields, for equations of arbitrary degree, but much simpler results of a quite different kind hold in the special cases mentioned here. THEOREM 437. Let F denote the finite field of characteristic p (> 3) such that O(F) = q, further let n =

q-1

q+1 or

6

(178.1)

6

denote the integer nearest to 6 . Let the cubic equation

x3+ax+b,0

(178.2)

over F be given with the discriminant

D = -4a3+27b2#0.

(178.3)

If D is not a square in F, then (178.2) has exactly one root in F. If D is a square in F, and

t, (-27b2D'1) = 0 .

(178.4)

where

V(X)=l

2n

Jx2...+2n

(178.5)

then (178.2) has three roots in F. In the remaining cases (178.2) has no roots in F. In order to prove the theorem, let us denote by nz (= 0, 1, 3) the number

of roots of equation (178.2) lying in F. Then we have m = 1 if, and only if, the left-hand side of (178.2) splits into the product of two irreducible factors over F. This is the case, according to STICKELBERGER'S theorem

(Theorem 421), if, and only if, D is not a square in F. Hence we have shown that m = I if, and only if, D is not a square in F.

704

GALOIS THEORY

Henceforth we suppose that D is a square in F, whence m = 3 or 0. It suffices to prove that (178.4) is true if, and only if, m = 3.

Let Fe denote the field of 6" degree over F, for which we then have O(F6) = q6. We denote by F2 and F3 the subfields of F of degrees 2 and 3, respectively, over F. Let x1, x2, x3 be the roots of (178.2) in F6. But since

m = 0 or 3, so (178.6)

xI, x2, x3 E F3.

Since a -+ aq is a primitive automorphism of F6 I F, if m = 0 we can choose

the notation so that (178.7)

X1= x2, X2= X3, x3= x1. Because of the definition of n,

q=6n+e

(e=±1).

(178.8)

Certainly F2 (possibly also F) contains the radical ,/-3, therefore, also, the

third roots of unity

o2(-1+

,

022(-1-%J-3).

(178.9)

Furthermore, because of the equivalence of the propositions

OEF, x2+x+lIx°-1-1, x3-1Ixq-1-1, 31 q-1, the rule

EFae=1

holds.

(178.10)

We now consider the Lagrange resolvents y = x1 + 0X2 + 02x3 , y' = x1 + 0`'x2 + 0x3.

(178.11)

These lie in Fe. Just as in (176.13), (176.19) now

y3=2 (- 27b +

-27D), y'3=2 (- 27b -

27D). (178.12)

Here also y, y' are different from 0, for otherwise we have (27 b)2 = -27 D, a = 0, although we have restricted ourselves to the case a 0 0. We compute yq, where we distinguish, according to the different

values of m (= 0, 3) and e (= 1, - 1), four cases. We prove that

yq=o2y (m = 0, e = 1); yq = oy' (m = 0, e = - 1) ; yq=

(178.13)

y(m=3,e= 1) ; yq= y'(m=3,e= -1). (178.14)

EQUATIONS OF THIRD AND FOURTH DEGREE

705

If m = 0, according to (178.7) and (178.111), we have yq = x2 + ogxs + o2gxl = o2q(x1 + ogx2 + o2gx3)

But, according to (178.10), we have oq = o or oq = 02, according as e = 1 or e = -1. Hence, and from (178.11), the assertion (178.13) follows. If m = 3, x1i x2, x3 E F, then yq = x1 + ogx2 + 02gx3 .

If the cases are similarly distinguished as formerly, then (178.14) follows. From (178.13), (178.14) we obtain proper equations if we exchange o for o2 and y for y'. By division of the corresponding equations we obtain (yy'_1)q-I=o(m= 0,e= 1)

;

(yy'_I)q_1= 1(m=3,e= 1);

(yy'-I)q+1= o-1(m=0,e= -1); (yy'-1)q+1= 1(m=3,e= -1).

Accordingly, m = 3 if, and only if,

(yy'-1)q_e

= 1. This condition, by

(178.8) and (178.12), may be represented in the form

27b +

- 27D 2n=

27b-

-27D)

1

and then in the form (1 +

227b2 D- I)2n - (1 -

- 27b2D_ 1)2" = 0

.

Hence, after applying the binomial theorem and dividing by 2,/ - 27 b 2 D-1 we obtain equation (178.4). Consequently Theorem 437 is proved. THEOREM 438. Let a biquadratic equation

x4 + axe + bx + c = 0

(b & 0)

(178.15)

without multiple roots be given over a finite fielal F of characteristic p (> 3).

Let m and m' (< m) denote the number of the roots of the cubic resolvent

ys + 2ay2 + (a2 - 4c)y - b2 = 0

(178.16)

lying in F, and the number of those roots of (178.16) which are also squares in F, respectively. If m > m', then (178.15) has no roots in F; if, on the other hand, m = m', then (178.15) has exactly m + 1 roots in F. (It should be noted that for the pair m, m' only the following five possibilities may be taken into consideration: 0, 0; 1, 0; 1, 1; 3, 1; 3, 3.)

GALOIS THEORY

706

For the purpose of the proof, let n denote the number of roots of (178.15) lying in F. The theorem can be divided into the following part-assertions:

Assertion 1. n = 1 if, and only if, m = 0.

Assertion 2. n = 2 if, and only if, m = m' = 1. Assertion 3. n = 4 if, and only if, m' = 3. We denote the roots of (178.15) by x1, . . ., x4 in a suitable overfield of F. Because x1 + ... + x4 = 0, the three elements [cf. (176.25), (176.28)] Y1 = - (x1 + x2) (x3 + x4) = (x1 + x2)2, (178.17)

Y2 = - (x1 + x3) (x2 + x4) = (xj + x3)2, Y3 = - (x1 + X4) (x2 + X3) = (x1 + x4)2

are then the roots of (178.16). Hence it follows [cf. (176.31)] that F(x1i

.

. ., x4) = F(Jyl,

Y2,

Thus x1, ..., x4 E F if, and only if, we have Jy1,

.,/Y3)

Y2,

(178.18)

.

Ys E F. Thus Asser-

tion 3 is proved. We denote the left-hand side of (178.18) by G. Because of (178.18), the degree of G I F(Y1, Y21 Y3) is a power of 2. Thus the degree of G I F is divisible by 3 if, and only if, the same holds for the degree of F(y1, Y2, y3)1 F. Thus we have established Assertion 1. In order to prove Assertion 2, we first suppose that n = 2. We may even suppose that x1, x2 lie in F. Then 1

x3=- 2(x1+x2)+Jw, x4=- 2(x1+x2)-Jw 1

where w is an element of F, which is not a square in F. Consequently, accord-

ing to (178.17), we have z

Y1 = (xl + xz)2, Y2 = ( 2

(xl - x2) + ',/;) , Y3 = l2 (xl -

7e2)

-

2

Therefore m = m' = 1. Conversely, we suppose that m = m' = 1. Then among the yl, y2, Ys exactly one is a square in F. We may suppose that it is yl. According to (178.17) we then have x1 + x2 = r,

y1 = r2

(178.19)

707

EQUATIONS OF THIRD AND FOURTH DEGREE

where r is an element of F. After substitution in (178.16) we get

r6 + 2ar4 + (a2 - 4c)r2 - b2 = 0

(r # 0).

(178.20)

Also 4

i

(x - xI) = x2 - rx +

r3+ar+b IIx2 + rx + r3+ar- b ) 2r

2r

(178.21)

holds, since the right-hand side, because of (178.20), agrees with the lefthand side of (178.15). Because b 0 0 it follows immediately from (178.15) that the six sums x1 + xk (1 <- i < k 5 4) are distinct. Hence, and from (178.191), it follows that the two factors of the right-hand side of (178.21) must be equal to the products

f(x) = (x - x)(x - x2) , g(x) = (x - x3) (x - x,1) .

(178.22)

Since, accordingly, the polynomials (178.22) lie in F[x], their discriminants (x1 - X2)2,

(178.23)

(x3 - x4)2

are elements of F. Since, on the other hand, y2, y3 are the zeros of an irreducible polynomial of second degree over F, (Y2 - Y3)

is an element of F, but not a square in F. Further according to (178.17) and

xl + ... + x4 = 0 we have Y2 - y3 = (XI + x3)2 - (XI + x4)2 = (x1 - X2) (;3 - x4) , thus

(Y2 - Y3)2 = (XI - x0)2 (x3 - X4)2.

Since the group F* is cyclic and of even order, it follows that of the two factors of the right-hand side exactly one is a square in F. These factors are the discriminants (178.23) of the polynomials (178.22), so that of these two polynomials f(x), g(x) E F[x]) 'bxactly one is reducible. That means that n = 2, by which Assertion 2 and Theorem 438, too, are proved. EXAMPLE. We see that Theorem 437 determines the number of roots of (178.2) lying in F. These roots themselves, of course, may be computed according to the Car-

dan formula, where possibly an extension of the fundamental field is needed; but in the majority of cases it is possible in another way, too. For instance, if (178.2) has exactly one root in F (i.e., if D is not a square in F), then this root corresponding to the cases 3 1 q - 1, 3 1 q + 1 is equal to q+2

- 3a-1 (w 3 + w'

q}2 3

) or w

20-1

2q-1 3

+ w'

3

(178.24)

GALOIS THEORY

708 respectively, where

ru

b

3

lE

+

Y

}

`3 (178.25)

bi_

_

(A 2)+

3

E

.

((

=3)

The formulae (178.24) are radical free, thus applicable within F, since after replacement by (178.25) the odd powers of the radical disappear owing to the forming of sums. Cf. also REDEI (1946-48b). EXERCISE. Prove the assertion concerning (178.24).

§ 179. Geometrical Constructibility

We shall see that the problem of geometrical constructibility (with compass and ruler) is solved by GALoIs theory. It will be shown that this problem may be reduced to the theory of solvability of equations. As a preliminary we need the following theorem, very important also in itself. THEOREM 439. Every finite p-group has a centre and is solvable.

Let c0 denote a finite p-group. Since 0 (4) is a power (>->- p) of p, so in

the class equation (42.32) for cPt the number of the terms equal to I is divisible by p and is consequently at least p. Hereby the first assertion of the theorem is proved. As the centre is a normal divisor, the second assertion follows by induction.

Let a Euclidean plane be given where we admit imaginary points, but exclude infinite points. When we speak of points or sets of points, these will always belong to the Euclidean plane. We consider a fixed set C of points,

which consists of at least two points. Then we consider the straight lines which pass through at least two points of G5, and the circles which have a point of G5 for centre and pass through at least one other point of G. The set of points which belong to at least two of the above straight lines or circles is denoted by `G' 1. Here, G51 ? G5 holds trivially. We repeat the above process with 61 instead of 6, so obtaining a set G52 of points, etc. This leads to a chain of sets G5 9- S19- G29- ...

whose union

S=G UG51U...

(179.1)

is called the set of (geometrically) constructible points of the points of G.

Note that every point of 9 belongs to a s,,, so that its geometrical construction may be carried out in finitely many steps. The straight lines and

GEOMETRICAL CONSTRUCTIBILITY

709

circles occurring when determining C5, further the lines, arcs, angles and plane

figures (triangles, etc.) determined by the former, are also called constructible. Different authors define geometrical constructibility differently. We can assume as given elements straight lines, circles, lines and angles in addition to points, and further admit, while constructing, the help of arbitrarily chosen points. Within this general framework we can, e.g., undertake the task of constructing for a given straight

line g the parallel straight line through a given point P. This task may be brought within our definition by a slight modification so that we give the straight line g by two

points A, B of it, then try to define in the set S = a fourth point Q such that the lines AB, PQ are parallel. We have purposely chosen as far as possible a simple formulation for the definition of constructibility, but it is sufficient, as the above example shows, for all the usual construction problems. For the sake of brevity we shall not carry out completely the simple elementary geometrical reasonings to be applied. For this, we refer the reader to BIEBERBACH (1952).

Our first aim is the algebraic characterization of the points of S. With this end in view we introduce rectangular point coordinates with the same metric on both axes x, yin our plane. As the origin of the (say right-oriented)

coordinate system and as unit point of the x-axis we take points of S. Then we adjoin to .70 both coordinates of all the points (x, y) of e, by which we obtain a subfield F of 9o)(i). We call F the field belonging to the set (5 of points. This is permissible, since F is uniquely determined by C. If we

choose, as above, another pair of points of e for fundamental points of the coordinate system, this implies a coordinate transformation in the determining equations of which only elements of F occur as coefficients, therefore the field F is again generated by the "new" coordinates of the points of e. We adjoin to F the square roots (E .9o)(i)) of all the elements a of F, by which we obtain a field FI. If we repeat this process for F1 instead of F, etc. then we obtain a chain of fields

F(= Fo) S F1 S F2 c

... (c

According to Theorem 47, the union of the i lds

F=FUF1U...,

(179.2)

exists and is called the 2-closure of F. It is evident that r is that minimal field

between F ando,(i), which contains the square roots of its elements. (This means, in other words, that all the quadratic polynomials f(x) over F are reducible.) In particular, F (and even F1) contains the element i = J -1 . The algebraic characterization of S then reads as follows:

710

GALOIS THEORY

The set e of the points, constructible from the points of G5, consists of those points both of whose coordinates belong to the field F.

From the definition of G and F it follows by elementary geometrical

reasoning that the coordinates of an arbitrary point of 6 are radical expressions over F, where only square roots occur as radicals, thus they belong to F. We have still to show that, conversely, every point with coordinates in F belongs to S. The proof depends upon the following propositions I, II, where a, j9 denote complex numbers: 1. A point (a, /1) lies in S if, and only if, (a, 0), (fl, 0) lie in G. II. When (a, 0), (fl, 0) (j 0) lie in ig, then

(a - fl, 0), (4-1, 0), (N/«, 0)

,

also lie in G. We shall show, first, that from I and II follows the previous statement,

viz., that every point (a, fE) with a, 9 E F lies in 8. Let P, a.... be the coordinates of the points of C. It follows from I that all the (e, 0), (a, 0), ... lie in G. Since, on the other hand, we have F = Y0(e, a, ...), it follows Further it from one part of II that all the (x, 0) with 1c E F(= FO) lie in follows from II that if for some n (= 0, 1, ...) all the (IC, 0) with K E F. lie in 8, then all the (ic, 0) with x E F,,+1 also lie in G. It follows by induction

that all the points (K, 0) lie in 9, in which K belongs to one of the fields F, F1, ... This implies, according to (179.2), that all the (x, 0) with K E F lie in ig, whence the assertion follows, according to I. I and II themselves have to be proved only for the case where C, apart from the points (0, 0), (1, 0), contains only the point (a, j9) or the points (a, 0), (f, 0), respectively. The proof is known from elementary geometry. We now prove the following THEOREM 440. Let a set G of points, consisting of at least two points, be given in a Euclidean plane extended by the imaginary points. Take two arbitrary points of G as zero point and positive unit point, respectively, on the

abscissa-axis of a rectangular coordinate system and extend .5 by the adjunction of the coordinates of the points of G to afield F. In order that a point (a, j9) of the plane should then be constructible from the points of it is necessary and sufficient that a, fl be algebraic over F and that the degree of the Galois field of F(a, fi) I F be a power of 2.

According to I it is sufficient to prove the theorem for one point of the special form (a, 0). First, we suppose that ((x, 0) is constructible. Then

a E F, where F denotes the 2-closure of F. This implies that a is a radical expression over F, wherein only square roots occur as radicals. Accordingly a is algebraic. Let N be the Galois field of F(a) over F. N is obtained from F by adjoining all the conjugates of square roots in N which

GEOMETRICAL CONSTRUCTIBILITY

711

occur in a. These adjunctions may be ordered so that the degree of the field

is doubled at every step. Hence it follows that [N : F] is a power of 2. Now we suppose, conversely, that a is algebraic and the Galois field of F(a) I F has a power of 2 as its degree. Its Galois group is a 2-group and consequently, according to Theorem 439, is solvable. Hence, it follows, according to the second part of the principal theorem for solvable equations (Theorem 432), that a. is a radical expression over F, in which only square roots occur as radicals. According to what has been stated above, the point (a, 0) is therefore constructible, thus Theorem 440 is proved. There is a particularly important special case of this theorem when the given set G5 of points consists of only real points and the constructibility of

another real point is required. In this "real case" of the theorem we can restrict ourselves to a real Euclidean plane. This may be understood as a Gauss number plane, so that we can identify the point (x, y) with the complex number x + yi. The special choice of the coordinate system described in the theorem is to ensure that the numbers 0, 1 (i.e., the corresponding points) should belong to C. In addition, we have to make the slight modification that we take F(i) for fundamental field instead of the above F. (This is quite permissible, since the point i, i.e. the positive unit point of the y-axis, is constructible.) When we again write F instead of F(i) then F may also be obtained from . by adjoining the complex numbers belonging to the points of G, their conjugates and also i. If now a + bi is a point of the Gauss plane (a, b E then, according to Theorem 440 it is constructible if, and only if, the field F(a, b) I F is algebraic and its Galois field has a power of 2 for its degree. However, these conditions may be replaced by the similar conditions relating to the field

F(a + bi) I F. The simplest way to recognize this is that a, b are radical expressions composed of square roots over F if, and only if, the same holds

for a + bi, and so, because i E F, also for a - bi. The result is expressed in the following THEOREM 440'. Let a set S of points, which among others shall contain the points corresponding to the numbers 0, 1, be given in a Euclidean plane considered as a Gauss number plane. Let F denote ;he field, which arises from .2 after the adjunction of i, of the complex numbers corresponding to the points of G5 and their conjugates. Then in order that an arbitrary point of the plane should be constructible from the points of S, it is necessary and sufficient that for the corresponding complex number a the field F(a) I F should be algebraic and its Galois field should have a power of 2 for its degree. EXAMPLE 1. We examine for which numbers is (> 3) a regular n-gon can be constructed. This construction problem is known as the cyclotomy problem. The problem is defined as follows: let two points 0, A be given. It is required to discover whether a regular n-gon can be constructed geometrically, one vertex of which is A and whose remaining vertices all lie on a circle with centre O. We proceed according to Theorem

712

GALOIS THEORY

440'. Since we take 0 and A for the points 0 and I of a Gauss plane, F = .57'o(i) is the fundamental field, and the angles of the regular n-gon are given by the complex roots of the equation

X"- I = 0.

These are of the form 1, a, ..., a"-', where a is a primitive nt root of unity, i.e., a root of the n`h cyclotomic equation F"(x) = 0. Of course, it is sufficient to consider the case in which a is constructible. Now .91-h(a) I F

is the n" cyclotomic field and this, according to Theorem 423, is a Galois field of degree 99(n). Because F = F"0(i) it follows from this, and from Theorem 419, that F(a) I F

is a Galois field of degree q9(n) or p(n). It now follows according to Theorem 440' that the regular n-gon is constructible if, and only if, p(n) = 2k

(179.3

for some integer k. This condition, on account of (84.22), implies that n has no multiple

odd prime factors and its odd prime factors are all of the form (s = 1, 2, ...).

2' + 1

There s must be a power 2' of 2, for ifs had an odd divisor d (> 1), then 2°-'' + 1 would be a proper divisor of 2' + 1. The prime numbers of the form 22' + 1

(179.4)

are called Fermat prime numbers. The final result now reads as follows: only those regular n-gons (n > 3) are constructible whose number of sides is of the form

n = 2'p, ... pk

(e > 0, k > 0),

(179.5)

where Pt, ..., Pk are different Fermat prime numbers. This is the famous theorem which Gauss proved in his youth. It should be noted that at present only those Fermat prime numbers are known which are given by t = 0, 1, 2, 3, 4; these are: 3, 5, 17, 257, 65537.

(179.6)

The case t = 5 of (179.4) is a number divisible by 641, thus not a prime number. According to (179.5) and (179.6), those n (3 < n < 20) for which the regular n-gon is constructible are the following: n = 3, 4, 5,-6, 8, 10, 12, 15, 16, 17, 20. EXAMPLE 2. By trisection we understand the problem of dividing an angle by geometric construction into three equal parts. This is equivalent to the following: let a complex number w (96 1) be given, where I co I = 1, furthermore let a = /w, where it is immaterial which of the three values of this radical are taken. We want to know whether the point a can be constructed from the points 0, 1, co. (According to the well-known de Moivre formula the construction of the third part of a given angle

is involved.) In order to clarify this problem by applying Theorem 440', We take F = Y0(i, J- 3 , co) as fundamental field. This is permissible, since we can construct

the point J- 3 . Then the polynomial X3 - W

(179.7)

over F is either irreducible or splits into linear factors. Correspondingly, the degree of the Galois field of F(a) I F is divisible by 3 or equals 1. Thus the trisection for a

GEOMETRICAL CONSRUCTIBILITY

713

given co is possible if, and only if, (179.7) is reducible over F. Since this is certainly not the case for a transcendental co, it follows that trisection is, in general, insolvable, but, at the same time, infinitely many particular cases are solvable, e.g., for all the co

a

_ I

i

s

a-1

(a E .7o).

EXAMPLE 3. The duplication of the cube or the Delian problem requires the construction of a cube with double the volume of a given cube, in other words, the construction of a segment of a straight line of length .J2 of a given unit segment is desired. This is like the preceding example, where now the polynomial

x3 - 2

(179.8)

occurs instead of (179.7). Since (179.8) is irreducible over .F-0 (1), a cube cannot be duplicated by a geometrical construction. EXAMPLE 4. The quadrature of a circle (squaring the circle) requires the construction of a square which has the same area as a given circle. As in the preceding example

this involves the construction of a straight line of length .n of a given unit straight line, where n (= 3.14. . .) i; the Ludolph number. Since ,z, according to the wellknown theorem of LINDEMANN, is transcendental, the quadrature of a circle, because of Theorem 440', is insolvable.

§ 180*. Remarkable Points of the Triangle We take a Euclidean plane without infinite points, containing all the geometric figures with which we deal. The elements constructible from the set of vertices of a triangle (points, straight lines and circles) are called constructible elements of this triangle. As triangles we only admit real triangles, but where necessary, we consider imaginary points, too. Certain constructible points of the triangle, in the construction of which no vertex is given preference, one usually calls "remarkable points of the triangle". Early examples are the centroid, orthocentre and circumcentre of the triangle; indeed we see that, e.g., the three medians, alti-

tudes and perpendicular bisectors of the sides of the triangle cut each other in these points. As a further, instructive example we consider the centres of the four tangent circles of the triangle, which we may obtain as points of intersection of the (internal and external) bisectors of the angles yet we now observe that of these four points, according to our subsequent general definition, one, namely the centre bf the inner tangent circle is called a remarkable point, and all four are called its "conjugates". From the beginning of the last century about twenty other remarkable points were discovered and their properties thoroughly investigated, but no doubt on account of some surprising difficulties, a systematic theory of remarkable points of the triangle has not been established, until now, nor has a general definition been given for them. This gap will now be filled up as briefly as possible. We shall make use of some well-known geometrical concepts and the theory of point sets without defining them.

OALOIS THEORY

714

Some preliminary considerations of geometrical constructions are also necessary.

We shall introduce for the natural numbers a, b, c, d, the following five symbols : 1 abed

(a, b, c, d different),

1(ab)(cd)

(aob; cod; aoc),

lab(cd)

(a96 b; c96 d),

2(ab)(cd)

(aob,c; d0b,c),

2ab(cd)

(a, b, c, d different),

(180.1)

which we call construction steps. The first three and the last two are called, more precisely, single-valued and two-valued construction steps, respectively. Thus an n-valued construction step (n = 1, 2) always occurs under the symbols nabcd , n(ab) (cd) , nab(cd) ,

(180.2)

of which the first is to be considered only for n = 1. In any case we call max (a, b, c, d) the order number of the construction step. We shall also consider an ordered set S, of v points (v = 2, 3, ...) in our plane which may include also imaginary finite points. The kte element of S,, is denoted by Ak. For two different points P, Q we denote by PQ the straight line through P and Q, and by (PQ) the circle through Q with centre P. Let q(n) be an n-valued construction step with an order number < v. According as q(n) is of the form (180.21), (180.22) or (180.23), we define the set S, ' ) as the difference set (Aa Ab fl A,. Ad) - S,,, ((Aa Ab) fl (Ae Ad)) - S,., (Aa Ab) n (Ae Ad)) - S

respectively. If this set S, (") is neither empty nor infinite, it consists of at most two points, which lie outside S, and are constructible from S,. If SF") consists of exactly n points, then we say that the ordered set S, is admissible for the construction step (E(n). Further on we shall conconsisting of n sider an S, of such a kind; then we consider the set points as a (somehow) ordered set, unless otherwise indicated. The latter condition for n = I is superfluous, since this set then consists of one point, which may be regarded (uniquely) as an ordered set. However, if n = 2 the definition of our ordered set is two-valued. Thus, in both cases: if S, is admissible for (E(n), then the (ordered) set S?(") has exactly n different meanings.

Let A denote a triangle which is non-degenerate, i.e., its vertices do

not lie on one straight line. By an ordered triangle we understand

REMARKABLE POINTS OF THE TRIANGLE

715

an ordered set of three points not lying in one straight line. A triangle thus leads to six ordered triangles. We often assume triangles as ordered without always expressing this in the notation (thus we also apply A as the notation of ordered triangles) and we even omit the adjective "ordered" where there can be no ambiguity. For any elementary ordered sets S', S" we denote the ordered set obtained

from them by juxtaposition by (S', S"). By a construction process $ for an ordered triangle A we understand a finite sequence C, ..., Sir (r >-- 1) of construction steps such that the ordered sets of points A1, ..., Ar, defined recursively by

Al = A, Ak+1 = (Ak, Akk)

(k = 1, ..., r - 1),

(180.3)

exist, i.e., Ak is admissible for +k (k = 1, . . ., r - 1), and also Ar is admissible for 19r, taking the many-valuedness of the A2, ..., Ar into consideration in all cases. It should be noted that if the construction step lZk is nk-valued, then Ak has exactly nI ... nk_1

(180.4)

distinct meanings and consists of 3 + n1 + ... + n1,_1 points. Therefore this

number must necessarily be an upper bound for the order number of

%k (k= 1,...,r- 1).

In (180.4) we particularly emphasize that the number of all the Ar is equal to

n1 ... nr_I .

(180.5)

In any case the set Air, now considered as unordered, consists of nr points. The whole set of

nl ... nr

(180.6)

points (which are not necessarily different) are constructible points of A.

We say that they are obtained from A by the construction process Their set is denoted by A% and called the set of conjugate points arising from A by $. We also say that these points are the conjugates of any one of them. It should be emphasized that, because of the (necessary) generality

of the concept of construction steps, every constructible point of A is contained in a suitable A$. To illustrate this, let us consider the construction process $ (for p) consisting, in order, of the construction steps 212(13), 212(23), 2(34)(43), 2(36)(63), 1881011

(so that now r = 5, n1 = n2 = ns = n4 = 2, n., = 1). The reader must realize that A consists of the centres of the four tangent circles of L and from p each of these four points is obtained exactly four times by $, but that all this holds only for those p which do not have two equal sides meeting at either the first or second vertex.

716

GALOIS THEORY

C5

will always denote an everywhere dense and similarity-invariant

set of ordered triangles; by the first of these two properties we understand that for three arbitrary circles 01, 02, 03 there exists at least one triangle A (E Cam) whose kte vertex lies in Ok (k = 1, 2, 3). If $ = 3c is simul-

taneously a construction process for all the Q (E G), then we call it a general construction process for triangles. This is said to be regular, if the number of elements of A (i.e., that of the conjugates) is the same for

all the A (E t) and if these elements occur each time in equally many Q(gr A '. We call them totally real, if for the A (E C5) all the

consist of only real points [the 91, ..., (9r and A 1, ..., A. mean the same as in (180.3)]. Further we call our general construction process $ continuous, if the correspondence A --> A (A E Cam) is continuous and we then extend the notion of the set A of the conjugates to all the ordered triangles A with the definition

Q = lim A) k- oo

(180.7)

... is a sequence from C5 converging to A. If the sets A consist of an equal number of points for all the ordered triangles A

where A(l), A(2),

then we call 3 non-degenerate. Lastly we call 3 irreducible, if no further general construction process 3' exists such that A W c A$ for all the ordered triangles A. We now make the following definition: for a correspondence A --->. PA of points P. to the real triangles A of a Euclidean plane without infinite points, Pa is said to be a remarkable point of the triangle A, if the following axioms are fulfilled :

1. Exactly one PA belongs to every Q.

II. P. is real. III. The correspondence A - . P. is continuous. IV. The correspondence A -p. P. is similarity-invariant. V. P. is a constructible point of A. VI. For the elements A of an everywhere dense and similarity-invariant

set S of ordered triangles, PA is obtained from A by a general construction process is regular. is totally real. is continuous. is non-degenerate. XI. is irreducible.

VII. VIII. IX. X.

XII. The correspondence Q - A is similarity-invariant. Of these axioms I-V concern only the remarkable point Pp itself, VI-XII refer also to the conjugates of Pp. Our axioms are not independent, for II is contained

REMARKABLE POINTS OF THE TRIANGLE

717

in VIII, and XII follows from the other axioms. But III is not contained in IX nor is IV in XII. As we shall see, III is designed to separate Pp from its conjugates. For a generalization of the remarkable points see Examples 3, 4.

Our further aim is to determine the remarkable points on the basis of their above definition. By this we mean an analytical determination, where, at first, we shall make use of rectangular coordinates, but later for the sake

of elegant results we shall pass over to invariant, barycentric triangular coordinates.

We take a rectangular coordinate system in our plane. Let (x, Y E .o); Y # 0)

A (x, Y)

(180.8)

denote the ordered triangle with vertices at (0, 0), (1, 0), (x, y). We denote the 2-closure of an arbitrary field F by F and consider the fields, rings, field or ring elements occurring as substructures or elements, respectively, of (oi (I, X, tj, Zit ..., Z Z, t, ql, q2, q3)

(S

0),

where g, ..., q3 denote indeterminates. (We shall decide on the number s of the zI, . . ., z, later.) We put F=.o

(180.9)

We make use of the 2s sequences

(e) = (el, . . ., e,)

(el, . . ., e, _ ± 1),

(180.10)

from a sequence [dependent on (e)]

el o-cl, ..., e,

(180.11)

of square roots (supplied with the factors el, ..., e,

MI = a, a, = li + cel aI, a3 = b + eel

+ (f + gel 1 J e2

1) defined by:

al +

(a, li, ... E F)

with (independent of el, ..., e,) fixed a,.... b, (The reader will understand

the general law of formation of the al, ..., a, even without further explanations.) It should be noted that depends only on el, . . ., ek-1. We then take two elements, also dependent on (e):

_(e) _

.f ((, t7, el

(e> _ 9 (X, t), eI

al, ..., e,

/al, ..., e. a,),

718

GALOIS THEORY

where f (g, t), z1, ..., z,), g (g, tj, zr, ..., z5) are fixed polynomials, linear with respect to each Zk, from F[z1, . . ., z,], thus their total degree with respect to all the z1, . . ., z, is equal to s. Call the pair of elements

(3e, D) = (3ecer J(e))

(180.12)

a (2s-valued) general (constructible) point. Of the two values of /u- for a complex number u (96 0) we call that with

. The main value of /0 must of course be 0. Thus, in particular, the main value of the square root of a positive number is positive. If we carry out, in (180.12), the substitution the argument x (0:5 rc < n) the main value of

g -> x, t) -> y

(x, Y E

o); Y 96 0),

(180.13)

while giving their main value to all the square roots which occur, we denote the 2s, not necessarily different, points so obtained by

((x, Y), ¶(x, Y)) = ('(e) (x, Y), `ce) (X, M.

(180.14)

According to the previous paragraph these are constructible points of A (x, y), which we consider as assigned to this triangle. [Of course, (180.14)

may become meaningless owing to the vanishing of denominators, but we shall deal only with general points (180.12), where such exceptional cases do not occur.] After these preliminaries we consider a remarkable point P. We take A itself as in (180.8) in the form A = A (x, y), so that for P,, we may also write PD(x,Y). Further we consider a general construction process $ = $.y determined according to VI. This may consist, with the same notation as above, of the construction steps (Y1, ...,r so that (Yk is nk-valued

(k = 1, .., r). From our triangles we temporarily consider only the A = A(x, Y) E 3.

(180.15)

For Qk [cf. (180.3)] we write, correspondingly, Ak(x, y). In order to determine the set A(x, y)$ of the conjugates of P,(.,, Y) we have to take care how the coordinates of each of the nk points of the nl ... nk_1 sets [cf. (180.4)] Ak(x, Y)°k

(k= I,.. ., r)

are to be computed one after another. For this we suppose that

nk,=... =nk,=2

(1
(180.16)

REMARKABLE POINTS OF THE TRIANGLE

719

and all the remaining nk are equal to 1, whence

nl...n,=2S follows. Evidently (180.16) then consists of one point with coordinates of the field 7o (x, y) for each k = 1, . . ., k1 - 1. (Here and in the following

we shall take (180.3) into consideration.) First of all in k = k1 a twovaluedness occurs in the calculation by the occurrence of a square root from an element of the former field, which must be adjoined. It continues

in this way, where for (and only for) k = k2, ..., k, a new square root is adjoined. Take these square roots, taking into consideration their twovaluedness, in the form

el / , ..., e, J

(el,

..., e, = ± 1),

where every term depends upon the preceding one. The , / always indicates

the main value. This short discussion already shows that we can give a general point (Y., J) = (X(e) SD(,,)) defined in (180.12) such that if (180.15)

holds then the 2S points (180.14) are exactly the (n1 ... n, _) 2S points [calculated in (180.6)] of all the n1 ... n,_1 unordered sets A,(x, y)ae. Hence, A(x, y)% is defined as the union of these sets. From VII it also follows that, as in case (180.15), the number of conjugates of PA(X, y), i.e., that of the different points occurring in (180.14) is a fixed divisor 21

(0 < l < s)

(180.17)

of 2S and these points, i.e., the elements of the set A (x, y)'P, all have the same multiplicity 2S-'

(180.18)

among the points (180.14).

We want to deduce from this result a formula which yields all the points of A(x, y)*, and only these, and each of them once only. For this purpose, using the indeterminates z, t we form, first, the 2S linear expressions (180.19)

(e) + J(e) s,

then the principal polynomial, which has these as all its zeros,

F(t) _ fl (t - Y.(e) - lD(e) z), (e)

(180.20)

where we have to multiply over all the 2' sequences (e). For fl we may (e)

write fl ... fl, where every e,, assumes both values ± 1. If we then carry e,

24 R.-A.

e,,

GALO[S THEORY

720

out successively the multiplication for e ..., e1, we see at once that this product lies in 70 (K, l) [z, t], so that [cf. (180.9)] we can write F(t) = FQ, l , z, t) E F [z, t].

(180.21)

According to our results, in the substitutions (180.13) [cf. (180.9)] satisfying (180.15), F(t) turns into the 2s-t-th power of an element from .52(O)(i)[z, t]. (Because of VIII, i might be omitted here, although this is unimportant at this stage.) Since the points satisfying condition (180.15) are everywhere dense on the plane, this implies that F(t) = G(t)21-'

(180.22)

G(t) = G(X, t , z, t) E F [z, t]

(180.23)

where

is a principal polynomial of degree 21 with respect to t. We shall show that G(t) is irreducible over F(z). Otherwise the elements (180.19) would have a degree < 2' over F (z). From this, it would follow that a general construction process V = $,, exists such that A(x, y)$' c c A(x, y) for all the A(x, y) satisfying (180.15). Then, because of IX and XII, AV e A$ would hold for all triangles A. Since this contradicts XI, the irreducibility of G(t) is proved. It now follows from (180.20), (180.22), (180.23) and the irreducibility of G(t) that there are exactly 2' distinct elements (180.19), and these are all the zeros of G(t) and conjugate over F(z). Therefore our general point (3c, P) = (ke), Dte) (especially, e.g., for el = ... =e,=I) may be expressed by a sequence such as

Jw1,

w, (E F)

a, . .

of square roots (instead of the previous

Cok E F (Jwl, ... 11IN-1)

(180.24) .,

(k = 1,

a,.), for which ., 1)

(180.25)

..., G).

(180.26)

. .

and

FcF

cF

/w2) c

... c

F (Jw1,

Because of (180.25) and (180.26), we say that (180.24) is an irreducible square root sequence over F. After this transformation we retain for our general point the previous notation (3r, J), where now 3r, SJ are certain elements of F. [In the former meaning we no longer use (., J). ] The 2' points arising from (X, J) by the substitution (180.13), taking into considera-

REMARKABLE POINTS OF THE TRIANGLE

721

tion all the possible many-valuednesses of the substitution values of (180.24),

are denoted (without expressing the many-valuedness) by (3(x, Y), ¶'1(x, A.

(180.27)

We shall show, for these 2' points (180.27), that they are real and distinct for every A (x, y) and just constitute the set A (x, y)V of conjugates of Po(x. vY

This is clear for the case (180.15) according to the above result, because

of VIII, (180.22) and the definition (180.20) of F(t). For the remaining A (x, y), the assertion follows at once from IX, X and definition (180.7). [It should be noted that in the second case even the existence of the points (180.27) has only now been proved.] In order to prove a further property of our (X, SD) let us now consider from XII that the correspondence A -> A is similarity-invariant. Since A(x, y) becomes A(x, - y) by reflection in the x-axis, it follows that

A(x, y)`t' and Q(x, -y)$ are obtained from each other by the same reflection. This means the equality of the two sets consisting of the points (180.27) or of the points

(T(x, -Y), - J(x, -Y)), respectively. We now assign to the points (180.27) the 2' different linear expressions formed by the indeterminate z: (x, Y) +

)(x, Y)

Z.

Y

The set of these expressions is, according to the above, invariant with respect to the substitution y -y. Therefore, if using the further indeterminate 1, we form the relative norm

NO = H(X, t), z, t) = NcIF(z)T t

-

-

z

(180.28)

which with respect to i is a principal polynomial of degree 2', then the validity of H(x, y, zz, t) = H(x, - y, z, t)

follows for all the real x, y (y 36 0). Consequently

H(X, , z, t) = HQ, -t), z, t), which we express briefly, but clearly, by saying that H(XC, t), z, t) involves (besides 2), z, t) only p`

GALOIS THEORY

722

On the other hand, because of (180.28), H(t) has the zero

whence its irreducibility over F(z) also follows. [We understand the latter property most easily by the fact that it holds for the polynomial (180.23) and we have H(g, t, t)z, t) = G(g, t, z, t).] From both these results follows the possibility of such a choice of the square root sequence (180.24) that [(180.25), (180.26) remain valid and] in

wl, ..., Jwr, E,

(180.29)

only 112 is involved (besides ). The importance of this apparently simple result will become evident in connection with barycentric coordinates

to which we now turn. The vertices and sides of the ordered triangle . are denoted in order by A1, A2, A3 and 91, 92i 93i so that Ak, gk are opposite to each other. By the barycentric coordinates x1, x2, x3 with respect to , of a point P we mean the quotients

xk =

P9k

(k=1,2,3),

Ak9k

where numerator and denominator denote the distance, together with sign, of the point P or Ak from the side gk, such that xk is positive if, and only if, these points he on the same side of gk. We Put P = (x1, x2, x3). As is well known,

x1 + x2 + x3 = 1 .

(180.30)

We denote the squares of the lengths of the sides opposite to the angles AI, A2, A3 of Q in order by q1, q2, q3 and for , write more precisely (180.31)

A = x(q1, q2, q3) .

Accordingly, q1, q2, q3 always denote positive numbers subject to the triangle-inequalities q1 I +

q2 I + I

q3 I > 2 I ,I qk I

(k = 1, 2, 3) .

(180.32)

Such numbers are, briefly, called admissible. Our aim is to give PA and its conjugates for the triangle (180.31) with barycentric coordinates as functions of the side-length squares qI, q2, q3.

REMARKABLE POINTS OF THE TRIANGLE

723

It is a paradox of remarkable points that - as we shall see - in their analytic description the side-length squares q1, q2, q3 are suitable parameters but the side lengths themselves are unsuitable. We obtain the transformation formulae between the rectangular coordinates il and the barycentric coordinates x1, x2, x3 of the triangle A (x, Y)

x5, x3= Y

x2=

Y

Y

Hence, and from (180.27), it follows that the 21 conjugates of Pots. yt, expressed by such barycentric coordinates, are given by

(x, Y) + (x - 1) I(y, Y) (x, Y) - g

(y, Y) ,

+1

(y, Y) l

,

(180.33)

.

Since these points arise in the substitution (180.13) from the triple

+(N-

+

(180.34)

(180.34) may be regarded as another form of our general point (3r, J), namely that adapted to the barycentric coordinates related to Q(x, y). Since every triangle is similar to a Q(x, y), because of XII, we can easily generalize the result (180.33) to all triangles instead of Q(x, y). For this purpose we take the arbitrarily ordered triangle Q as in (180.31) as Q(q1, q2, Q. The A(x, y), similar to this, are (taking into consideration the arrangement of the vertices) defined by the conditions

(x - 1) 2+ Y2

= qI

q3 '

ti+

x2

q2

2

Y-

=

q3

'

which we can also write as X

q2 + q3 - qI 2 q3

,

y2 =

2(g1g2 + g1g3 + g2g3) - g1 - qz - qs 4qi (180.35)

The numerator in (180.352) is sixteen times the square of the area of A (q1, q2, q3), and so is positive. Therefore we always obtain exactly two solu-

tions x, y as was obvious from the outset. The solutions differ only in the sign of y, but this difference does not affect the following arguments.

GALOIS THEORY

724

Keeping the relation (180.35) in view, in the general point (180.34) we make the substitution q2 + q3 - q1

2q3 2

2(g1g2 + g1g3 + g2g3) - ql - q2 -

q32

(180.36)

4q3

%%

where q1, q2, q3 are three new indeterminates; for the "homogeneous" rational field over 70, generated by them, we introduce the notation

Q0 = -ro

ql

q2

gl

qg3 _

g3 17-0

q3

q3

1 q2

q2

,

q1

ql

'YO

01 : q2 (180.37)

Since because of (180.29) only l 2 is involved (besides Xr) in (180.34), no "new" square roots occur after the substitution (180.36) in (180.34); the only change is that new indeterminates q1, q2, q3 will appear in (180.34) after

this substitution. More precisely we have the following: if, after the substitution (180.36), we introduce for our general point (180.34) the notation (21, £2,

2(q1, q2, q3), X3(gl, q2, q3)) (180.38)

3) = (31(g1, q2, q3),

and retain the notation,Jcol, ...,

Y coo

introduced in (180.24), for the

square roots occurring in (180.38) we now have X1, X2, X3 E Qo (/w1,

., Jam,) ,

(180.39)

then [instead of (180.24), (180.25), (180.26)]

Jwl, .. , Jw! E Qo .Jwk E Qo (Jw1,

..., ,Jcok-

(180.40)

(k = 1, ...,1),

(180.41)

Qo C Qo (Jw1) C Qo (Jwl, Jw2) c ... c Qo (Jwi, ..., JWt) (180.42) now hold. From the above it is now clear that by the substitution 3

q1 -- q1, q2 --,, q2, q3 -->- q3 (qk > 0; m'1

4m I > 2 1 Jgk I ; k = 1, 2, 3) (180.43)

we obtain from (180.38) exactly 2' different real points, which we denote by

REMARKABLE POINTS OF THE TRIANGLE

725

barycentric coordinates related to A(g1,g2,q3) (without expressing the many-valuedness) such as (180.44)

(E1(g1, q2, q3), X2(g1, q2, q3), .3(q1, q2, q3)),

and that these points constitute exactly the set A(ql, q2, q3)` of the conjugates of PD(q, q, q,J. [It should be noted that this result refers to all

triangles, for all may be taken in the form A(q1, q2, q3); the fact that A(q1, q2, q3) is only defined up to position by q1, q2, q3, has, because of XII,

no disturbing effect, since the barycentric coordinates are invariant with respect to similar mappings. I To distinguish it from the previous (X, J) we call (180.38) a barycentric general (constructible) point. (Of course this concept is capable of wider generalization but this is unnecessary for our purposes.) It is now necessary to examine how the Po(q q,, q,) itself can be separated from the currently existing 2' conjugates (180.44) of PA(q,, q,, Q. The answer to this important question is unexpectedly simple. According to what has been proved concerning (180.27), the square roots (180.40) occurring in

(180.38) assume only real values other than 0 in all the substitutions (180.43) (considering all the possible many-valuednesses). The substitution values, to which the remarkable point Po(q , q,, Q in (180.44) always belongs,

must then keep their sign because of III. Then we find that these substitution values, belonging to Po(q q,, Q, are always positive, for this only requires that the square roots (180.40) occurring in (180.38) have suitable signs, which is permissible because of the irreducibility of the square root sequence (180.40). From now on this will be assumed. Then, according to (180.44), Po(gl, q,, q.) = (X I (q1, qs, q3)+,

Y-2(gl, q2, q3) +,

X3(g1, q2, q3)+)

,

(180.45)

where "+" indicates that only positive substitution values are to be taken for the square roots which occur. (This notation will be applied later in similar cases.) Finally, we consider that, according to I, exactly one Po belongs to every A, but, by (180.45), a Po(q,, q, q,) was ascribed to every ordered triangle A(q1, q2, Q. Therefore the Po(q,', q.', q,) belonging to the six permutations

2 3l 1'2'3' 1

(180.46)

must be equal to one another. This means the validity of 3Ek'(gl, q2, q3)+ _ 3k(gl, q2, q3)r

(k = 1, 2, 3)

GALOIS THEORY

726

for all permutations (180.46) and for all admissible q1, q2, q3. Hence, it follows that X2(gl, q2, q3)+ _ Y-1(g2, q3, q1)+, 2e3(g1, q2, q3)+ = 3e1(g3, q1, q2)+,

X1(g1, q2, q3)+ = 31(gl, q3, q2)+

Obviously even the rules 2E2(g1, q2, Q3) = 3E1(g2, q3, q1),

(180.47)

-T3(g1, q2, Q3) = 'f1(g3, q1, Q2) ,

Y1(Q1, q2, Q3) = X1ig1, q3, Q2)

are then valid.

In order to formulate this result conveniently we shall use homogeneous barycentric coordinates Yl, Y2, Ys (Y1 + Y2 + y3 # 0), and stipulate that the point determined by them shall be denoted by (yl, y2, y3) and defined

as usual as 1

e=

(Y1, Y2, Y3) _ W11 QY2, US)

l

,

Y1 + Y2 + Y31

where "ordinary" barycentric coordinates are intended on the right-hand side.

As further preparation we introduce the field ( D QO)

Q = -70'(gl, q2, Q3)

and consider an arbitrary element 97 = 9'(q1, g2,g3) of Q together with its minimal polynomial 0(t) = 0(q1, q2, q3, t )E Q[t]

over Q. If 0(t) lies in [q1, q2, Q3,t], then we call the element integral. If, in addition, there are two integers j (> 0), k (>_ 0) such that the polynomial O(ql, q2, g8, tk) is homogeneous, then we call the element 99 homogeneous. If further P1, 9'2, P3 are elements of the field Q with non-vanishing sum Cr,

then we call the relative degree [Y0(g1, q2, q3, a-1 991, a-I 922,

a-1

q'3)

:

01, q2, q3)]

the degree of the triple ON, 9221 993). This is a power of 2 and evidently has the property that in the substitution (180.43) at most as many different points are obtained from (991, 9'2, 993) as the degree of this triple.

REMARKABLE POINTS OF THE TRIANGLE

727

THEOREM 441. In order to give the remarkable points of a triangle, take a homogeneous integer element q'(Q1, q2, q3) of the 2-closure of the rational function field Q = Y0(Q1, q2, Q3) such that (180.48)

9901, q2, Q3) = 901, Q3, q2) and

9'(q1, q2, q3) + 9'(Q2, q3, q1) + 9'(Q3, qI, q2) , 0;

(180.49)

further, suppose that for (9,(ql, q2, q3) , 9;02, q3, q1)

9'(Q3, ql, q2))

(180.50)

9'(q3, q1, q2)) ,

(180.51)

,

the number of different real points (9' (ql, q2, q3),

9, (q2, q3, q) ,

(where homogeneous barycentric coordinates are being used) arising by any substitution (180.43), is equal to the degree of (180.50). Then the point PA = (9%(ql, q2, q3)+, 9'(q2, q3, q1)+ ,

9p(g3, q1, q2)+)

(180.52)

assigned to the triangle A = A(q1, q2, q3) with the side-length squares q1, q2, q3

is a remarkable point of A, and all the remarkable points of the triangle arise

in this way. The points (180.51) are there exactly the conjugates of PA. [For the signs "+" in (180.52) see the remark after (180.45).] The assertion of the theorem that every remarkable point PA and its conjugates may be given by (180.52) or (180.51), respectively, is deduced from the above results. With this end in view we have to consider that, according to (180.39), 2E1 = 3E101, q2, q3) is an element of the 2-closure Q0 of the field Qo defined in (180.37). Therefore we put X1(81, q2, q3) _

9'(q1, q2, Q3)

(Q1, , qsQ

,

(180.53)

where p and r are homogeneous integer elements of Q. It may be assumed that Z lies in the fundamental field Q, and therefore in the polynomial ring o[ql, q2, q3], where we can restrict ourselves to a symmetric homogeneous polynomial T. [Since in the substitutions (180.43) no denominator can vanish,

in 31 we may suppose moreover that neither do the substitution values r(q1i q2, q3).] Afterwards the denominator r may be cancelled, in so far as we now employ homogeneous (barycentric) coordinates. This proves, according to (180.38), (180.44), (180.45), (180.47) that Po and its conjugates

arise as described in the theorem. The easy proof of the other assertion of the theorem that a remarkable point and its conjugates are always furnished by (180.52), (180.51) is left for the reader. 24/a R.-A.

GALOIS THEORY

728

If the degree of (180.50) is equal to 1, then we call the remarkable point Pp rational.

These remarkable points are thus characterized by the circumstance that they have no conjugates other than themselves. [Consequently, we may assume (180.51) for them instead of (180.52).] They are easily given, and are furnished by those homogeneous polynomials 9901, q2, q3) over .F70, which are subject only to (180.48) and the con-

dition that the sum of the coordinates in (180.51) vanishes for no A. In the general case it would be desirable to complete Theorem 441 by examining the condition with regard to the diversity of the conjugates (180.51), but this appears to be no easy task. No matter which remarkable point PA is concerned, it is clear that, for the particular case of an isosceles or equilateral triangle A, this lies on the axis of symmetry or at the centroid of A, respectively. In the following few examples we restrict ourselves almost exclusively to cases already known (cf. ALTSHILLER- COURT (1952).)

EXAMPLE 1. For 92 = 1, qi - (q2 - q3)2, ql (q1 - q2 - Q9 q, the centroid, vertex, circumcentre and Lemoine point of the triangle are obtained from (180.52). EXAMPLE 2. For T = ti/ 4i, y4i - \/q2 - vg3, q1 - (v/q2 - N /Q2 the cen-

tres of the four tangent circles, the four Nagel points and the four Gergonne points, are obtained from (180.51) in order. From among each of these, sets of four conjugate points only one is a remarkable point, namely that given by (180.52). EXAMPLE 3. An easy generalization of the remarkable points arises if we admit exceptions in (180.51). To these belong 4' _ (q1 - q2) (q1- q3), (q1 + qz - q3 (q1 + q3)) (q2, + q3 - q2 (q1 + q3)), VIQ

01 - q2 - q3) v

where El indicates the numerator of (180.362). Here the Steiner point, Tarry point and the two isodynamic points of the triangle are the points under consideration. In each case only equilateral triangles are exceptions. EXAMPLE 4. A further generalization will arise if we admit reducible radical expres-

sions for T in Theorem 441. This includes the case 99 = q1(q1 - q2) 01 - q3) (q2 - q3)2 ((q2 + q3) (q1 - q2) (q1- q3) +

+ N/01 - q2)2 (q1 - q3)2 (q2 - Q2), which furnishes the two Brocard points. (Equilateral triangles again consitute an exception, since then the two Brocard points coincide. On the other hand isosceles triangles obviously do not form any "essential" exceptional cases.) EXAMPLE 5. If 99, y, are two elements of Q, satisfying the conditions of Theorem 441,

to which rational remarkable points belong, then the same applies to

wtV*p+(1-w)p*tp

(WE,9-p)

where tV* _ Y_

q2., q3.),

V* = E

q0,

where we have to sum over the three even permutations of the form (180.46). Hence it follows that on any straight line through two rational remarkable points of a triangle there exist infinitely many rational remarkable points of the same triangle and these are equally dense everywhere on the straight line. If we proceed to three remarkable points not lying on one straight line, then it follows by repeated application that all the remarkable points of a triangle without equal sides are everywhere dense on the plane - a surprising and disillusioning fact.

REMARKABLE POINTS OF THE TRIANGLE

729

ExERCISE. Define by suitable axioms the remarkable straight lines and circles of a triangle and construct for them the analogue of Theorem 441. PROBLEM. How can we define analytically those remarkable points which are definable by a construction process consisting only of single-valued construction

steps? (Of the centroid, orthocentre and circumcentre of the triangle only the second is included here.)

§ 181. Determination of the Galois Group of an Equation Over a field F we consider a not necessarily irreducible equation such as

f(x)=0

(181.1)

without multiple roots and want to elaborate a process for the determination of its Galois group c4. Of course, this is also suitable for the determination of the Galois group of an arbitrarily given Galois field, since this may always be given as a Galois field of a suitable equation. For our purpose we put

f(x) = (x - a) ... (x -

(G = F(ai, ...,

(181.2)

where G I F is the Galois field of (181.1). Furthermore we have recourse to the polynomial ring G [u1, . . .,

(181.3)

consider in it the element

I

(181.4)

and form with the help of a new indeterminate z the polynomial

F(z) _ fi (z - s ,O) ,

(181.5)

su

where s has to run through all the permutations of the set
C11,.. ., «n. Consequently, it follows from the main theorem for symmetric polynomials (Theorem 268) that F(z) lies in the polynomial ring

F[z],,= F[z,u1,...,u,,].

(181.6)

In this, let the irreducible factor decomposition

F(z) = Fi(z) ... Fk(z)

(181.7)

GALOIS THEORY

730

hold, where we may assume that the factors are principal polynomials and

z - 0 FF(z).

(181.8)

For every s,, let s,, denote the corresponding permutation of the set
a.>. (More precisely sa and s,, applied to or , respectively, will give rise to the same permutation of the indices.)

We consider those permutations s with respect to which the F1(z) is invariant. The group consisting of them is denoted by co. and we prove that the permutations sx, corresponding to the elements of C4,,, constitute exactly the Galois group 44 of the equation (181.1), hence 4 c.Pt . With this end in view we first show that the product of all the (181.9)

is equal to F1(z). For we have z - su# I F1(z)

if, and only if,

z - 01su1F1(z).

(181.10)

The right-hand side is, because of (181.5) and (181.7), equal to an Fk(z) for every s,,. Hence, and from (181.8), it follows that (181.10) has the same meaning as the condition s,-,1F1(z) = F1(z), which can also be expressed

s with this property constitute exactly the

as

group c4,,, F1(z) is in fact the product of the polynomials (181.9).

Now we show that the product of the polynomials

z - so

(sE4)

(181.11)

is similarly F1(z). We denote this product by P(z). If we apply an arbitrary

element of 4, the polynomials (181.11) undergo a permutation, so that the coefficients of P(z) He in F. More precisely, P(z) E F[z],,.

Furthermore, according to (181.11), z - 0 is, in particular, a divisor ofP(z), whence, because of (181.8), and the irreducibility of F1(z), follows the divisibility Fi(z) I P(z) .

(181.12)

Again, because of (181.8), all the divisibilities z - sO I sF1(z)

(181.13)"

GALOIS GROUP OF AN EQUATION

731

also hold. But since the coefficients of F1(z) lie in the fundamental field F, s may be cancelled on the right-hand side of (181.13). Consequently, (181.13) means, on account of the definition of P(z), that P(z)!F1(z). This and (181.12) prove that F1(z) = P(z). It now follows immediately from (181.4) that, for an arbitrary s and the corresponding sa, sx(su$) = 0 always holds, i.e., s ,,O = s, 1t'. Hence it follows that the polynomials (181.9), apart from the order of succession, coincide with the polynomials z - sa 1'+9'

(s, E (4.) ,

where co. denotes the group of permutations corresponding to the s, (E This result may also be expressed so that the polynomials (181.9), apart from the order of succession, coincide with the polynomials

z - sat

(S« E Cjx)

According to the above, these coincide, apart from the order of succession, with the polynomials (181.11). This means that co = 4., whereby the above assertion is proved. The direct application of the process so obtained for the determination of the Galois group of an equation involves almost insurmountable difficulties of calculation, in return for which we easily prove by its help the following very useful THEOREM 442. Let R denote a ring with prime decomposition and n a prime element of R, for which the factor ring R = R/(n) is zero-divisor-free and the homomorphism

R -.R(a `a)

(181.14)

holds, where a denotes the residue class a (mod n). The quotient fields of R and R are denoted by F and F. Let f(x) (E F[x]) be a principal polynomial without multiple factors and with coefficients from R, furthermore let f(x) (E F [x]) be its image under the homomorphism (181.14). If this has no multiple factors, then the Galois group 4 of f(x) = 0 (regardless of the meaning of the permuted elements) is a subgroup of the Galois group d off(x) = 0. In the proof we use some of the above notations. According to GAUSS'S theorem (Theorem 207) the factors of the right-hand side of (181.7) lie in the

polynomial ring

R[z]n= R[z,u1,...,un] Furthermore, after the application of the homomorphism (181.14) a not necessarily irreducible factor decomposition F ( z ) = Ft (z)

... Fk (z)

(181.15)

GALOIS THEORY

732

is obtained from (181.7) in the polynomial ring

R[ZIU = R[z, u1,..., Since the permutations s (E dL) leave the polynomial Fj(z) fixed and the remaining permutations of the set
This theorem, for R = 7, n = p, turns out to be particularly simple, since then F = R = gyp, thus the Galois group c, is, according to Theorem 306, cyclic. As an application, we shall show that over Jr, there are affectless equations of degree n for every natural number n. (In this respect Theorem 433

only says in somewhat weaker form that over every rational function field of at least n variables there are affectless equations of degree n. Against which, over finite fields, for the same reason, there are no affectless equations

whatever of degree >_ 3.) To prove this assertion some preparation is needed.

A permutation group 9 of a set S is called transitive, if for every pair of elements a, b of S there is a permutation in 9 which carries a into b. From Theorem 329 it follows that the Galois group of an irreducible (separable) equation is transitive. We show the following: if a transitive permutation group 99 of the set

<1, ..., n> contains a transposition and a cycle of order n - 1, then .53 is the full permutation group of degree n.

We may assume that (1 2

... n - 1) E -0.

(181.16)

Since 93 contains a transposition and is transitive, it follows from Theorem 82 that it also contains a transposition of the form (in). According to the same theorem, and on account of (181.16), 9s contains all the transpositions (1 n), .., (n - 1 n). Therefore, because

(r s) = (r n) (s n) (r n)

(r, s = 1, . . ., n - 1),

it contains all the transpositions. Hence the theorem is proved.

We can now construct for n > 2 an affectless equation f(x) = 0 of with coefficients in 7 as follows : let us take three distinct degree n over prime numbers P1, P21 P3 (say Pl = 2, P2 = 3, p3 = 5) and three polynomials f1(x), f2(x), f3(x), of degree n over,7 such that the first is irreducible mod pl,

the second has an irreducible factor mod P2 of degree (n - 1), and lastly among the mod p3 irreducible factors of the third, one factor is of degree 2 and the remaining factors are of odd degree. All this is possible according

GALOIS GROUP OF AN EQUATION

733

to Theorem 306 (cf. also § 133, Example 1). Then take a principal polynomial

f(x) of degree n over Y with

f(x) - f (x) (mod pi)

(i = 1, 2, 3) ,

which is likewise possible according to the Chinese remainder theorem (corollary of Theorem 198). Let &, denote the Galois group of the equation

f(x) = 0 over . and cdc the Galois group of the equation f(x) = 0 over Jr, (i = 2, 3). Since fi(x) is irreducible mod pl, so f(x) is all the more irreducible over 5, thus 4 is transitive. Since, furthermore, 42i 4t3 are cyclic, it follows by hypothesis that they contain a cycle of order (n - 1) and a transposition, therefore, according to Theorem 442, the same holds for c.P. Consequently, co is, according to the above, the full permutation

group of degree n which is the same as saying that the equation f(x) = 0 is affectless. VAN DER WAERDEN (1933) has shown that among the equations with integers those without affect prevail (in a certain sense). It is a so far unsolved problem, whether equations with integer coefficients exist whose Galois group is prescribed. With regard to this cf. NOETHER (1918). As regards equations without affect see also BAUER (1932).

§ 182. Normal Bases In § 147 we have already defined the normal bases of separable norma I fields of finite degree, i.e., of the Galois fields and, in Theorem 365, proved their existence for the case of finite fields. THEOREM 443. Every Galois field N I F has a normal basis.

According to what has been stated above, the case where N is infinite still remains to be considered. In order to prove this case we need the following PaoposITTON. If N I F is an infinite Galois field of degree n and Al, ...,

A are all the elements of its Galois group, then for a polynomial f(xl, ..., xn) over N f (Ala, . . .,

0

(x E N)

(182.1)

always holds if only f(xt, ..., xn) = 0 . In order to prove the proposition, we take a basis coi, ..., w,, of N I The elements of N may then be taken in the form

a = aiool + ... + anwn

(a,, ..., an E

It follows that A,a = > ajA,wi /=I

(i = 1, ..., n).

F) .

F.

734

GALOIS THEORY

We introduce the polynomial

g(Yi, ...,

(E N [Yi, ...,

=.f(E [y1A¢Ai , ..., E yiAnwi) i i

(182.2)

where we have to sum over the j = 1, ..., n. (The coefficients A,ai are deliberately placed on the right side.) Because of (182.1),

(a,,...,a,, E F).

g(al,...,a.) = 0 From this, because of Theorem 206,

0

g(Y1, ...,

arises from f(xl, . . ., follows. But g(yl, . . ., the linear transformation

xi -

(182.3)

because of (182.2), by

(i= 1, ..., n)

yyA,pii

(182.4)

whose matrix is (A,i). On account of (145.10) the square of the determinant I A,wi I is equal to the discriminant of the basis elements w1, ..., w,,. This discriminant is, according to Theorem 362, different from 0, therefore the same also holds for A,wi I. This means that the matrix (Ap) is invertible. also arises by a linear transformation Since, according to this, f(xi, ..., from g(yl, . . ., it follows from (182.3) that the proposition is true. We can now prove the theorem. Keeping the former notations we write A;A; = A(i.i)

(i, j = 1, . . ., n) ,

(182.5)

where (1, j) denotes the function of i, j with the range <1, . . ., n> just defined by (182.5). Using this function we form the determinant

f(xl, ...,

I xy, i) I

(E F [x1, . . ., x )) .

(182.6)

For every a (E N), according to (182.5) and (182.6),

.f(Ala, ..., AA) = I A(i, )a I = I AiA,a 1.

(182.7)

But it follows from (182.5) that the condition (i, j) = 1 between i (= 1, . , n) and j (= 1, ..., n) produces a one-to-one relation. This means, because of (182.6), that f(l, 0, ..., 0) is a determinant which contains in each row and column exactly one element other than 0 and this element is equal to 1. Such a determinant is different from 0 (and equal to 1 or -1). 0 0, it follows from the Since accordingly we have, a fortiori, f(xl, . . ., proposition, and from (182.7), that an a (E N) exists such that I AiA,,x 1

0.

(182.8)

NORMAL BASES

735

The square of the left-hand side is, according to (145.10) the discriminant of the elements Ala, ..., Ana. (182.9) Since this discriminant, because of (182.8), is different from 0, the elements (182.9) constitute, because of Theorem 362, a basis, and this is evidently a normal basis of N I F. For the above proof cf. BOURBAICI (1939). For another proof, which also holds for finite fields, cf. PICKERT (1951).

EXAMPLE 1. Let N I F be a Galois field of degree 2 and of characteristic p (Z 0).

If p 96 2, then N = F (,) for a suitable element d from F. Now .1

+ ,Fd ,

1 - Jd

is a normal basis of N I F. If, on the other hand, we have p = 2, then, according to Theorem 429, N = F(8), 02 + 0 + a = 0 for a suitable element a of F. Now

0,1+0 are conjugate; consequently they constitute a normal basis of N I F. EXAMPLE 2. In the n`h cyclotomic field F (of characteristic 0) the primitive nth roots of unity 91, ..., e9,(")

(182.10)

constitute a basis, thus a normal basis if, and only if, n has no multiple prime factors.

We first prove the "if" part. This is true if n is either 1 or a prime number. If n = = pl . . . pk (k > 2) is the product of different prime numbers pl, . . ., pk, then it follows from the preceding statements and from Theorem 292 that the products

al...ak

(182.11)

constitute a basis of F, if a; runs through the primitive PC roots of unity (i = 1, . . ., k). But these products (182.11) now coincide with the elements (182.10), so that the assertion is also true in this case. In order to prove the "only if" part, we denote the n1h cyclotomic polynomial by F"(x). If d is the product of the different prime factors

of n, then F"(x) = Fd(x° '").

If n > d it follows that the sum of the elements (182.10) vanishes, therefore these elements now constitute a basis. This completes the proof. For an application cf. REDEI (1959, 1960).

EXAMPLE 3. The solution of the Exercise in § 172 is obtained as follows. Let us take an element 0 of G whose conjugates constitute a normal basis of G I F, together with a primitive automorphism s of G I F and a primitive n`h root of unity a (E F). The Lagrange resolvent [cf. (172.2)]

A = 0 + e(s6) + ... + e"-'(s"-'0) is then different from 0, moreover (as quoted above) sA = e''7. and A" E F. Accordingly

G=F(.jx)for rc=A" (E F).

CHAPTER XII

FINITE ONE-STEP NON-COMMUTATIVE STRUCTURES A non-commutative structure is called one-step non-commutative when its proper substructures (of the same kind) are all commutative. The determination of these structures is a difficult task, and has only been solved for groups, rings and semigroups; furthermore, we have to note WEDDER-

theorem (Theorem 318) which implies that no finite one-step non-commutative skew fields exist. Finite one-step non-commutative structures are of great importance because every finite non-commutative BURN'S

structure contains at least one one-step non-commutative substructure. § 183.* Finite One-step Non-commutative Groups In compliance with the above general definition a non-commutative group

with only commutative proper subgroups is called a one-step

non-

commutative group. All the finite groups of this kind are fully determined by the following two theorems. THEOREM 444. The finite one-step non-commutative p-groups are: the quaternion group (of order 8), every group (of order pn,+n+l) for given m, n (E .f') defined by the equations Apm = 1, BO = 1, C° = 1, AC = CA, BC = CB, BAB-' = AC (m>_n>_ 1)

(183.1)

and every group (of order p,n+n) defined by the equations

Ae = 1, B°" = 1, BAB-' =

Al+vm-'

(m >_ 2, n > 1; pm+n > 8). (183.2)

These groups are not isomorphic. THEOREM 445. The remaining finite one-step non-commutative groups are obtained as follows: Given two different prime numbers p, q and a natural number n, put (183.3) m = o(p (mod q)) , 736

GROUPS

737

take the finite field F such that

O(F) = p"

(183.4)

and from the group F* an arbitrary fixed element co such that

o(w) = q.

(183.5)

Then a group (of order pmq") exists such that by the help of certain special elements

Px(a E F), Q (Pi, the unity element; o(Pj = p for at

0; o(Q) = q") (183.6)

all its distinct elements may be written uniquely in the form P,.Qi

(aE F; i=0,...,q"- 1)

(183.7)

and the rule for multiplication of elements is

(a, P E F; i, k = 0, ..., q"- 1).

P«Qi PpQk = PQ+.r# Qi+k

(183.8)

Taking for each of the above triples p, q, n such a group, we just obtain the mutually non-isomorphic finite one-step non-commutative groups which are distinct from the p-groups.

It has to be proved that every finite one-step non-commutative group is isomorphic with one of the groups given in Theorems 444, 445 and that, conversely, these groups are one-step non-commutative and not isomorphic with each other. We begin the proof with the first of these assertions. We denote by 4 a finite one-step non-commutative group and show first that it is not simple. For the proof we assume c.P to be simple. We first remark that for two proper subgroups a°, . Y of c4, for which 2c° is maximal and .2' not contained in :72, "'

1 (15Y= 1. For, because of the supposition,

'Xj

C4

Since moreover

, W' are Abelian, ' (1 r is normal in JT and 5Y

and therefore also in 4. Because of the simplicity of 4 the truth of the remark follows. Since c.P is not commutative, 0(4) is a composite number. Let a maximal subgroup of 4 be denoted by A' and its index by h = O(C

:

Let Al, .., Ah

(183.9)

738

FINITE ONE-STEP NON-COMMUTATIVE STRUCTURES

be a left representative system of 4 mod .Y. We show that all the conjugates

1i = A;,7,4; I of fT.

(i= 1, ..., h)

(183.10)

are pairwise distinct. We assume that Xi = X, for two different

i, j. Itsfollows that Aj 1A,YoAr 1 Aj

so that

=

°,

is normal in the subgroup

_{

,A.'A,}

of 4. But since X is maximal, we have 4.l = 4. Since, according to this, Y is normal in 4 and 4 is simple, it follows that ' = 1. This contradicts the maximal property of Y, so that we have proved that the conjugates (183.10) are pairwise distinct.

Hence, and from the above remark, it even follows that

tf210;= 1

(1 <-i<-j_
Thus 2C1i ...,h, because of (183.9) and (183.10), contain altogether exactly

h(O(Jf) - 1) = 0(4) - h

(183.11)

different elements (# 1) of 4. We take a further element (0 1) of 4. This is contained in a maximal

subgroup .7 of 4. We put

k = 0(4 :

(183.12)

Here 7 is, according to its definition, different from all the X1, . . ., Xh If what was said above is applied to 7 instead of AT, then it follows that the k conjugates 1, . . ., Ak of contain altogether exactly

0 () - k

(183.13)

different elements (0 1) of 4. Since furthermore, as has been said,'' is different from the conjugate subgroups °l, ..., °h, so the same holds for its conjugates X1, . . ., Xk. It follows from the above remark that

min a,,= 1

(i= 1,...,h; j= 1,...,k).

According to this and (183.11), (183.13) all the X1, Xj contain altogether

20(4) - h - k elements of 4 different from 1, so

20(4) - h - k < 0(4)

,

GROUPS

739

and so

h + k > 0(c4). But no natural number is smaller than the sum of two proper divisors, therefore we have proved by contradiction the non-simplicity of 4. We show that cP is even solvable. Let l° denote a maximal proper normal

subgroup of 4. The factor group c4/a° is then simple. Thus, according to what has been proved above, it is certainly not one-step non-commutative. On the other hand, since its proper subgroups are commutative, it is itself commutative. Since it is simple, it is, consequently, of prime order. Since

furthermore r is Abelian, 4 is in fact solvable. We show that in c0 there are two elements A, B with

AB # BA.

(183.14)

such that for the commutator

C = A'1BAB-1

(183.15)

AC= CA

(183.16)

the equation holds. We take a normal divisor cY,

of 4 of prime index and an element

B from the difference set 4 - 7°. Since :c° is Abelian and 4 =

B},

2C has an element A which does not commute with B. Since, together with A and BAB-1, C also lies in (183.16) follows. This proves the assertion. Because of Theorem 195 we may also suppose that o(A), o(B) are prime powers :

o(A) = p', o(B) = q' !

(183.17)

where the prime numbers p, q need not be different. Because of (183.14),

d4 = {A, B).

(183.18)

We denote by 4' and Z the commutator group and the centre of 4. Henceforth we distinguish two cases. First let us consider the case

C4, g Z.

(183.19)

Because of (183.15) we have C E c4', then C E Z. Accordingly, we now have

BC = CB.

(183.20)

We now need the following useful theorem whose proof can easily be carried out by induction by the reader.

740

FINITE ONE-STEP NON-COMMUTATIVE STRUCTURES

THEOREM 446. If A, B, C are three elements of a group with the properties (183.15), (183.16), (183.20) then the formulae

BkA,B-k = A'C`k , (AiBkC)r = AirBkrClrtttkr(r-1)

(183.21) (183.22)

hold for all integers i, k, 1, r. We may suppose for the above A, B that APB = BAP, ABq = BqA,

(183.23)

for whenever; if one of these equations does not hold, then we may replace A, B by AP, B or A, Bq, respectively, where (183.14) remains satisfied and

in (183.17) one of the exponents m, n is reduced by 1. By sufficiently many substitutions of this kind we achieve our result. From (183.21), (183.23) we have C° = 1, Cq = 1. Because C 0 1 this gives

o(C) = p

(183.24)

and p = q, so that (183.17) becomes o(A) = pm, o(B) = p".

(183.25)

From (183.15), (183.16), (183.20), (183.24), (183.25) it follows that all elements of c0 may be written in the form A`BkCi

((i i=0 = O,,..

1;

k = 0, ..., p" - 1;

1 = 0, ...

,p-

1).

(183.26)

Of course, these elements need not be different. Every element (183.26) has, according to (183.22), a power of pas order, so that cfd is a p-group. In the sequel the case where 4 is the quaternion group will be disregarded. We choose the A, B above so that

m+n

(183.27)

is minimal. We then show that with a suitable choice of A, B

{A} n {B} = 1.

(183.28)

For, if it were false, then there would be a power other than 1 of A which is equal to a power of B. When we replace A by a suitable primitive element of {A}, which is evidently permissible, we can assume an equation such as

Ai''m' =BP""(o 1) (0<m'<m; 0
(183.29)

GROUPS

741

Since A and B may be interchanged with one another, we can assume that

(0<)m'5n'. For the element

A' = AB-

y,.-m.

(183.30)

(183.31)

we have

A'p'"' = C-

(183.32)

according to (183.22), (183.29).

If the right-hand side is equal to 1, then o(A') I pm'. Since, further, A'B & BA', because of (183.15) and (183.31), A may be replaced by A'. But since we have m' + n < m + n, we are contradicting (183.27). The case still remains, where the right-hand side of (183.32) is different from 1, i.e., the exponent of C is not divisible by p. This means, because of (183.30), that

p=2, m'=n'= 1.

Hence and from (183.24), (183.32),

o(C) = 2, A'2 = C, o(A) = 4. Because of (183.31) we have A'B 0 BA', thus because of (183.27) we have o(A) 14. But as m > m', we have m > 2 and so, by (183.251) ,

o(A) = 4. Because of (183.31) A' = AB-1. Since we now have (AB-1)2 = C, it follows from (183.15) that

A2 = B2, o(A) = o(B) = 4. Therefore if our assertion is false, then any two elements of order 4 of cP which do not commute have the same square. Since A', B are such elements, because A'2 = C we also have A2 = B2 = C. Hence and from (183.15) it follows that BAB'1 = A3. This means that 4 is the quaternion group. As we have excluded this case, the assertion (183.28) is true. Accordingly we have in any case 0(44) > p'+", thus, since C4 is a p-group,

p'+" 10(4). Consequently 0(c4) = pm+"+1 or 0(4) = pm+",

(183.33)

according to whether the elements (183.26) are all different or not. For the case (183.331) we may suppose that m > n, since otherwise we could interchange A, B. We see from (183.15), (183.16), (183.20), (183.24), (183.25) that the equations (183.1) are fulfilled. Since, on the other hand,

FINITE ONE-STEP NON-COMMUTATIVE STRUCTURES

742

the group defined by (183.1) is evidently of order at most p"'+'+', (183.331) means that cP is this group. For the case (183.332) the elements (183.26) are not all different. Hence and from (183.28) it follows that C is equal to an A'Bk. Because C E Z, p1 l, k. From (183.22), (183.24) it also follows that p'"-I 11 and p"-1 I k. Accordingly an equation such as C = APO-" B""-' y (183.34)

holds, where both exponents must be divisible by p: pI

p"'_ix, pn_1y.

(183.35)

We prove that after we have suitably chosen A and B one of x, y will be divisible by p. Namely if we have p I' x, y, then it follows from (183.35) that m, n >_ 2. Symmetry allows us to assume that

mznZ2. We determine a solution z of the congruence yz

x (mod p)

(183.36)

and put B' = AJ "n-"zB .

As AB'

(183.37)

B'A, (183.27) implies the necessity of o(B') z o(B). Since,

on the other hand, because of (183.22), (183.25) and n >_ 2 ,

B'P"= 1

,

it follows from (183.252) that o(B') = o(B). Accordingly we may interchange B and B'. Because of (183.15), (183.16), (183.37),C remains unaltered.

Moreover, according to (183.22), (183.34), (183.36), (183.37), -IPM -'y

B'P"-'y = CC2

-Iy-1)

If the second factor on the right is 1, i.e., C = B"'"-'y, then the assertion is true. If the above-mentioned factor is not equal to 1, then the exponent in it is not divisible by p. This means that

p = 2, m=n=2. Then we have

o(A) = 4, o(B) = 4.

(183.38)

GROUPS

743

Further, because of (183.34), we have C = A2B2 .

Hence and from (183.15) A2B2 = A-1BAB-1. Thus, because of (183.38), (AB)2 = 1. We have found an element of order two, non-interchangeable with A. But this contradicts (183.27) because of (183.382). This contradiction proves that it may be assumed in (183.34) that p I x or p I y. Then C belongs to {B} or to {A}. If A, B are interchanged, then according to (183.15), (183.16), (183.20) C changes into C-1, so that we can put C E JA}.

If a suitable primitive element of {B} is substituted for B, we can show, according to (183.21), (183.24), that even

C = A""'-'.

(183.39)

Moreover, because of (183.14) and (183.20) we must have m Z 2. We see from (183.15), (183.25), (183.39) that the equations (183.2) are satisfied. But as the group defined by (183.2) is evidently of order at most pm+", so (183.332) implies that & is exactly this group. Consequently we have

proved our assertion for the case (183.19), and have shown that in this case c4 occurs among the groups enumerated in Theorem 444. Secondly we consider the case where c4' is not contained in Z. We denote, as above, a normal divisor of cP with prime index by A. In the above proof of the assertions concerning (183.14), (183.15), (183.16) we have seen that we may also require that

A E r° a (In the previous case keeping to this additional requirement would have been superfluous and even prejudicial.) Furthermore (183.23) may also be assumed, since the proof was independent of assumption (183.19). We show that in addition to (183.231) we may also assume that A° = 1

.

For since according to (183.15) we have BAB-1 = AC, it follows from (183.16) that

BA°B-1 = ARC". This results, because of (183.23,), in

C°= 1 On the other hand, we have

.

BC0CB,

since from BC = CB and (183.15), (183.16) it would follow that cj' _

744

FINITE ONE-STEP NON-COMMUTATIVE STRUCTURES

_ {C} c ;x which contradicts our assertion. Since, accordingly, we have

4={C,B} and also C E ', we only need to take C instead of A in order to ensure that the requirements for A shall be satisfied. If we write Q for B we can express what has been said so far as follows : In c0 there are two elements A, Q such that AE

(183.40)

o(A) = p .

(183.41)

O(Q) = q",

(183.42)

f

AQ 96 QA ,

cP={A,Q} AQQ = QQA .

(183.43) (183.44)

(183.45)

[These are not independent, (183.43) and (183.44) being equivalent, while (183.45) follows from (183.40) and (183.42), since T is an Abelian normal divisor of index q in c &..l

We put

(i=0,1,...; A0=A).

Ai=Q`AQ-I

(183.46)

According to this we have (i, k = 0, 1, ...) .

QkAiQ-k = Al+k

(183.47)

From (183.45), (183.47) it follows that

A,=AI

(i=j(mod q)).

(183.48)

From (183.41), (183.46),

o(A) = p

(i = 0, 1, ...) .

(183.49)

By (183.40) and (183.46), A. E

'

(I= 0, 1, ...) .

(183.50)

Because of (183.50) the Al generate a subgroup Iwo

= {A0, AI, ... }

(183.51)

GROUPS

745

of which is thus likewise Abelian, moreover because of (183.49) it is an elementary p-group. From (183.44), (183.46) it follows that o is a normal

divisor of 4. In the sequence A0, Al, ... there is a first term Am, for which we have Am E {Ao, ..., Am_i} , and, because of (183.48), certainly

1 <_ m S q .

(183.52)

We choose the starting element A (without altering OL , Q) so that

m is minimal.

(183.53)

From the assumption it follows, because of (183.47), that Am+' E {Ai,

..., Am-1+1} f

then by complete induction

Am+' E {A0,.. ., A.-,I

for all i = 0, 1,

.... Accordingly Xo = {Ao, ..., Am_i} .

(183.54)

Further, as A0, ..., Am_1 are independent, they constitute a basis of whence

O(fo) = p'a .

(183.55)

In the following we use polynomials f(x), g(x), F(x) ... in an indeterminate x with integral coefficients, which are considered mod p, so that they are also to be understood as polynomials over the prime field F,,. To every polynomial such as

f(x)=ax'+...+ao we assign the element AX) = Aar

.. Aa,o

of moo. This correspondence is unique, because of (183.49). Moreover it is evident that o consists of all the different f(x). Evidently f(x) + g(x) = MAX)

(183.56)

0 = 1, - f(x) = (Ax))-'

(183.57)

c f (x) _ (f(x))`

(183.58)

and more generally (c E Y) .

746

FINITE ONE-STEP NON-COMMUTATIVF STRUCTURES

From (183.47) we have

(k = 0, 1, ...) .

xkf(x) = Qkj(x) Q-k

(183.59)

Thus the formula

(x - 1) f(x)

(Q, Ax)) = Q AX-) Q-1(f(x))

(183.60)

holds for the commutator. Since, further, according to (183.48), AA, -l

=

= 1, it follows that

x9-1=1.

(183.61)

Because of (183.50), (183.54), (183.55) we have

A. Am_i ... Ao = 1

(183.62)

with exponents uniquely defined mod p. Correspondingly we introduce the polynomial (183.63) F(x) = x' + cm._.1xin-1 + ... Co. Then

F(x) = 1

(183.64)

.

We prove that

(x - 1)11

(r=0, 1,...).

(183.65)

For this we denote the left-hand side by B,. Here, B0

B1=A1Aa1

1.

Then let r>- 2 and assume the assertion for smaller r. According to (183.60), QB1-1 Q-1Bi l 1 = BI

(i = 1 , 2, ...) .

Since, because of the supposition, B,_1 1, we have B,_2 Q that d4 _{B,_2,Q).

QB,_2, SO

Since the commutator B,_1 of these generators Q, Br_2 is interchangeable furthermore since dj' does not lie with B,_2 (because B,_1, Br_2 lie in in Z, it follows that Br_i, Q cannot be interchangeable. This implies that Br 0 1, so proving (183.65). Since, in FP[x], xP - 1 = (x - 1)P, it follows from (183.61), (183.65) that

p# q.

(183.66)

GROUPS

747

We prove that F(x) is irreducible. For this purpose we consider a factor decomposition (f(x), g(x) E FP[x]) F(x) = f(x)g(x) where x*_1

f(x) = xs + ds_l

+ ... + do

(s > 0) .

Because of (183.64), (183.65) we may assume that

f(x)#x- 1. It is sufficient to prove that necessarily s = m, since from this the irreducibility of F(x) follows because of (183.63). We therefore write

B = g(x),

Bi = Q'BQ-`

(i=0,1,... ; Bo = B) .

Because of the assumption we have

F(x) 0 (x - 1) g(x) .

From this, because of (183.60) and the definition of F(x) it follows that

QBQ-'B-' + 1, and so

di ={B,Q}. Since, furthermore, B E A o and so o(B) = p, B satisfies the requirements for A in (183.40) to (183.45). But we also have s

F(x) =

=0dkX'AX)

whence it follows, because of (183.56), (183.58), (183.59), (183.64), that

Bs Bds'l s-1

Bd. 0 =1

therefore, because of (183.53), s>-m, i.e., s = m. This proves the.irreducibility of F(x). Now, because of (183.56), we have a homomorphism Fp[xl+ ,.,

(f(x) -)-f (x))

,

(183.67)

the kernel of which we denote by a. Then

f(x) E a a f(x) = 1

.

(183.68)

We show that a is an ideal of FP[x]. By definition a is a submodule and so because of (183.59), (183.68) an 'ideal of Fp[x].

748

FINITE ONE-STEP NON-COMMUTATIVE STRUCTURES

0 1, a o Fp[x] follows from (183.67). FurtherSince we have more F(x) is, according to (183.64), (183.68), an element of a. From both, because of the irreducibility of F(x): a = (F(x))

(183.69)

follows.

Because of (183.61), (183.65) and (183.68),

x4-IEa, x-11a. Hence and from (183.69) owing to the irreducibility of F(x), it follows that x9 - 1 F(x)

X- 1

'

(183.70)

so that for the degree m of F(x), by § 133, Example 3,

m = o(p(mod q)).

(183.71)

We make use of the finite field F over FP such that

O(F) = p'

.

(183.72)

Since F(x) is irreducible and of degree m, we may put F = FP(w)

(183.73)

F(w) = 0 .

(183.74)

where

Because of (183.70), wQ = 1, co 0 1 also holds, thus o(w) = q .

(183.75)

is normal in d d, the elements of ci, because of (183.44), may be Since written in the form .f(x) Q' . From (183.68), (183.69),

f(x) = g(x) «f(x) = g(x) (mod F,(x)) . But, according to (183.73), (183.74),

(x) = g(x) (mod F(x)) a f(w) = g(w) .

GROUPS

749

Since, accordingly, the arbitrary element (,-r) of 2l° is completely deter. mined by the element f(co) of F, we may write the elements of X0 in the form f(x) = Pn.)-

(183.76)

Then the PP

(183.77)

(a E F)

are all the different elements of Ta. Because of (183.42), according to the above, the (a E F; i =0,...,q" - 1) (183.78) P. Q' are all the different elements of 4. Since from (183.59), (183.56) -f(X)Q' . g(x)Q" = f(x) + x'g(x)Q' +k

,

after writing the elements in the form (183.76) we get P,Q i

k - Pa+w'fQi+k PeQ-

(183.79)

Thus the elements of 44 have to be multiplied according to this rule. With regard to (183.71), (183.78), (183.79) we see that (4 is completely characterized by the determining elements p, q, n, co. The elemdht co of F according to (183.75), (183.74) is there subjected to the conditions o(co) = q, F(co) = 0. The second of these conditions may be neglected, i.e.,

we may take for co in (183.79) a quite arbitrary element of F such that o(w) = q, because from this, as will be shown immediately, we obtain groups isomorphic with 4. All the co such that o(w) = q may be given by any arbitrary one of them in the form wr (r q - 1) Now according to (183.79), P«Qri

PpQri =

Pa+wrrJ5Q('+k)

.

The comparison with (183.79) shows that the substitution of Qr for Q has the consequence that in the multiplication rule (183.79) the element to is exchanged for co". This establishes the required isomorphy. From this last proof and from (183.66), (183.71), (183.72), (183.78), (183.79) we see that c4 is in fact one of the groups given in Theorem 445. It still remains to be proved that, conversely, the groups given in Theorems

444, 445 are one step non-commutative and are not isomorphic with each other.

First of all it is evident that the quaternion group is one step noncommutative. Since, furthermore, condition (183.28) is not satisfied here, it is not isomorphic with any of the remaining groups given in Theorem 444.

750

FINITE ONE-STEP NON-COMMUTATIVE STRUCTURES

The groups defined by (183.1) and (183.2) are denoted by co, and 4 2 respectively. First of all we consider coil. It is evident that its elements may

be written as

(i==0,...,p' - 1;k=0,...,p"- 1;1=0,...,p- 1),

A'BkCI

(183.80)

If we show that equality holds here, it follows so that O(d4j) that all the different elements of coil are given by (183.80). For this purpose we consider the set G of triples i, k, I

of integers i mod p', k mod p", 1 mod p. From the full permutation group .9(S) we take the following three permutations

_

i

kl

B'

_ ik / C'-lik1+1)' = ik+1i+1)' r

k

l

(183.81)

which are evidently of orders p', p" and p. According to (1 83.81)

B'A'B" _

i

k+1 i+l

_

i

k1

i+1k+11+l+1,i+1 k1+1,

A'C'

'

and also A'C' = C'A', B'C' = C'B'. Accordingly, the equations (183.1) are satisfied by (183.81), from which by Theorem 101 the homomorphy

dil x {A', B', C) follows. Therefore it is sufficient to show that the group on the right-hand side is of order at least p"'+"''. It is not commutative and because

B'P =

I

k l)

ik+pl,

it contains the direct product {A'} ® (B'P} ® {C'}. This is of order p'+" and its index in (A', B', C) is at least p. Hence the assertion is proved. Since, according to this, the elements (183.80) are distinct, it follows also that cPil is not commutative. In order to show that c4JI is one-step noncommutative we have to demonstrate that any two non-interchangeable elements X = A`BkCI , Y = A"Bk'('c'

GROUPS

751

of 4, are already generators of this group. Here Ymay be replaced by any X'Y which, because of (183.1), allows us to restrict ourselves to the case where i

or i' is equal to 0. We can put i' = 0. The assumption and Theorem 446 implies that p,f' V. If we then replace X by a suitable XY', we also obtain k = 0. Since two elements of the form X = A'C' ,

Y = B"C" ,

where p,j' i, k', are evidently generators of d41, so the assertion has been proved.

The corresponding results may be proved for dg, as follows: What we have proved concerning co, did not necessitate the assumption m >_ n, and so remains true for all m >_ 2, n >_ 1. As APm-', C are central elements of dul, so {APm-' C-1} is a normal divisor of order p of 4g,. A glance at (183.1) and (183.2) shows that `42

zt; 41 / {APm-` C-1} ,

implying that co, is in fact of order p'"+" and one-step non-commutative. (This proof holds even for the case p°'+" = 8, although we have not yet made use of this fact.) We see from Theorem 446 that in c& the commutator group {C} is prop-

erly contained in a cyclic subgroup only when p = 2 and m = n = 1, i.e., O(c.Pg,) = 8. Since for 42, on the other hand, O(c42) > 8 and its commutator group {APm-'} is properly contained in {A}, it follows that no d%g, can be isomorphic with any c42. Both in 4,, if 0(ddl) > 8, and in 42 because of Theorem 446 the maximum of the orders of the elements is equal to p,a,(m,") Hence it is obvious that neither the d41 nor the 442 are isomorphic with each other. Finally, we consider the group defined in Theorem 445 which we denote

by cog. As opposed to the previous cases we have here to prove first the existence of cog, for cog is not defined by equations like cogs and 42, but by the

multiplication rule (183.8). We make the observation that all integers i, k may be admitted in (183.8) because o(w) = q and o(Q) = q". For the product of any three elements of c4 we have, according to (183.8), (PPQ1'PPQk) PyQI =

p.'d'3Q'-1k pYQl

-

Pn+w"/3+uat+k'Q1+k+1

P«Q'PP"u,'Qk+1 _ P.+w,(P+ruky)Qi+k+l

Hence the multiplication (183.8) is associative. Evidently P0 Q° is the unity element. The left inverse of PgQk is P_,0_k8Q-k. Thus we have proved the existence of the group jog. 25 R. - A.

752

FINITE ONE-STEP NON-COMMUTATIVE STRUCTURES

It is obvious that cQ,l is not commutative. We still have to show that c4 is generated by any two non-permutable elements

X=P«Q', Y=p'Qk. Since we may replace X and Y by XY' and X'Y, respectively, we may assume, e.g., that k = 0. Equation (183.5) and the assumption imply that q,j'i. Since X may be replaced by every primitive element of {X}, we may assume that i = 1. Since now

X=P,Q, Y=P,j, 0, we have fro ni (183.8)

where

(i=0,...,m - 1).

X'YX-l = Since

cof,

...,

O)m-I Q constitute th(e'

a basis of F, it follows that the group

P. (o E F). It then also contains P_,X = Q, con{X, Y} contains all sequently it is equal to L. Accordingly, dj is one-step non-commutative. Because of the diversity of the orders, df cannot be isomorphic to either ddI or c1762.

Since, according to (183.8),

Q'PIQ-` Pl I = P.,-1

(i=0, ..,m- 1),

the P. (a E F) make up the commutator group of c0,1, which is then of order

p'. From this it evidently follows that no two of the d3 are isomorphic. Consequently we have proved Theorems 444, 445. For finite one-step non-commutative groups cf. MILLER- MORENO (1903), SCHMIDT (1924), REDEI (1950a, 1958b). Further literature and applications see in GOLFAND (1948), IT6 (1955), REDEI (1951a, 1955-56), SUZUKI (1957).

EXAMPLE 1. From Theorem 445 it follows that there is at least one one-step non-commutative group of order pq" (p, q different prime numbers, m, n > 1) if, and only if, one of the conditions in = o(p (mod q)), n = o(q (mod p)) is satisfied. There are two such groups if, and only if, both these conditions are satisfied.

There are never more than two such groups. EXAMPLE 2. The centre of the group given in Theorem 445 is the cyclic subgroup (Q°} of order q"-1. The only centre free case is that with n = 1. In this case the commu-

tator group is the one and only one proper normal divisor. EXERCISE 1. A one-step non-commutative group (of order 8) is defined in the (excluded) case p'+" = 8 by the equations (183.2), which is, however, isomorphic with one of the groups given by (183.1). EXERCISE 2. Every finite one-step non-commutative p-group is the homomorphic image of a group defined by (183.1).

GROUPS

753

EXERCISE 3. Every finite one-step non-commutative group is a homomorphic image of the group defined by the equations ABTA-1 = Bt+1,

B;B, = Bi$,

(i,.% E J).

PROBLEM. Are there infinite one-step non-commutative groups?

§ 184.* Finite One-step Non-commutative Rings

In virtue of our general definition a one-step non-commutative ring means a non-commutative ring with only commutative proper subrings. The finite rings of such a kind are determined by the following two theorems.

As a preliminary, let us consider elements e, a from a ring R and a polynomial from 7[x]:

f(x) = c + g(x)

(c E J ; 9(x) E Y [x] = xxY [x]).

Though f(e) is then in general meaningless, the "product" f(e)a = Ca + A00, may always be defined. (The reader will easily see that this is an operator product and that this is essentially a special case of the construction in § 62, Exercise 5.) Then f(e)e is the substitution value of f(x)x for x = e. THEOREM 447. In order to determine the non-nilpotent finite one-step non-commutative rings, we define for every prime number p and every natural number m a ring

RI = {e, a}

(184.1)

p"', = 0, pa = 0, e2 = e, ea = or, ae = 0, a2 = 0;

(184.2)

by the equations

further, for any (not necessarily distinct) prime numbers p, q and natural numbers m, e, n (n < q) we define the ring

R11={e,a}

(184.3)

by the equations

e =0, pa=0, F(e)e=0, F(e)a=0, are = epa,

a2 = 0

(P = p" °e-,),

(184.4)

where F(x) (E J [x]) denotes a fixed arbitrarily chosen irreducible principal polynomial mod p of degree qe the particular choice of which is of no importance since the R11 belonging to fixed p, q, m, e, n and different F(x) are

754

FINITE ONE-STEP NON-COMMUTATIVE STRUCTURES

isomorphic. The rings RI (belonging to all the p, m), the rings opposed to these and the rings RII (belonging to all the p, q, m, e, n) then constitute a complete system of non-isomorphic non-nilpotent finite one-step noncommutative rings. There, RI is of order O(RI) = pm+l,

(184.5)

it has the basis elements e, a, where

o+(e) = p', o+(a) = p,

(184.6)

and

(ae + ba) (ce + do) = ace + ado,

(a, b, c, d E -.7),

(184.7)

for the product of two elements. RII is of order O(R1I) = p(m+1)WW ;

(184.8)

its elements may be uniquely given as

a(e) + b(e)a

(a(x) E 7[xlo, b(x) E 7[xl),

(184.9)

where a(x), b(x) run through a representative system of J [xb mod (pmx, F(x)x) and of J' [x] mod (p, F(x)), respectively. The rule

(a(e) + b(e)a) (c(e) + d(e)a) = a(e)c(e) + (a(e)d(e) + b(e)c(e)0a (a(x), c(x) E J' [x b, b(x), d(x) E J [x]) (184.10)

holds for the product of two elements. THEOREM 448. The nilpotent finite one-step non-commutative rings are the rings (184.11)

Rn1

defined by the equations

pme=0, p"o=0, er=0, a,=0, o2a = eoe = ae2,

(184.12)

eat = aoa = ate, pea = pae

and their non-commutative homomorphic images; here p is a prime numbe

and m, n, r, s are natural numbers such that m S n and r, s

2. R111 is of

order O(R1I1) =

p1+m (r- 1) s+ n (s- 1)

(184.13)

755

RINGS

with basis elementsoa - o, e'a'(01) for which

o+(Pa-aO)=p; o+(ea')=p' (1 <-i
o+(aj)=p"(1 <j<_s- 1), and here the rule

(a(er - aee) + t,j E au 'a) (b(Pa - aN) + Yk,lbkl?ka) _ (184.15)

_ -aoibio(Na - erg) + E aljbkl°'+kaj+l i,j.k.l

holds for the product of two elements where a, a,j, b, bkl denote integers and we have to understand finite sums. NOTE. The equations (184.12) are invariant with respect to substituting e, m, r for a, n, s. Hence it follows that we obtain all the different RIII, when we

prescribe that if m = n then r < s. It would be easy to show that then the RIII are not isomorphic with each other. However, if we wished to state Theorem 448 in a similar perfect form, like that of Theorem 447, then it would be necessary to solve the homomorphy problem for the rings Rill, but it seems to be a difficult problem. PROOF OF THEOREM 447. We first make these trivial observations concern-

ing one-step non-commutative rings: These rings are characterized by the fact that we can generate them by any two non-interchangeable elements.

The homomorphic images of these rings are commutative or one-step non-commutative. Every such ring is directly indecomposable, thus in the case with which we are concerned, i.e., in the finite case, it is, because of Theorem 197, necessarily a p-ring. The commutator a4B - floc of two ring elements a, j3 is always denoted by [a, 9]. We denote by R a non-nilpotent finite one-step non-commutative ring, which therefore is a p-ring. We want to prove that R is either a ring RI or a ring opposed to an RI ring or a ring R,I. We denote the radical of R by n. Because of the second part of Theorem

259, n is nilpotent. Hence n s R. According to Theorem 260 the factor ring R/n is radical free, thus, as a finite ring, semisimple. Consequently from the first structure theorem of WEDDERBURN - ARTIN (Theorem 256) the

direct decomposition R/n = Fl (D

... q)

Fk

(184.16)

follows, where the F1 are full matrix rings over (finite) fields of characteristic p.

Since the F1 are homomorphic images of R, they are partly commutative,

756

FINITE ONE-STEP NON-COMMUTATIVE STRUCTURES

partly one-step non-commutative. But as a full matrix ring of rank > 1 over a field always contains non-commutative proper subrings, so F1, . . ., Fk must be fields.

We show that k = 1. For this we suppose that k >_ 2 and denote by RI,

.

. ., Rk those subrings of R, for which

R,/n = F;

(i = 1, . . ., k).

(184.17)

For every i we take an element er from Rf such that the residue class

Ic,=ei+tt

(i= ],...,k)

(184.18)

is a generator of the field F,. Let.fi(x) be the minimal polynomial of x,.

(We suppose here that the coefficients of f(x) are integers considered mod p but fixed somehow.) The constant term of f, (x) is denoted by c,. 0, so p x c,. We put As Ici g,(x) = f ,(x) x = c,x + hi(x)

(i = 1, ..., k)-'

(184.19)

where the h,(x) are polynomials divisible by x2. Because of (184.18) and g,(rc,) = f (rq)IC; = 0 we have

gi(e) E it

(i = 1, ..., k).

(184.20)

Since the RI, ..., Rk, because k >_ 2, are proper subrings of R, they are commutative. Since, further, by (184.17), they contain it, it follows from (184.20) that

[g,(e,), e;] = 0

(i, j = 1, . . ., k).

(184.21)

On the other hand, it follows from (184.16) that icircj = 0 (i 0 j), i.e., because of (184.18), ere; E it (i 0 j). Since it is an ideal of R, it follows that ej E

Q

i,j= 1,...,k; a,b>_ 1).

Therefore, since it c R, and by the commutativity of R,: c-1 c-1 ere, = erC-1 ere; = ereier = eer er = e,er (c This, (184.19), (184.21) and x2 I hi(x) result in

0 = [crer + hi(ei), e,] = [tier, e,] = cr[er, e,]

2; i, j = ], ..., k). (i, j = 1, .

.

., k).

Since p,f'c and R is a p-ring, it follows that [ei, e;] = 0 (i, j = 1, . . ., k). Since, on the other hand, according to (184.16), (184.17). (184.18)

R={el,...,ek,tt}, R,={er,n}

(i= 1,...,k)

RINGS

757

and the R, are commutative, R is also commutative. This contradiction proves the assertion that k = 1. From this and (184.16), R/n is a finite field. We write R/n = F.

(184.22)

We take an element o of R, for which the residue class e + it is a generator of the cyclic group F*, and so, a fortiori, of the field F:

F = {e + n},

o(o (mod n)) = O(F) - 1.

(184.23)

From (184.22), (184.231) it follows that R = {o, n}.

We show that the degree of F is a prime power (>_ 1), i.e. (Q = q`; q prime, e >_ 0).

O(F) = pQ

(184.24)

Otherwise F would contain two proper subfields F1, F2 such that F = {F1, F2}.

We denote by R1, R2 the proper subrings of R determined by R1/n = F1,

R2/n = F.2 ,

whence R = {R1, R2}.

(These F,, F2, R1, R2 have nothing to do with those in (184.16), (184.17).)

Any two elements 01, Q2 of RI and R2, respectively, may be taken in the form 01 = fl(o) + P1 ,

e2 = f2(o) + v2 ,

where f1(x), f2(x) are polynomials from 7 [x]o and I. 1'2 elements from it. Since R1, R2 are commutative and contain it, so [f1(e), v2] = 0, Since, further,

(f2(o)3 v11 = 0,

[v1, v2] = 0.

[f1(e),f2(o)] = 0 trivially, it follows that [o1, 021 = 0.

We have shown that R is commutative, a contradiction by which we have proved that the degree of F is a prime power. From (184.232) and (184.24) it follows that PQ 10

= o (mod n).

(1 84.25

758

FINITE ONE-STEP NON-COMMUTATIVE STRUCTURES

Because R = {o, it) there is an element a of it such that [o, a]

0

(184.26)

whence (184.27)

R

We define recursively

(f = 1, 2, ...).

al = [e, al, ai+I = [e, (1,]

(184.28)

As it is an ideal of R, so, together with a E it, all the statements

a, al, a2i ...E IT

(184.29)

a, al, a2, ... # 0.

(184.30)

hold.

We prove that

0, therefore, a fortiori, a

According to (184.26), (184.281) al

0.

In order to prove next that a2 0 0 we assume that a2 = 0. This means according to (184.28) that

o(oa - an) - (oa - ao)o = 0, i.e.,

ao2

= 2nao - o2a.

Hence, generally,

an' = io1-lao - (i - 1)o'o

(i = 2, 3, ...),

(184.31)

since

ao2 - (i - 1)olao =

ao'+i = ao`o =

= 2iotao - io'+1a - (i - 1)o'ao = (i + 1)0'ao - io'+Ia. On the other hand from (184.25) it follows that npkQ

- o (mod n)

(k = 1, 2, ...).

As it is commutative and contains a, from this and from (184.26) it follows that [QpkQ,a]

0

(k= 1,2,...).

However, if we apply (184.31) to some i = pkQ with pkQ i = 0, then we obtain aopkQ = opkQa,

which is a contradiction. This proves that a2 write [o, [o, a]] 0.

0, for which we may also

RINGS

759

In order to complete the proof of (184.30) we suppose that a, e 0 has been proved for some i (>_ 2). Because [o, a,_1] = a, 0 0 (and (184.29)) we may apply the result now obtained to a,_1 instead of a. Then we obtain [o, [e, a,-I]] # 0, i.e., [because of (184.28)] a,+1 generally.

0. Consequently (184.30) is proved

We show that among the a defined in (184.26) there is one with

a2 = 0, pa = 0.

(184.32)

Since it is commutative and contains the elements or, a1, Oa1, pe, we have

a1 = eaa1 - aQOa1 = Oaa1 - gala = Qaal - eaa1 = 0,

pal=P[A,a]= [peg a]=0. These mean that (184.32) is satisfied for a1 instead of a. Since according to (184.28), (184.30) we also have [O, all = a2 96 0, al is an element equivalent to a so proving the assertion concerning (184.32). Later we also assume (184.32) and show that for the principal ideal

(a) (C n)ofR (a)2 = 0 and p(a) = 0.

(184.33)

From the commutativity of (a) it follows that (a)2 = (a2). According to this and (183.321), (184.331) is true. From (184.322) (183.332) follows trivially.

Here we distinguish two cases and deal first with the easier case, where among all the possible pairs e, a there is at least one such that aO = 0 or

ea = 0. We show that R is then an RI or anti-isomorphic with an RI which is just a part of the above assertion. It is sufficient to show that from the supposition ae = 0

(184.34)

it follows that R is equal to R1. Because of (184.25) and the nilpotence of it, (e - el Q)k = 0 for some k >- 1. This equation, because pQ >- 2, may be written in the form ek

25/a R. -A.

= ekf(e)

760

FINITE ONE-STEP NON-COMMUTATIVE STRUCTURES

where f(x) is a polynomial from 7 [x]0. Hence 12k = Qk(f(e)),

(i = 1, 2, ...).

If we apply this with 1 = k + 1, we obtain Qk = Q2kg(Q)

(184.35)

for some polynomial g(x) from 7 [.Y]o. Since, according to this, we have QkQI = Q2kg(o)Qi

(i = 1, 2, ...),

it follows immediately that Qkg(Q) = Q2k(g (9))2.

This and (184.34) imply that for the element o = Qkg(Q)

of R,

o2 =Q ',

(184.36)

aQ'=0.

(184.37)

We show that

Q'a 0 0.

(184.38)

If Q'a = 0, then from (184.35), (184.36) we have Qka = 0. We denote by d the least natural number with the property Qda = 0.

(184.39)

From (184.26), (184.34) it follows that Qa 0 0, therefore d other hand, from (184.34)

2. On the

a(o - QPQ) = 0.

Because of (184.25) both factors lie in n, therefore, since It is commutative, (Q - QPQ )a = 0.

Then, a fortiori,

(9d-1

-

Qd+PQ-2 )a

= 0.

Because of (184.39) and since pQ >- 2, we have Qd-1a = 0. Since this contradicts the minimal property of d, we have proved (184.38). We put a,

= e'a.

Because of (184.33), (184.37) and (184.38) we then have

a'2=0,pa'=0, Q'a'=a'00, a'go'=0.

(184.40)

RINGS

From this [o', a]

761

0, and so

R={o,

(184.41)

We now put o+(e') = pm

(184.42)

for some natural number m. From (184.37), (184.40), (184.41), (184.42) it follows that R is a homomorphic image of the ring Rl defined by (184.1) and (184.2): RI - R.

Since, further, R is not commutative, so from (184.42) it follows that pm+l 10(R).

But, on the other hand, (184.1) and (184.2) imply that every element of RI

may be written in the form ao +ba (a=0,...,pm-1;b=0,...,p- 1), thus we have O(RI) I pm+'.

From these three relations the required isomorphism of R with RI follows. For the other far more difficult case, where for all the possible pairs o, a oa

0 and ae

0,

(184.43)

we shall prove that R is then a ring R11 First, we show that o'aoj

0

(i, j = 0, 1, ...).

(184.44)

For this we suppose that (184.44) is false for some pair i, j, i.e. o'aoj = 0,

where, according to (184.43), i + j z 2. If i Z I then

o'-lae1+1= 0 since otherwise a' = o'- aej would be an element of it with

oa'=0, a'o56 0. [o,a']00, which is impossible because of the supposition in (184.43). One similarly

proves that if j z 1 o'+t aoi-1= 0.

762

FINITE ONE-STEP NON-COMMUTATIVE STRUCTURES

By induction it follows that all the equations

ease`+j_p=0

(a=0,...,i +j)

hold. On account of the definition (184.28) this gives ai+j = 0. Since this is impossible according to (184.30), we have proved (184.44). Since al is, as mentioned already, an element of it equivalent to a, it follows from (184.44) that e'aiej 0, i.e., (184.45) (1,.1 = 0, 1, ...). [e, eaoj] # 0

We denote by F(x) the minimal polynomial of the generating element + it of F [cf. (184.231)], for which therefore F(e + n) = 0;

(184.46)

with regard to (184.24), F(x) is then a principal polynomial of degree Q, which we may suppose to be an irreducible element of.7 [x] modp. Of course,

the coefficients of F(x) (apart from the leading coefficient 1) are determined mod p, but we shall later choose them suitably fixed. For the sake of brevity we write (184.47) G(x) = F(x)x. Then from (184.46) (184.48) G(e) E n.

We prove that by suitably choosing a the equation

G(e)a = 0

(184.49)

holds.

Because of the nilpotence of It the existence of a natural number k with (G(e))ka = 0

follows from (184.48). If k = 1, (184.49) follows. If k z 2 we assume the assertion for smaller k. If 0, [e, (G(e))k-'al

then a' _ (G(e))k-la is an element of it equivalent to a; since we have G(e)a' = 0, the assertion concerning (184.49) is now true. In the other case we have

0 = [e, (G(e))k-la] _ (G(e))k-'[e, a] = (G(e))k_la1.

Thus if we replace a by the (equivalent) element a1, then instead of k we get a smaller number. Because of the assumption this proves the assertion concerning (184.49).

RINGS

763

From (184.48), (184.49) and the commutativity of n it follows that aG(e) = 0.

(184.50)

Because of (184.47), (184.49), (184.50),

F(e)eae = 0,

eaeF(e) = 0.

Thus if we replace a by the element eae which, according to the case i = j = I of (184.45), is its equivalent, we even have [over and above (184.49), (184.50)]

F(e)a=0 and aF(e)=0,

(184.51)

which we now also assume. We show that e>1

(184.52)

[which, according to (184.24), means that the common degree Q of F and F(x) is at least q].

If e = 0, then O(F) = p according to (184.24), thus F(x) = x + c for an integer c (which is prime to p, though this will not be considered). Hence and from (184.51) it follows that

(e+c)a=0, a(e+c)=0, and then ea = ae. This contradiction proves (184.52). As a refinement of (184.51) we prove for elements a' (# 0) of the ideal (a) and polynomials f(x) from ..7 [x] the rule :

J(e)a' = 0 a a'f(e) = 0 a f(x) - 0 (mod p, F(x)).

(184.53)

For this we interpret the ideal (a) of R as an Y [x]-double-module such that we define for any r (E (a)) and J(x) (E -' [x]) the left- or rightoperation, respectively, by

f(x)* = f(e)r, rlx) = 'Cf(e). We then denote by I and r the annihilating left and right ideals, respectively,

of the element a', which are thus ideals of 7[x]. Both contain, because of (184.332) and (184.53), the maximal ideal (p, F(x)) of 7[x]. On the other hand, they are different from 7[x], since a' 0 0, thus both are equal to (p, F(x)). This proves (184.53).

764

FINITE ONE-STEP NON-COMMUTATIVE STRUCTURES

For the time being we write F(x) explicitly :

F(x) = xQ + a1xQ-' + ... + aQ (a,, ..., aQ E .Y).

(184.54)

We have recourse to the so-called Horner polynomials

Fi(x) = xJ + alxj-1 +

... + a,

(j = 0, ..., Q; Fo (x) = 1, FQ (x) = F(x))

(184.55)

and put at = aeQ-1 + F1(e1aeQ-2 + ... + F0_1(e")a

(i = 1, ..., Q - 1).

(184.56)

Al] these r, are elements of the ideal (a). Since according to (184.55) we have

(1= 0, ..., Q - 1).

xF,(x) = Fj+1(x) - aj+l it follows from (184.56) that

o"r, = (Fi(ed) -a

j)aeQ-1 +

i.e.,

e,"rt = rte - aF(e) + F(el)o

... + (FQ(e") -aQ)a, (i = 1, ...,

Q-

1).

The last two terms vanish because of (184.53), therefore Tie = e°fTi

(i = 1, ..., Q - 1).

(184.57)

We prove that not all the rl, ..., TQ_1 vanish. For let us assume that r1 = ... = TQ_1 = 0 and put

Q - 1),

ei = e°,

j=aeQ-j_1

(j=0,...,Q- 1).

(184.58)

(184.59)

Because of (184.56) the assumption then reads as follows :

o + F1(e)51 + ... +

0

(i = 1, ..., Q - 1).

(184.60).,

The matrix of the coefficients of this homogeneous system of linear equations is

1 FI(e)

... FQ_1(e)

1 F1(eQ_1) ... FQ_1(QQ-1)

RINGS

765

After cancelling theft column a square matrix is obtained whose determinant is denoted by Dj(I = 1, . . ., Q). From (184.60), according to CRAMER'S rule (Theorem 172), it follows that (184.61)

0.

According to (184.55) suitable multiples of the columns may be subtracted from subsequent columns in DQ so that the Vandermonde determinant, consisting of the rows

1, el'...,

eQ_2

(i = 1, ..., Q

is obtained. After a similar process DQ_1 becomes the determinant consisting of the rows

1, or, ..., PQ-$, PQ-1 + atoQ-2

(I= 1, ..., Q - 1).

Hence (§ 115, Exercise 1),

DQ_1=(ei+... + oQ_1+a1)DQ thus according to (184.59), (184.61), DQ(aw + (got + ... + QQ_1 + at)a) = 0.

(184.62)

According to the above, because of (184.58), we have:

Do = d(o) with 4(x) _

fl

(xP' - xP).

15I<j5Q-1

But as F(x) is irreducible mod p and is of degree Q, on account of Theorem 308, 4(x) # 0 (mod p, F(x)). Hence and from (184.62) it follows, according to rule (184.53), that

ao +

(e1 + . .

. + oQ_ 1 + a1)a = 0.

(

4.63)

Because of (184.54) and Theorem 311:

x + x° + ... + xP°-' + a1 = 0 (mod p, F(x)). On account of rule (184.53) and according to (184.58),

(o+Ql+ ...+oQ-1+a1)a=0, thus (184.63) becomes ae - ea = 0. This contradiction proves the assertion that not all the rl, . . . 7.Q_ I vanish. For these elements we further prove that if aI # 0 then [o, c;l

0.

(184.64)

766

FINITE ONE-STEP NON-COMMUTATIVE STRUCTURES

The validity of (184.64) is, according to (184.57), equivalent to (eP, - Q)TI # 0.

Furthermore, because r, ,-E 0 and by rule (184.53),-this is identical with

0 (mod p, F(x)).

XP, - x

Since the latter follows from (184.231), (184.46) and from Theorem 308, the assertion in (184.64) is proved. Of the TI, ..., TQ-1 we now take a T, other than 0 (0 < t < Q). Because of (184.64) this is not interchangeable with n and because of (184.56) it lies in (a), therefore, a fortiori, in n, consequently it is an element equivalent to or. When we replace the previous or by this Ti retaining for it the notation a, we may now assume that ao = e"`a.

(184.65)

[The reader will see that (184.26), (184.27), (184.29), (184.30), (184.32), (184.33), (184.44), (184.45), (184.51), (184.53) remain valid.] We prove that

q`-1 I t.

(184.66)

As a preliminary we show that R = (ok, n}

R

(184.67)

for every natural number k. From the right-hand side the left-hand side follows trivially. Now assume that the right-hand side is false, so that ok and a are interchangeable with each other. Hence it follows, because of (184.27), that ok lies in the centre of R. Furthermore, since it is commutative, IQ', u} is also commutative, i.e., properly contained in R. Hence (184.67) is true. The proof of (184.66) will be obtained by the following transformation of

(184.67). Its left-hand side, because of (184.22), means that {ek, n}/n = F. It also means that the residue class Qk + it = (o + n)k is a generator of F. This, because of (184.23.), (184.24) and Theorem 307, means that

pq` - I I' k(p9`-' - 1). On the other hand, according to (184.23), (184.24), (184.46), (184.53), (184.65) and according to § 133, Example 9, we have R

(okpr

- ek) a 0 O q xkPl - xk - 0 (mod p, F (x)) ok,a} - [nk,a] 0 q to xk(Pl-1) - 1 (mod p, F(x)) pq` - 1.}' k(p` - 1).

767

RINGS

According to this the rule (184.67) means that pq`-1

-1=

(pqe

- 1,p` - 1),

whence (184.66) follows.

Since, according to (184.24) we have Q = qe, we may write t, because

of (184.66) and 0 < t < Q, in the form t = n qe-1, where n is a natural number and n < q (exactly as in Theorem 447). Then (184.65) may be written as

Qa=QPa,

(184.68)

where P has the same meaning as in (184.4). We prove the generalization of (184.68): af(e) = (.f(Q))Pa

(Ax) E 5' [x]).

(184.69)

From (184.68) we know that af(O) = f(o' )a and hence, because of rule (184.53), (184.69) also follows. From (184.27), (184.321), (184.65) it follows that all the elements of R are expressible in the form

a(e) + b(Q)a

(a(x) E 7 [x]o, b(x) E 7[x]).

(184.70)

For this we prove the rule a(e) + b(O)a = 0 - a(e) = 0,

b(O)a = 0.

(184.71)

To do so we assume the left-hand side. It is sufficient to prove that b(O)a = 0. 0 (mod p, F(x)), whence

If this is false, then according to (184.53), b(x)

we can infer the existence of a polynomial f(x) from Y [x] such that f(x)b(x)m 1 (mod p, F(x)). Again according to (184.53), f(Q) b(O)a = a. Hence

and from the assumption we have f(e) a(e) + a = 0. Because [Q, a] 0 0 we have got a contradiction, therefore (184.71) is now proved. All the a(x) from . Y [x]o with the property a(Q) = 0 constitute an ideal of . Y [x]o, which we denote by ao. This definition is expressed in the rule a(e) = 0 c* a(x) = 0 (mod Q.

(184.72)

Hence and from (184.53), (184.71) we obtain the rule

a(e) ± b(O)a = 0 .* a(x) = 0 (mod ao), b(x) = 0 (mod p, F(x)).

(184.73)

Before determining ao we prove that from the assumption (f(x'))P * f(x) (mod p, F(x))

(f(x) E .7[x]o)

(184.74).

768

FINITE ONE-STEP NON-COMMUTATIVE STRUCTURES

follows the existence of a polynomial g(x) such that

g(f(x)) - x (mod ao)

(g(x)

According to (184.53), (184.74), (f(o))"a (184.69), it follows that [f(o), a]

0,

(184.75)

f(n)a. Hence, according to

R = {f(o), a},

E {f(o), a}.

Because of these relations, by similar reasoning to that by which we obtain ed (184.70), we obtain the equation o = g(f f(°)) + h(e)a

(g(x) E 7 [x]o, h(x) E 7 [x]).

In this equation the last term must vanish according to the rule (184.71); then from this and from rule (184.72) the validity of (184.75) follows. We now want to determine the ideal ao; this work will be somewhat bothersome. According to the third paragraph of § 123 we have ao = xa

(184.76)

for some ideal a of 'Y [x]. Since R is a p-ring, 0+ 0) = Pm

(184.77)

for some natural number m. Then p'"o = 0, thus from (184.53) and (184.76) it follows first that pmx E ao and then that

p E a.

(184.78)

From (184.47), (184.48) and the nilpotence of it it follows that (F(o)g)" = 0 for a natural number u. From (184.72) (F(x)x)" E ao, therefore according to (184.76) afortiori (184.79) (F(x)x)" E a.

On the other hand, we can show that

a c (p, F(x)).

(184.80)

If a(x) E a, then according to (184.76), a(x)xE ao therefore, according to (184.72), a(p)e = 0, consequently a(o)oa = 0. Hence it follows, according to (184.53), that a(x) x E (p, F(x)), a(x) E (P, F(x)). This proves (184.80).

RINGS

769

Because of (184.79) a is a primitive ideal of 7 [x], which according to (184.78) contains a power (> 1) of p. A simple application of the KRONECKER - HENSEL theorem (Theorem 286) shows that

a = (p`',pel GI(x), ...,pek-IGk-I(x), Gk(x))

(k ? 1),

(184.81)

where G.(x) is a principal polynomial of degree nj from _'Y [x] and

eo>...>ek_1>O; 0
(i = 2, ..., k).

(184.82)

Hence and from (184.79) we get at once (F(x)x)" __ 0 (mod p, GK(x)). Since therein F(x) and x are irreducible principal polynomials mod p of different degree, it follows that Gk(x) _- (F(x))°x'(mod p)

(184.83)

for non-negative integers v, w. According to (184.81), (184.82), (184.83) we have

a c (p,(F(x))°x').

(184.84)

Hence and from (184.80), because a # 7[x] we have V

We prove that

v=1 and w=0.

(184.85) (184.86)

If (184.861) is false, then, according to (184.85), v > 2, whence from (184.84) a c (p, (F(x))2), so that according to (184.76) ao c (px, (F(x))zx) .

(184.87)

We take a polynomial a(x) with

(a(x))' * a(x) (mod p, F(x)), a(x) E.' [xb

(184.88)

[because of (184.232) we could take, e.g., a(x) = x], further a polynomial b(x) with

a'(x) + F'(x)b(x) = 0 (mod p, F(x)), b (x) E 7 No

(184.89)

and put

f(x) = a(x) + F(x) b(x).

(184.90)

770

FINITE ONE-STEP NON-COMMUTATIVE STRUCTURES

This, because of (184.88), satisfies condition (184.74). Thus there exists a

polynomial g(x) satisfying (184.75), for which, because of (184.87), a fortiori g(f(x)) = x (mod px, (F(x))-x). After differentiation we get

f'(x) g'(f(x)) = I (mod p, F(x)). But, on the other hand, according to (184.89), (184.90) we have

f'(x) = a'(x) + F'(-x) b(x) = 0 (mod p, F(x)). These two congruences are contradictory, so proving (184.861). In order to prove (184.862) notice that because of (184.232) (cf. § 133, Example 9)

x2tP-1t *I (mod p, F(x)), i.e., (184.74) is satisfied by f(x) = x2. According to (184.75) there is thus a polynomial g(x) with g(x2) = x (mod ao),

g(x) E -7 [x]o .

Hence by (184.76), (184.84), g(x2) = x (mod px, x`°+1), so that (184.862) follows.

According to (184.83), (184.86) we have Gk(x) = F(x) (mod p). But since

F(x) is only determined mod p, so after a suitable choice we have even Gk(x) = F(x).

(184.91)

We show that necessarily k = 1. For this purpose we assume that k ? 2. Then according to (184.82) and (184.91) Gk_I(x) is a non-constant principal polynomial such that F(x) = 0 (mod p, Gk - 1(x)). But since F(x) is irreducible

mod p, we have a contradiction and so we have proved that k = 1. According to this and (184.81), (184.91), a = (pep, F(x)), therefore from (184.76) ao = (pe°x, F(x)x).

On the other hand, from (184.72), (184.77) it follows at once that m is the least natural number such that pmx E no, thus necessarily eo = in. This means that ao = (p'"x, F(x)x);

this is the desired determination of ao.

(184.92)

RINGS

771

Because of (184.72) and (184.92), pmQ = 0 and F(o)o = 0. Hence and from (184.27), (184.32), (184.511) and (184.68) it follows that R is a homomorphic image of the ring R11 [defined by (184.3) and (184.4)]: R11 - R.

(184.93)

From (184.70), (184.73) and (184.92) we have (184.94)

O(R) = p(m+1)¢

where it is taken into consideration that F(x) is of degree Q = qe. On the other hand, it follows from (184.3) and (184.45,6) that the elements of R11 may be written in the form (184.9), moreover because of (184.41,2.3.4) it is clear that every element of R11 arises (at least once), when a(x), b(x) are restricted to the representative systems listed according to (184.9), so that 0(RI1) <

p(m+l)qc.

(184.95)

From (184.94) and (184.95) it follows that the homomorphy (184.93) is an isomorphy, as was asserted for this case.

Summarizing, we have so far proved the assertion of Theorem 447 that every non-nilpotent finite ring is either a ring R1 or a ring opposed to a ring R1 or a ring R11.

In order to prove the remaining assertions of Theorem 447, let us consider first the ring R1 [defined by (184.1) and (184.2)]. We have recourse to the (free) ring SJ'I = {x, y}

(184.96)

with the free generators x, y. (We assume the subring {x} of 91 to be identified

with 7 [x]0.) Because of (184.1) and (184.2), the isomorphism R1 czz; 91/r (O -- x + r, a -- y + r)

(184.97)

holds for the ideal r of R1, where

r = (p'"x, py, x2 - x, xy - y, yx, y) It is obvious that the elements

ax + by (a=0,...,pm- 1; b=0,...,p- 1)

(184.98)

represent all the residue classes ofd{ mod r. We show that here each class will be represented only once. It is sufficient to prove that in particular

only for a=b=0.

ax + by E

(184.99)

FINITE ONE-STEP NON-COMMUTATIVE STRUCTURES

772

By the substitution x = 1, y = 0 the relation a E (p') is obtaineu from (184.99), where on the right is meant an ideal of 7, so that a = 0. According to this, (184.99) is reduced to by E r. If we carry through the substitution

x=E11=

f0

y=E12=1o

01'

off,

where we have matrix units over the prime field of characteristic p, then we obtain b E12 = 0, thus b = 0. Consequently we have proved that every residue class mod t of R is represented exactly once by (184.98). Since the number of these representatives is prntl, the assertion (184.5) follows from (184.97). As a result of (184.97) the representatives (184.98) themselves give by

means of the substitution x = e, y = a all the different elements of R1. This means that e, a are basis elements of RI and (184.6) holds. The validity of (184.7) follows trivially from (184.23.4.5,6) Because of (184.62), a is diffej ent from 0, thus RI according to (184.24,5)

is not commutative. In order to prove that RI is one-step non-commutative, it is thus sufficient to show that RI is generated by any two noninterchangeable elements

a=aee+ba, j9=cO+da

(a,b,c,dEJ).

By hypothesis, [a, fi] = ap - fSa 0 0. This, according to (184.7), means

the same as (ad - bc)a # 0, whence according to (184.62) it follows that p f' ad - bc. This means, by (184.6), that a, fi are basis elements, indeed, generators of RI. Therefore RI is one-step non-commutative. According to (184.23) and (184.61), 0 is an idempotent element other than 0, so that RI is not nilpotent. Because of (184.5), p and m are invariants of RI. Hence it follows that none of the RI belonging to the pairs p, m are isomorphic.

RI has, according to (184.7), the p left annihilators ba (b = 0, ..., p - 1), but only the right annihilator 0. This, together with what has

been said above, means that among all the R, and the rings opposed to them, none is isomorphic. Consequently we have proved the assertions of Theorem 447 referring to the rings R,. We still have to consider the rings RII [defined by (184.3) and (184.4)].

In order to prove (184.8) to (184.10) we will construct a skew product of the factor rings A

=,Y[x]o/(pn,X'F(x)x),

F = Y[x]/(p, F(x)).

(184.100)

Here,

0(A) =

p,nq'

,

O(F) = pqe

,

(184.101)

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furthermore F is a field, since (p, F(x)) is a maximal ideal of 7 [x]. We denote by R' the set of all the (ordered) pairs (a, fl)

(184.102)

((x E A, fi E F)

and adopt the convention that (a(x), b(x))'

(a(x) E,7 [x]o, b(x) E 7 [x])

(184.103)

shall mean the element (184.102) of R' formed from the residue classes

a = a(x) + (pmx, F(x)x), f = b(x) + (p, F(x)).

(184.104)

Then we define addition and multiplication in R' by (a(x), b(x))' + (c(x), d (x))' =

= (a(x) + c(x), b(x) + d(x))',

(184.105)

(a(x), b(x))' (c(x), d(x))' =

= (a(x) c(x), a(x) d(x), + b(x) (c(x))P)',

(184.106)

where a(x), c(x) and b(x), d(x) are polynomials of Y [x]o and 3' [x], respectively, and P denotes the same power of p as in (184.4). Because of (184.100) it is evident that both compositions (184.105), (184.106) are unique. Moreover, because of (184.102), O(R') = O(A) O(F).

(184.107)

Next we show that R' is a ring. Because of (184.105) R' evidently constitutes a module. In order to verify

the associativity of the multiplication, we denote by A and B the first and second factor, respectively, on the left-hand side of (184.106), further we take a third element C = (f(x), g(x))' of R' where f(x) E,7 [x]o, g (x) E E 3[x]. Then, according to (184.106), (with abbreviated notations)

AB = (ac, ad + bc1')',

BC = (cf, eg + df P)',

(AB)C = (acf, acg + adfP+ bc"f P)' = A(BC). This proves the associativity of multiplication. The distributivity is seen at once from (184.105) and (184.106) since (u(x))P + (v(x))"

(u(x) + v(x))P (mod p)

Accordingly R' is a ring.

(u(x), v(x) E

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FINITE ONE-STEP NON-COMMUTATIVE STRUCTURES

We show that R' is generated by the two special elements (x, 0)', a, _ (0, 1),

(184.108)

i.e. that R'

o' a'),

(184.109)

and show that the arbitrary element (184.103) of R' may be expressed in the form (a(x), b(x))' = a(o) + b(e)a. (184.110)

Namely, if b(x) lies even in 7[x]o, then, according to (184.105) and (184.106), the left-hand side of (184.110) is equal to (a(x), 0)' + (b(x), 0)'(0,1)'. Again, from (184.105), (184.106) and (184.1081),

(u(x), 0)' = u((x, 0)') = u(Q)

(u(x) E .7[x]o).

According to this and (184.1082) the previous expression becomes the right-

hand side of equation (184.110), which now proves this. In the general case we may put b(x) = bo(x) + k

(bo(x) E 7[xlo, k E 7).

According to the previous special case of (184.110) and (184.1082) (a(x), b(x))' _ (a(x), bo(x))' + k(0, 1)' =

= a(o) + bo(o) a + ka = a(o) + b(o)a. This completes the proof of (184.110). For the elements (184.108), from the definition of (184.103) and from (184.105), (184.106), (184.110) we have the following:

pm = (Pmx, 0)' = 0, P a' = (0, P)' = 0,

F(o') o' _ (F(x)x, 0)' = 0, F(o') a' _ (0, F(x))' = 0,

UV _ (0, 1)' (x, 0)' _ (0, x")' _ (x", 0)' (0, 1)' = o' Pa

a'2=(0,1)'2=(0,0)'=0. Since these equations agree formally with (184.4), the homomorphy relation as in (184.93) with R' instead of R follows from (184.3) and (184.109). The order relation similar to (184.94) likewise holds with R'

instead of R because of (184.101) and (184.107). From these two relations R' which is no longer

and from (184.95) we obtain (the isomorphy R

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of interest, and at the same time) the validity of (184.95) with "_" instead i.e., that of (184.8) and, furthermore the validity of the part of of the assertion about (184.9) which remained unproved before (184.96), (namely that we have uniqueness as asserted in (184.9)). The rule (184.10) follows at once from (184.42,5,6).

For the proof of the remaining assertions about R11 we need some preparation.

The following rules hold for the elements of RII with the notations used in (184.9):

a(o) + b(o) a = 0 = a(o) = 0, b(o)a = 0, a(o) = 0

(184.111)

0 (mod p'" x, F(x) x),

a(x)

(184.112)

b(o)a = 0 -tz> b(x) = 0 (mod p, F(x)).

(184.113)

These follow from the uniqueness proved for (184.9). Since (p, F(x)) is a prime ideal of ..9'[x], we have

f(o)a 0 0, g(o) a 0 0 =f(o)g(o) a

(184.114)

0,

for polynomials f(x), g(x) from.Y[x] because of (184.113). For the commutator of two arbitrary elements a = a(o) + b(e) a, fl = c(o) + d(o)a (a(x), c(x) E _7 [x]o; b(x), d(x) e 7 [x])

(184.115)

of R11 we have from (184.10) that

[a, R] == ((a(e) - a(o)' (d(e) + b(o)(c(o)' - c(o)))a,

(184.116)

whence according to (184.113)

[a, j9] = 0 p (a(x)' - a(x))d(x) = b(x)(c(x)1 - c(x))

(mod p, F(x)). (184.117)

We now want to prove that R11 is one-step non-commutative. According

to (184.45) we have [o, a] = (o - o")a. But according to Theorem 308

x° - x

0 (mod p, F(x)).

From these and from (184.113) [o, a] 0 0, so that RII is not commutative. We consider two arbitrary non-interchangeable elements a, 9 of RI1 in the form (184.115), in order to show that then necessarily R1, = {a, #1,

(184.118)

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FINITE ONE-STEP NON-COMMUTATIVE STRUCTURES

whereby we shall have proved that R11 is one-step non-commutative. We carry out the proof so that we first reduce the assertion by several steps to a simpler case. By interchanging x and j9 only the two sides of the congruence in (184.117) are interchanged. As the order of succession of these elements is immaterial and this congruence itself, because of the assumption [x, fl] 0 0, is not satisfied [thus at least one side of it is # 0 (mod p, F(x))], so we may assume that

a(x)' * a(x) (mod p, F(x)).

(184.119)

Because of (184.119) a polynomial f(x) exists such that f(a(x)) = x (mod p, F(x)),

f(x) E 7 [x]o

.

(184.120)

For, on account of Theorem 310 the assertion is in the first place true with 7[x] instead of 7[x]o, but then obviously (184.1202), too, may be required. As a first reduction step we show that in the proof of (184.118) we may

restrict ourselves to the case c(x) = 0 (mod p, F(x)).

(184.121)

For, we deduce from (184.1201) the existence of a polynomial g(x) _

= c(f(x)) such that g(a(x)) = c(x) (modp, F(x)).

(184.122)

With this g (x) we form NI = fB - g(x).

(184.123)

We write this element of RI, [as in (184.1152)] in the form

f1 = cllo) + di(ct)a

(c1 (x) E ,Y[x]o ,

d1(x) E 7'[x]).

(184.124)

Since according to (184.123) {x, #1} = {x, 13}, the assertion (184.118) means the same as R11 = {x, #1;. Since, moreover, from (184.123) and 0 evidently follows, it is sufficient from [x, j3] 0 the inequality [r, NI] to show that (184.121) is satisfied for c1(x) instead of c(x). To do this we bear in mind that it follows from (184.1151) that g(x) = g(a(O)) (mod o) (in R1i). Similarly the congruence g(a(o)) = c(s) (mod pp, a)

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follows from (184.43) and (184.122) [because g(a(x)), c(x) E J [x]6]. Because of (184.1152) and (184.123), Nl =_

0 (mod po,a).

According to this and (184.42,5,2), Pr = 0. Because of (184.46) and (184.124)

we can replace this by c1(o)a = 0. Since, by (184.113), this means that (184.121) is satisfied by cl(x) instead of c(x), we may in fact restrict ourselves to the case (184.121). Because [a,#] 0 0 it follows from (184.117) and (184.121) that

d(x) * 0 (mod p, F(x)).

(184.125)

As a second reduction we show that we may restrict ourselves [in the proof of (184.118)] to the case d(x) = 1.

(184.126)

a' = d(o)a.

(184.127)

For this purpose we put Because of (184.125) there is a polynomial u(x) (E,7 [x]) with u(x)d(x) __ 1 (mod p, F(x)). According to (184.42,4), u(o)d(o)a = a. Hence and from (184.127) it follows that

a = u(o)a'.

(184.128)

Because of (184.3), (184.127), (184.128) we have R11 = {o, a} = {o, a'}. Again, it is evident that the equations (184.42.4,5,() are satisfied for a' instead

of a. Both together mean that o, a' and o, a are equivalent pairs of generators of R11. Since (184.115), according to (184.127) and (184.128), may be represented in the form a = a(o) + b(o)u(o)a', /1 = c(o) + a',

this means [after comparison with (184.115)] that we may in fact restrict ourselves to the case (184.126). Accordingly, (184.115) transforms [with unaltered a(x), c(x), but with new b(x)] into (184.129) a = a(e) + b(o)a, fl = c(o) + a.

As a final reduction step [for the proof of (184.118)] we show that we may restrict ourselves to the case b(x) = 0.

(184.130)

778

FINITE ONE-STEP NON-COMMUTATIVE STRUCTURES

With this end in view we put

al =

(184.131)

with some polynomial h(x) from J [x]o, to be soon defined more explicitly below. Since from (184.131) [for any h(x)] the relation {al, #19 {a, #1 follows, it will be sufficient to prove our assertion (184.118) for 0c1

instead of a. In order to choose h(x) suitably we first transform (184.13 1). According to (184.1291) we have h(a) = h(a(Q) -}- b(Q) a) .

Hence, according to (184.45,6), h(a) = h(a(e)) + v(o)a

(184.132)

for some v(x) from r [x]. Because of (184.42, 4) and (184.121), c(o)o = 0. Thus, according to (184.10), (184.1292), (184.132) h(a)/3 = h(a(o))c(p) + h(a(o))o.

From this and from (184.1291), (184.131) we obtain

a.1= a(o) - h(a(o))c(o) + (b(o) - h(a(e))a.

(184.133)

We now choose h(x) so that the part multiplied by a on the right-hand side vanishes. Since this requirement, according to (184.113), means that b(x) - h(a(x)) = 0 (mod p, F(x)), it is, according to (184.120), sufficient if we take h(x) = b(f(x)). With this choice of h(x), (184.133) becomes (184.134) al = a1(o) for some polynomial a1(x) from -X[x]o, for which, according to (184.121),:

al(x) = a(x) (mod p, F(x)).

(184.135)

Comparison of (184.134) with (184.129) shows that we may in fact restrict ourselves to the case (184.130). When we also take (184.135) into consideration, the result obtained means that in the proof of (184.118) for [a, 14] 0 0 only the case where a = a(o), 9 = c(o) + a

(184.136)

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remains, where (184.119) retains its validity for the "new" a(x); since # has not been altered in the last reduction step, (184.121) also holds unaltered. (Because of (184.118), (184.119) and (184.121), [x, #] 96 0 is also valid

for (184.136), but this is not required in the part of the proof to follow.) Now it is easy to prove directly (viz., without further reduction) the assertion (184.118) for (184.136). It is sufficient to show that e E {a}, for, from this and from (184.1362), it follows that a E {a #}, thus {e, a} C C {a, #} which, because of (184.3), is equal to (184.118). The remaining assertion that p E {a} requires the existence of a polynomial k(x) from 5' [x]o such that e = k(a). Because of (184.1361) and the rule (184.112) we have thus only to verify the possibility of fulfilling the conditions x =_ k(a(x)) (mod p'", F(x)),

k(x) E

[x]o .

(184.137)

(This is sufficient since it already follows, from (184.1362) and the fact that a(x) E 7 [x]0, that x I k(a(x)).) Because of (184.120) the condition (184.137) without the exponent m is to be satisfied. Suppose that (184.137) holds for an exponent I (= 1, .. m - 1) instead of m. Then we have

x =_ k(a(x)) + p'w(x) (mod F(x)) for some polynomial w(x) from.7 [x]o, thus according to (184.120)

x = k(a(x)) + p'(w(f(a(x)))) (mod p1+1, F(x)).

This means that (184.137) can be satisfied for I + 1 instead of m, thus by induction also (184.137) itself. This completes the proof that R11 is one-step non-commutative.

From (184.112) it follows that all the Q' (i = 1, 2, ...) are different from 0, so that R11 is not nilpotent. We now turn to the assertion of Theorem 447 that R11 does not essentially

depend on the special choice of F(x), i.e. we obtain a ring isomorphic with it when F(x) is replaced by another, irreducible mod p principal poly-

nomial F1(x) (E 7[x]) of the same degree q0. It is sufficient to verify the existence of an element jol in R11 with R11 = {pig a},

(184.138)

for which the equations (184.41,2..,5) are satisfied with of and F1 instead of a and F. As a preliminary we show that there is a polynomial a(x) with

F(a(x)) = 0 (mod p', F(x)),

a(x) E,7[x]0

(184.139)

780

FINITE ONE-STEP NON-COMMUTATIVE. STRUCTURES

and note that for this a(x) the relations

F;(a(x)) * 0 (mod p, F(x)),

(184.140)

a(x)" * a(x) (mod p, F(x)) (184.141) necessarily hold. Without the exponent m (184.139,) can be satisfied, according to Theorem 312, with an a(x) from J' [x]. Evidently (184.1392) may also be prescribed. Further from (184.139), according to the same theorem, (184.140) and (184.141) also follow. Suppose that (184.139) is satisfied for an exponent !(= 1, . . ., m - 1) instead of m. We show that then

F,(a(x) + plb(x)) = 0 (modp'+', F(x))

has a solution b(x) (E7[x]o), so proving our statement by induction. The last congruence is identical with

Fl(a(x)) + p'F,(a(x))b(x) = 0 (modp'+I, F(x)). Because of the supposition and because of (184.140) this has a solution of the required type. Consequently the assertion is proved. We now form with the help of a polynomial a(x) from (184.139) the element

ei = a(e) of R,I and show that for this element (184.138) and the properties postulated there hold. Since, according to (184.139), we have

Fi(a(x)) a(x) = 0 (mod p'"x, F(x)x) .

it follows from the rule (184.112) that

FI(a(e)) a(g) = 0 . From (184.139) itself

FI(a(e)) or = 0 according to the rule (184.113). According to (184.10)

aa(e) = a(e)" a. From (184.41) pa(o) = 0. These four equations mean that the equations (184-41,3,4,5) are satisfied by e, and F, instead of o and F. Furthermore it follows from (184.113) and from (184.141) that a(g)pcr 0 a(g)o r.

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RINGS

Thus according to what has been said above a(e)a 0 aa(e), i.e., [a(e), a] 0. Hence (184.138) follows. This proves the assertion of Theorem 447 concerning the neutrality of the choice of F(x). We now prove that p, q, m, e, n are invariants of R11; hence it will follow that of the RII given in Theorem 447 none are isomorphic. According to what has been proved concerning (184.9), RII has the basis elements e`

(i= 1,. ..,qe),

e'a(i=0,...,qe- 1),

and the first or last of these have the additive order p' or p, respectively. Hence it follows that p, m, q, e are invariants of RII. The invariance of n has still to be demonstrated. On account of (184.3) and (184.45.6) it is sufficient to prove the following: If e', a', t are two elements of R and an integer, respectively, with

R _ f e', a' }

(184.142)

,

a'2 = 0,

aq' = e' 'a'

(184.143)

(184.144)

,

0
(184.145)

then t = nq` -1 .

(184.146)

For the proof we write e' = a(e) + b(Lo)g, a' = c(e) + d(e)a (a(x), c(x) E -'Y [xb ; b(r), d(x) E ..-' [x]) .

(184.147)

From (184.46) and (184.1472) it follows that a'a = c(e)a. According to this, (184.10) and again (184.1472), a12 or = c(o)2a. Because of (184.143) this gives c(e)2 a = 0, whence, from the rule (184.114) even

c(e)a = 0.

(184.148)

By (184.142), [o; a'] + 0. From formula (184.116) and (184.147), (184.148) we then have (184.149) (a(e) - a(e)") d(e)a 0 . But, from (184.10), (184.147), and (184.148) we obviously have

a'o' = a(e)c(e) + a(e)"d(e)a, o °`a' = a(e)"c(e) + a(e), d(o)a

.

782

FINITE ONE-STEP NON-COMMUTATIVE STRUCTURES

According to (184.144) and (184.111) we therefore have

(a(o)' - a(o)") d(e)o' = 0 . This and (184.149) give, according to rule (184.114):

(a(Q) - a(P)P) r

0, (a(P)" - a(o)°`) cr = 0.

These imply, by rule (184.113), the same as

a(x)" * a(x) (mod p, F(x))

,

a(x)°` - a(x)P (mod p, F(x))

.

Consequently, if on account of (184.1002) we take a generator x of F, where F(K) = 0, then it follows that (a(K)) P

a (K) ,

(a(K))" = (a(K))P .

Because of (184.1012) it follows from the first that a(K) is also a generator

of F, and from the latter that t = n qn-I (mod qe). Because of (184.145), it then follows that the equality t = n qe-I, i.e., (184.146) holds. We have proved that p, q, m, e, n are invariants of RI,, thus in the RII given in Theorem

447 there are no isomorphic rings. When we have proved that no ring RII is isomorphic with a ring RI or with one of the rings opposed to the R1, we shall have proved Theorem 447. The truth of this assertion follows from the fact that RI has left annihilators

other than 0, as noted above, while obviously there are no left or right annihilators different from 0 in RII. Consequently the proof of Theorem 447 is now completed. PROOF OF THEOREM 448. This needs some preparatory work. LEMMA 1. If for a ring R and a subring S of R

R = R2 + S

,

(184.150)

then

R = Rk + S

(k = 2, 3, ...) .

(184.151)

Therefore, for a nilpotent ring R, (184.150) can be satisfied only by S = R. Let us assume (184.151) for some k (>_ 2). Hence and from (184.150)

R=(R2+S)k+S c Rk+I+Sk+S= Rk+I+S. This implies that (184.151) holds for k + t instead of k, and so generally. LEMMA 2. In a minimal system of generators 01, w2, ... of a nilpotent ring R, no co; may be replaced by an w; coy or coj w; (j arbitrary).

Let S denote the (proper) subring {co, ..., co;_,, w;+I.... } of R. If the assertion is false, we have R = {S, w; ww} or R = {S, w;coif'

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for some i, j. Then the subring R2 + S of R contains a system of generators of R, whence (184.150) follows. Therefore, according to Lemma 1, S = R. This contradiction proves Lemma 2. In order to prove Theorem 448, let us first consider a nilpotent finite

one-step non-commutative ring R. We have to prove that this is a

homomorphic image of a ring Rill. Because of the assumption, R is necessarily a p-ring, therefore there are two elements e, a in it with i.e., {e, a} 56 0.

(184.152)

We put

o+(e) = P', o+(q)=p", o(e) = r , o(a) = s ,

(184.153)

where m, n z 1 and r, s z 2 necessarily hold. Because of the nilpotence of R, r and s denote the least natural numbers for which e` = 0 and a' = 0, respec-

tively. Of all the possible pairs e, a we choose one for which m + n is minimal. ThenPe, a-is not a system of generators of R, and so we have p [e, a] = [Pe, a] = 0, i.e., pea = Palo .

(184.154)

In all our assumptions the order of succession of e, a is immaterial, therefore

we can also assume that m 5 n. Because of (184.1521), Lemma 2 implies that all the four products o2, ea, ae, a2 are interchangeable with both a and a. Therefore e2a = eae = ao2

,

eat = aea =ate .

These equations and (184.1521), (184.153), (184.154) imply that R is a homomorphic image of the ring R1; [defined by (184.11) and (184.12)], as stated. Conversely, let us consider the ring RIII. In order to prove assertions (184.13) to (184.15), we have again recourse to the free ring 91 _ {x, y}, introduced in (184.96). For its ideal r = (Pmx, P"y, x', y', x2y - xyx, xyx - Yx9, xy2 - YxY, YxY - y2x,

p(xy - yx))

(184.155)

we have according to (184.11) and (184.12) the isomorphism

Rlll : SR/r(e-x+r,a-.y+r). 26 R.-A.

(184.156)

784

FINITE ONE-STEP NON-COMMUTATIVE STRUCTURES

Because of (184.155) and since m < n, the elements

a =0,...,p -1, au=0,...,p'-1;i>0

ayx + Y_ a;jx'y'

.i

(184.157)

a01=0,...,p^-1

represent all the residue classes mod r of !1, where we have to sum over the i, j for which

0
r-1; 0<_j<_s-1; i+j>1.

We show that each class is represented here only once. It is sufficient to prove that (184.157) lies in r only if a and all the a,1 vanish. In the following proof we shall use the concepts "degree", "homogeneous"

and "homogeneous component" in the free ring 91 with similar meaning as in polynomial rings. We assume that (184.157) lies in r. Since r, s Z 2 and because of (184.155) r S (px, py, x2, xy, y2). Consequently it follows by hypothesis that ayx + a10x + aoly E (px, py, x2, xy, y2) .

Since the ideal on the right-hand side has homogeneous generators, it must contain (cf. § 67, Examples 6, 7) the homogeneous components of the left-

hand side, and, in particular, ayx E (px, py, x2, xy, y2)

From this it follows that ayx E (px, py), thus p I a. But since a only takes the values 0, ..., p - 1, we have

a=0.

(184.158)

We still have to show that all the a1 are equal to 0. We suppose that this is false and denote by u (0 < u <_ s - 1) the largest integer with the property

that all the au such that j < u are equal to 0. If (184.157) is multiplied from the right by ys-1-", then the product so obtained lies in the ideal r, thus, because of (184.158) Y_

ar,.x'ys-1-"+j E r

.

e..i

Here, according to the definition of u, the terms for which j < u disappear. Since, further, according to (184.155) we have y' E r, the terms for which s - 1 - u + j > s also disappear, i.e., the terms for which j > u may be omitted. After this only the terms for which j = u remain : a, x'ys-1 E r .

RINGS

785

(According as u = 0 or u > 0, one has to sum here over i = 1, ...,p - 1 or i = 0, . . ., p - 1, respectively.) The terms of the left-hand side are of different degree. But t has, according to (184.155), only homogeneous generators, so that the mentioned terms must each lie in t. Among these terms

there is, according to the definition of u, one which is different from 0. Suppose that ar" 0. According to what has been said we have armxrys-I E t .

Because of (184.155), army 'YS -1 E (pmx, p"y, xr, .3's, xy - Yx) .

since (after adding xy - yx) the cancelled generators of t lie in the principal

ideal (xy - yx) . Because i < r and s - i < s it follows that a;" xry"-1 E (pmx, p"Y, xy - yx)

(184.159)

.

The substitution y = x results in a,,,x`+s'' E (P-x, P"x) , where we now have an ideal of .7 [xb on the right. Because m _:!9 n it follows that Pm

a:"

But since for i > 0 a;,, only takes the values 1, ..., p-1, because of (184.157) and since a1 # 0, i = 0 (and u > 0). In the remaining case i = 0, (184.159) reads as follows: aouys-1 E (pmx, p"y, xy - yx)

After the substitution x = 0, y = x we obtain

.

E (p"x), thus

p' I a... But since, because of (184.157) and because ao, 0 0, aou only takes the values

1, ...,p"-1, we. have proved by this contradiction that the elements (184.157) represent all the different residue classes of . mod r. The number of these representatives (184.157) is 1

,

whence, because of (184.156) the validity of (184.13) follows. From (184.11), (184.123, 4, s.6) it follows at once that the elements 'Oo' -

eY, mentioned before (184.14), are generators of the module Ri1I+. On the other hand, from (184.1 21, 2, 7) and m < n it follows that for these elements equations (184.14) hold with "<-" instead of °°_". But since the

786

FINITE ONE-STEP NON-COMMUTATIVE STRUCTURES

product of the right-hand sides of equations (184.14) is exactly the right-hand side of equation (184.13) as already proved, we see at once that the generating elements mentioned constitute a basis of R111, i.e., of and all the equations (184. 14) are true.

If we multiply oa - aO from the left or right by a or a, then according to (184.125,6) we always obtain 0. Hence (184.15) follows.

Because of (184.125,6) the module of R1, (k 3) has the generators o'a"(i + j > k) . Hence and from (184.123,4) we have Rii1S-1 = 0 and so RI,I is nilpotent. Lastly we wish to show that the ring RIII is one-step non-commutative. Since the element ga - op, according to (184.141), is different from 0, RIII is not commutative. Therefore it is sufficient to prove that R111 is generated by any two non-interchangeable elements a, fl. For the proof we take these two elements a, j3 as the first or second factor, respectively, on the left-hand side of (184.15). It then follows from (184.15) that [a,

P]=ap-fa=la0a0(0a-ago). bl,

1

Because of (184.141) and the assumption that [a,#] 0 0, the determinant on the right-hand side is not divisible by p. Therefore a and a may be written as a sum of two elements, of which one belongs to the submodule of R11, generated by a and 8, the other to the ring RII1. Since; according to (184.11), p, a are generators of R111, we obtain R111=S+ RI11,

where S denotes the subring {a, f} of R111. Because of the nilpotence of Rm, if we apply Lemma 1, we obtain S = 8111. This means that R111 is generated

by a and f, i.e., it is one-step non-commutative. Consequently Theorem 448 is proved. Cf. R>DEI (1957).

EXAMPLE. The ordinary quaternion field is a one-step non-commutative skew field.

§ 185.* Finite One-step Non-commutative Semigroups

According to the general definition a one-step non-commutative semigroup means a non-commutative semigroup whose proper subsemigroups are all commutative. THEOREM 449. In order to give the finite one-step non-commutative semigroups, we define first for natural numbers r, a (r > a > 1) the semigroup

S1 = (, a}

(185.1)

SEMIOROUPS

787

by the -equations er = ea , a2 = ,7' Ua = (?,

2

2

,,

(185.2)

secondly for the natural numbers r, a, e (r > a > 1) and for a prime number p we define the semigroup (185.3)

S11 = {Lo, a}

by the equations

or = e, ape+I _ a, ape = eap, LOaiI = e, L02ak = eakQ = ake2

(k = 1, 2, ...) ,

(185.4)

thirdly for the natural numbers r, a, s, b (r > a > 1, s > b > 1) we define the semigroup Sin = {Lo, a}

(185.5)

by the equations Qr = 9a, or'

oure = aLo2 ,

Dr

(185.6)

a2LO = aqa = LOa2.

In addition, we denote by S1V a semigroup consisting of two right units. These semigroups S1, . . ., S1V, and their non-commutative homomorphic and

anti-homomorphic images and, finally, the finite one-step non-commutative

groups defined in Theorems 444, 445 are, apart from isomorphy, all the finite one-step non-commutative semigroups. S1 has order

O(S1) = r + 1

(185.7)

ee'(i= 1,...,r - 1),a,ago,

(185.8)

O(Sn) = pe(r + p - 1)

(185.9)

and consists of the elements

S11

has order

and consists of the elements ahoak, oiak, a1+k

(i=2,...,r- 1; h=0,...,p- 1; k=0,...,pe- 1),

(185.10)

finally, 5111 has order 0(S111) = rs

(185.11)

and consists of the elements

Loiak(i+k>-,1; i=0,...,r- 1; k=0,...,s- 1),aLo. (185.12) 26/a R.- A.

788

FINITE ONE-STEP NON-COMMUTATIVE STRUCTURES

NoTE. In order to give a complete system of non-isomorphic finite one-

step non-commutative semigroups, we still have to solve the homomorphy problem for the semigroups S, S11, 5111. This is simple for S1 and Sn, but for S111 it is an extremely fatiguing task. In order to prove the theorem, first of all we have to take into considera. tion the fact that, according to Theorem 33, a finite group has only groups

as subsemigroups. It follows that the only groups occurring among the finite one-step non-commutative semigroups are the finite one-step noncommutative groups. Consequently our proof should consider only those semigroups which are not groups. We now wish to prove the assertions listed in (185.7) to (185.12). From the defining equations (185.2), (185.4), (185.6) for S1i 511, S111 it follows that all the elements of these semigroups occur among the elements quoted in (185.8) (185.10) and (185.12) respectively. Therefore it is sufficient to

prove that these elements are all distinct in the three cases, for (185.7), (185.9) and (185.11) then follow by counting the elements. The fact that the mentioned elements are different, will be proved by Theorem 101, when we consider three semigroups belonging to the three equations (185.2), (185.4) and (185.6), respectively, for which we retain the notations (185.1), (185.3), (185.5), and show that the relations (185.7), (185.9), (185.11) hold for them

with Z instead of =. In all three cases we shall define the generators 'o, a as certain mappings of a set S1, S11 or S111, respectively, into itself.

For non-negative integers t, in, n (0 < m < n) we denote by (t);, that integer, for which

(t)M' = t (t < n) or

(t)om,-I (mod n - m), m 5(t)7
We now accomplish the definition of the semigroup S1 = {p, a}, starting from the set

S1=<0',1',0,...,r- l>

(where 0, 1' are to be regarded as pure symbols) and defining the mappings e, a of this set into itself by

QO'=1, 0l'=2, gx=(x+1)Q (x=0,...,r- 1), and

a0'=0', al'= 1 ' ,

F

ax =x (x=2,...,r- 1).

From this definition we have P'0' = (i)Q, O'1' = (i + 1)Q, c'x = (x + i)Q

(x=0,...,r - 1; 1 = 1,2,...);

SEMIGROUPS

789

and in particular

e'= e Further it is easy to show that

a2=a, ea= e,

ae2= e'

-

This means that SI belongs to the equations (185.2). Since, moreover, the images

eb=i(i=l,...,r- 1), aO=O', aeO=1' of the element 0 of SI are different, it follows that 0 (S1) z r + 1. For the definition of S11 = { e, a) we denote by S11 the set of elements x, (Y, X), [u, z]

(x=0,...,pe; y=0,...,p- 1; z=0,...,pe- 1; u=2,...,r- 1), which are to be regarded as pure symbols without any particular meaning, and define the mappings e, a of this set into itself by

ex = (0 , (x),) ,

a(y, z) _ [2, (y + z),e] ,

e [u, z] _ [(u + 1)a, z]

and

ax = I + (x)p<, a(y, z) _ ((y + 1)p, (y + z + 1 - (y + a[u, z] = [u, (z + 1)pe] ,

where (t)e denotes the least non-negative residue oft mod c (t = 0, 1, ...; c = 1, 2, . . .). Then the following relations hold: e`X = [(i)a, (x),,e], e'(Y, z) = [(i + 1)4 , (y + z),O], e,[u, z] _ [(u + i)a, z]

(i=2,3,...), akx = 1 + (x + k - 1)p"

ak(y,z)=((y+k)p,(y+z+k)-(y+k)p)_), ak [u, z] = [u, (z + k)p,]

(k= 1,2,...). Hence we see that for i > a, j z a, i m j (mod r - a) we always have ej = e', and in particular,

e'= a.

790

FINITE ONE-STEP NON-COMMUTATIVE STRUCTURES

Moreover ape+i

Further, it follows that apx = 1 + (x + p - 1),e ,

ap(y, z) _ (y, (z + P),) ,

ap[u,z]=[u,(z+P)p], and then apox = (0, (x + P),,) = eapx,

ape(y, z) = [2, (y + z + p)] = ea°(Y, z) , ape [u, z] = [(u + 1)a, (z + P),] = eae [u, z] ,

so that

ape= eap.

We also obtain ap°X = 1 + (x + Pe

ap`(y, z) _ (y, Z),

ap`[u, z] = [u, z] .

Hence

oa,'x = (0, (x),e) = ex, ea°`(y, z) = e(y, z)

,

eap1[U, z] = e[u, Z],

so that

oap`=o.

Finally we obtain o2x = [2,(x),]

,

e2 (y, z) = [(3)a, (y + z),] , o2[u, z] = [(u + 2)a, z] ,

oakx = (0, (x + k)p.) , Qak(y, z) _ [2, (y + z +

k),9]

,

eak[u, z] = [(u + 1)a, (z + k),,] (k = 0, 1, ...) , and then e2akx

= [2, (x +

eakex

= ake2X ,

e2ak(Y, z) = [(3)a, (Y + z + k)] = oake(y, z) = ake2(y, z)7,

02ak[u, Z] = [(u + 2)a, (z + k),,] = eake[u, z] = ake2[u, z] ,

so that e2ak = eake = ako2 (k = 1, 2, ...)

791

SE IIGROUPS

Consequently S11 belongs to the equations (185.4). Since the images

akeakO=o'(0,k)=(h,k) (h=0,..1; k = 0,...,pe- 1), e'akO= e'--'(O, k)= [i, k] (i=2,...,r- 1; k=0,...,pe- 1) .

ai+k 0 = 1 + k

(k = 0, ..., pe - 1)

of the element 0 of S15 are different, O(S51) >- pe (r + p - 1), as was required. For the definition of SIIl we denote by S1II the set of elements

(x,y)

(x=0,...,r- 1; s=0,...,s- 1), 9,

which are again to be regarded as pure symbols, and denote the mappings

a of this set into itself by 60(x, y) = ((x + 1)p, y),

eQ = (2, 1)

and

a(x, y) = (x, (y + 1)b) (except for x = 1, y = 0)

,

a(l,0)=d2, crQ=(1,2). Then

(i = 2, 3, ...) .

60'(x, Y ) = ((x + i)Q, y), o'Q = ((1 + i)a, 1)

and

(k=2, 3, );

ak(x,Y)=(x,(Y+k)b). akSa=(1,(1 +k)%) so that 60' = 60° , as = b Moreover we have the following relations 60 2(x, y) _ ((x + 2)a, y) ,

602Q

((3)a, 1)

,

a2(x, y) _ (x, (y + 2)b) , a2Q _ (1, (3)b) ,

ea(x, y) = ((x + 1)a , (y + 1)b) , eaQ = (2, 2) ,

so that we obtain easily e2a(x, y) _ ((x + 2)0 , (y + 1),) = eae(x, y) = ae2(x, y) , e2a.Q = ((3)a ,

2) = eaeQ =

ae2.Q

and

a2e(x, y) = ((x + l )y , (y + 2)b) = aea(x, y) = ea2(x, y) , a2eQ = (2,(3)b) = aeaQ = ea2Q .

792

FINITE ONE-STEP NON-COMMUTATIVE STRUCTURES

Hence

e2d'=eae-cie2, atLo =uea=Lo a2. Consequently SI1I belongs to the equations (195.6). Since the images

e'ak(0, 0) = e'(0, k) = (1, k)

(i+kZ 1; i=0,...,r- 1; k=0,...,s- 1), and

ae(0,0)=cr(1,0)=Q

of the element (0, 0) of SIII are different, 0 (SI11) >_ rs. Consequently the assertions in (185.7) to (185.12) follow. We denote the semigroups defined in (185.1) to (185.6) again by SI, SII, SIR, and now prove that they are one-step non-commutative. (The same holds for SIv.) It is sufficient to prove that the semigroups mentioned have non-interchangeable elements and are generated by any two such elements. Of all the elements (185.8) of SI only a and a, according to (185.2), consitute an (unordered) pair of non-interchangeable elements. From (185.1) it follows that they generate SI. Of all the elements (185.10) of S11, according to (185.4), only the

K=a"eo°, a=aw

(u=0,...,p- 1; v=0,...,pe- 1; w= 1,. ,pe;

P,f'w)

constitute non-interchangeable pairs of elements. If we determine a natural number t by wt = 1 (mod pe), then according to (185.4)

a=0.W1 =R'

0=

ap,Loap'=O,Pe_"KO'P'

=

= )(P'_u)1K;i.(Pe_v)t.

Hence, and from (185.3) it follows that K, ? are generators of SII. Among the elements (185.12) of SIII, according to (185.6), only a and Cr

constitute a non-interchangeable pair of elements and, by (185.5), they generate S111. Since, by the above, Si, . . ., S,v are finite and onestep non-commutative, the first part of the Theorem is proved. In order to prove the second part, we consider a finite one-step non-commutative semigroup S which is not a group. We may assume this to be of the form S

(185.13)

SEMIGROUPS

793

We have to show that S is a homomorphic or an anti-homomorphic image of one of the semigroups S1,. .., SIV. First of all we prove some propositions about S. PROPOSITION 1. If pS c S and

a, b, c, d, e,...

(185.14)

is a finite non-empty sequence of natural numbers with then

a+c+e+...?4,

(185.15)

eaabQcad ... _ pa+c+... ob+d{...

(185.16)

Before giving the proof we make the observation, important also for what follows, that oS is a subsemigroup of S and so by virtue of the assumption, commutative. We now denote by k and K the number of terms in the sequence (185.14) and the left-hand side of (185.16), respectively.

For k = 1, 2, the assertion (185.16) is trivial. For k = 3 we distinguish two cases : if c 2, then paab and p` lie in pS, thus we have K=

paab

pe = p'paab ,

so that (185.16) holds. In the other case, when c = 1, according to (185.15), we have a >_ 3. Hence, by an analogous reasoning, we obtain K = p2 . pa

tab p1

= pa

tab pe+Q,

so that we have reduced this case to the preceding one. The case k = 4, since

K= may be reduced to the case k

paob pc . Off,

3. Finally the case k >_ 5, since

paab . p°Qd oe = eeadpe+aab

may be reduced to a case with smaller k. Hence Proposition 1 is proved. PRoPosmoN 2. If eS e S, then OiOkp!_e'+Jok

(185.17)

For jk = 0 this is trivial. We suppose that (185.17) does not hold for some triple i, j, k of natural numbers. This means that the elements peak, pi are not interchangeable, i.e., S = (p'ak, ell .

794

FINITE ONE-STEP NON-COMMUTATIVE STRUCTURES

In particular, a can then be expressed as a finite product or = e"a°eW ...

(u, v, w.... Z 1).

(185.18)

Since e, a are not permutable, the case a = e" is excluded in (185.18), i.e., at least v must exist in addition to u. Let us substitute for a factor a of the right-hand side of (185.18) this right-hand side itself. Then we again obtain

an equation of the form (185.18), where u + w + ... >_ 2. After a further such substitution we even have u + w + ... >_ 3. Then Proposition 1 may be applied to both the products (ea =) a"+Ia°ew ..., (ae =)

a"a"e"

... e,

by which their equality is established, i.e., ea = ae. This contradiction proves Proposition 2. PROPOSITION 3. If Se = S, then e°(e) is a right identity for S.

From the assumption it follows that S = So = See = ... Since S is finite, so the mapping at -* aek ((x E S) is a. permutation of S for every natural number k. Consequently there are two natural numbers k, 1(k < 1) such that ao' = aek for all at. If we write this equation in the form aek er-k = = aek and consider that Sek = S, then we see that eI-k is a right unit for S. Hence, in particular, e'-k+I = Q. Let n denote the least natural number

(therefore existing) with a"+' = e. Then e" is a right unit for S since Se = S but clearly n = o(e). Consequently Proposition 3 holds. PROPOSITION 4. If eS c S, Se = S, then all the powers e, e2, ... lie outside the centre of S.

This is obvious from eS, e2S,... c S and Se = See = ... = S. PROPOSITION 5. If there exists a natural number n with a"+' = or, then

ea`'s aye

((k,n)= 1; k= 1,2,...).

(185.19)

For the proof take some 1(>_ 1) with kl =_ 1 (mod n). From ea ae and a = (ak)I the assertion follows. PROPOSITION 6. If, for a fixed natural number n and for all the natural numbers k,

eak = ake a (k, n) > 1

(185.20)

holds, then n is a prime power (>_ 1).

If the assertion were false, we could write n = kl, where k z 2, 1 > 2, (k, 1) = 1. From (185.20) e would then be interchangeable with ak and a1, and so also with ak+I. But according to (185.20) and (k + 1, n) = 1, this is impossible, so Proposition 6 is true.

SEMIGROUPS

795

We now want to prove the assertion stated after (185.13). Not all the

eS, Se, cS, Sa can be equal to S, for then from (185.13) S = aS = Sa would follow for all elements a of S, which is contrary to the assumption that S is not a group. Since the order of succession of e, a, and an anti-automorphism of S are immaterial we may assume that (185.21)

OS C S.

In the following proof we distinguish four cases.

1. Let Se = Sa = S. According to Proposition 3 (applied also with a instead of e) S contains two right units of the form (m, n ? 1),

9m, an

from which we obtain em+1

= e

a+1 = a.

If we have eman = an

em.

i.e. em = a", then [partly from (185.21) and Proposition 21 we have

ae = an+ie = em . ae = emae = em+l a = U. But since this is false, we must have ?n

a tae. n

IT m

Hence S contains two non-permutable right units, which are therefore necessarily generators. Thus S contains no further elements whatever and is, consequently, isomorphic with Siv.

2. Let Se = S, Sa c S. According to the Propositions 3, 4, S contains a right identity em which is not a central element. Hence from (185.13) ema # aem.

Because eS c S, and Se = S, we also have eS c S; Sem = S, so that we may take em instead of e. In other words this means that a may be assumed to be a right unit. Then every element of S is of the form ak

(k = 1, 2, ...) or ea! (1= 0, 1, ...).

Next we suppose that there is a k (> 2) for which eak

A ake.

From this it follows that S = {e, ak} so that or E {e, ak}, and a is of the form

a=ak' (x

1)

or a =eak-v (p

0).

FINITE ONE-STEP NON-COMMUTATIVE STRUCTURES

796

But the second case is impossible because p is a right unit for which ea 0 a, consequently the preceding case remains the only possibility. Because kx Z 2 there are thus natural numbers n for which

a"+1=a. We assume n minimal. Then kx 1 (mod n), thus (k, n) = 1. This result, together with Proposition 5, implies that the rule (185.20) holds. According to Proposition 6 we thus have n = pe (e z 0) for some prime number p.

If e z 1 then (p, n) > 1. Since (185.20) now holds and a is a right unit, we have

eaP+I =

ooP

, a = aPe , or = aP , a = aP+1 = aP+ie.

But this contradicts (185.20), so only the case e = 0, n = 1, a2 = a remains. By what has been said above all the elements of S occur among

a, e, ea. Then

aea

e

e = ea,

whence it follows that Sa = S. But as this contradicts the supposition, all the equations

(k = 2,3,...),

eak=ake must hold, in particular,

eat=ate=a2. From the above we must also have e2 = e,

ae = a.

The finiteness of S implies that

a' = a"

(for some r > a ? 1).

Here necessarily a > 1, since otherwise ea = ea' = a'e = ae would follow, which is false. We have found that all the equations hold which result from (185.2) after interchanging a and a. Consequently S is now a homomorphic image of S1.

3. Let Se c S, Sa = S. If aS c S, then after interchanging a and a [with regard to (185.21)] the assumptions of the preceding case are satis-

797

SEMIGROUPS

fled; therefore it is sufficient to consider the case aS = S. For the sake of convenience we summarize the suppositions:

eS,SocS; aS=Sa=S. These are self-dual, a fact which we have to consider repeatedly in the sequel. According to Proposition 2 we have

(i,j= 1, 2, .. .; k=0, 1,...),

of+Jak = Qiakef = ak`O1+J

and, in particular, 02ak = oako =

ako2

(k = 1, 2, ...).

From Proposition 3 (applied with a instead of o) ao(, is the unity element of S, thus a"+1 = a,

a"o = oa" = o

(n = o(a)).

We consider a k (>_ 2) for which oak # ako. Then it follows that S = {L0, ak}

and a E {o, ak}, so that we have an equation of the form a = ak-o''akZ, where x + y + z > 0. Since oa ao, necessarily y = 0. Consequently we can write a = a' (t > 0). Since here we have kt Z 2, it evidently follows that kt = I (mod n), thus (k, n) = 1. This result and Proposition 5 imply that rule (185.20) holds and so, according to Proposition 6, we have n = p° (e _>_ O, p prime). But because oa ao, n = 1 is not possible, so it necessarily follows that e >_ 1. Further, because of (185.20), we have gap = apLo-

On account of the finiteness of S an equation of the form Q

r = Qa

holds for r > a >_ 1. But here we must have a > 1, since the assumption a = 1, because r Z 2, gives the contradiction oa= o'a = ao' = ao The result is that all the equations (185.4) are satisfied, so that S is a homo-

morphic image of S. 4. Let So, Sa c S. If we have a S = S, then [with regard to (185.21)] after dualization and interchanging of o and a we obtain the case before last. Accordingly, we have only to consider the case aS c S. We now have the (self-dual) suppositions eS, So, aS, Sa c S.

798

FINITE ONE-STEP NON-COMMUTATIVE STRUCTURES

According-to Proposition 2 it follows that 92a

= o e = ae2 and

°.2e = QLoa = Nat.

From the finiteness of S we must have equations of the form Lo"=N

o'=o'b,

where r > a z I and s > b ? 1. But here a > 1, b > 1 must hold, since e.g., if a = 1 we obtain the contradiction Loa = 'a = aLor = a Lo. Since according to this all the equations (185.6) are satisfied, S is a homomorphic image of Sn1. We have now confirmed the assertion in all four cases. Consequently Theorem 449 is proved. EXAMPLE. The (infinite) semigroup defined by the equations

eEa = eae = aez, ate = aea = ea: consisting of the (distinct) elements

e'ak(i+k>0; i,k=0, 1,...), ae, has e, a as its one and only one non-permutable pair of elements. Consequently it is one-step non-commutative. EXERCISE. Solve the homomorphy problem for the semigroups SI, SE, Sm of Theorem 449.

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Szeged 20, 233-237. REDEI, L. (1960) Halbgruppe and Ringe mit Linkseinheiten ohne Linkseinzelemente, Acta Math. Acad. Sci. Hung. 11, 217-222. REDEI, L. (1965a) Ein Uberdeckungssatz fur endliche abelsche Gruppen im Zusammenhang mit dem Hauptsatz von Hajbs, Acta Sci. Math. Szeged 26, 55-61. REDEI, L. (1965b) Logische Dualitat der Frobenius-Stickelbergerschen and Hajbsschen Hauptsatze der Theorie der endlichen Abelschen Gruppen, Acta Math. Acad. Sci. Hung. 16, 327. REDEI, L. (1965c) Theory of Finitely Generated Commutative Semigroups. Pergamon Press, Oxford. REDEI, L. (1965d) Powers in groups, Amer. Math. Monthly 73. REDEI, L. and STEINFELD, O. (1952) (Jber Ringe mit gemeinsamer multiplikativer Halbgruppe, Comment. Math. Hely. 26, 146-151. REDEI, L. and SZELE, T. (1947) Algebraisch-zahlentheoretische Betrachtungen fiber Ringe, I, II, Acta Math. 79, 291-320. REDEI, L. and SZELE, T. (1950a) Algebraische-zahlentheoretische Betrachtungen fiber Ringe, I, II, Acta Math. 82, 209-241. REDEI, L. and SZELE, T. (1950b) Die Ringe "ersten Ranges" Acta Sci. Math. Szeged 12a, 18-29. REDEI, L. and TuRAN, P. (1959) Zur Theorie der algebraischen Gleichungen uber endlichen Korpern, Acta Arithmetica 5, 245-247. REMAK, R. (1911) Uber die Zerlegung der endlichen Gruppen in direkte unzerlegbare Faktoren, J. reine angew. Math. 139, 293-308. RENYI, A. (1947) On the minimal number of terms of the square of a polynomial, Hung. Acta Math. 1, 30-34. SACHS, H. (1956-57) Untersuchungen uber das Problem der eigentlichen Teiler, Wiss. Z. Martin Luther Univ., Halle-Wittenberg, Math. Nat. Reihe 6, 223-259. SANDS, A. D. (1957) On the factorisation of finite Abelian groups, Acta Math. Sci. Hung. 8, 65-86. SANDS, A. D. (1959a) The factorization of Abelian groups, Quart. J. Math. Oxford. Ser. (2) 10, 81-91. SANDS, A. D. (1959b) On the factorization of finite Abelian groups, 11, Acta Math. Acad. Sci. Hung. 13, 153-169. SANDS, A. D. (1962) The factorization of Abelian groups, (II), Quart. J. Math. Oxford. Ser. (2) 13, 153-169. SANDS, A. D. (1964) Factorization of cyclic groups, Proc. Colloq. Abelian Groups (Tihany 1963), Akademiai Kiad6, Budapest. SCHMIDT, O. J. (1924) Uber Gruppen deren samtliche Teiler spezielle Gruppen rind, Recucil Math. d. Moscow 31, 366-372. SCHMIDT, O. J. (1928) Uber unendliche Gruppen mit endlicher Kette, Math. Z. 29,34-41.

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INDEX Abelian field 663 Abelian group 35 Abel's theorem 674

Algebraically independent 516 Algebraically independent set 516 Alphabet 163 Alternating group 136 Alternating ring 280 Alternating units 280 Amicable double homothetism 196 Anharmonic group 90 Annihilator ideal 396 Annihilator left ideal 396 Anti-automorphism 117 Anti-isomorphism 117 Archimedean ordering 580 Archimedean valuation 618 Artin ring 430 theorem 575 Ascending chain 424 Associated elements 310 Associated factor systems 178 Associated function pairs 178 Associated function quadruple 191 Associative 30 Associativity conditions of the structure constants 243 Autohomomorphism 100 Automorphism 100 group 101

Absolute algebraic element 481 Absolute algebraic field 481

Absolute conjugate 529 Absolute degree

481

Absolute different 556 Absolute discriminant 556 Absolute field 481 Absolute transcendental element 481 Absolute transcendental field 481 Absolute valuation 586 Absolute value 583, 612 of a complex number 612 Absolutely irreducible 449 Absorption rule 214 Abstract structure 89 Addition 28

mod m 65 theorem for determinants 286 Additive commutator 83 Additive order 59 Adjoint 287 determinant 290 matrix 289 Adjunction 107, 481 of indeterminates 259 Admissible factor structure 225 Admissible substructure 223 Affect of equation 688 Affectless Algebra

system 181

47, 152, 153, 240, 245 representation 152, 153, 364 transformation 277, 279 vector 273

688

Basis

34, 241

Algebraic closure

494

Algebraic element 480 Algebraic equation

Between-field

290

Algebraic equation system 290 Algebraic field

479

Bicontinuous isomorphism Bimultiplication 198 Binomial coefficient 62 Binomial equation 672 Binomial theorem 62 Biquadratic 264 equation 697, 705 Bolzano's theorem 603

613

Algebraic hull 516 Algebraic number 613 Algebraic number field 613 Algebraically closed 494, (516) Algebraically dependent 515 Algebraically equivalent 520 am

27/a R.-A.

176

Automorphism-free Schreier extension

594

INDEX

810

Bolzano-Weierstrass theorem 610 Boole algebra 222 Boole lattice 222 Boole ring 222 Breakable semigroup

Commutator ideal 207 Commutator ring 83 Comparable elements 19

Cancellation 45

Central Schreier extension 182 Centralizer 132 Centrally isomorphic subgroup 234

Complementary lattice 221 Complementary theorems to the quadratic reciprocity theorem 665 Complete field 541 Complete group 185 Complete induction 11 Complete invariant system Complete prime ideal 320 Completely reducible 151 Complex character 614 Complex field 297

Centre of a structure 82

Complex number 610

Centre-free 84 Chain 3 Character 377 group 377 relations 614

Complex number field 610 Complex ring 297 Complexes 118-21 Component 143 representation 143 Composite number 50 Composition 28, 78

Cantor's theorem 17 Cardan's formula 699

Cardinal number 16 Cauchy's theorem Cayley's table 41 Cayley-Hamilton theorem subgroup 294

Central automorphism 234

sum 615 Characteristic of a structure 64 Characteristic matrix 293 Characteristic polynomial

293, 551

Characteristic polynomial of an algebra element 295 Characteristic series 229, 232 Characteristic subgroup 184 Characteristic submodule 230 Characteristic subring 198 Characteristically simple 229 Chevalley -Jacobson density theorem 424 Chinese residue theorem

Choice axiom 8 Choice function

8

Class 9

Class composition 97 Class equation 133 Classification 9 Closed 70, 223

Codeterminant 291 Coefficient 33, 57, 259 Coefficient ring 259 Column rank 419 Column vector 269 Combined product 119 Commutative 32 Commutator 83 Commutator group 83

332

Compatible 98

Complement 287 of an element 221

series

211

Concomitant classification 11 Concomitant matrix 553 Congruence 154 Congruent elements 154 Conjugate 54, 121, 528 Conjugate complex 121, 612 Conjugate element 54, 528 Conjugate fields 528 Conjugate quaternions 300 Conjugate subgroups 121 Conjugates of an element 54, 528, Constant 259

Content of a polynomial 342 Continuation of a mapping 5 Convergent infinite product 593 Convergent infinite series 592 Convergent sequence 587 Coset 122 Countable set 15 Countably infinite set 15 Cramer's rule 291 Cross-cut 2 Cross product 252 Cubic 264 Cubic equation 694, 703 Cubic resolvent 699 Cycle

55

555

811

INDEX

Cyclic determinant 561 Cyclic field 663, 672 Cyclic group 54, 360 Cyclic matrix 507 Cyclic module 59, 396 Cyclic permutation 55 Cyclic ring 394 Cyclotomic field 508, 669 Cyclotomic polynomial 509 Cyclotomy 711

Dedekind ring (example) 315 Defining equation of a field 487 Definition domain 488 Degree of an element 293, 480 Degree of a polynomial 260, 264 Degree of a rational function 439 Degree of a skew field 482 Degree of incompletability 544 Degree of inseparability 541 Degree of transcendence 520 Degree valuation 651 Deli problem 713 Delsarte function 382 Delsarte inversion formula 382 Delsarte's theorem 382

Dense set of mappings 424 Density theorem 424 Derived structure 96 Descending chain 423 Determinant 282 Determinant divisor 399 Determinant of a linear equation system 291

Determinant polynomial 284 Diagonal 269 Diagonal matrix 269 Difference 58 Difference field 162 Difference module 162 Different 556 Differential quotient 434

Differential ring 162 Differential set 2 Differential skew field 162 Differential structure 162 Dihedral group 169 Dimension 239 Direct component 141 Direct composition 141 Direct decomposition 148 Direct factor 141 Direct product 105, 141

Direct sum 105, 141 Direct summand 141 Dirichlet mapping 379 Discrete valuation 627 Discriminant of an element 556 Discriminant of a polynomial 451 Disjoint sets 2 Distributive 33, 219 Distributive lattice 220 Divisibility 50, 304 Division 57 Division ring 37 Divisor 59 Domain of definition 7

Dorroh extension ring 110 Dorroh's theorem 110 Double algebra 246 Double endomorphism 195 Double homothetism 195 Double homothetism ring 196 Double module 244 Double operator domain 228 Double vector space 245 Dual 38, 215 zeta function 391 Duality 39, 215 Duality principle 39 of determinants 285 of finite Abelian groups 378 of lattices 215 Duplication of a cube 713 Dyck's theorem Eisenstein's theorem 463 Element I Elementary divisor

402

Elementary group 366 Elementary symmetric polynomials 442 Elementary transformation 402 Embedding 108 Endomorphism 100 Endomorphism ring 113, 115 system

189

Equation 8 system

8

Equipment 14 Equivalence axioms 10 Equivalence classes 10 Equivalence relation 9 Equivalent elements 9 Equivalent Everett extensions 191 Equivalent extension fields 480 Equivalent extensions 182

INDEX

812 Equivalent Schreier extensions 183

Finite field 499 Finite one-step non-commutative group

178,

Equivalent sets 14 Equivalent valuations 593 Euclidean algorithm 328 Euclidean ring 325 Euclid's theorem 330 Euler function 333

736

Finite one-step ring 753 Finite one-step semigroups 786 Finite set 15 Finitely generated structure

First main theorem for finite Abelian

Euler's theorem 436 Even permutation 135 Everett extension ring 187 Everett's fundamental theorem 188 Everett's sum 188

groups

Form 264

Formal degree 261 Formally real 572 Four-group 90 Fraction 57 Fractional linear group 279 Frattini substructure 81 Free field 172

33, 47

of an element 535 of a field 535 Exponent valuation 620 Expression 78 Extension 5, 106 Extension field 71

Free generators 166 Free group 166

of a finite degree 488 Extension group 71 Extension module 71 Extension of the fundamental ring

Free module 172 449

Extension of an ordering 576 Extension of a valuation 594

362

Factor decomposition 49 Factor field 99 Factor group 98, 126 Factor module 98 Factor ring 98 Factor semigroup 98 Factor skew field 99 Factor structure 98 Factor system 116, 175, 188

Factors of normal series 211 96

F-basis 482

Fermat prime numbers 712 Fermat -Euler theorem 334 Field 36

of algebraic number 613 of complex numbers 610 of prime number characteristic of rational functions 438 of real numbers 602

Free product 172 Free ring 164

Free semigroup 164 Free structure 166 Free sum 173 Frobenius classes 54 Frobenius's theorem 406 Frobenius-Stickelberger main theorem

Extension ring 71 Extension semigroup 71 Extension set I Extension skew field 71 Extension structure 71

Faithful representation

369

First structure theorem 427 Fixed element 4

284

Expansion of a determinant

Exponent

80

First general isomorphy theorem 200

498

Full automorphism group 101 Full double endomorphism ring 196 Full endomorphism ring 113, 115 Full invariant series 229, 230 Full invariant subgroup 229 Full linear group 279 Full matrix ring 274 Full permutation-group 111 Full set lattice 214 Function 7 Function pair 175 Function quadruple 188 Fundamental domain 242 Fundamental field 242, 478 Fundamental ring 242 Fundamental skew field 242 Fundamental solutions 418 Fundamental theorem for elementary number theory 329 Fundamental theorem for finite generable Abelian groups 406

813

INDEX

Fundamental theorem for the Galois theory

Icosahedron group Ideal

658

140

128

Ideal of a semi gr oup Galois correspondence 656 Galois field 655 of an equation 659 of a field

133

Ideal of zero sequences

591

Idealizer 132

Idempotent 48

681

Identical mapping

Galois group 655

Identical operator

5 34

Galois theory 655 Gauss rings 335

I mage 4 Incomplete field Indecomposable

Gauss theorem 342

Independent

Gaussian integers 154 General algebraic equation 688 General valuation 626 Generalized associativity 42 Generating elements 77 Generating system 77 Generators 77 Geometrical constructibility 708 Greatest common divisor 310 Greatest common left divisor 308

Independent transcendental elements 517 Indeterminate 259 Index 122

of an equat ion

681

,

729

Greatest lower bound 216 Gratzer-Schmidt theorem 577 Group 35, 51 of units 307 ring 255

Haj6s product 369 Haj6s's main theorem 369 Hamilton's principle 105 Hasse diagram 216 Hausdorff-Birkhoff theorem 25 Hensel's lemma 635 Hilbert's basis theorem 464 Holomorph of a group 184 Holomorph of a ring 198 Homogeneous 264 Homogeneous components 265 Homogeneous linear equation system Homogeneous linear rational function 439

Homomorphic invariant 93 Homomorphic mapping 87 Homomorphic structure 92

Homomorphism 92 Homomorphy 92 Homomorphy problem 96 Homomorphy theorem 99 Hua's theorem 102 Hypercomplex ring 232 Hypercomplex system 241 Hypermatrix 422

Index set

541 151

152, 153

21

Induced congruence 156 Induced equivalence relation 10 induced mapping 5 Inertia theorem of a finite Abelian group 388

Infinite product 42, 592 Infinite series 592 Infinite set 15 Infinitely large 581 Infinitely small 581 Inner automorphism 101, 102 Inner automorphism group 101, 102 Inner double homothetism 195 Inner product 271 inseparable 532 Integral domain 64 Integral quatemion 345 Integrity domain 64 Interchangeable 7, 46, 120 Interpolation 458 Intersection 2 invariance field

Invariance group

656 656

Invariance of the greatest common divisor

339

Invariants of a finite Abelian group 364

Invariants of a module 409 Inverse

29

Inverse complex 120 Inverse composition 57 Inverse image 4 Inverse mapping 5 Inversion 135 Invertible element 29 Invertible matrix 279 Irreducible case 702 Irreducible element 310

INDEX

814

Irreducible equation 487 Irreducible factor decomposition 310 Irreducible radical 673 Irreducible radical expression

681

Isbell's theorem 80 Isobaric polynomial 446 Isomorphic invariant 89 Isomorphic mapping 87 Isomorphic normal series 211 Isomorphic structures 87 Isomorphism 87 of a field 527 Isomorphy 87 Isomorphy principle 89 Isomorphy theorems for groups 203 Isomorphy theorems for rings 204 Iterated polynomial 262

Jacobi's theorem 358 Jordan-Holder theorem 212 Juxtaposition 17

Left inverse 44

Left module 230 Left multiple 50 Left multiplication 232 Left operator 33 Left principal ideal ring 321 Left principal series 232 Left regular 45 Left representative system 123 Left residue polynomial 338 Left side cancelling rule 45 Left unity element 43 Left vector space 239 Left zero divisor 45 Left zero element 44 Legendre symbol 665 Letters 164 Levi's theorem 571 Lexicographical ordering 265 Limit 587 Linear algebra

395

Linear diophantian equation system

Kernel of a homomorphism 95 Konig-Rados theorem 507 Kronecker symbol 275 Kronecker's factor decomposition 560 Kronecker's rank theorem 422 Kronecker-Hensel normal form 472 Kronecker- Hensel theorem 471

Krull's theorem 206 Kuratowski-Zorn lemma 22

406 Linear equation system

Linearly independent Lagrange resolvent

Lagrange's formula of interpolation

Logarithm 608 Liiroth's theorem

458

Lattice 214

Leading coefficient 261 Leading term 261

Least common left multiple 308 Least common multiple 310 Least upper bound 216 Left annihilator 45 Left associated elements 306 Left cosets 122 Left distributivity 59 Left divisor 50 Left Euclidean division 325 Left Euclidean ring 325 Left ideal 129

153, 153, 395,

410

674

Lagrange's theorems 122, 348 Laplace's expansion of determinants Laplace's theorem 287

291. 414

Linear form module 265 Linear group Linear mapping 267 Linear polynomial 264 Linear rational function 439 Linear transformation 274 Linearly dependent 410 Linearly equivalent 411

287

525

Main theorem for finite Abelian groups 369 for finitely generated Abelian groups 406

Main theorem for solvable equations 684 for symmetric polynomials 442

for zeta functions for finite Abelian groups 386 Mapping 4 Matrix 65, 269 Matrix units 276 Maximal condition 424 Maximal element 19 Maximal ideal 205 Maximal left ideal 205

Maximal normal subgroup 205

INDEX

Maximal set 2 Maximal subfield 72 Maximal subgroup 72 Maximal submodule 72 Maximal subring 72 Maximal subsemigroup 72 Maximal sub-skew-field MacCoy's theorem 432 Meromorphism 100 Minimal condition 423 Minimal element 19 Minimal generating system 80 Minimal polynomial 480 Minimal set 2 Minimal subgroup 73 Minimal submodule 73 Minimal subring 73 Mixed group

321

Modular lattice 219 Module 15, 57 of a congruence

of a ring

155

59

Monomial algebra 254 Monomial basis 254 Monomial double algebra 254 Monomial ring 254 Monotonicity 219, 569 of zeta functions 390 Mobius function 385 Mobius inversion formula 385 Mobius theorem 385 Multiple divisor 440 Multiple factor 440 Multiple root 440 Multiple zero 440 Multiplication (6), 28

Multiplication mod m 65 Multiplication table 41 Multiplication theorem for determinants 285 Multiplication theorem for discriminants 453 Multiplication theorem for resultants 450

Natural group ring 256 Natural homomorphism 98 Natural numbers 11 Natural operator 47 Natural polynomial ring 263 Natural semigroup ring 256 Negative

568

Neutral element 29

815

Newton formulae 453 Newton's method of interpolation process 459

Nil left ideal 430 Nil ring 64 Nilpotence degree 49 Nilpotent 49, 423 Non-Archimedean ordering 580 Non-Archimedean valuation 618

Norm 551 Normal automorphism 234 Normal basis 560, 733 Normal divisor 125, (183) Normal field 496 of an algebraic field

498

Normal form of a matrix 402 Normal series 211 Normal subgroup 125, (183) Normal subring 129 Normalizer 132 of a ring

133

Normed class 307 Normed discrete valuation 627 Normed elements 307 Normed polynomial 338

Norming 307 n-tuple

17

Null module 38 Null operator 34 Null ring 38

Odd permutation 135 One-sided ideal 129 One-step non-commutative

structure

736

Operation 33 Operator 33

automorphism 224 compatible classification domain 33

endomorphism 224 Operator group 229 homomorphism 224 isomorphism 224 module 230 ring 231 simple 220 structure 40, 223

Opposed structure 117 Order ideal 399 Order of an element 48 Order of a structure

38

Order preserving automorphism 576

INDEX

816

Order preserving embedding 576 Order preserving isomorphism 576 Order preserving substructure 576 Orderable 571 Ordered Abelian group 607 Ordered field

Ordered module 568 Ordered ring 568 Ordered set 19 Ordered skew field 568 Ordering relation 19 Ordinary quaternion field 299 Ore polynomial ring 559 Ostrowski's first theorem 633 Ostrowski's second theorem 644

55, 111

Pole

575

653

Polynomial 259 Polynomial coefficient 61 Polynomial ring 259, 266 Polynomial theorem 61 Polynomial with integer coefficients Positive 568

Positive kernel 572, 574 Positivity domain 568 Potency

14

Power 46 Power set 2

Overstructures of the same kind p-adic field 631 p-adic hull 631 p-adic number 631 p-adic number field 631 p-adic series 631 p-adic valuation 630 p-adic number 632 p-adic number field 632 p-adic series 632 p-adic unit 632 p-adic valuation 632 p-basis 544 p-component 331 p-dependent 543 p-equivalent sets 544 p-group 331 p-independent 543 p-independent set 543

630 195

Partial product 392 Partial sequence 610

63

Polynomial vanishing differential quotient 435

Overfield 71 Overgroup 71 Overmodule 71 Overring 71 Overset 1 Overstructure 71

Pair 17 of mappings

group

Pickert-Szele theorem Place valuation 653

568

n-adic valuation

Periodic group 331 Periodic kernel 331 Permutable elements 46, 120 Permutation 5

71

Power sum 453 Primary component 331 Primary element 335 Primary group 331

Primary number 335 Primary prime number 335 Primary quaternion 355 Prime 312 Prime decomposition 312 Prime element 312

Prime factors 314 Prime field 477 of a skew field 478 Prime ideal 320 Prime number 50, 329 Prime polynomial 339

Prime power 314 Prime power basis 364 Prime power decomposition 314 Prime power factor 314 Prime power module 408 Prime residue class 333 Prime simplex 369 Primitive element 482 Primitive ideal 466

Partial sum 592

Primitive number mod w 394

Peano's axioms 14 Peirce decomposition 233 Perfect field 592 Perfect hull 594 Perfect valuation 592 Period 48, 671

Primitive polynomial 342 Primitive quaternion 345 Principal class 122 Principal ideal 131 Principal ideal ring 322

with prime decomposition

323

INDEX

Principal left ideal 131 Principal left ideal ring 321 Principal minor 287 Principal polynomial 339 Principal series 229, 232 Product 28

of mappings 6 of matrices 271 of sets 2 Projection 145 Proper degree 261 Proper left divisor 308 Proper subgroup 72

Proper submodule 72 Proper subring 72 Property of finite character 24 PrUfer group 66 Pure inseparable 538 Pure transcendental 517 q-polynomial 558 Quadratic 264

Quadratic equation 693 Quadratic reciprocity theorem 665 Quadratic residue 664 Quadrature of a circle 713 Quadruple 17 Quasi ideal 133 Quaternion

69, 300

with integer coefficient 347 Quaternion field 69, 299 Quaternion group 70 , 298 Quaternion ring 299 Quaternion units 300 Quotient 57 Quotient field 158 Quotient group 158 Quotient ring 158 Quotient semigroup 157 Quotient structure 157

Radical exponent 49 Radical expression 681 Radical free 430 Radical of an element 49

Radical of a ring 430 Radical over a field 672 Range (of a function) 7 Rank 239, 412, 419 of a linear equation system 415 of a matrix. 399 Rank theorem for matrices 419 Rational function 439

817

Rational function field 438 Rational number field 40, 163 Rational numbers 163 Real number 602 Real number field 602 Real valuation 617 Really closed field 615 Recursive definition 12 Reduced degree of an element 530 Reduced degree of a field 530 Reduced norm 296, 301 Reduced trace 296, 301 Reducible element 310 Reducible radical 673 Refinement of a normal series Reflexive relation 3

Regular multiplication

211

31

Regular norm 296 Regular representation 112, 295 Regular semigroup 33 Regular trace 296 Related classifications Relation 3

132

Relative automorphism 480 Relative conjugate 529 Relative different 556, 557 Relative discriminant 556, 557 Relative field 232, 478 Relative isomorphism 182, 480 Relative meromorphism 480 Relative norm 551 Relative trace 551 Relatively prime 324 Remak decomposition 235

Remak-Krull-Schmidt theorem 235 Remarkable points of the triangle 713 Representation 96 Representation problem 96 Representative 9 Representative system 9, 123 Residue 325 Residue class 124 Residue polynomial 338 Residue system 124 Residue theorem for polynomials 340 Resultant 444 Riemann zeta function 391 Ring 36, 59 defined by equations 167 of integer quaternions 345 of integers 40, 329

with greatest common left 308

divisor

INDEX

818

Ring with irreducible factor decomposition 310 with least common left multiple 308 with prime decomposition 313 with unity element 108, 336 Root 8, 49, 262, 290 of unity 509 Root exponent 49 Root system 291 Row rank 419 Row vector 269 Ruffini-Abel theorem 688 Scalar

272

Scalar factor

272

Scalar matrix

276

Schlicht product 120

automorphism-free

Schreier extension, 181

Schreier extension, factor-free 181 Schreier extension, group 174 Schreier extension, semigroup 183 Schreier's fundamental theorem 176 Schreier's main theorem 211

Schur's lemma 423 Second general 200

isomorphy

theorem

Second main theorem for finite Abelian groups 369 Second maximal subgroup 140 Second structure theorem 428 Semifield

36

35, 40 defined by equations 167

Semigroup

of a ring 59 Semigroup ring

255

with factor system 256 Semilattice 222 Semimodule 35 Semiordered set 19 Semiordering relation Semiring 36

Set

24

Set composition

119

extension

Simplex 369

Single-value mapping Skew field 36, 67 Skew product 105 Solution 8, 290 Solution formula 415, 682 Solution system 291 Solution vector 418 Solvable equation 680 Solvable field 663 Solvable group 214 Splitting Everett extension 194

Splitting field of polynomials 490 Splitting Schreier extension 181 Steinitz's first main theorem 494 Steinitz's interchange theorem 411 Steinitz's principle 89 Steinitz's second main theorem 520

Steinitz's theorem 494 of exchange 411 Stickelberger's theorem 663 Structure 34 belonging to equations 167 defined by equations 167 Structure axioms 38 Structure constants 241 Structure extension 106 Subdeterminant 286, 399 Subdirect product 151 Subdirect sum 151

40

Subsemigroup 70 Subset 1 Sub-skew-field 70 Substitution 262

principle of polynomials 262 theorem for polynomials 263 value 262

Substructure

70

of the same kind

1

Set chain

Simple structure 95 Simple transcendental field 483, 523

Submodule 70 Subring 70

19

of natural numbers 40

Separable elemen3 532 Separable hull 538 Sequence 17, 21

Simple algebraic field extension 483, 523 Simple field extension 482, 549

Subfield 70 Subgroup 70 Submatrix 270

of non-negative integers

Sensisimple 427 Semi-skew-field 36 Semistructure 37

Set lattice 220 Simple adjunction 482

71

Substructure problem 96 Subtraction 58

INDEX

Sum 28 of matrices 271 Sylvester determinant 444 Symmetric polynomial 44l Symmetric relation 3 System 1, 19 Szekeres normal-form 467 Szekeres's theorem 466 Szendrei's theorem 335

Taylor's theorem 438 TeichmUller-Tukey lemma 24

Theorem of inertia of finite Abelian group 388 Topological isomorphism 594 Topologically equivalent 594

Torsion free ring 331 Torsion group 331

272

Transposition 55 Triple 17 Trisection 712

Trivial homorphism 95 Trivial ring 64 Trivial solution 418 Trivial valuation 586 True representation 96 Tschirnhaus transformation of an ideal473

Tschirnhaus transformation of a polynomial

691

Union of structures 76 Union set 2 Uniqueness theorem for finite generable Abelian groups 409 Unit

306

Unit factor 314 Unitary module 230 Unity element 29 Unity group 38

Unity ideal 324 Unity matrix 275

Unity of a lattice 219 Unknown 3, 80 Upper bound 22 Vahlen-Capelli theorem 675 Valuated field 585 Valuation 585 Valuation factor field 623 Valuation ideal 523 Valuation ring 622 Value field 623 Value' module 627 Value of an element 585 Vandermonde determinant 457 Variable 7 Vector space

Trace 551 Transcendence basis 520 Transcendental element 480 Transcendental field 480 Transfer rule of equivalence 10 Transfinite induction 26 Transformation 278 Transformed matrix 280 Transitive permutation group 732 Transitive relation 3 Transpose of a matrix

819

239

Vector space over a skew field 412 Waring's formula 454 Wedderburn-Artin structure theorems 426, 428

Wedderburn's theorem 513 Weierstrass's upper bound 610 Weight 446

Well-ordered set 21 Well-ordering theorem 25 Word problem 172 Word sum 164 Words

164

Zassenhaus's lemma 208 Zassenhaus's subgroup 208 Zassenhaus's subring 208 Zermelo's postulate 8 Zermelo's theorem 25 Zero class 124 Zero divisor 45 Zero-divisor-free 64 Zero-divisorless 64 Zero element 32

Zero element of a lattice 219 Zero endomorphism 114 Zero of a polynomial 262 Zero polynomial 260 Zero ring 64 Zero sequence

587

Zeta function for finite Abelian groups 385

Zsigmondy's theorem

394

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