North-Holland Mathematical Library Board of Advisory Editors:
M. Artin, H. Bass, J. Eells, W. Feit, P. J. Freyd, F. W. Gehring, H. Halberstam, L. V. Hormander, M. Kac, J. H. B. Kernperman, H. A. Lauwerier, W. A. J. Luxemburg, F. P. Peterson, I. M. Singer and A. C. Zaanen
VOLUME 5
NORTH-HOLLAND PUBLISHING COMPANY- AMSTERDAM LONDON AMERICAN ELSEVIER PUBLISHING COMPANY. INC. - NEW YORK
Algebra of Polynomials HANS LAUSCH Department of Mathematics Monash University Clayton, Australia
and
WILFRIED NOBAUER Technische Hochschule Wien IV. Instituf fur Mathematik Wien,Osterreich
1973 NORTH-HOLLAND PUBLISHING COMPANY- AMSTERDAM LONDON AMERICAN ELSEVIER PUBLISHING COMPANY, INC. - NEW YORK
@ North-Holland Publishing Company
- 1973
All rights reserved. Nopart of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic,mechanical,photocopying, recording or otherwise, without the prior permission of the copyright owner.
Library of Congress Catalog Card Number: 72-88283 North-Holland ISBN for the series: 0 7204 2450 x North-Holland ISBNfor this volume: 0 7204 24550 American Elsevier ISBN: 0 444 10441 0
Published by: North-Holland Publishing Company -Amsterdam North-Holland Publishing Company, Ltd. -London
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Printed in Hungary
INTRODUCTION
Polynomials are a classical subject of mathematics. The first steps towards the abstract concept of a polynomial were the investigation of algebraic equations and the theory of real and complex functions f of the form f ( x ) = a,$+. . +alx+ao. The introduction of the abstract notions of a field and ring at the beginning of this century subsequently brought about the development of the abstract concept of a polynomial over a commutative ring with identity. While polynomials over the fields of real or complex numbers play an important r61e in analysis and numerical mathematics, the algebraic properties of polynomials have also been a research object for a great number of papers and under various different points of view. There are, however, features in the algebraic theory of classical polynomials that have been treated inseveral papers to some extent, but so far have not been given a coherent representation. Among all these aspects we think the most important ones to be the connection between polynomials and polynomial functions and the properties of polynomial functions. In particular, the so-called permutation polynomials over finite fields (i.e. polynomials which represent permutations) have suggested plenty of interesting algebraic and number-theoretical investigations. Moreover we think that an important and interesting part of the theory consists of the composition of polynomials. Questions concerning the decomposition of polynomials into indecomposable factors, permutable polynomials, or congruences which are compatible with the composition operation, have been tackled by various authors. It may appear somewhat strange that polynomials over commutative rings with identity have been dealt with quite extensively while polynomials Over other classes of algebraic structures, such as groups, semigroups, lattices etc. have been given little attention. Those papers on polynomials Over classes other than rings, fields, and maybe Boolean algebras, are Scarce and scattered, and so far not even a general agreement on basic definitions has been achieved. The first author who has endeavoured to
.
1X
X
INTRODUCTION
use some concept of polynomials over arbitrary algebraic structures, is G. GRATZER in his book on universal algebra, but these polynomials have turned out not to be a straightforward generalization of the classical polynomials. With this state of the theory of polynomials in mind, this book will serve two different purposes. First we want to give a general approach to the notion of a polynomial via universal algebra in such a way that the classical concept of a polynomial becomes a special case within the general theory, hence we will endeavour to extend the classical theory to generalized polynomials. Secondly, our object is to give a representative sample of results on polynomials over special classes of algebraic structures, such as commutative rings or groups, which refer to the connection between polynomials and polynomial mappings or to the composition of polynomials. This book consists of six chapters. Chapter 1 gives first of all a short exposition of those facts of universal algebra that are needed in this book, then the definitions and theorems on polynomial algebras and the algebra of polynomial functions that are fundamental for all the subsequent chapters. Finally these definitions and theorems are illustrated by special algebraic structures like rings, groups, lattices, and Boolean algebras. Chapter 2 deals with systems of algebraic equations and related topics such as algebraic extensions and algebraic dependence for arbitrary algebras, and finally specializes to systems of equations over groups. Chapter 3 is based merely on Chapter 1 and investigates the composition of polynomials and polynomial functions for universal algebras in general, then for multioperator groups in particular, and, again by specializing, for rings and groups. The last sections are devoted to polynomial vectors and polynomial function vectors over arbitrary universal algebras, and furthermore, to polynomial permutations, permutation polynomial vectors, and permutation polynomials. Chapter 4 is based just on Chapters 1 and 3 and deals with polynomial composition, polynomial vectors, permutation polynomials and polynomial permutations over rings and fields while Chapter 5 which again depends only on Chapters 1 and 3, does the same for groups. Each chapter concludes with a section “Remarks and Comments” which gives references to the sources being used for that chapter, mentions papers related to that chapter (which have been completed by summer 1971), and states some open problems (the number of which could be easily enlarged by a lot more problems, a task we leave to the reader),
INTRODUCTION
xi
We have tried to make this book comparatively “self-contained”. This goal should be achieved by Chapter 6 , Appendix. Here we have collected all those definitions and theorems (without proof) of classical algebra which are used in this book but can also be found in lots of textbooks in more detail. Furthermore concepts and results that cannot be found in textbooks but are necessary for understanding various proofs in the book are treated in detail. We hope that this book will stimulate the interest in a field which still has lots of open problems that are often easily accessible and, in part, not too difficult. Moreover it was our goal to contribute to the development of a more “universal” approach to algebraic problems, i.e. considering the problems not only for a single class of algebraic structures but for a number (if not all) classes of structures simultaneously. Finally the authors wish to express their deep gratitude to all those who have contributed to the completion of this book by their kind help and assistance. Our special thanks go to Mr. D. G. GREENwho did the proofreading, to Prof. C . WELLSand Dr. R. LIDLfor helping us with compiling the bibliography, to Herr G. EIGENTHALER who also read the proofs and prepared the drawings, and to Frau A. CISKOVSKY who typed the manuscript with care and endurance. Moreover we owe our thanks to the North-Holland Publishing Company and its staff, in particular, to Drs. H. J. STOMPS and Mr. E. FREDRIKSSON whose invitation to publish this book, appointing a deadline for its completion, and kind advice on preparing the manuscript have substantially influenced our work. The book is partly based on our previous research work on this subject which, to a large extent, has been done a t the Mathematisches Jnstitut der Universitat Wien, Vienna (Austria), and partly at the Department of Mathematics, Institute of Advanced Studies, Australian National University, Canberra (Australia). Consequently we express our gratitude to Prof. N. HOFthese institutions and their heads, Prof. E. HLAWKA, REITER,and Prof. B. H. NEUMANN. Last not least we are grateful to those mathematicians and students who attended our lectures on subjects of this book and, by their interest, have given us encouragement and moreover frequently valuable suggestions. Clayton, Vienna, April 1973
HANSLAUSCH WILFRIED NOBAUER
CHAPTER I POLYNOMIALS AND POLYNOMIAL FUNCTIONS
1. Basic concepts of universal algebra
1.1. Let M be a non-empty set and n a positive integer. An n-ary operation o on the set M is a mapping from the Cartesian power M" to M . That means that o assigns a well-defined element wa,a, . . . a,, of M to each ordered n-tuple (al, a,, . . ., a,,)of elements of M . For n = 2, which is a very important special case, we often use infix notation : The operation symbol w stands between the two elements or may be omitted. A 0-ary operation on the set M means singling out a fixed element of M . This particular element is also denoted by the symbol o of the 0-ary operation. A universal algebra or, briefly, an algebra is a pair ( A ; S)where A is a non-empty set and SZ = {mi I i E I} is a family of operations on A being indexed by the set I of all ordinals L < 0, o being an arbitrary ordinal. If no confusion can arise, we will denote the algebra ( A ; 12)by A. The family T = {ni I wi is an ni-ary operation, i E I} is called the type of SZ or also the type of ( A ; Q). Any two algebras A , B of the same type are called similar. For similar algebras A , B we can use the same operation symbol for those operations of A and B respectively which are indexed by the same i E Z. If A is an algebra, the cardinal I A I of the set A is called the order of A . An algebra A of finite order is called a finite algebra. Thus the order of a finite algebra A is the number of distinct elements of A . Classical algebra gives us quite a lot of examples of algebras, e.g. a lattice V with the operations U, n yields the (similar, but distinct) algebras (Y;U, n) and ( V ; U) of type (2,2}.
n,
1.2. Let ( A ; S)be a universal algebra, and U a non-empty subset of A such that, for all i E Z, wia,a, . . . a,,<E U whenever a,, a,, . . ., ani E U and ni =- 0, and oiE U if ni = 0. Then the restriction of any mi to U yields an operation on U which we will denote again by wi.The algebra (U;9 )is called a subalgebra of A , and A an extension of ( U ; SZ). Clearly, any non-empty intersection of subalgebras of A is again a subalgebra of A. If S is a non-empty subset of A, then the intersection of all subalgebras of A containing S as a subset is called the subalgebra of A 1
2
POLINOMIALS AND POLYNOMIAL FUNCTIONS
CH.
1
generated by S, and will be denoted by [ S ] .If [S] = A , then S is called a generating set for A . 1.3. Let A and B be similar algebras. A homomorphism from the algebra A to the algebra B is a mapping z? : A B such that 8w,al.. . a,, = w16a16a,. . . 6unt for ni > 0 and 8wi = mi, for ni = 0, for all i E I. If 6 is injective (surjective), 6 is called a monomorphism (epimorphism), and if 6 is bijective we call it an isomorphism. In case B = A , 6 is an endomorphism of A ; and if 19 is an isomorphism, we call it an automorphism of A . The algebra B is called a homomorphicimage of A if there exists an epimorphism from A to B, and B is isomorphic to A if there is an isomorphism from A to B. In this case we write B z A. As immediate consequences of the definitions we get: If 6 is a homomorphism from A to B and 7 is a homomorphism from B to C, then 178 is a homomorphism from A to C ; if 6 is a homomorphism from A to B, then (#A; Q) is a subalgebra of B ; if 8 is an isomorphism from A to B, and r] is an isomorphism from B to C , then 7.8.is an isomorphism from A to C ; if 6 is an isomorphism from A to B, then 8-l is an isomorphism from B to A. Thus the relation “is isomorphic to”, in any set of similar algebras, is an equivalence relation. If A is finite, then any monomorphism from A to A and any epimorphism from A to A is an automorphism of A . Let 6 be a monomorphism from A to B. Then there exists an algebra B Y B such that A is a subalgebra ofB: WLOG, we may assume A n B = 4. Then we get such an algebra B if we replace every element 8a E B by the element a E A , but do not change other elements in B. This procedure is called an embedding of A into B, and the isomorphism E :B + fi defined by E6a = a, for a E A, Eb = b, for b E B 6 A , is called an embedding isomorphism from B to B. -t
-
1.4. Let ( A ; Q) be an algebra. An equivalence relation 0 on A is called a congruence on the algebra A if, for any miwith ni=-0, nlOb1, . .,anf?bni
.
implies (w,al. . . anJO(oibl . . . b,J. Let F be the set of all equivalence classes under 0, and C(a) the class of a E A. On F, we define, for any i E I, an n,-ary operation w iby wiC(al) . . . C(a,,J = C(wial wi = C ( W , ) ,
.. . a,,,),
=- 0,
for
ni
for
ni = 0.
(1.4)
01
BASIC CONCEPTS OF UNIVERSAL ALGEBRA
3
The operations coi on F a r e well-defined since the result of any operation does not depend on the particular choice of the representatives. The algebra (F;Q) is called the factor algebra of A with respect to the congruence 0 and will be denoted by A I @. (1.4) implies that the mapping 8 :A A /0 defined by 8a = C(a)is an epimorphism. 8 is called the canonical epimorphism from A to A ] @ .Thus AIO is a homomorphic image of A . -+
1.5. Up to isomorphism, the factor algebras are just all the homomorphic images of the algebra A . This is an immediate consequence of the following theorem : 1.51. Theorem(Homomorphism Theorem). Let y :A -* B be an epimorphism of algebras. Then there exists a congruence K on A and an isomorphism y :B -+ A I K such that y y = 8, the canonical epimorphismfrom A to A I K.
Proof. Let b E B and y-l(b) be the set of inverse images of b under y . Then A = U (y-l(b)I b E B ) is a partition of A. Let K be the corresponding equivalence relation on A , then K is a congruence, for let a,Kb,, Y = 1,2, . . ., n,, then ya, = yb,, v = 1,2, .. ., n,, hence wipla, . . . yan6= wiybl . . . yb,,, thus (wial . .. ay)K(wibl .. . b,,J Moreover b -+ y-I(b) is a bijective mapping y from B to A I K. It is also a homomorphism, for, let b, = yap,and n, z 0, then ywibl .. . b,,, = yp-l(wib1. .. b,J = y-l(ywia1
. . . a,,)
... a%) = wiC(al) .. . C(cr,,)
= C(w,al
= wiy-l(bl)
= wiybl
.. . p-l(b,J
.. . yb,,.
If ni = 0, then ywi = cp-l(wi) = C(wi) = wi. Thus y :B -c A1 K is an isomorphism. Furthermore, i f a E A, then yya = cp-l(ya) = C(a) = 8a which completes the proof. 1.52. If y :A B is an algebra homomorphism then, by changing the range from B to P A , we obtain an epimorphism from A to y A which is -f
4
POLYNOMIALS AND POLYNOMIAL FUNCTIONS
CH.
1
also denoted by 9. As in the proof of Th. 1.51, the set (pl-'(b)Ib EVA} constitutes a partition of A which induces a congruence K on A . K is called the kernel of pl and we write K = Ker V. Every algebra A has at least two congruences, the congruence where the classes consist of only one element, and the congruence whose only class is A itself. These are the trivial congruences. In case there are no further congruences we call A a simple algebra. The algebra A is simple if and only if every homomorphism of A is a monomorphism or maps A onto an algebra of order 1. Indeed, if A has only homomorphisms of this kind, and if 0 is a congruence on A , then the canonical epimorphism 8 :A -+ A / @ is an isomorphism or I A 101 = 1, thus 0 is a trivial congruence. Conversely, if A has only the trivial congruences, v is a homomorphism of A, and Ker 9 = K, then, by Th. 1.51, there is an isomorphism y such that yp = 6, the canonical epimorphism from A to AIK. Since K is a trivial congruence, 6 is an isomorphism, or [ A I R [ = 1. Thusplisamonomorphism, or [ P A / = 1. 1.6. Let 2(A) be the set of all congruences on A . On 2(A) we define a binary relation =sas follows: @I e O2 if and only if the set-theoretical inclusion 01 C O2 holds; 01,O2 considered as subsets of the Cartesian product A X A. 1.61. Theorem. The set 2(A) is a complete lattice with respect to the relation 4, the so-called congruence lattice of the algebra A .
Proof. It is clear that G is a partial order relation. It remains to show that every non-empty subset of 2(A) has a greatest lower bound, for $(A) has a greatest element. Let M = { @ i l i E Z } be a non-empty set of congruences on A , and d = n(OiI i E I), the set-theoretical intersection of the subsets Oi of A X A. Clearly, d is a congruence on A, and also the greatest lower bound of M. Hence 2(A) is a complete lattice. 1.62. In order to obtain a description of the least upper bound of M we proceed as follows :Let @ be the binary relation on A defined by u@bif and only if there are congruences Oil, . ., a,.,in M and elements cl, . .,cr-l E A such that
.
.
§I
5
BASIC CONCEPTS OF UNIVERSAL ALGEBRA
We call (1.6) a chain of length r from a to b. One can easily verify that 4j is an equivalence relation. Suppose now that ag@b,, for some integer g , 1 =zg =sn,. Then there exists a chain of the form (1.6) from a, to b, which yields a chain from q a l .. . ag.. .anito wial . ..b, . . . a,. Since@ . . is transitive, a,@b,, Y = 1, . . ., n, implies wial . . . ani@'oibl. . . b,,, hence @ is a congruence, and, of course, the least upper bound of M . Let P be a subset of A X A , then the intersection of all congruences on A containing P as a subset is called the congruence on A generated by P. 1.7. We want to know the connections between the congruences on an algebra A and those on its factor algebra A 10.These are given by 1.71. Theorem (Second Isomorphism Theorem). Let A be an algebra, 0 a congruence on A , C(a) the congruence class containing a E A, and 5D the sublattice of 2 ( A ) consisting of all congruences @ ==@ on A . Then a) If @ € 9,then there is a congruence @I@ E 2(Al@) dejned by: C(a)(@ 10)C(b)$and only if a@b. b) If YE ,€!(A(@),then there is a congruence p C % l dejned by: n p b if and only if C(a) YC(b). c) The mapping a :sb 2 ( A j 0):defied by cz@ = @ j 0 is a lattice isomorphism, and a-lY = !P. d) Al@ 2 (A/@))(@[@),fora n y @ € % .
-
Proof. a) @ /@iswell-defined since a@b,a@a, b@bimply al@a, bl@b and hence alObl. @ 10 is a congruence on A1 0 by definition of AIO. b) By definition of A 10,p i s a congruence on A . Moreover aOb implies C(a) = C(b), hence == 0. then O p = O s i n c e a m b is equivalentto
[email protected], c) Let@ E 9, for Y € 2 ( A 1 O), 10 = Y.Thus a is bijective. Since @I =s!Dz implies 10 =sG 210 and __ vice versa, cz is a lattice isomorphism. The last assertion follows from @ I@ = 0. d) Let @ be a congruence of 9, C,(a) the congruence class of @ containing a, and Cz(a)the congruence class of @ I @ containing C(a). Then C,(a) -+ C2(a)is an isomorphism from A I@ to ( A 10) 10). For, by a), this is a well-defined mapping and is bijective since @I@ = @. That this mapping is a homomorphism follows from straightforward calculation.
I(@
1.72. Remark. A straightforward argument shows: Let @a 0 be congruences on A and C(a), C(a) the congruence classes of @, @ resp., con-
6
POLYNOMIALS AND POLYNOMIAL FUNCTIONS
CH.
1
taining a. Then C(a) C(a)is an epimorphism from A 1 0 to A 1 @ which is called the canonical epimorphism. +
1.8. Let {(G,; 0)I v E Z} be a family of similar algebras, and D the Cartesian product of the sets G,. Its elements are families of the form {a(.)}, v E I , where a(v) E G,, for every v. We define the operations w iE Q on D by wi{al(v)}. . . {a,i(.)} = {wial(v) . . . u,,(v)} if q =z 0 and mi = {wi(v)} if ni = 0 and o i ( v )is the uji of G,,. The algebra (D; a)is called the direct product of the algebras G, and is denoted by n ( G vJ v E Z), or, for I = { 1,2} by GIX Gz. The definition of n ( G , I v E I) = D implies that, for fixed p E I, the mapping { ~ ( v ) } a(p) is an epimorphism z,from D to G, being called the projection of D onto G,. A subalgebra S of D is called a subdirect product of the algebras G, if zS , = G,Lfor all p E I. +
1.9. Let Z be the set of all ordinals v < y , y being an arbitrary ordinal. A family of similar algebras ( G Y ;Q), v E I is called an ascending family of algebras, if G, is a subalgebra of G,, whenever a -= p. We set G = U (G, I v E Z) and define the operations wi E Q on G by wi = wi of Go if ni = 0, while if n, > 0 and p is the least v E Z such that a,, a2, . . ., a,:€ G, (which exists, since I is well ordered), we put wial . . . a,; = wial . . . aniof G,. The algebra (G; Q) is called the direct 0). limit of the ascending family of the (G%,; It is easy to see that every G, is a subalgebra of G. 2. Varieties 2.1. Let Q = {oi Ii I } be a set of elements mi, indexed by the set Z of all ordinals L < o where o is an arbitrary ordinal. Moreover, for all i E I , let ni be a non-negative integer, and X = {xi\j E J } be a set which is disjoint from SZ. The elements of X will be called “indeterminates”. We define “words in X over Q” as follows : Words of rank 0 are the wi such that tii = 0, and the xiE X . A word of rank k + 1 is either a word of rank k or an expression of the form wiwl . . . wni, ni =. 0, and w,,words of rank k , v = 1: . . ., ni. The smallest rank of a word w is called the minimal rank of w. Induction on the minimal rank shows that every word is composed by means of a finite number of elements of QUX. Let w be a word, then a “subword” of w is w if w is of rank 0, and if w =
§2
7
VARIETIES
wiwl . . . wniis a word of minimal rank k+ I , then the subwords of w are w and the subwords of w,,,v = 1, . . . , n,. Induction on the minimal rank of w shows that every subword of a subword of w is again a subword of w. Let W be the set of all words in X over Q, and i E I. Then we can define an n,-ary operation wi on W by: wiwl... wni is the word wi
is the word
w i w l . .. writ,
for
n, =- 0,
wi,
for
n, = 0.
(2.1)
The algebra ( W ;SZ) = W ( X )is called the word algebra in X over Q. 2.2. Let ( A ; a)be an algebra, w = w(xjl,. . ., xjn)a word of W containing no other indeterminates than xi,,. . ., xi,, and ail,. . ., ajmelements of A . By replacing each xjyin w by aj$,,we obtain a well-defined element w(aj,,. . ., ajn)E A if we consider the w, occurring in w as operation symbols on A , as one readily finds by induction on the minimal rank of w.(2.1) shows that if {aj(j € J ) is any system of elements of A then the mapping 6 : W ( X ) -+ A defined by w(xj,,. . ., xjn) w(aj,,. .., ai,) is a homomorphism of algebras. -t
2.3. Let (wl(xjl, . . ., xjJ w2(xjl,. . ., xi,)) be a pair of words in X over s2. Such a pair is called a law over Q. We say that the law (wl(xil, . . ., xj,,), w2(xj1,. . ., xjn)) holds in the algebra ( A ; s2) if w,(uj,, . . ., aja)= w2(aj,,. . ., uj+J, for arbitrary elements ail,.. ., ajnE A. For the sake of convenience, we sometimes write wl(xjl,. . ., xi,> = w2(xj,,. . ., xi,) instead of (wl(xjl, . . ., xJ, wz(xj,,. . ., xJ).
Let @ be a set of laws over Q. The class %(Q) of all algebras A of the same type as 8,such that all laws of hold in A , is called the variety defined by @. One and the same variety can, however, be defined by different sets of laws : If % = %($) and we adjoin a law to $ holding in all algebras of %, we get a new set of laws which also defines %. Every variety contains the algebra of order 1 of the type of 8which is unique up to isomorphism. A variety which contains no other algebra is called a degenerate variety.
2.4. Most of the well-known classes of algebras are varieties. A few examples shall illustrate this :
8
POLYNOMIALS AND POLYNOMIAL FUNCTIONS
CH. 1
a) The class of semigroups is a variety if we consider semigroups as algebras ( A ; 01) of type ( 2 ) . b) The class of groups is a variety if we consider groups as algebras ( A ; wl, 0 ~ ~ 0 3 of ) type (2, 1,0} where 01 is the group operation, w2 the operation of forming the inverse, and w 3 the identity. With the same operations, the class of abelian groups is a variety. c ) The class of rings is a variety if we consider rings as algebras ( A ; 01, 0 2 , w 3 , 0 4 ) of type (2, 1,0,2} where w1 is the addition, w 2 the operation of forming the additive inverse, w3 the zero, and w4 the multiplication. d) The class of commutative rings with identity is a variety if we consider these rings as algebras ( A ; 01, 02,0 3 , 0 4 , 0 5 ) of type (2, 1,0,2,0> where ol, 02,w 3 , 04 are defined as in c) and 05 is the identity. e) The class of lattices is a variety if we consider lattices as algebras ( A ; w1, w g ) of type (2,2} where 01 is the union and w 2 is the intersection. f) The class of Boolean algebras is a variety if we consider Boolean algebras as algebras ( A ; w1, w 2 , u j 3 , w4,w5) of type (2,2,0,0, 1) where a1,w 2 are the operations of e), w 3 is the zero, w 4 the identity, and w5 the operation of forming complements. From elementary algebra, it is well-known that for each of these classes there exists a set of laws defining just this class, and thus these classes are varieties. 2.5. An important result for varieties is provided by
8.Then every subalgebra and every homomorphic image of A are also in %, and if {A>,}is a family of algebras in 8,then the direct product n A v is also in %. 2.51. Theorem. Let % be a variety and A an algebra in
Proof. Suppose % = %($) and A is in %. If U is any subalgebra of A, the laws holding in A also hold in U , thus U is in 8,Let B be a homomorphic image of A and cp : A B an epimorphism. If (wl(xjl, . . ., xJ, w2(x,,, . . ., xj,)) is a law of $ and b,, . . ., b, are arbitrary elements of B, then there exist elements a,, . . .,a, E A such that bi = yai, i = 1, . . .,n. Since wl(al, . . ., a,) = wz(a1, . . ., a,) and 'p is a homomorphism, we get w d h , . . ., b,) = wz(b1, . . ., b,) which means that B is in 8.The last assertion is a consequence of the definition of n A Y . +
03
FREE ALGEBRAS, FREE UNIONS A N D FREE PRODUCTS
9
2.52. We have just seen that every variety is closed with respect to forming subalgebras, homomorphic images and direct products. It is true that the converse also holds, i.e. a class -of algebras closed with respect to these processes is a variety - which we are not going to prove since this result will not be required later on. 2.53. Proposition. Let 23 be a variety and {GYIv E I } an ascending family of algebras in 23. Then the direct limit of this family is also in 58. Proof. This is a consequence of the fact that every G , is a subalgebra of the direct limit. 3. Free algebras, free unions, and free products 3.1. Let 9 be a class of similar algebras, s2 its family of operations. An algebra F of Q is called a free algebra of 9 with free generating set X = {xiI i E I } if X is a subset of F generating F and if, for any algebra A of Q, every mapping from X to A can uniquely be extended to a homomorphism from F to A . A free algebra of 9 with the free generating set X will be denoted by F(X, The structure of F(X, Q ) depends only on the cardinality of X , i.e.
a).
3.11. Proposition. Let F(X, 9) and F(Y, 9) be free algebras of P and 1x1 = IYl, thenF(X, Q) 2 F(Y, Q).
-.
Proof. Let X = { x i ]i E I}, then we may write Y as Y = {yiji E Z}. Let 8: F(X, a) F(Y, 9) be the unique extension of the mapping xi yi to an algebra homomorphism, and 7 : F(Y, Q ) F(X, Q ) the unique extension of the mapping yi xi to an algebra homomorphism. Then q8 :F(X, 9) F(X, 9)extends xi xi to an algebra homomorphism,and 8 ~F(Y, : Q) F(Y, 9) extends yi yi to an algebra homomorphism. By the uniqueness of extensions, q8 and 8~ are the identity homomorphisms of F(X, 9) and F(Y, 9),resp. Thus 8 is an isomorphism. -t
+
+
--t -+
-. +
3.2. Not every class of algebras has free algebras. The following theorem, however, will ensure the existence of free algebras in vaneties and thus is of great importance.
10
POLYNOMIALS AND POLYNOMIAL FUNCTIONS
CH.
1
3.21. Theorem. Any non-degenerate variety ‘23 has free algebras with free generating sets of arbitrary cardinality. Proof. Let ‘23 = 23($), X = { x j l j E J } be a set of indeterminates of arbitrary cardinality, and W ( X )the word algebra. Let P be the subset of W ( X ) x W ( X ) consisting of all elements a) (w, w), w E W ( X ) , b) (wl(ul, . . ., u k ) , w2(u1, . . .,.,I) and (w2(u1, . . ., uk), wl(ul, . . .,u,)), where w,(yl, . . ., y,) = w2(y1, . . .,y,) is a law of $ and ul, . . ., u, E W ( X ) . Let 0 be the binary relation on W ( X )defined by VOWif and only if there is a finite chain v = zo, zl,. . .,z, = w of elements of W ( X ) such that, for t = 1, . . ., r, one can obtain zt from ztP1 by replacing a subword of ztPl which is the left-hand term of an element of P by the right-hand term of the same element-such a chain will be called, in short, a “chain from z1 to w ” . 0 is, of course, an equivalence relation on W ( X ) .Moreover, since any set of chains from v, to w,, v = 1,2, . . .,ni,gives rise to a chain from wivl . . . vni to wiwl . . . w , ~ ,0 is also a congruence. We claim that W ( X )I 0 is a free algebra of 8 with the free generating set {C(xj)1 j E J } = X and 1 XI = I J 1. In order to prove the latter assertion we have to show that C ( x m ) # C(x,), for any pair of distinct indices m,n in J . Suppose, by way of contradiction, that xmOx, for m # n. This means that there exists a chain from x, to x,. If B is an arbitrary algebra in 23 and {bj1 j E J } a family of elements of B, then from this chain we get a chain of elements of B consisting of equal links by replacing each x j by bj, j E J . Hence b, = b,, for all b,, b, E B, i.e. IBI = 1, and 23 would be degenerate, a contradiction. Since X generates W ( X ) , X is a generating set for W ( X ) I 0 of cardinality IJI. Let wl(yl, . . .,y,) = w2(y1, . . .,y,) be a law of &, and ul, . . ., u, E W ( X ) ,then wl(C(ul), . . .,C(u,)) = C(wl(ul, . . ., u,)) = C(w,(ul, ..., u,)) = w2(C(u,), . . .,C(u,)), hence W ( X )10 is an algebra of 8.Suppose now that Ais an algebra of ‘23, and y : X A an arbitrary mapping. We define a mapping p : W ( X )I 0 A by vC(w(xj,, . . ., xi,)) = w(yC(xj,),. . ., yC(xjn)).p is well-defined since w(xjl, . . ., xi,) @wl(xj,, . . ., xi,) implies w(yc(xjl),. . .,yC(xin))= wl(@(xj,>, . . ., yC(xjn)), by replacing every xi by @(xi) in a chain from w to wl. Clearly p extends y, and putC(wl) . . . C(W,>= pC(oiwl . . . w,,> = wipC(wl) . . . pC(w,>. Therefore p is a homomorphism and is clearly uniquely determined by being an extension of y . -+
-+
$3
FREE ALGEBRAS, FREE UNIONS AND FREE PRODUCTS
11
3.3. Let 9 again be a class of similar algebras, SZ its family of operations, and {A,/I E L } a family of algebras of 9. A pair A , {pi\1 E L } ( A being an algebra of 9 and p, :A, -+ A, I E L, an algebra homomorphism) is called a free union of the algebras A, in 8 if, for any algebra B of R and any family of algebra homomorphisms y, : A, B, I € L, there exists a unique homomorphism e : A B such that y, = ep, for all I E L. The free union of the algebras A, in 9, so far as it exists at all, is unique up to isomorphism, which is a consequence of +
-f
3.31. Proposition. Let A , {pi} and A, {ql} be free unions of the algebras A, in 9, then there is an isomorphism e :A A such that @, = pp, for all IEL. -f
Proof. By definition of free unions, there exist homomorphisms e : A + A a n d @:A-Asuchthatp?,=~p,andp,=@,,forallIEL. By substituting these equations into one another, we get pi = @pi and = @pi. But the uniqueness part of the definition of a free union forces @ and e&ito be the identity mapping of A and A, resp. Hence e is an isomorphism.
@,
3.32. Remark. A straightforward argument shows that if A, {pi} is a free union of the algebras A, in 9and oI : A, A, are isomorphisms, for all I E L, then A, {pro,}is a free union of the algebras 2,in 9. -f
3.33. If $ is a variety, then there exists a free union, for any family of algebras in P. A proof can be found in COHN[l]. 3.4. Let { A , ]1EL} be a family of algebras of 9. A free union A, {p, I I E L } of the algebras A , in 9 is called a free product of the A, in 9 if every pI is a monomorphism and U (p,A, 11EL) is a generating set of A . This
definition together with Prop. 3.31 implies that if the family {A,IIEL} has a free product in 9, then every free union of this family in 9 is a free product. If there is no danger of confusion we will call the algebra A a free product of the A,. There exist varieties R and families of algebras of R such that the free union of these families in R is not a free product. Conditions for the existence of the free product in R of a family of algebras in R are provided by GRATZER [3].
12
CH.
POLYNOMIALS AND POLYNOMIAL FUNCTIONS
1
4. Polynomial algebras 4.1. Let A be an algebra of the variety % = %($) with Q as its set of operations. To each element aj t A , we will assign a symbol Zj and A will denote the set of these symbols. Clearly, A and Q are disjoint. Let
X = {xi[i E I} be a set of indeterminates disjoint from QUA, and w(AUX) the word algebra over Q. Then W ( X ) is a subalgebra of W(AUx). As in the proof of Th. 3.21, we consider a subset P of w ( A l Jx) xW (JUX) which again will give rise to a congruence 0. Let p be the set of all elements : a) (w, w), w E W ( AUX I , b) (wl(ul, . .., uk), w2(uI, . . . u,)) and (w2(u1, . . ., uk), wl(ul, . . ., u,)) where wl(yl, . . . , y k ) = w2(y1, . . . , y k ) is a law of $, and u1 . . . ukE E W(AUX) c) (mi, 2 ) and (5, mi) where wi is a 0-ary operation and wi = a in A, and (wiZl . . . Z,, 5) and (5, wiZl . . . Z,i) where mi is an ni-ary operation, n, =- 0, and wial . . . ani= a in the algebra A . The corresponding congruence 0 yields an algebra W ( A U X )[ 0 of 23, as in the proof of Th. 3.21. By c), the mapping p :A W(AU X ) [ 0 defined by pa = C(5) is a homomorphism. q is even a monomorphism, for let 5,05,, then there is a chain from 5, to 5,. By replacing each xi in this chain by some arbitrarily chosen ai E A and each Zj by aj, we get a chain of equal elements of A , hence a, = a,. By way of embedding we may thus identify A with FA, and if we write xi instead of C(x,)-and admit the possibility xi = xj for i # j (see the subsequent subsection)-we obtain an algebra A(X, %), with a generating set A U{xi/i E I}, isomorphic to W(AU X ) [ 0 which we will call the 8polynomial algebra over A in the set of indeterminates X, and its elements will be called %-polynomials in X over A , or just “polynomials”. In 9 4.3 we will show that A(X, %) does not depend on the particular choice of $. We summarize these considerations in
-
4.11. Proposition. The polynomial algebra A(X, 23) is an algebra of %,
and A is a subalgebra of A(X, %). The set A U {xiI i C I } is a generating set of A(X, 23). 4.2. The problem arises under what conditions it may happen that two elements of the generating set A U {xi/i E I} are equal, i.e. a, = x,, or
§4
13
POLYNOMIAL ALGEBRAS
x , = x,,, for rn # n. If this is the case, then in W ( A U X ) I O , we have C(ii,) = C(x,,) or C(x,) = C(x,,),whence i@x, or xmOx,,. In a chain 6 from GI to x , or from x , to x,,, we replace each xi by an arbitrarily chosen ai E A and each iij by aj, which shows that necessarily I A ( = 1, so suppose A = {a}.If B is an algebra of 23 containing a subalgebra of order 1, then we can embed A into B and thus obtain an algebra B1 s B having A as a subalgebra. By replacing each iij by aj, x , by a and x,,by b E B1 in the chain 6, and taking arbitrary elements of B, for all the other indeterminates, we obtain a chain of equal links. Thus b = a and I B11 = 1 B 1 = 1. This means that % is a variety such that no algebra A of 23 of order I A1 # 1 contains a subalgebra of order 1. Such a variety will be called semidegenerate. Conversely, let A be an algebra of order 1 of the semidegenerate variety %. Then, since A is a subalgebra of A(X, %), we have j A(X, %)I = I , and in particular all the elements of A U{xiI i E I } coincide. We state this result as 4.21. Proposition. If 1 A 1 = 1 and 23 is a semidegenerate variety, then all the elements of AU{xiliEZ} E A(X, 23) coincide. Otherwise all the elements of AU{xili E I } are distinct,' and in particular, we may identifv X with {xiI i E I}. 4.22. There do exist semidegenerate varieties which are not degenerate. We give two examples : a) The variety of rings with left identity E considered as algebras of type {2, 1, 0,2,0} as in § 2.4. Indeed, if an algebra B of this variety has a subalgebra of order 1, then E = 0 and hence IB I = 1. b) The variety of lattices with zero and identity considered as algebras of type {2,2,0,0}, by the same argument as in a).
-
4.3. Let A(X, 23) be a polynomial algebra, then A(X, 23) is in 23, and A is a subalgebra of A(X, 23). Hence p1 : A A(X, 23) defined by pla = a is a monomorphism. Let F(X, 23) be the free algebra of 23 with free generating set X , if 23 is not degenerate, otherwise the algebra of order 1 of 23. Let q2 :X A(X, 23) be the mapping defined by &xi = x,, then q2 can be uniquely extended to a homomorphism pz: F ( X , 23) A(X, 23).
-
-.
4.31. Theorem. A(X, 'Is), {PI,p ~ is} a free union of fhe algebrm A and F(X, 23) in 23.
14
POLYNOMIALS AND POLYNOMIAL FUNCTIONS
-
CH. 1
Proof. Let B be an algebra of %, and YI A B, Y z : F ( x , -+ B algebra homomorphisms. Let e : A(X. %) B be the mapping defined by ew(ail, . . .,ajk,xi,, . . .,xi,) = w(ylai,, . . . Ylai,, ~)2xj,,. . .,yzxj,). Since A U X is a generating set of A ( X , %), @ is actually defined on the whole of A(X, 8).e is also well-defined, for let wl(ai, ,. . .,aik,xjl, . . .,xi,)= w,(ail, . . ., a,*, xi,, . . ., xi,) in A ( X , %I. Then wI@w2 in w(AUX), by construction of A(X, %), hence there is a chain from w1 to w2. By replacing each iij by ylaj and each xi by yzxjin this chain, we get a chain of elements of B whose links coincide since B is in % and yl is a homomorphism. Thus ewl = e w , . e is, of course, a homomorphism, by definition. Moreover, eplai = y p i , for all q E A , and, if f E F ( X , a), f = w(xjl, . . ., xj,), then e y 2 f = W ( Y ) ~ X ~ ., ,. .,y2xjk)= y,.C i.e. epl = ~ 1 Finally, let o : A(X, %) B be any homomorphism such that yl = opl, 1 = 1, 2. Since 5plnj = ylai = eplai, we have aai = pai, for all nj E A, and ap,xj = y2xj = ep2xjimplies axj = exi, for all xj EX. Thus o and e coincide on the generating set A U X of A(X, %), hence o = e. +
+
4.32. Corollary. The polynomial algebra A(X, %) is independent of the set @ of laws defining 23. Proof. Let $, $* be two sets of laws, each defining 23, and A(X, %), A ( X , %)* the polynomial algebras constructed as in 4.1. Since both A(X, %), {TI,pz} and A(X, %)*, {p;, p l } are free unions of A and F(X, 53)in 23, there is an isomorphism e :A(X, 23) A(X, %)* such that py = epl, i = 1 , 2 by Prop. 3.31. In particular, e fixes AUXelementwise, hence A ( X , 23) = A(X, %)*. +
4.33. Remark. Since p1 is a monomorphism and plAUp2F(X, 23) 2 AU{xiliE I } which generates A(X, %), the free union A ( X , a),
vz
{PI,pz} is a free product if and only if is a monomorphism. But, by no means, is this always the case: Let A be an algebra of order 1 in a
semidegenerate, Lon-degenerate variety %, and let / X I =- I . Then IA(X, %)I = 1, but IF(X, %)I =- 1. 4.4. Let B be an algebra,
D the family of its operations, A a subalgebra and U a subset of B. We will write A ( U ) for the subalgebra [AU U ] of B. If A is a subalgebra and U, V are subsets of the algebra B, then A(UUV) = A(U)(V).
4.41. Lemma.
.
04
15
POLYNOMIAL ALGEBRAS
Proof. A ( U U V ) is a subalgebra of B containing AUU and V , hence A ( U U V ) 2 A ( U ) ( V ) .Also A ( U )( V )is a subalgebra of B containing A and U U V , hence A(U)(V) 2 A ( U U V ) . 4.42. Let A be an algebra of the variety %. An algebra B of % containing
A as a subalgebra is called a %-extension of A . In particular, every %-polynomial algebra over A is a %-extension of A . The role of 8-polynomial algebras over A for %-extensions of A can be seen from 4.43. Lemma. I f A ( U ) is a %-extension of A and X = {xu[ u E U } is a set of indeterminates, xu # x, for u f v, then there is an epimorphism p : A(X, %) A ( U ) such that pa = a, for all a E A , and px, = u, for all xu E X . -+
Proof. The mapping yl : A A ( U ) , yla = a, is a homomorphism. Moreover, the mapping xu u from X t o A(U) extends to a homomorphism y2 : F(X, %) + A(U). By Th. 4.31, A(X, %), { q ~912) , is a free union of the algebras A and F(X, 23) in %, hence there is a homomorphism e :A(X, 23) -4(U) such that y~~= p q l , and y2 = ev2.By definition of p1, p2,we have pa = a, for all a E A , and exu = u, for all x,,EX. Since AU U generates A(U), and A U U E pA(X, %), p is an epimorphism. -+
-+
--f
4.5. Proposition. Let A be an algebra of the variety 23 and I9 :A B an epimorphism. Then there exists exactly one extension of 8 to an epimorphism p : A(X, %) B(X, 23) jixingx. g, in particular, 8 is an isomorphism, then p is an isomorphism. --t
+
Proof. B(X, 2.3) is an algebra of %, and B is a subalgebra of B ( X , %), hence y1 = 8 :A -+ B(X, %) is a homomorphism. Let y z be the extension of the mapping x, xi from X to B(X, %) to a homomorphism from F ( X , 23) to ( B , 8). Since A(X, 23), {pl, p2}is a free union of A and F(X, %), there is a homomorphism p : A ( X , 8).+ B(X, 8)such that y,, = py,,, v = 1,2. Hence pa = 8a, for all a t A , and ex, = xi,for all xi EX. p is an epimorphism since B ( X , 23) = [ B U X ] ,A(X, '23) = [ AU X ]implies that p is unique. If I9 is an isomorphism, then 8-1:B A is an epimorphism Thus op and thus extends to an epimorphism o :B(X, 23) A(X, 8). and pa are the identity mappings, implying that p is an isomorphism. -+
-+
+
4.51. Remark. The unique epimorphism p so constructed will also be denoted by 6(X , %), 8 [ X ]or 8 ( X ) if it is necessary to indicate the dependence of p on 8.
16
cn. 1
POLYNOMIALS AND POLYNOMIAL FUNCTIONS
4.6. Theorem. Let Y , Z be disjoint sets of indeterminates and X = Y U Z . Then there exists an isomorphism 9 : A(X, %) A(Y, 23) (2,%) such that p~ = u, ,for all u E A UX. -+
-
Proof. Clearly A(X, a) = A ( X ) . By Lemma 4.41, A(X, %) = A ( Y ) ( Z ) . By Lemma 4.43, there is an epimorphism e : A ( Y )(2,8) A ( Y )(2)fixing A ( Y ) and Z elementwise, and also an epimorphism 7 :A(Y, %) -, A(Y) fixing A and Y elementwise. By Prop. 4.5, 7 can be extended to an epimorphism cr : A(Y, 23) (Z, 2.3) A ( Y )(Z, %), fixing Z elementwise. Therefore x = eo : A(Y, %) (2,%) A ( Y )(2)is an epimorphism fixing 2, Y , and A elementwise. Conversely, A(Y, %) (Z, %) = A(Y, 8) (2)= A(Y)(2)= A(X). By Lemma 4.43, there is an epimorphism p :A(X, %) A(Y, %) ( Z , Q fixing A, Y , Zelementwise. Hence xp :A(X, 23) A(X, %) fixes A U X elementwise, thus is the identity mapping. Similarly q-x is the identity mapping of A(Y, %)(Z,%). Hence 9 is an isomorphism fixing A UX elementwise. --f
-
-.
+
4.61. Corollary. Let X,, . . ., X,, be pairwise disjoint sets of indeterminates and X = X I UX , U . . . UX,,. Then there exists an isomorphism9 :A(X, %) &XI, 23) . . . (X,,, 23) fixing A UX elementwise. +
Proof. By induction on n, using Prop. 4.5.
4.62. Corollary. If Y is a subset of a set X of indeterminates, then there is a monomorphism from A(Y, 8)to A(X, %), fixing A U Y elementwise. Proof. This is a consequence of Th. 4.6., putting Z = X - Y and restricting p-l to A(Y, %).
5. The lattice of polynomial algebras over an algebra 5.1. Let A be an algebra, SZ the set of its operations, and X a set of indeter-
minates. The polynomial algebra A(X, %) depends, of course, on the variety 8,and this dependence is now to be further investigated. The @-polynomial algebra A(X, $'3) is defined for every variety 23 containing A . Therefore let M be the set of all varieties containing A (M is a set, indeed,since every law over SZ involves just a finite number of indeterminates ;thus taking a countable set Y of indeterminates, we can find every
95
17
THE LATTICE OF POLYNOMIAL ALGEBRAS OVER AN ALGEBRA
law over a in W ( Y ) x W ( Y ) ,and thus every variety with the set l2 of operations is determined by a subset of W ( Y )X W ( Y ) ) . We define a partial order C on M by: B1G 'iE2 if and only if every algebra of Bl is an algebra of B2. 5.11. Proposition. The partially ordered set M = (M; G )of all varieties containing A is a complete lattice, the so-called lattice of A-varieties. Proof. Let {%"I v E I } be a set of varieties of M, and for each v, @, the set of all laws which hold in every algebra of 23". Then 23" = a(&,). %( U($, 1 v E I ) ) and %( ($, 1 v E I ) ) are both varieties of M, and these are the greatest lower bound and the least upper bound, resp., for the given set of varieties. As usual n(B,, I v E I ) will denote the greatest lower bound and U ( % v / v E I )the least upper bound for {Bvl.€I}.
n
5.2. Let P = {A(X, 8) I 8 E M} be the set of all polynomial algebras in Xover A , and =sbe the partial order on P defined by: A ( X , 81) =sA(X, 232) if and only if there is an epimorphism y : A(X, 2 3 2 ) A(X, 231) fixing A UX elementwise. Indeed, reflexivity and transitivity of G are immediate. The antisymmetry of =sfollows from
-
5.21. Lemma. Let A(X, Bl)and A(X, 5 3 2 ) be polynomial algebras in X over A , and 01, O2 those congruences on W ( A U X ) which are used for constructing A(X, Bl)and A(X, %2), resp., according to 4.1. Then A(X, Bl)s A(X, B2)ifand only if012 02.
?ZS) -
Proof. Let A(X, BS,) s A(X, 232) and y : A(X, A(X, %I) an epimorphism fixing AU X elementwise. Then yw(ai, xi) = w(ai, xj). Hence w(q, xj) OZu(iil,x j )in W ( A U X )implies w(iii, xj) 01v(Zi, xJ, thus 0 2 01. Conversely, suppose that 0, G 0,. Since w(a,,xj) = v(a,, xj)in A(X, Bs,) implies w(gi, xj)8,v(iii,3) in W ( A U X ) and hence w(iii, xj)@,v(Ci,xi), we conclude that w(ni, xi) = v(ai,xi) in A(X, BJ7 and V'W(ai7x j ) = w(q, xi) is a well-defined epimorphism y : A(X, B2,) ,4(X, fixing A U X elementwise. Hence A(X, 81) =sA(X, Bz).
-
-
5.22. Theorem. The mapping 23 v:(M; C > (P; s).
--t
A ( X , %) is an order epimorphism
18
CH.
POLYNOMIALS AND POLYNOMIAL FUNCTIONS
1
Proof. Let $,, i = I, 2, be the set of all laws holding in 5Bi,thus Bi= 23(Gi).Let BlG B2,then G12 IfP, G W ( A U X )X W(AUX), i = 1,2, is the set of elements being used for defining Oias in $4.1, then PzC PI, and hence O2 C 01. Lemma 5.21 implies A(X, 231) =sA(X, 232).
a2.
5.23. Remark. In general, g, is not an isomorphism. If, for example, IAl = 1 and is a semidegenerate variety, then A(X, 231) = A(X, B2), for any C %I2.Another, not so trivial, example we will give in 5 9.4. That is not, in general, an isomorphism is, to a large extent, due to
a2
5.24. Lemma. If %1 C ‘23.2, then A(X, ‘$31) = A(X, 232) A(X, Z2) is an algebra of 81.
if and only if
Proof. The “only if” part is obvious. Suppose now that A(X, 8 2 ) is in %I. Since A ( X , %2) is a B1-extension of A , by Lemma 4.43 there is a unique epimorphism x: A(X, %I) A(X, %,) fixing A U X elementwise. By Th. 5.22, there is also an epimorphism y : A(X, %z) A(X, 8 1 ) fixing A U X elementwise. Then yx: A(X, 81) A(X, %I) and xy: A(X, 2 3 2 ) -+A(X, are epimorphisms fixing A U X elementwise, and hence are the identity mappings. Thus y is an isomorphism, and A(X, 231) = A ( X , 232). -+
a2)
-
+
5.3. Proposition. In the partially ordered set (P ;=s),every non-empty subset has a greatest lower bound. If n (B, I v E I ) denotes the greatest lower bound for the subset {BL,j v E I } of P, and p the order epimorphism of Th. 5.22, then v(n(DVjvF 11) = n(p%,,Iv t I ) . (5.3)
This is the consequence of 5.31. Lemma. Let {%%,Iv E I } be a set of varieties of M, and 0, the congruence on W ( A U X )corresponding to %”, v E I. I f 0 is the congruence corresponding to n(%I v E I ) on W ( X X ) , then 0 = U (0?, 1 v E I ) , the least upper bound in the lattice of congruences 011 W ( A U X ) . Proof. Let 8,be the set of all laws in a,,, v t I. By Prop. 5.11, fl(8” I v E I ) = %( U (&, 1 v E I ) ) . VOWholds if and only if there is a chain P are the sets being used for defining O,, 0 as in 4.1, from 2: to w. If P,,,
95
THE LATTICE OF POLYNOMIAL ALGEBRAS OVER AN ALGEBRA
19
then P = U (P,I v E I ) . Hence VOW if and only if there are elements wlr . . ., w , - ~E W ( A U X ) and congruences Oil, . . ., O,, i, E I, such that vOi,wl, w1Oi,wZ,. . ., W , _ , ~ ~ , W . By 3 1.62, this holds if and only if u( U (@,I v EI))w. Hence 0 = U (@,I v E I ) . The proof of Prop. 5.3 can now be established. Let { A ( X ,23J j v E I } be a subset of (P; =s). Since n(233,1vE Z) C Bv, by Th. 5.22, A(X, f’ (a,,1 v E I ) ) s A(X, %,), v E I . Let A(X, 23) E P such that A(X, 23) =G A(X, 23J, for all v E I , and 0 be the congruence corresponding to 8. Then, by Lemma 5.21, 0 2 O,, for all v E I , thus 0 1U (0, Iv E I). Again, by Lemma 5.21, and Lemma 5.31, A ( X , 23) =s A ( X , ri(23vl 1’ E I ) ) . Hence A(X, (23,12, E I ) ) = ( A ( X , 23”)I E I ) .
n
n
5.32. Theorem. The partially ordered set (P; 6 )of all polynomial algebras in X over A is a complete lattice. Proof. By a well-known theorem of lattice theory, a partially ordered set is a complete lattice if it has a greatest element, and every non-empty subset has a greatest lower bound. By Prop. 5.3, the second condition holds for (P; s ) .Moreover, if 9 A %(4)is the variety defined by the empty set 4 of laws - A E D obviously-then A(X, D)is a greatest element of (P; ==). 5.4. Remark. So far we have been interested in what happens to the polynomial algebra A(X, 23)if the variety 23 varies. But we may also vary the Q) family SZ of operations and ask what happens to A(X, 23). Let D = (D;
be an arbitrary algebra withnas its family of operations, and@a subfamily and if is a set of laws over Q, &, shall of Q. We set D, = (D;@), denote the subset of consisting of the laws where only operations of @ areinvolved. Then 23($,)is a variety over@,and if A is in then A, is in A(X, contains A , as a subalgebra, hence A d , ( X ) is a subalgebra of A(X, B($)), belonging to By Lemma 4.43, there is an epimorphism from A,(X, to A,(X), or, as we may put it, a homomorphism from A,(X, B(&,)) to A(X, %(&)), fixing AUX.
%?(a,).%(a)),
%(a), %(a,). %(a,))
20
POLYNOMIALS AND POLYNOMIAL FUNCTIONS
CH.
1
6. Functions and polynomial functions on algebras 6.1. Let A be an algebra, 9 its family of operations, and k a positive integer. By Ak we will denote the Cartesian product of k copies of A . A k-place function on A is a mapping from Ak to A . The set of all k-place functions on A will be denoted by F,(A). On Fk(A) we now define the operations wi E Q by "fl
..
*
P&lr
. . .,
= w;y&,,
..
0
,
ak)
. . . PEi(U1, . . ., a J ,
for every(a,, . . ., ak) E Ak, if ni =- 0, and wi(u,, . . ., ak) = wi, for every (al, . . ., ak) E Ak, if n, = 0. We call ( F J A ) ; Q) = Fk(A) the full k-place function algebra over A . Clearly, Fk(A) 2 n ( A I v E Ak), the direct product of IAkJcopies of A. Hence Th. 2.51 implies 6.11. Proposition. I f A is an algebra o f t h e variety 23, so is Fk(A). 6.12. The function p E F,(A) defined by ~ ( u , ,. . ., ak) = c, for all (al, . . ., ak) E Ak, is called the constant function with value c, denoted by x,. c ~t,is a monomorphism sc : A F,(A). Thus the set C,(A) of all constant functions of F,(A) is a subalgebra of F,(A), and A can be embedded into F,(A). Henceforth Fk(A)will always be understood as the algebra obtained from this embedding. +
--t
6.2. Let Ei E Fk(A), i = 1,2, .. ., k be the functions defined by [,(a,, . . .,a,) = a;, for all (a,, . . .,uk) E Ak. ti is called the i-th projection of Fk(A). A([,, . . ., 5,) is a subalgebra of Fk(A), called the algebra of k-place polynomial functions on A . We will write Pk(A)for A ( [ , , . . ., tk). By Prop. 6.11 and Th. 2.51, we get 6.21. Proposition. I f A is an algebra of the variety %, so is Pk(A). 6.22. Remark. It follows from the definition of Fk(A)that Fk(A,)=Fk(A),. Since P,(A,) is the subalgebra of Fk(A,) which is generated by A, and the projections and P,(A), is a subalgebra of Fk(A), = F,(A,) which contains A , and the projections, Pk(A,) is a subalgebra of Pk(A),. 6.3. Let A be an algebra of the variety %, X = {xl, . . .,x k } a set of indeterminates, and A(X, %) = A(xl, . . ., xk,%) the %-polynomial algebra
06
21
FUNCTIONS AND POLYNOMIAL FUNCTIONS ON ALGEBRAS
in X over A. Let B be a %-extension of A and (b,, . . ., bk) a k-tuple of elements of B. Let y1: A B, y l a = a, and y2: F(Xy23) B be the extension of the mapping x, b,, from X to B, to a homomorphism. Using the same argument as in the proof of Lemma 4.43, we see that there is a homomorphism p: A ( X , 8) B such that pa = a, a E A, and ex, = b,, u = 1, . . ., k. We summarize this in +
-+
--f
-.
6.31. Proposition (Substitution principle). Let A bean algebra of the variety 8,(x,, . . ., xk} a set of indeterminates, and (b,, . . .,bk) a fixed k-tuple of elements of the %-extension B of A. For every polynomial p E A(x,, . . .,xk, %), define e p E B as: e p = w(di,b,, . . .,bk) where p = w(di,x,, . . ., xk) is any representation of p as a word in elements di E A and xi. Then pp is well-defined and the mapping p pp is a homomorphism from A(xl, . . ., x k , %) to B.
-.
6.32. If p E A(x,, . . .,x,, %), then pp E B is called the value of the polynomial p at the place (bl, . . ., bk), and is denoted by p(b,, . .., bk). 6.4. In Prop. 6.31,k t B = Pk(A) and (b,, . . ., bk) = (&, . . ., &), then we obtain a homomorphism a: A(x,, . . ., xk, %) Pk(A) such that oa = a, a E A, and axi = &, i = 1, . . .)k. Since Pk(A)is generated by AU{t,, . ..,Ek}, a is an epimorphism. Let p = w(di,x,, .. ., xk) be a representation o f p as a word, then, by Prop. 6.31, and the definition of Q on Fk(A), we have (up)(a,, . . . ak) = w(d,, 5,, . ..,t k ) (al, . ., ak) = w(di,a,, . . ., ak) = p(a,, . . ., ak). Hence we have proved
-
.
6.41. Proposition. The mapping a: A(x,,
. .., x k , %)
-
Pk(A)defined by
is an epimorphism, the so-called canonical epimorphism.
6.42.One may raise the question: Can it happen that rs is an isomorphism? Since A(X, 23) is defined for every variety 23 containing A , we have to consider the lattice M of these varieties. Let C?+!be the set of all laws in the countable set Y of indeterminates holding in A , then clearly %((21)is the least element in M.
22
POLYNOMIALS AND POLYNOMIAL FUNCTIONS
-.
CH.
1
6.43. Lemma. ZJ for some %€EM, the canonical epimorphism a1 : A(x,, . . ., xk, 23) Pk(A) is an isomorphism, then the canonical epiPk ( A ) is an isomorphism, too. rnorphism a2 : A(xl, . . .,xk,
%(a)) --f
Proof. By Th. 5.22, there exists an epimorphism y :A(X, %) A(X, which fixes A.UX elementwise. Then a2yo;' : P k ( A ) -.. pk( A ) is an epimorphism which fixes AU{E,, . . .,&} elementwise. Hence o2yai1 is the identity mapping. Thus a2 is a monomorphism, hence an isomorphism. +
%(a))
6.44. Proposition. If the set of 0-ary operations on A generates A , then the -* Pk ( A ) is an isomorcanonical epimorphism (T : A(x,, . . .,xk, phism.
%(a))
Proof. Let p = w(a,, x,l . . ., xk) and p1 = w,(ai, xl, . . ., xk) be polynomials of A(X, such that ap = op1, By definition of a, w ( q , tl,. . ., t k ) = wl(a,, tl, . . .,&). By assumption, the elements ai can be represented as words in the 0-ary operations of A. Performing this substitution, we obtain an equation v(&, . . .,t k ) = vl(t13 . . 6 k ) where w, v1 are words over SZ in El, . . .,t k . We evaluate this polynomial function for (al7 . . ., ak)E Ak. Then w(a,, . . ., ak) = zil(al, . . ., uk), for all (al, . . .,ak) E Ak, i.e. ~ ( y , ,. ., y k ) = vl(yl, . . .,y k ) is a law of A and thus is in (21. Since A(X, is an algebra of v(x,, . . ., xk)= vl(xl, . . ., xk) in A(X, %(U)),and since A is a subalgebra of A(X, %((21)), we have w(a,, xl, . . .,xk) = wl(a,, xl, . ., xk). Hence (T is an isomorphism.
%(a))
- 7
.
%(a))
%(a), .
7. Normal forms of polynomials 7.1. Let A be an algebra of the variety 23, and A(X, %) the %-polynomial algebra in X over A . By Prop. 4.11, A(X, %) = [ A U X ] ,so every polynomial p E A(X, 23) has a representation p = w(a,, xj) as a word in elements a , € A and x j € X . In general, p will have various different representations of this kind. For practical reasons, in order to control the computations in A(X, 23) completely, one wants a set % of words w(ai, xj) where each element of A(X, %) is represented exactly once. But this would still be too little for computations since one also wants to find, in a finite number of steps, the representation of the element
08
POLYNOMIALS OVER COMMUTATIVE RINGS WITH JDENTITY
23
wiwl . .. wnrf A(X, 23) by a word in ’$2, for all operations wi, and all words wl, . . ., wniE %. Thus we define : A set % of words in W ( A U X ) is called a normal form system for A(X, 23) if a) every p E A(X, 23) is represented by exactly one word in 8, b) for any ni-ary operation mi E 9 and every n,-tuple of words wl, . . .,wniE 8,it is possible to find the word in % representing wiwl. . . wniE A(X, 23), in finitely many steps. If p is a polynomial of A(X, 23), the word w(ai,xj) in % representing p is called the normal form of the polynomial p (with respect to %).
Having a normal form system of A(X, 23) at hand, we can master the computation in A(X, 23) completely. If we know even the normal forms of the polynomials ui and xi, we can find the normal form of p E A(X, 8) in a finite number of steps, for any presentation p = w(ai,xi)as a word in a, and xi. Thus we can always decide in a finite number of steps whether or not two words of W ( A U X ) represent the same element of A(X, %), and so have solved the “word problem” in A(X, 23). The subsequent sections will list some normal form systems for various important polynomial algebras A(X, 23) by use of 7.11. Lemma. A set % of words in W ( A U X ) is a normal form system of A(X, 23)provided a) for every representation p = w(ai,xi) of an element p E A(X, ‘23) us a word in certain elements of A U X , one canfinda word in ‘$2 representing p , in a finite number of steps; b) any two dzrerent words of ‘$2 represent dlfSerent elements of A(X, 8). The proof of this lemma is clear, and follows straight from the definition of a normal form system.
8. Polynomials over commutative rings with identity
8.1. Let 9 be the family of operations {w1,w2,w3,w4, wg} of type (2, 1,0,2,0}, and 23 the variety of commutative rings with identity, these rings being considered as algebras ( A ; 9 )as in 5 2.4. Thus o1is the addition, w 2 the operation of forming the additive inverse, w3 the zero,
24
CH. 1
POLYNOMIALS AND POLYNOMIAL FUNCTIONS
the multiplication, and w5 the identity. As usual, we write +, -, 0, -, and 1 for these operations, and use infix notation for and .. Let R be an arbitrary commutative ring with identity, and X = { x } a oneelement set of indeterminates. The polynomial algebra R(X, %) will be denoted by R [ x ] ,X3 being kept fixed. Our objective is to obtain a normal form system for R [ x ] . For the sake of notational simplicity we define inductively: If (for any X ) wl, w2, . . ., w, are words in W ( R U X ) , then w,+ . . . +w, means (w,+ . . . +w,,-,)+w,, and wl... w, means (w, .. . wnP1)w,. If w1 = . . . = w, = w, we write wl... w,,= w", and set wo = 1 . Then ( a n y ) + . . . (alx)+ao, a, E R, v = 0, . . . , n, is a well. . . +alx+a,. defined word of W ( R U X )which we will write as a,?+ 04
+
+
8.11. Theorem. Let % betheset of all words a n y + . . . +a,x+ao where n a 0, a,€ R , for t = 0, . . ., n, and a,, # 0. Then %U{O} is a normal form system of R [ x ] . Proof. We will show that % satisfies the conditions of Lemma 7.11. a) We have to show that, for every representation p = w(a,, x ) of an element p E R [ x ] ,we can find a word in % representing p , in a finite number of steps. We proceed by induction on the minimal rank r of w ( a , x). The words of minimal rank 0 are 0, 1, a, and x, a E R, and since x = l x f O in R [ x ] , our assertion is true for r = 0. Suppose, that the assertion has been proved for words of minimal rank r G m. Every word of minimal rank m+ 1 is of one of the types w1+w2, -w1, wlwz where w1, w2 are words of minimal rank less or equal to m. By induction hypothesis, we can find, in a finite number of steps, words vl,v2in % representing the same elements of R[x]as w1, w2,and applying the laws of %, we can find, in a finite number of steps, words in % representing the same elements of R [ x ] as ul+vz, --G~,and vluz. b) We have to show that any two different words of % represent different elements of R [ x ] .For this purpose, let S be the set of all infinite sequences (ao,al, . ..) of elements in R where a, # 0 just for finitely many indices v. In S we define an operation + by
+ is a binary operation on S . Furthermore, we define an operation .on S
by where
.
25
POLYNOMIALS OVER COMMUTATIVE RINGS WITH IDENTITY
08
(a,,
0 1 ,a2,
. . .) (bo, bl, b2, . . .) = (co,CI, c2, . . .)
c, = aob,+a,b,.-,+a,b,-2+
. .. +a,b,,
v=0,1,2,
...
is really a binary operation on S since c, = 0, for Y > r+s, if a,, = 0, for v =- r, and b, = 0, for v =- s. With respect to +, -,S is a commutative ring with identity : Clearly ( S ; ) is an abelian group, and is associative, commutative and distributive with respect to + ; (1,0,0, . . .) is the identity. Thus S is an algebra ( S ;9) of 23. The mapping 6 : R S , #a = (a, O,O, . . .) is a monomorphism of algebras, hence we can perform the embedding of R into S and obtain a ring S1 S which contains R as a subalgebra. Let 5 = (0, 1 ,0 ,0 , . . .)E SI. Induction on n shows that E" = (0, 0, . . .,0,1,0, . . .), the sequence with 1 a t the place with index n and 0 elsewhere. Then
.
+
-
(u,, a,, a2, . . .,a,,, 0, 0,
. . .) = a, + alE + a,S2+ . . . +a,,?,
i.e. S , = R(5).By Lemma 4.43, there is an epimorphism such that QX = E and pa = a, a E R. Hence ~ ( a , , ~ + a , , - , x ~ -.~. .+ +a,x+a,);=
a,,5"+an-,E"-l+
e : R [ x ] S1 +
. . . +a,~+a,,.
Let e : S -,S1 be the embedding isomorphism. Then e-'e(a,x"+
a,,-,X"-l+
...
+a,x+ a,)
= (ao,a,,
. . ., a,,, 0, 0, . ..).
We conclude that different words in %U{O} represent different elements of R [ x ] . 8.2. Let R again be a commutative ring with identity, considered as a n algebra of 93 and suppose that X = (xl, . . .,x k } .R(X, 8)will be denoted by R[x,, . . .,xk].We want to find a normal form system for R[x,, . . .,xk], but first of all simplify our notation. Let N be the additive semigroup of non-negative integers equipped With its natural total order 4, and Nk the direct product of k copies ofN, lexicographically ordered by s.Then it is well known that through G, Nkalso becomes a totally ordered semigroup. If L = (i,, . . ., ik) EN^, we will set C(L) = i,+i,+ . . . +ik. and if (x,, . . ., xk) = F is a k-tuple of arbit. . . x$ is well-defined, we will write rary elements for which ~ 1 x 2 . . x'Ix'2 . .. x t = F'. Thus, if x,, . .., x k are elements of a commutative semigroup with identity and 1,;. E Nk, then zL@=
26
POLYNOMIALS AND POLYNOMIAL FUNCTIONS
CH.
1
8.21. Theorem. The set % of all words C(an$I il E P)where P is an arbitrary jinite subset of Nk, a, E R , aA # 0, for all ;1E P, the summands written with decreasing order of A, is a normal form system for R[xl,x,, . . ., xk] where z(a,$ I ;1 4 ) = 0 is additionally dejined. 8.22. Remark. For k = 1, this normal form system is different from the system of Th. 8.1 1 in so far as the words a,xv, where a, = 0, of 8.11 do not appear in % while a. of Th. 8.11 is replaced by a d . 8.23. Proof of Th. 8.21. We are going to show that the conditions a), b) of 7.1 are satisfied for %. That b) holds is a consequence of the laws in 23. We will prove by induction on k, that every p E R[x,, . . .,xk]is represented by exactly one element of %. Th. 8.1 1 implies that our assertion holds for k = 1. Suppose the theorem is true for k-1. Since our theorem is true for one indeterminate, one can find a normal form system for R[x,, . . ., Xk-11 [xk] of the form c(aikxkIik E Q) where Q is a finite subset of N , aikE R[xl, . . .,xk-,], ui, # 0, and the summands are written down with decreasing order of ik. By induction, R[x,, . . ., X k - , ] has a normal form system of the kind the theorem asserts. Hence if we replace each aik by that word of this normal form system which represents the same element, we see that every element of R[xl, . . .,Xk-11 [xk] can be represented by exactly one word of %. But, by Th. 4.6, %is also a normal form system for R[x,, . . .,xk]. 8.3. The normal form system of Th. 8.21 allows us to introduce some frequently-used concepts : a) A polynomial f E R[xl, . . ., xk] is called monic in x1 if, in its normal form x(a,$IA E P),the first summand is 1xpxi . . . x i . b) A polynomial f E R[x,, . . .,xk] is called a form of degree m if, in its normal form c(a,$ I A E P),.(A) = m, for all il E P. The polynomial 0 is a form of arbitrary degree. One can easily verify that the product of two forms of degree m and n resp. is a form of degree m n. Forms of degree 1, 2, 3 resp. are called linear, quadratic and cubic forms. Every polynomial f # 0 of R[x,, . . ., xk] can be represented uniquely as a sum of forms qi # 0 of different degrees i. This is immediate from comparing the normal form for f and the forms qi.
+
09
21
POLYNOMIALS OVER GROUPS
c) Let f E R[xl, . . ., xk], and C(aAxA I A E P) its normal form. We define the degree [ f ] off by [ f ] = max (a(A)l L EP). [f ] is a well-defined non-negative integer, for all f # 0. As a consequence of this definition when regarding the laws of '$3, we get that f # 0, g # 0, andf+g # 0, f g # 0 resp. implies tf+gl== max ([fl, [gl), [fgl [fI+ [gl resp. A ~ 0 1 ~ nomial of degree 1 is called linear. 8.31. Proposition. If R is an integral domain, so is R[xl, for any two polynomials f # 0,g # 0, [f g ] = [f ] [g].
+
. . .,xk], and then,
Proof. Using Th. 8.21 and the laws of '$3, R[xl]is an integral domain, hence, by induction, R[xl, . . ., xk-l][xk]is an integral domain, and so is R[xl, ..., xk], by Th. 4.6. Let [ f ] = m, [g] = n, so that . f = q,+ qm-i,+ . . ., g = qn+qn++ . . . represent f and g as sums of forms q, # 0 and qv f 0 of different degrees v. Since q, # 0, ijn $ 0, and qmqnis a form of degree m+n, fg can be written as fg = qmqn+ . . ., i.e. as a sum of non-zero forms of different degrees, for R[x,, . . .,xk]is an integral domain. Hence [ f g ]= [ f ] [gl.
~,+,-,,+
+
9. Polynomials over groups 9.1. Let SZ be the family of operations (w1, w z , 023) of type (2, 1,0}, and % the variety defined by the laws for groups as in 2.4, the groups being regarded as algebras ( A ; SZ). Thus w1 is the group multiplication, w2 the operation of forming the inverse, and w3 the identity. As usual, we will 1 for these operations and use infix notation for and a-1 write , for -'(a). Let G be a group and X = {x} a set of indeterminates consisting of one element x. As in $ 8 we will write G [ x ] for G(X, %), and, for words wl, . . ., w, in any indeterminates, give w1 . . . w,, and w", n a 0, the same meaning as in 6 8. Moreover, if n -= 0, then we will write w" = (w-l)-". a,
9.11. Theorem. The set 2 of all words a,x'"alx"2 . . . a,_,x".a, where r is a non-negative integer, each ni a non-zero integer, a, E G, t = 0, 1, . . .,r, and a, # 1, t = 1,2, . . .,r - 1, is a normal form system of G [ x ] . Proof. We will prove the theorem by means of Lemma 7.11.
28
POLYNOMIALS AND POLYNOMIAL FUNCTIONS
CH.
1
a) We show by induction on the minimal rank of the word w(a,, x) that, for every representation p = w(a,, x) of an element p E G[x], we can find a word in % representing p , in a finite number of steps. If r = 0, by x = lx 1 this is certainly true. Suppose the assertion holds for r -= -- m. Every word of minimal rank m + 1 has the form wlw2 or w;‘ where w1, w2 are words of minimal rank =s m.Using the induction hypothesis and the laws of 23, we can find words in % representing w1w2 and wF1, in finitely many steps. b) We have to show that no two different words in %represent the same element of G [ x ] . For this purpose, let S be the set of all sequences (ao, n,, a,, n2, . . ., a,-,, n,, a,) where Y 0, every n, is a non-zero integer, a, E G , t = 0, ..,r, a, # 1 for t = 1, . . . ,P- 1. We define a binary operation on S : (ao,n,, p17 b17 . . . ,b,-,, P,, bi) is . . .,a,-,, n,, that element of S we get from “reducing” the sequence (ao, n,, a,, . . ., a,-,, n,, a,, b,, p,, b,, . . ., b,-,, p,, b,). Here “reducing” means that we apply the following rewriting process: If (ao,n,, . . ., n,, arbo,p,, . . ., bs-,, p,, b,) is an element of S , then this is the reduced sequence. Otherwise, arbo = 1 and we rewrite this sequence as (ao,n,, . . ., n,+p,, b,, . . ., bs). Either this is an element of S , then this is the reduced sequence, or n,+p, = 0. Then we rewrite the latter sequence as (ao,n,, . . ., a,-,b,, . . ., bs), and continue this process which has to terminate after a finite number of (rewriting) steps. We obtain then a well-defined element of s. Clearly, the sequence (1) is the identity with respect to ., and (a;,, -1 -n,, a,-,, . . ., -n,, a;,) is an inverse of (ao,n,, . . ., a,-,, n,, a,). S will be a group as soon as the following lemma is established :
.
9.12. Lemma. The operation
- on S is associative.
Proof. We have to show that
(u1u2)u3
= ul(uZu3), for arbitrary elements
u, = (ao7 n2, * * * n,, u2 PI7 b,, p27 . ’ ‘ 3 ’ S - 1 , P S , and u3 = (co,q,, c,, q2, . . ., c,-~, qr, c,) of S . Suppose the lemma holds for u2 = tc, u2 = p, and all ulr u3. Then [u1(a,f?)Iu3= [(ula)Blu3= (ula)(&) = U I [ C C ( ~ U ~=) ]u1[(a@)u3], i.e. the lemma holds for u2 = tcp and all ul, us. Suppose now that the lemma is true for all 242 = (bo), all 242 = (l,p,, l), and all u1, us. We claim that the lemma holds for all u1, u2,u3. For s = 0, the lemma holds by hypothesis. Since (bo,PI,b l ) = [(bo)(l,p l , 1)](61), the preceding argument proves the case s = 1 . 9
09
29
POLYNOMIALS OVER GROUPS
Suppose the lemma is established for s = 0-1, then (bo,pl, bi, . . . p,, b,) = (bo,p,, . . .,p,-,, b,-,) (1, p,, b,) and the preceding arguments prove the case s = a. Let u2 = (bo), then ( ~ 1 ~ 2 ) ~ (3 ~ 0 n,, , . . ,arboco, 41, . . .,C,) ~ 1 ( ~ 2 ~ where 3 ) the arrows denote the rewriting process. Hence ( ~ 1 ~ & 3= ~ l ( ~ Z u 3Let ) . uz = (l,pl, 1). Four cases are to be distinguished: a) a, z 1, c0 # 1. Then (~1uz)u3= ( ~ 0 n1, , . Co, 41, . . C,> =
-
---t
-
+
-3
ul(uZu3)-
-
b) a, = 1, co f 1. Then u p 2 = (ao,n,, . . ., n,+p,, 11, for n, # -ply and u,uz = (ao,n,, . ..,n,-,, a,-,), for n, = -pl. Thus (uIuZ)u3 (ao,n,, . ..,n,, a,, p,, co, . . ., c,) ~ 1 ( ~ + 3 ) since u2u3 = (1, pl, co,
-
. . .,c,).
c) a, # 1, co = 1. We apply a similar argument as in b). d) a, = 1, co = 1. Then ulu2 = (ao,n,, a,, . . .,n,+p,, I), for n, f -pl, and uluz = (ao,n,, a,, . . ., n,-,, a,-,), for n, = -pl. Furthermore uZu3 = (I, pl+ql, cl, .. .,q,, c,) if q1 F -pl, and u2u3 = (~1,q 2 , czy . . CJ if q1 = -pl. Thus (u1u2)u3is obtained from (ao,n,, . .., n,+pl, 1, ql, . . .,q,, c,) by reduction, for n, # -pl, and from (ao,n,, . . .,ar-13 q1, c,, . . ., q,, c,), for n, = -pl. u , ( u ~ u ~is) obtained from (ao,n,, . . ., nrr Lp,+q,, c1, . . .,qt, c,) by reduction, for q, # -pl, and from (ao,a, . . * a,-,, n r , ~ 1 ,* * .> qt, c,), for q1 = Comparing (uIu2)u3with u1(u2u3)in all the four possible cases, our equation holds, and Lemma 9.12 is proved. .?
9
9.13. We complete the proof of Th. 9.1 1. Let us regard the group S as an algebra ( S ; Q) of 23. a + (a) is a monomorphism from G to S , thus we can embed G into S , and obtain a group S1 s S containing G as a subalgebra. Let (1, 1, 1) = 5. Then clearly (ao,n,, a,, n2, . . ., a,-,, n,, a,) = uopal$2 . . . pa,. Hence S, = G(5). By Lemma 4.43, there is an epimorphism e : G[x] S, such that ex = 5 and eu = a, a E G . Then e(u0xn1. . . x"~'a,)= a0Y1. . . p a , whence, if F : S-+ S1 is the embedding isomorphism, e-'e(aox"1 . . . x"~,)= (ao,n,, . . . ,n,, a,). This establishes the theorem. -+
9.2. Let G be a group we will regard as an algebra of 23 and suppose X = {x,, . . ., xk}. We write G(X, 53) = G[x,, . . ., xk] and want to find a normal form system for this algebra. Again we first simplify our nota-
30
POLYNOMIALS AND POLYNOMIAL FUNCTIONS
CH.
1
tion. Let M be the additive group of integers and M k the direct product of k copies of M . (0, 0, . . .,0) E M k will be denoted by 0. An ordered pair (i,, . . ., i,), (Z, . . ., 1,) of elements of M k is called reducible if, for some index v, i, f 0, iv+l = iV+,= . . . = ik = ll = 1, = . . . = lvPl = 0. If L = (il, . . ., i k ) EM^ and (x,, x,, . . .,xk) = t: is a k-tuple of arbitrary elements such that xp . . . xk is well-defined, then we write xt . . . xp = F'.
9.21. Theorem. Let % be the set of all words aoealp. . . ar-lx a'ar where r is a non-negative integer, A, EMk, A, # 0, v = 1, 2, . . ., r, a,cG, v = I , . . ., r, and a,, = 1, for 0 -= v -= r only if A,, is not reducible. Then % is a normal form system for G[xl, . . .,xk]. The proof is along the same lines as that of Th. 8.21. Condition b) in the definition of a normal form system is satisfied for as a consequence of the Iaws of 8.By induction on k, we prove that every p E G[x,, . . . ,xk] is represented by exactly one word of %. Th. 9.11 implies that the assertion holds for k = 1. Suppose this assertion is true for k - 1. Then we can find a system !Jt1of words representing each element of G[xl, . . .,xke1][xk]exactly once, namely is given by the set of all words aox;fiu,$ . . . a,-,x",.a, where r is a non-negative integer, ni are non-zero integers and every a, runs over the normal form system for G[xl, . . ., x , - ~ ]of our theorem which exists by induction, but a, # 1 for t = 1, . . . r-1. By inserting factors x: at suitable places in these words we obtain, for each element of G[xl, . . ., Xk-11 [xk],a word in % representing this element. Let us assume that different words W I , w2 in % represent the same element of G[xl, . . ., Xk-11 [xk]. Omitting all factors x: and xyxi . . . and inserting a factor a, = 1 in front of every factor x;I where v # 0 in these words, we get words vl, v 2 of %I representing the same elements as w l , w,. Hence v1 = v 2 which is a contradiction since w1 # w2 implies v1 # vz. % is therefore a normal form system for G[xl, . . . ,Xk-11 [x,] and, by Th. 4.6, also one for G[x1, . . ., xk].
9.22. Corollary. Let X = {xl, . . ., xk}and y,, q ~ , be the homomorphisms as dejined in $4.3. Then G(X, %), {pl, yz>is a free product of the group G and the free group F(X, 8)in the variety 8 of groups. Proof. As stated in 4.33 we have just to prove that v Z is a monomorphSince F(X, 8)is generated by X,we have v1 = ism. Let 211, v z E F(X, 8). xy;$; . . . $;. Thus we can write v1 = 1 ~ " l ~. " . . ~ ' 3 1 where A,, E Mk,
0 10
POLYNOMIALS OVER LAlTICES AND BOOLEAN ALGEBRAS
31
.. ., s, and no pair A", AYfl. is reducible. Similarly, vz = . lgF1lxp*. . . ~"'1.By Th. 9.21, rpzvl = rp2vzimplies w1 = 'u2.
2, # 0 , v = 1,
9.3. Let be the family of operations of 8 9.1, 2'33 the variety of abelian groups regarded as algebras ( A ; and X = {xl, . . .,xk}. The problem of finding a normal form system for A ( X , D), A E D, is solved by
a),
9.31. Proposition. The set % qfall words form system for A(X, 23).
4, 1 E Mk,a E A, is a normal
Proof. That, in a finite number of steps, every representation p = w(ui,xi) of an element p E A(X, B) leads to a word of IXz representing p , is a consequence of the laws of B. That different words of % represent different elements of A(X, B) follows from an argument similar to that in the proof of Th. 9.11 by considering the abelian group of all pairs (a, A), a E A , ilE M k , with respect to componentwise composition, i.e. the direct product A x M k . 9.4. We are now in the position to give another example-as
announced in 5 5.23-that the epimorphism of Th. 5.22 need not be an isomorphism. Let A be the group of order 1, then A belongs to '2%and also to %, and D c 8.A normal form system for A(x, %) is given by the set of all words 1Y1l,by Th. 9.1 1, nl z 0 integral. Hence A(x, %) is also in B, and Lemma 5.24 implies A(x, 23) = A(x, G).
10. Polynomials over lattices and Boolean algebras 10.1. Let SZ be the family of operations (wl,wz,0 3 , 0 4 ) of type {2,2,0,0}, and % the variety of distributive lattices with zero and identity considered as algebras ( A ; Q); here w1 is the union, w2 the intersection, 0 3 the zero, and o4the identity. As usual, we use infix notation and write U , n, 0, 1 for these operations, while == shall mean the partial order relation on these lattices resulting from these operations. Let D be a lattice of %, X = { x } , and set D(X,%) = D [ x ] . 10.11. Theorem. The set 2' 2 of all words ( a n x ) U b . a, b E D , a a b, is a normal form systemfor D [ x ] .If w1 = ( a l n x ) U b l and w2 = ( a 2 n x ) U b 2 are the representations of w1, w2 E D [ x ] in %, then w1 s wz if and only if a1 s a2 and bl s bz.
32
POLYNOMIALS AND POLYNOMIAL FUNCTIONS
CH.
1
Proof. We will use Lemma 7. I 1 to prove the theorem. a) We show by induction on the minimal rank of w(ai, x) that, in a finite number of steps, for every representation p = w(a,, x) of an element p E D[x], we can find a word of r32 representing p . Let dE D, then d = (dnx)Ud, and x = (1 nx)UO, and thus the assertion holds for minimal rank 0. Suppose the assertion is true for words of minimal rank s rn. Every word of minimal rank m f 1 has the form wlUw2 or w1 w2 where w1,w2 are words of minimal rank G m.By induction, in a finite number of steps, we can find words (alnx)Ubl and (a2nx)Ub2 of C,n representing the same elements as w1 and wz,resp. Using the laws of %, we have [(ad7 x)Ubl]U [(a2nX)Ub2] = [(alUa2)nx]U(blUb2) and a ~ U Z= a blUb2 ~ since al == bl, a2 3 b2. Furthermore [(al nX)UWnm2n x)ub2i = = [[(al na2)u(bl
n a2)u(alnb2)1n
u(bl nb2)
=[[(~~~b~)n(a~~b~)1nx]~(~~nb~)
and
= [(al
ulna2
na2)nXI u(bl nb2)
blnb2.
b) We have to show that no two different words of r32 represent the same element of D[x]. Let o :D[x] -+ P1(D) be the canonical epimorphism. By 6 6.4, ax = 5, the identity mapping of D and ad = d, d E D. Let (a1nx)Ubl = (a2nx)Ub2 in D[x]. Then (aln5)Ub1 = (a2n5)Ubz, for all 5 f D. By substituting 5 = 0, we get bl = b2, and, for 5 = 1, a,lJb, = a2Ub,, but bj =sai, i = 1,2, and hence a, = a2. c) Let w1 = (alnx)Ubl and w 2 = (a2nx)Ub2be two elements of Dfxl represented by words in 8.Then w1 Uw z = [(alUa2) fl XIU (bl Ubz), by a), which is also the normal form for wlUw2. Since w1 e w2 if and only if w 1 U w 2 = w2,the second assertion of the theorem is true.
10.12. Remark. a : D[x] -+ P1(D) is, in fact, an isomorphism. This is an immediate consequence of Th. 10.11 and part b) of its proof. Since the 0-ary operations are, in general, far from being a generating set for D, we see that the condition of Prop. 6.44 is not necessary for u being an isomorphism.
10.2. Let D again be a distributive lattice with zero and identity, regarded as an algebra of 23, and let us write D(x,, .. ., x,, 23) = DIx,, . . ., xJ.
0 10
33
POLYNOMIALS OVER LAlTICES A N D BOOLEAN ALGEBRAS
We want to find a normal form system for D[x,, . . ., x k ] . For this purpose, we first introduce some new notations: As before, we define w, U w, U .. . U w, and w, nw, n . . . nw,, inductively. Let P be the power set of the set (1, 2, . . ., k } and C the set theoretical inclusion. Let Q(r) be the subset of P consisting of all subsets of (1, 2, . . ., k } of cardinality r, 0 =s r =s k. Each set of Q(r) is ordered by the natural order of its numbers. This yields a total order e on Q(r) when injecting Q(r) into the totally ordered direct product N' of r copies of the semigroup of non= n(xiI i B. S,), negative integers. Let S, = {Il, . . .,I,} f Q(r), and the xi written down according to the natural order of the numbers i. Here n ( x J i f 6 )means the element 1 . With this notation we state
n($,)
10.21. Theorem. A normal form system of D[xl, . . ., xk]is given by the set 112 of all words of the form Ulk_oU (aNrnn ( N , . ) N, E Q(r)) where the N,EQ(r) are written down according to the total order relation =s on Q(r), and ah: 3 aN,+*, for N, C N,,,. If w, w E D[x,, . . ., xkl are represented by the words of 112 with coeflcients aN, and b , , resp., then v =s w if and only v a r y ,=s b , , for all N , f P.
1
Proof. By induction on k . By Th. 10.11, all assertions are true for k = 1. Suppose that all the assertions are true for k - 1. a) Let p = w(a,, xi) be any representation of p E D[x,, . . ., xk]. We have to find, in a finite number of steps, a word of 112 representing p. By Th. 4.6, there is an isomorphism p : D[x,, . . ., x k ] D[x,, . . ., xk-J [xk]fixing DU {xl, . . ., xk}elementwise. By Th. 10.1I , we can find a representation p = (a,n xk)U b, where u,, b, 6D[x,, . . .,xk-,], a, == b,, in a finite number of steps. By induction, in a finite number of steps, we can find representations of a,, b, in the 112 belonging to D[xl, . . ., xk-,]. By substituting these representations into p = (a,nx,)U b, and applying the laws of 23,we get a representation of p by a word of 9. b) No two different words of 112 represent the same element of D[x,, . . ., xk],forletp = UFE0 U (aN,nn ( N , ) I N , f Q(r))be therepresentation of p by a word in 112 and let bi = 0, for i EN,, bi = 1, for i 6N,, then p(b,, b,, . . ., bk) = aN,. Thus any two different words of 112 represent different polynomials. c) The proof of the second statement runs along the same lines as part c) of the proof of Th. 10.1I . -L
34
POLYNOMIALS AND POLYNOMIAL FUNCTIONS
CH.
1
10.3. Let l2 = (w1, w 2 , w3, w4, wg} be the family of operations of type (2, 2, 0, 0, l}, and 23 the variety of Boolean algebras regarded as algebras ( A ; l2) as in 9 2.4. Thus w1 is the union, w 2 the intersection, w3 the zero, wq the identity, and w:, the operation of forming the complement. We write U for wl, and n for w 2 , and use infix notation, 0 for w3, 1 for 0 4 , and a-' for w5a. Moreover, we set u1 = a, thus ( d ) j = uii, for every pair i , j = f 1. As before, we define w,U w,U . . . U w, and w, n w,n . . . n w, inductively. Let M be the set { - 1, l} with total order -1 =s 1, and M k the lexicographically ordered Cartesian product of k copies of M . For ;1 = (Il, ..., l k ) c M k ,set f = x t n . . . nxk. Let X = {xl, . . ., xk} be a set of indeterminates and B(X, 23) = B[x,, . . .,xkl the polynomial algebra in X over an arbitrary Boolean algebra B. Then 10.31. Theorem. The set % of all words U(u,n$l ;lEMk), a,tB, and the elements under the union symbol arranged according to the order relation on the index set M k , is u normal form system for B[xl, . . ., xk].
Proof. a) We first show for k = 1 that, in a finite number of steps, we can find a word of % representing p E B[x,, . . ., xk], for every representationp = w ( q , xi). Since a = (anx-1)U(anx) and x = (0nx-l)U (1 nx), we can use induction on the minimal rank of w. By the laws of '8,we have
[(an x- 1) u (bnx)l u n x-1) u (dn = [(UUC) nX- 11u [(bud )nXI, [(anx-1) u (bn x)l n [(cnx-l) u ( d n x)] = [(anC) n x-11 u [(bnqnXI, n x- 1) u (bnXI]- 1 = (a- 1ux) n(b- 1ux- 1) = (a-lnb-1)u(a-lnx-l)u(b- in x) = (a-ln x-l) u (b-ln x) u (a-ln b-1 n x-1) u (a-w b - i n x) = (a-lnx-yu (b-lnx). As in the proof of Th. 10.21, we now proceed by induction on k. b) Different words of represent different elements of B[xl, . . ., xk]: For let p = U (a,n$\ 2 €&Ik) be a representation of p by a word of %, A = (Zl, . . ., lk), and bi = lit, i = 1, 2, . . ., k, then d b , , . * -,b,) = a,. 10.32. Remark. Th. 10.31 and part b) of its proof again show that the canonical epimorphism o : B[x,, . . ., xk] Pk (B) is an isomorphism.
-.
11. Polynomially complete algebras 11.1. Definition. An algebra A is called n-polynomially complete if P,(A) = F,(A), i.e. every n-place function on A is a polynomial function.
0 11
POLYNOMIALLY COMPLETE ALGEBRAS
35
11.11. Proposition. If A is n-polynomially complete, then A is also m-polynomially complete for all m n. Proof. Let m e n, and v E F,(A). Let y E F, ( A ) such that y(al, . . .,a,) = y(a1, . . .,a,), (al, . . .,a,) E A". Since, by hypothesis, y E Pn( A ) , there exists a word w(g,, x,, . . ., x,) such that y = w(gi, El, . . ., t,). Let b,+l, . . .,b, be arbitrary fked elements of A , then ~ ( a , ,. . ., a,) = ~ ( a i *, . a, b,+l, . . ., b,) = W(g,, a,, . . ., a, b,+,, . . ., b,) = w(sj75 1 7 . . ., 5,, b,,+l, . . ., b,) (al, . . ., a,), for all (q, . . ., a,) E Am. Hence y j = w(gi, El, . . .,Em, b,+l, . . .,b,), i.e. v E P, (A). - 7
11.2. Theorem. If an algebra A is 2-polynomially complete, then A is n-polynomially complete, for n = 1 , 2 , 3 , . . .. Roof. We have to show that A is n-polynomially complete for n =- 2, by Prop. 1 1 . 1 1 . We distinguish two cases : a) 1 A I is finite. Since J A 1 = 1 implies [ F, ( A ) 1 = 1 , we may assume I A1 > 1. To tackle this case, we need two lemmas, which hold under the hypothesis of Th. 11.2. 11.21. Lemma. Let n == 2, a # b E A, and p E F, ( A ) such that
where (ul, . . ., u,) E A" arbitrary, then rp E P, (A). Proof. By induction on n. For n = 2, this is our hypothesis. Suppose the lemma holds for n - 1 instead of n. Let y E F,-l (A) be the function defined by
36
POLYNOMIALS AND POLYNOMIAL FUNCTIONS
CH.
1
for some word w3.Thus a, E P,( A ) . 11.22. Lemma. Let n a 2,
e E P, ( A ) , and
y E F, ( A ) such that
. . * > gn), for
(g1,
y k , , * * * , g n#e(g,, ) - - - > g , )for ,
(gp
v(g1, *
*
- 9
8,) = dg,,
- . - > gn) f (~1,. 0
.
0
,
g,)
=@1,
*
~n),
.-:,u,),
for some (ul, . . .,u,) E A", then y E Pn(A).
proof. Suppose that y(u,, . . .,u,) = c, and let a f b E A . Let a, E Fn(A) be defined as in the hypothesis of Lemma 11.21. Then a,€P,,(A),thus a,(g,, . . .,9,) = v,(aj, g,, . . ., gn), for some word wl. Let x E Fz ( A ) be defined by v, for u z b
x(u,v> =
c, for u = b.
11.23. We complete the proof of the theorem for case a). By definition of P,(A), every element of A is in P,(A). By Lemma 11.22, every function of F,(A) which differs from a constant function only for one (gl,. ..,g,) E A" is in P,(A). By repeating this argument, every function of F,(A) which differs from a constant function only for finitely many (gl, . . .,g,) E A" is in P,(A). Since A is finite, A" is also finite, hence F,(A) = P,(A).
§ 11
31
POLYNOMIALLY COMPLETE ALGEBRAS
11.24. b) Let 1 A1 be infinite. Then we use induction on n. For n = 2, the assertion holds by hypothesis. Suppose A is (n - 1)-polynomially complete, for some n z 2, and g; E F,(A). Since n is a finite cardinal, I A"-'l = 1 A Inp1 = 1 A ] , hence there is a bijective mapping y : A"-, A , and y-l : A + A n p l is also a bijection. Let X E Fz(A) be defined by x(u, 4 = d y - l u , 8). Then P(&, . . g,) = x(v(g1, . * g , - A g,), for all (gl, . . .,g,) E A". Since y E F,-, ( A ) , by induction, we have y E P,-l ( A ) , thus tp(gl, . . .,gn-l) = w,(a,, g,, . . ., g,-l), for some word wl. Also 3: E P 2 ( A ) by hypothesis, thus ~ ( uv), = w2(bj,u, v), for some word w2. Hence q(gl, . . ., g,) = w2(bj,wl(ai,g17 . . .)g,-l>, g,), i.e'p = w2(bj,wl(ui, El, . . ., &-J,En) is in P,(A). This completes the proof of the theorem.
-.
- 7
* ?
11.3. Proposition. Let A be I -poljwomially complete, then A is simple. Proof. By way of contradiction, suppose that A is not simple. Then there is a non-trivial congruence 0 on A . Thus 0 has a congruence class C1 such that 1 CI I =- 1, moreover there is another congruence class Cz # C1. Let a, b EC1, a # b, c EC2, and y E K ( A ) such that y(a) = a, y(b) = c. Since aOb implies y(a)Op(b)-by induction on the least minimal rank of the words representing p-for all y E P1( A ) , we conclude that y $ PI( A ) , i.e. A is not I-polynomially complete.
11.31. Proposition. Let the algebra ( A ; 52) be 1-polynomiully complete, and D finite or countable. Then A is finite. Proof. This is an immediate consequence of 11.32. Lemma. Aizyalgebra(A; l2)suchthat jAl is infinite and lQl is not 1 -polyizomially complete.
==
IAl
Proof. We have I F1(A) 1 = I A =- I A I. It suffices to show that J PI( A )I s I A / . Let W = W ( A U { x } ) be the word algebra over 9, and w E W . The number of elements of DU AU { x } involved in w, each element counted with the multiplicity it appears in w, Will be called the length of w. Being of the same length is an equivalence relation on W. Let C , be the class consisting of all words of length n. C , is a subset of ( Q U A U {x}>", hence IC,] =s I(Q'JAU{x})"J= J Q U A U { x } l "= I Q U A U { x } l = (QI+
38
POLYNOMIALS AND POLYNOMIAL FUNCTIONS
CH.
1
+ / A J + l = I A ] .Since W = i ; ( C , I n ~ l ) , w e h a v e I W j = C ( j C , ] I n ~ l ) (I A 1 In 2 I) = / A I. As every element of P,(A) can be represented as a word of W , there is an injection from P1(A) to W , hence IPl(A)I s j W / =s / A / . s
12. Some examples of polynomially complete algebras 12.1. Let A be an arbitrary algebra. Then the results of 5 11 show that the following three cases are possible: a) A is n-polynomially complete for all n. b) A is n-polynomially complete for n = 1, but for no IZ =- 1. c) A is n-polynomially complete for no n. In case a) we say A is polynomially complete, in case b) A is polynomially semicomplete, and in case c) A is polynomially incomplete. For some important varieties, we are going to investigate now in what way the algebras of these varieties distribute over these three cases. We will consider just algebras A such that I A 1 # 1, for j A j = 1 implies that A is polynomially complete for all varieties. These algebras will be called non-trivial algebras. 12.2. Let be the variety of commutative rings with identity as considered in 3 2.4. By 3 11.3, every 1-polynomially complete non-trivial algebra of 23 is simple and finite, i.e., as we know, a finite field. Conversely, let Q be a finite field of order q. Then the number of distinct polynomial functions of the form a,-
g;-I
+
a,-,5:-2
+ ... +altl+ao,
aVEQ,
~ = 0 ..., , q-1,
is qq. For, if any two such functions are equal, we get a polynomial function ug-l&-lf . . . ful(l+uo, by taking their difference which is the constant function with value 0. This is only possible if uo = u1 = . . . = ug-l = 0, else the polynomial U , - ~ X ~ - ~ +. . . u,x+ uo would have at most q-1 different roots, a contradiction. Hence ]Pl(Q)l== q4 = IFl(Q)I,i.e. Q is 1-polynohially complete. The functions P,-~&) t~-l+p,&) 5;-,+ . . . +pl(El>5,+pO(tJ, P,.(&)E PI@), 2’ = 0, . . ., q - 1, are functions of P,(Q). These functions are pairwise distinct, for assume
+
c 0
v=q-l
P,.(51)
5;
c 0
=
v=q-1
6;.
r”(E1)
§ 12
SOME EXAMPLES OF POLYNOMIALLY COMPLETE ALGEBRAS
Let a € Q be arbitrary, then
0 ,,=q-l
p,(a) ti
0
=
v=q-1
39
r,(a) ti, and by the
argument from above we conclude p,(a) = rv(a), v = 0, . . ., q - 1, for all a E Q , i.e. p,(5,) = r&), v = 0, . . ., q- 1. Hence we have IPz(Q)( q4’ = IFz(Q)l, i.e. Q is 2-polynomially complete. Hence 12.21. Theorem. The finite jields are the polynomially complete algebras in the variety of commutative rings with identity. All the other algebras of this variety are polynomially incomplete. 12.3. Lemma. Let L be a lattice and p E Pl(L). Then p is an order endomorphism of L, i.e. a G b in L implies q(a) =sgj(b).
Proof. By induction on the least minimal rank k of the words representing p. The lemma obviously holds for k = 0. Suppose the lemma is proved for k G m - 1, and let p be a polynomial function of least minimal rank m. Then p = ylUyz or p = y l n y z where yl, yzEP1(L) are of least minimal rank s m-1. By induction, for a 4 b in L, p(a) = yl(a)U Uyz(a) == yl(b)Uyz(b)= g(b) or e(a) = yl(a)nyz(a>== y l ( b ) n y z ( @ = q(b), respectively. 12.31. Let L be 1-polynomially complete, then, by Q 11.31, L is finite, thus L has a least element 0 and a greatest element 1. If x C P1(L) is a function ) 1, then, by Lemma 12.3, 1 = ~(0) 6 such that ~ ( 1 = ) 0 and ~ ( 0 = ~ ( 1= ) 0, i.e. lLI = 1. Therefore we get 12.32. Theorem. In the variety of lattices, every algebra is polynomially incomplete. 12.4. Let 23 be the variety of Boolean algebras as considered in 0 2.4. If B is an n-polynomially complete algebra of 8,then, by Prop. 11.11 and Prop. 11.31, B is finite. Let ( B l = r . By 10.32 and Th. 10.31, IP,,(B)1 = IB[x,, . . .,x,] 1 = r2n.Since I F,(B) 1 = rrn,we conclude r e 2. If, on the other hand, r =s2, then F,(B) = P,,(B). Thus we have proved 12.41. Theorem. In the variety of Boolean algebras, the algebra oforder 2 is polynomially complete, while all the other algebras are polynomially incomplete.
40
POLYNOMIALS AND POLYNOMIAL FUNCTIONS
CH.
1
12.5. Proposition. Let G be afinite group, N a non-abelian minimal normal subgroup of G, A a non-empty finite set, and F the group of all functions from A to N where the multiplication in F is dejined by (cpy)( a ) = p(a)y(a), a E A . Let H be a subgroup of F which satisfies a) every constant function of F belongs to H ; b) for every pair x,y E A , x # y , there exists p E H such that 2s f ~ y ; c) for all cp E H and ail r E G, the functiot? a r-’F(a) r .from A to N belongs to H . Then H = F. +
Proof. Let j =- 0 be an integer such that j
=s I A
I.
Aj-tuple (gl, . . . ,gj),
giE N , i = 1, . . .,,j, shall be called accessible if, for every j-tuple (ul, . . ., aj) of pairwise distinct elements of A , there exists Q; E H such that p(ai) = g,, i = 1, 2, . . . , j . By induction on j , we will show that every j-tuple is accessible. By a), this is true for j = 1. Suppose the assertion is true for k-tuples, k -= j. It suffices to show that all j-tuples (g, 1, 1, . . ., l), g E N , are accessible. For, let p i €H , i = 1, . . .,j, such that cpi(ai)= gi, rp,(ah)= 1, for h # i, 1 G h ~ j then , Q; = q1cp2 . . . q,j i = 1, 2, . . .,j.Thus let a = (al, . . ., uj) be a satisfies p(ai) = g,,
j-tuple of pairwise distinct elements of A , and N , the set of all g E N such that p(al) = g, cp(ai) = 1, i = 2, 3, . . . > j , for some cp E H . Since 1 EN,, N , is not empty. Since H is a group, N , is a subgroup and by c), N , is a normal subgroup of G . Hence we have only to show that N , # {I}, then the minimality of N implies N , = N . First of all, let j = 2. By b), we can choose p E H such that p(a1) # @(az). Let y be the constant function with value e(a2)-l and set x = ey. Then x € H and x(a1) # 1, X(a2) = 1 , hence N , # { l}. Suppose now j =- 2. Since N is non-abelian, we can find g l , g z E N such that glg2 # g 2 g l . By induction, every ( j - I)-tupleis accessible, hence thereexistyl, y 2 E H such that yl(a,) = gl, yl(a2) = yl(a,) = yl(a,) = . . . = yl(aj) = 1, y,(al> = g, , w2@J = y2(a4)= y2(a5)= . . . = y2(aj)= 1 . Then y 3 = y;1y;1y1y2 E H and w3(a1) = gl-’gF1g1gz # 1, y3(ai)= 1, i = 2, . . . , j . Hence N , + { l} and the proposition is proved. 12.51. Corollary. Every finite non-abelian simple group G is 2-polynomially complete. Proof. We apply Prop. 12.5 to the case where N
=
G and A = GXG.
41
REMARKS AND COMMENTS
Then F = F2(G).Conditions a) and c) are trivially satisfied for H = Pz(G). Let x, y E A , x # y , then x = ( g l , g2), y = (hl, h z ) where either gl f h l or gz f hz. Thus ex # ey is satisfied either by e = or p = &, and condition b) is satisfied. Hence P2(G)= F2(G).
[,
12.52. By Prop. 11.3 and Prop. 11.31, every 1-polynomially complete group must be a finite, simple group. Thus it remains to consider finite simple abelian groups G. Then 1 GI = p , a prime, and IK(G)I = p p The set S = {a[;laE G, 0 6 r p } is a subgroup of Pl(G) containing t1 and the constant functions. Hence S = P1(G), but every element of S is represented by exactly one a,!;, a E G, 0 == r c= p , thus JP,(G)I = p2. We conclude that G is 1-polynomially complete if and only if p = 2. Conversely, if 1 G I = 2, then, as before, P,(G) = {aE;ltglI a E G, 0 == ri-=2, i = 1,2}, I P2(G)j = 23. But I F2(G) I = 24, hence G is not 2-polynomially complete. Summarizing these results we state 12.53. Theorem. Zn the variety of groups, the finite non-abelian simple groups are polynonzially complete, the grozip of order 2 is polynomially semicomplete, and all the other groups are polynomially incomplete. Remarks and comments
$6 1-3. Universal algebra, the study of sets with arbitrary oFerations, was started off by G. BIRKHOFF about 1935 and since has advanced to an extensive theory. There are two excellent monographs on this subject, [3]. Moreover K U R O[2] ~ also one by COHN111 and one by GRATZER includes a transparent introduction into the fundamentals of universal algebra which, in particular, served us as model for some parts of Ch. 1. $ 4. Up to 1950 the notion of a polynomial was used almost exclusively in connection with rings. Subsequently papers appeared sporadically which introduced polynomials over algebras other than rings, e.g. over groups, lattices, and Boolean algebras, especially being tied up with systems of algebraic equations (see, for example, SCHUFF [l], [2], SHAFAAT [I]). A few papers (e.g. SCHUFF[ 3 ] ,SLOMINSKI [I]) consider polynomials even over arbitrary algebras. The first time, however, that the concept of a polynomial was used in universal algebra, was in GRATZER[3]. There the “polynomial symbols” are nothing else than the elements of
42
POLYNOMIALS AND POLYNOMIAL FUNCTIONS
CH.
1
our word algebra W ( X ) in Q 2.1, and “k-ary polynomials over A” are, in our terminology, the elements of the subalgebra G,(A) of the algebra F,(A) in 5 6 which is generated by the projections tl, E2, . . ., Ek and the Gary operations mi of A. Thus Gk(A)is a subalgebra of P,(A), and we call the elements of Gk(A) “Gratzer’s poiynomial functions” while our book. polynomial functions are called “algebraic functions” in GRATZER’S The definition of polynomials as it is used in our book is due to HULE [l], [2], and is-as we show in Q 8-a straightforward generalization of the classical concept of a polynomial over a commutative ring with identity. Our presentation of $5 4-6 follows partly HULE’Swork. In all examples for semidegenerate varieties we know, semidegeneracy is due to 0-ary operations. We do not know whether or not the existence of 0-ary operations is a necessary condition for semidegeneracy. In some sense dual to our Prop. 4.5 is the following statement: Let A be an algebra of the variety % and p : B A a monomorphism. Then p can uniquely be extended to a monomorphism from B(X, %) to A(X,%) that fixes X . In general, however, this is not true, as the following counterexample by L. G. KovAcs shows : Let % be the variety of all nilpotent groups of class ~ 2A,= F(yl, y2, %) the free algebra of % with free generating set {yl, y ~ } B , = A’ the commutator group of A. By H. NEUMANN [l], B is a free abelian group. Let g be any element of a free generating set of B and g-lx-lgx E B(x, %). Since B is free abelian, there exists a homomorphism from B to the dihedral group D, that maps g onto some element gl E D, where D, = [gl, gz]. Since 0,belongs to %, there is a homomorphism from B(x, 23) to D, that maps {g, x} onto {gl,gz} whenceg-’x-lgs f 1 in B(s, %)as 0,is non-abelian. In A(x, %), however, we have g E A(x, a)’, and since A(x, ‘8) is nilpotent of class s2,we have g-lx-lgx = 1. If we replace “monomorphism” by “homomorphism”, however, then the statement just considered is true. Moreover, there are varieties, e.g. the varieties of commutative rings with identity and groups, where this statement is true as it stands (the proof can be given by aid of the normal forms in $5 8,9). Thus it would be interesting to characterize all these varieties. The problem of determining all those homomorphisms (isomorphisms) from A(X, 23) to B(X, %) which extend a given epimorphism (isomorphism) 6: A B without necessarily fixing X has been studied just for [l]). special cases (see e.g. GILMER +
-
REMARKS AND COMMENTS
43
0 5. The lattice of polynomial algebras over a given algebra was studied the first time by HULE[2]. Among the open questions concerning this lattice, there is the problem of characterising all those algebras for which the epimorphism p of Th. 5.22 is an isomorphism and the question whether or not the equation (5.3) remains valid if 17is replaced by U .
8 6. For rings and fields, polynomial functions have been investigated long before the concept of a polynomial was introduced. In connection with the canonical epimorphism (T of Prop. 6.41, one can raise the question-which is closely related to the problem referred to in D6.42-how to characterize all those algebras of a variety 8 for which the canonical epimorphism (T: A(x,, . . ., xk, 23) P,(A) is an isomorphism. For the variety of rings with identity there exist a few results (ACZBL[I], Hosszrj [I]). Recently a characterization of those noetherian rings for which CT (in the case k = I) is an isomorphism, could be achieved by CAHENand CHABERT [I]. In 3 10 we show that (T is always an isomorphism for the variety of distributive lattices with zero and identity and for the variety of Boolean algebras. A central problem in the theory of polynomial functions is the “problem of interpolation” which can be stated as follows: How can one recognize a function q € F,(A) to be a polynomial function and if is a polynomial function, how can one construct a corresponding polynomial? A closely related problem is that of “local interpolation”: Let B be any finite subset of Ak and y : B A a function. Is there a way to see whether or not y can be extended to a polynomial function on Ak and if y is extensible, how can one construct a corresponding polynomial ? Both problems have been attacked mainly for k = 1 where, as is well[I]), for A being a field, a characterization of known (see VAN DER WAERDEN polynomial functions can be achieved by certain difference equations and the construction of the corresponding polynomials is possible by means of the interpolation formulae of NEWTON or LAGRANGE. For other algebras than fields, however, there are just scattered results (for rings 181, DUEBALL [l], REDEIand SZELE[I], [2], SPIRA[l], for see CARLITZ groups see SCHUMACHER [I], and for Boolean algebras SCOGNAMIGLIO [I]). +
-+
0 8. Whereas polynomial functions over commutative rings with identity, and even more in the case of fields, have been considered in algebra and analysis from time immemorial, the concept of a polynomial has
44
POLYNOMIALS AND POLYNOMIAL FUNCTIONS
CH.
1
not become relevant until the rise of modern algebra. In modern algebra texts, polynomials are usually defined by the procedure as in Q 8.11, that means by using certain sequences. It should be mentioned that a similar construction performed with all sequences (uo, ul, . . .) of elements of a commutative ring R with identity, again leads to some extension ring of R being called the ring of formal power series in x over R which is frequently denoted by R [ [ x ](see ] e.g. K E R T ~ S[I]). Z Some of our definitions for polynomials over commutative rings with identity make sense also for formal power series, and also some of our results have their counterparts for formal power series. Since, however, a generalization of formal power series to arbitrary algebras seems to be almost impossible, we have abstained from including formal power series in our book. For the same reason, the elements of the total quotient ring of R[x] (see ZARISKI and SAMUEL [l]), the so-called rational functions in x over R , have been used here only as a handy tool for some proofs. But there are possibilities of generalizing some definitions and results of [15], this book about polynomials to rational functions (see NOBAUER [16], LIDL[3]).
Q 9. Polynomials over groups have so far been dealt with only implicit-
ely in connection with systems of equations over groups (see the remarks and comments at the end of ch. 2.).
Q 10. For polynomial algebras over arbitrary lattices, a normal form system has been found by SCHUFF[2] which is defined recursively and looks rather complicated, but it seems hopeless at the moment to find a handy normal form system, and this makes the investigation of this polynomial algebra rather tedious (some results can be found in MITSCH [l], SCHWEIGERT [l], SHAFAAT [I]). For the varieties of distributive lattices and Boolean algebras, however, there are quite a few results (e.g. MITSCH [2], RUDEANU [4] and ANDREOLI [ l ] , RUDEANU [2], resp.). Polynomials and polynomial functions over Boolean algebras have important applications to several branches of mathematics, we mention probability theory, propositional calculus, and the theory of switching circuits.
Q 11. Several authors have defined concepts that are related to our notion of polynomially complete algebras, for arbitrary algebras as well as for certain varieties (for a survey on these concepts. see KAISER [l]). The general idea behind all these concepts is :
REMARKS AND COMMENTS
45
Let 9 be an arbitrary class of algebras and, for any integer k > 1 and any A in 9, S,(A) and Tk(A)a pair of subsets of Fk(A).such that S,(A) 2 Tk(A). A n algebra A of 9 is called k - ( S , T)-complete if S,(A) = T,(A), and A is called (S, 7’)-complete if A is k-(S, T)-complete for any k. Moreover, A is called locally k - (S, T)-complete if, for any q~E Tk(A)and any finite subset B of Ak, there exists some y S,(A) such that y(a,, . . . , ak) = q~(a,, . . ., ak), for all (al, . . ., ak)E B, and consequently A is called locally ( S , T)-complete if A is locally k - ( S , 7‘)complete for any k. KAISER[ l ] has also introduced a measure for the degree of incompleteness of an algebra A : If A € 9, then the least amongst the cardinals of all subsets U of Tk( A ) such that [Sk(A)UU ] 2 Tk( A ) is called the k-(S, T)-completeness defect of A . If R is the class of all algebras, Sk(A)= Pk(A) and T k ( A )== Fk(A), then we obtain just our concept of polynomial completeness. Other concepts of completeness were studied by FOSTER and his school in a long series of papers (see FOSTER [1]-[9], FOSTER and PIXLEY [l], KNOEBEL [l], PIXLEY [l], QUAKKENBUSH [I 1, WERNER [I]). There e.g. the following special cases of (S, T ) completeness were introduced : a) Sk(A)= G,(A), the algebra of GRATZER’S polynomial functions, Tk( A ) = Fk (A); the ( S , T)-complete’algebras are called primal. b) S , ( A ) = G,(A), T k ( A )the set of all “conservative” functions of Fk(A), i.e. the set of all those functions p E Fk(A)such that, for any subalgebra U of A, (al, . . ., ak)t U k implies q(a,, . . ., ak)€ U ; the ( S , T ) complete algebras are called semiprimal. c) S,(A) = G,(A) (Pk( A ) ) ,Tk( A ) the set of all “compatible” functions of Fk ( A ) 1i.e. the set of all those functions y, of Fk ( A ) such that, for any congruence @ on -4,a, Obi, i = 1, . . .,k , implies p(a,, . . .,ak)Op(b,, . . .,b,); the (S!T)-complete algebras are called hemiprimal (polynomially hernicomplete). Th. 11.2 and its proof are due to S I E R P I ~ S[I]. KI
3 12. Th. 12.21 ha, been proved by K ~ D Eand I SZELE[l]. In R ~ D Eand I YZELF [I], [Z] and NOBAUER [21], some more concepts of Completeness are investigated for commutative rings with identi?y. HEISLER[ 13 generalizes Th. 12.21 to non-commutative rings. For the variety of lattices, SCHWLIGLRT [I] has introduced the concept of 1-order-polynomial cornpleteness which is a special case of 1 - (S,T)-completeness when setting Si = PI, TI = 01 where 01(A) is the set of all order endomorphisms of
46
POLYNOMIALS AND POLYNOMIAL FUNCTIONS
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1
A, and the same author has given examples for 1-order-polynomially [11, [2] has studied polynomial hemicomplecomplete lattices. GRATZER teness for the varieties of Boolean algebras and distributive lattices. Our proof of Prop. 12.5 is due to MAURER and RHODES [l] who use this proposition to show that every finite simple nonabelian group is polynomially complete. By using different methods, FROHLICH [ 11 has proved earlier that these groups are 1-polynomially complete. These methods were also used by LAUSCH[4] to investigate 1-polynomially [11 has studied polynomial complete multioperator groups. ROUSSEAU completeness for algebras with a single operation.
CHAPTER 2 ALGEBRAIC EQUATIONS
1. Systems of algebraic equations 1.1. Let 23 be any variety, A an algebra of 23 and X = (xl,. . ., x,} a finite set of indeterminates. An algebraic equation over (A, 23) in the indeterminates xl,. .., x, is a formal expression of the form
P=q where p , q E A(X, 8). A system of algebraic equations over ( A , B)-in short, an algebraic system-in the indeterminates xl, . . ., x, is a family of algebraic equations over (A, 23) in x,, . . .,xk,indexed by an arbitrary (possibly infinite) set I : pi = q,, i E I . (1.1) In particular, we may regard a single algebraic equation as an algebraic system. Let B be an arbitrary 23-extenslon of A, then, by ch. 1, Prop. 6.31, p(b,, . . ., b,) is a well-defined element of B, for every (b,, . . ., bk) Bk and every p E A(X, 23). This allows us to make the following definition: The element (bl, . . .,bk)E Bk is called a solution of the algebraic system (1.1) if pi@,, . . ., b,) = ql(bl, . . .,b,), for all i E I. An algebraic system over ( A , %) is called solvable if there exists a %-extension B of A such that the system has a solution in B. 1.2. We want to obtain a criterion for the solvability of an algebraic system. First we prove 1.21. Lemma. Let X be an arbitrary set of indeterminates, P = {(pi,qi)I i E I } a subset of A(X, 8)x A(X, 23), and 0 the congruence on A(X, B) generated by P. Then f o g holds if and only if there is a chain f = zO,z,, . . ., z, = g of polynomials of A(X, 23) such that any two polynomials zj, zj+l are either equal or zj+, is obtainedfrom zj by replacing a subword equal top, of a representation of zj as a word by the corresponding qi or by replacing a subword equal to qi of a representation of zj as a word by the corresponding pi. 41
48
ALGEBRAIC EQUATIONS
CH.
2
Proof. We set,fOlg if and only if thereisachain from f to g as described in the lemma. Then O1 is a congruence on A ( X , 23) as one can easily see. Certainly O1 contains P. Also every congruence on A ( X , 8)containing P contains OI.This proves the lemma. 1.22. We define : A congruence 0 on A(X, 23) is called separating if, for any two elements a, b A , aOb implies a = b. With this definition we can now state
1.23. Theorem. The algebraic system pi = qi, i E Z , over (A, 23) in X = {xl, . . ., x k } is solvable If atzd only if the congruence 0 O H A ( X , %) generated by rhe subset {(pi,4,) j i t I } of A(X, 8)x A(X, 3 )is separating. Proof. Suppose the hypothecis of the theorem is satisfied. We can then perform the embedding of A into A ( X , %) 10,this yields some %-extension B of A . Let 2,. i = I , . . ., k , be the congruence class of x , under 0, then p,Oq, implies pl(Xl, . . ., X k ) = ql(-Y1.. . ., X k ) Hence (Z1,. . ., X k ) is a solution of the algebraic system in B. Conversely, suppose that the system is \olvable, and let (bl, . . . , bk)be a solution of the system in some %-extencion B of A. Then f 0 g implies, by Lemma 1.21, that f(b,, . . ., bk) = g(b,, . . ., bk), hence, for a, b E A , a0b implies a = 6. Thus 0 is separating. 1.3. Theorem. Let p , = q,, i E I , be any algebraic system over ( A , %) in X = {xl,. . ., xk}.Then one of the ,following statements is true: a) The system has at most one solzrtion in every 8-extension of A . b) For e i w y cardinal u, there exists some %-extension C of A such that the set oj all solutions oj the system in C has cardinality greater than U. Proof. Suppose that alternative a) is not true. Then there exists some %-extension B of A such that the system has at least two different solutions in B, say (bl, . . ., bk)and (cI, . . ., CJ. Let u be an arbitrary cardinal, L any set of cardinality 2u, Y = {xltl1 EL, I =s t =s k } a set. of indeterminates, and A that congruence on A(Y, 022) which is generated by {( . . , .Y/,J, q,(xil. . . . , x i k ) )1 i E t, I t L}. A is separating since, for a, b E A, a,lb yields a polynomial chain in A(Y, %) from a to b according to Lemma 1.21 which, by replacing each s/,by xt,yields a polynomial chain in A ( X , %) from a to b showing that neb, again by
81
SYSTEMS OF ALGEBRAIC EQUATIONS
49
Lemma 1.21. But 0 is separating, by Th. 1.23, whence a = b. Hence A can be embedded into C = A(Y, % ) / A ,a %-extension of A . If Z,, is the congruence class of xl,under A, then (XI1, . . ., 2,J E C k 1, a solution of the given system, for all IEL. We have to show that rn # 11 implies (&, .. ., Xmk) # (Xfll, . . ., X,J. Assume the contrary, then x m t h f l t , t = 1, 2, . . ., k . Lemma 1.21 yields a polynomial chain in A(Y, %) from x,, to xnt. If we replace each x,, by b, and each x,, by c,, for r # m, in this chain, this chain turns into a sequence of equal element5 of B whence b, = c,, t = 1, . . ., k . This is a contradiction. Hence there are at least j L I = 2u =- u solutions in C . 1.31. Remark. Unless %is semidegenerate and I A I = 1, either alternative of Th. 1.3. can occur, for arbitrary k . Indeed, let a i € A , then the system xi = ai, i = 1,2, . . ., k is an example for alternative a) while ai = ui, i E 1, represents alternative S). 1.4. Let ( S ) be any solvable algebraic system over ( A , %) in X = {x,, . . ., xk},and (b,, . . .,bk)a solution of ( S )in some %-extension B of A . The subalgebra A(b,, . . ., bk) of B.is called an (S)-root extension of A . A pair A(b,, . . ., bk), A(c,, . . ., ck) of (S)-root extensions of A is called equivalent if there exists an isomorphism from A(b,, . . ., h,) to A(c,, . . ., ck)fixing A elementwise and mapping b, to c,, i 1, . . ., k . A greatest (S)-root extension of A is an (S)-root extension A(c,, . . ., ck) of A such that, for any (S)-root extension A(b,, . . ., bk) of A, there exists a homomorphism from A(c,, . . ., cx) to A(b,, . . ., bk) fixing A elementwise and mapping c, to b,, i = 1, . . ., k . Clearly such a homomorphism is the only one with this property and is an epirnorphisrn. 1.41. Lemma. Every sohiable algebraic system ( S ) over ( A , %) it7 X = {x,, . . ., x k } has a greatest (5')-root extension, and any two greatest (S)-root extensioris of A are equivalent. Proof. The second statement is an immediate consequence of the definition of a greatest (S)-root extension. We claim that, if 0 is the congruence on A(X, 8) generated by the equations of CS),C the algebra resulting from embedding A into A(X. 2.3) 10,and X,, i = 1, . . .,k , the congruence class of x,in C,then C = A(%,, . . .,Zx)is a greatest @)-root extension. For let A(b,, . . .,bk)be an arbitrary (S)-root extension of A . If ~ ( u ,%,, . . .,.Tk) =
50
ALGEBRAIC EQUATIONS
CH.
2
?.(a,,-Tl,. . .,Xk), for two words w,v, then w(a,, x,, . . .,xk)Ov(a,,xl,. . .,xk) whence, by Lemma 1.21, w(a,, b,, . . ., bk) @(a,,b,, . . ., bk). Let fCA(i?,, . . ., ik) and f = w(q, Xl, . . ., xk) be any representation of f a5 a word, then cpf = w(a,, b,, . . ., bk) yields a well-defined mapping 9 from A(Z,, . . .,Xk) to A(b,, . . ., bk). Clearly 9; fixes A elementwise, maps .?, onto b, and is a homomorphism. Thus C is a greatest (S)-root exten\ion. 1.5. Let A be any algebra of %, X = {xl, . . ., xk}, and 2 ( A ( X , 23)) the congruence lattice of A ( X , 53).One can easily verify that the subset S of 2(A(X, ’a)) consisting of all separating congruences is a subsemilattice with respect to 0, i.e. the intersection of any two separating congruences is again separating. Now let A be any congruence on A(X, %). A k-tuple (u,, . . ., uk) of elements of a %-extension U of A is called a general root of A whenever p . l q holds if and only if p(ul, . . ., uk) = q(ul, . . ., uk). 1.51. Proposition. A congruence 11 on A ( X , 8)has a general root i f and only if A E 2. r f U is an arbitrary %-extension of A and (ul,. . ., uk)E Uk, then there is one and only one congruence A on A(X, %) such that (ul, . . ., uk) is a general root of A.
Proof. If A 6 2, then there exist a, bE A such that a # b but aAb, thus “I cannot have a general root in this cas?. If A c X , let U be the %-extension of A obtained from embedding A into A(X, 23) I A. If X, is the congruence class of xiin U, then (Zl,. . ., Zk)is a general root of A. Let (ul, . . ., uk)E Uk where U is now an arbitrary %-extension of A , and ,1 be the binary relation on A(X, 23) defined by: p A q if and only if p(u,, . . ., uk) = q(ul, . . ., uk). Clearly A is a congruence on A(X, 23) with (ul, . . ., uk) as a general root. By definition of a general root, A is uniquely determined by (zt,, . . ., ux). 1.6. Let (S,), (S,) be two solvable algebraic systems in X = {x,, . . . , xk} over (A,%). We define : (Sl)2 (Ss)means that every solution of (SI)is a solution of ( S 2 ) .( S l )and (S2) will be called equivalent if (S1) 2 (Sz)and (S2)
2 (S1).
1.61. Lemma. Let (Sl), (S,) be solvable algebraic systems in xl,. . ., xk
81
SYSTEMS OF ALGEBRAIC EQUATIONS
51
over ( A , 23) and Oi, i = 1,2, the congruences on A(X, 23) generated by the equations of ( S J .Then (S,) 2 (S,) fi and only if@, 2 0,. Proof. Suppose that (S1) 2 (Sz), and let B be the algebra we get from By the proof of Th. 1.23, the k-tuple embedding A into A(X, %)I@,. (z1,. . ., Xk) of congruence classes of the xi under 0, is a solution of (S,) and therefore a solution of (S,). Thus, for every equation pi = qi of (S,) we have pi(Xl, . . ., 2,) = qi(Xl, . . ., X k ) whence piO,ql, thus 0, 2 0,. Conversely, if 0, 2 0, and pi = qi is an equation of (S2),we have piO,qi. Hence, by Lemma 1.21, every solution of (S,) is a solution of (S,), thus (S1) 2 (S,).
1.62. Corollary. Two solvable algebraic systems (SI) and (S,) are equivnlent if and only if@, = 0, where Oi is the congruence on A(X, %) corresponding to (S,), i = 1, 2.
Proof. Obvious. 1.7. Let again 3be any variety, A ag algebra of %, and X = {x,, . . . , xk} a finite set of indeterminates. An algebraic inequality over ( A , 3)in the indeterminates xl, . . ., x, is a formal expression of the form f # g wheref, g E A(X, 8).A mixed algebraic system over (A, 23) in the indeterminates x,, . . ., xk is a family consisting of algebraic equations and inequalities where the equations are indexed by some set I and the inequalities by some set J disjoint from I. Thus a mixed algebraic system has the form p i = qi, i E I, 4 # gj, . j E J . (1.7) In particular, we may regard any algebraic system as a mixed algebraic system. Now let B be any %-extension of A . The element (b,, . . .,bk)E Bk is called a solution of (1.7) if pi(b,, . . ., bk) = qi(b,, . . ., b,), for all i E I , and fj(b,, . . . bk) zgj(b,, . . ., bk), for all j~ J . The mixed algebraic system (1.7) is called solvable if it has a solution in some suitable %-extension of A . Th. 1.23 can now be easily generalized to the case of mixed algebraic systems : ?
52
ALGEBRAIC EQUATIONS
CH.
2
1.71. Theorem. The mixed algebraic system p i = qi, i E I , fi # gj, j € J , over ( A , 8)in X = {xl, . . ., x k } is solvable if and only i f the congruence 0 on A(X, 23) generated by the subset {(pi, qi)1 i E Z} of A(X, 8)x A(X, 23) is a separating congruence such that J o g j holds for no index j E J .
Proof. Suppose the hypothesis of the theorem holds. Then we can embed A into A(X, %)I@ in order to obtain a %-extension B of A where the k-tuple (X1, . . ., X k ) consisting of the congruence classes of the xi under 0 is a solution of the given mixed algebraic system. Conversely suppose the system is solvable, and let (bl, . . ., bk) be a solution of this system in some 8-extension B of A . Since f o g implies f(b,, . . ., bk) = g(b,, . . ., bk),by Lemma 1.21, we see that 0 is separating, but &Ogj never holds. 1.72. Again we will call two solvable mixed algebraic systems equivalent if they have the same solutions. 2. Maximal systems of algebraic equations 2.1. Let 8 be any variety, A an algebra of 23, and X = {xl,. . ., xk} a finite set of indeterminates. An algebraic system ( S )over ( A , 23) in X i s called maximal if the congruence 0 on A(X, 23) generated by the equations of ( S ) is a maximal element of the partially ordered set (S, G ) of separating congruences, where == is the set-theoretical inclusion of congruences. 2.11. Theorem. Let (S) be a solvable algebraic system. Then the following conditions are equivalent: a) (S) is maximal. b) [ f p = q is any equation over ( A , 8)in X and (SI)= (S)U { p = qj, then either (S1)is equivalent to ( S )or (SI)is not solvable. c) Any two (S)-root extensions of A are equivalent. d) Every solution of ( S )is a general root of one and the same congruence A on A(X, 8). e) Every (S)-root extension of A is a greatest (S)-root extension. f ) If(S1) is any algebraic system over ( A , 23) in X , and ( S )and (SI)have one solution in common, then ( S ) 2 (S1). g) Every solvable algebraic system (SI)is either equivalent to ( S )or (Sl) has a solution which is not a solution of (S).
-
02
MAXIMAL SYSTEMS OF ALGEBRAIC EQUATIONS
53
Proof. a) b). Let ( S ) be maximal, 0 be the congruence generated by the equations of (S),(SI) as in the hypothesis of b), and 0 1 the congruence generated by the equations of (Sl). Then 01 2 0 whence 01 = 0 or 0,6 S.In the first case, (S1)is equivalent to (S), by Cor. 1.62 while in the second case, (S1)is not solvable, by Th. 1.23. b) 3 c). Suppose ( S ) satisfies b), but not c). Then Lemma 1.41 implies that A has a greatest @')-root extension A(c,, . . ., ck) and some other inequivalent @)-root extension A(b,, . . ., bk). The epimorphism from A(cl, . . ., ck)to A@,, . . ., bk) fixing A elementwise and mapping cito bi is therefore not an isomorphism whence, for some polynomials p , q E A(X, %), we have p(c,, . . ., ck) # q(c,, . . ., ck), but p(b,, . . ., bk) = q(b,, . . ., bk). Let (S,) = (S)U(p = q}. Then (S,) is not equivalent to ( S ) and is solvable, contradicting b). c) = d). Let (u,, . .., uk),(a, . . ., vk) be solutions of ( S ) and A,, A, the congruences according to Prop. 1.51 corresponding to these solutions, respectively. Then pA,q holds if and only if p(u,, . . ., uk) = q(ul, . . ., uk) which holds if and only ifp(vl, . . .,vk) = q(aV . . ., vk) which is equivalent to prlzq. Thus A1 = A,. d) e). Let A(c,, . . .,ck)and A@,, . . ., bk)be two @)-root extensions of A . Then (c,, . . ., ck) and (bl, . . ., bk) are both general roots of A. Let 8 : A(cl, . . ., ck) A@,, . . ., bk) be the mapping defined by: 8p(cl, . . ., ck) = p(b,, . . ., bk), for all p E A(X, 23). This is a well-defined mapping since p(c,, . . ., ck) = q(c,, . ., ck) implies prlq whence p(b,, ..., bk) = q(b,, . . ., bk). Clearly 8 is a homomorphism fixing A elementwise and mapping ci to bi. Thus A(c,, . . ., ck) is a greatest (5')root extension. e) f). Let (c, . . ., ck) be acommon solution of (S) and (S,). Then pi(c,, . . ., ck) = qr(cl, . . ., ck), for every equation p , = qi of (S,). Let (b,, .. .,bk) be any solution of ( S ) . Then, by Lemma 1.41, there is an isomorphism 8 : A(c,, . . ., ck) A@,, .. ., bk)fixing A elementwise and mapping ci to b,, i = 1, . . ., k. Hence pi@,, . . ., bk) = qi(b,, . . ., bk) and thus (b,, . . ., bk) is also a solution of (S,). f ) g). Suppose every solution of (S1) is also a solution of ( S ) i.e. (5') and (Sl)have some solution in common. Hence (S) 2 (S1) 2 (S), i.e. ( S ) and (S1) are equivalent. g) =+a). Let 0 be the congruenceon A(X, 23) generated by the equa2 0. The algebraic system (SI) tions of ( S ) and O l f S such that consisting of all equations p = q such that pOlq has O1as its correspond-+
.
-
-
-.
54
ALGEBRAIC EQUATIONS
CH.
2
ing congruence. By Lemma 1.61, (Sl) 2 (S),whence (Sl)has just solutions which are also solutions of (S). Thus (Sl) and ( S ) are equivalent. By Cor. 1.62, O1 = 0 showing that 0 is maximal in S. Therefore ( S ) is maximal. 2.2. Proposition. An algebraic system in X = {xl, . . ., xk} over (A, %) which has a solution (al, . . .,ak) Ak is maximal if and only if it is equivalent to the system xi = a,, i = I , . . ., k . Proof. Let ( S ) be a system equivalent to the system xi = a,, i = 1, . . .,k , then ( S ) has the unique solution (al, . . ., ak)in every %-extension of A. Thus every solvable algebraic system (S1) has either a solution which is not a solution of ( S ) or is equivalent to (S). Hence, by Th. 2.11 g), (S)is maximal. Conversely, let ( S ) be a system with the solution (al, . . .,ak)E Ak which is maximal. Then, by Th. 2.11 d), every solution of ( S ) is a general root of the same congruence A on A(X, 23). Hence xiAai, i = 1, . . ., k . Let (bl, . . ., bk) be an arbitrary solution of (S), then bi = ai, i = 1, . . ., k . Thus ( S ) is equivalent to the system xi = ai, i = 1, .... k. 2.21. Proposition. Let ( S ) be any solvable algebraic system in X over ( A , %). Then there exists at least one maximal algebraic system (Sl) 2 ( S ) . Proof. Let 0 be the congruence on A(X, %) generated by the equations of ( S ) , 6 the subset of S consisting of all congruences of S containing 0, partially ordered by the partial order of S,and {OiI i E Z } any chain of 6. Then Z = U{Oi/iEZ}is a congruence on A(X, %) such that if aZb, then aOib, for some i E I, thus a = b. Hence Z € S and even more, 2 E 6.By Zorn's Lemma, 6 has a maximal element A which is, of course, maximal in S. If (Sl)is the algebraic system consisting of all equations p = q such that p d q , then (Sl) 1 (S)and (S1) is maximal.
2.3. Lemma. Let be a semidegenerate variety, A any simple algebra of %, X = {xl, . . ., xk},0 any congruenceonA(X, 8)which is maximal in S and P the congruence on A ( X , 23) which has only one congruence class. Then there is no congruence 2 on A(X, %) such that 0 c Z c P. Proof. Let Z be a congruence on A(X, %) such that 2 3 0. Since Z 6 S, the congruence Zn ( A x A ) on A cannot be the congruence where every
§2
MAXIMAL SYSTEMS O F ALGEBRAIC EQUATIONS
55
congruence class contains only one element. Hence Zn ( A X A ) = A X A since A is simple. Under the canonical epimorphism from A(X, %) to A(X, %) 12,the subalgebra A of A(X, 23) is therefore mapped onto a n algebra of order 1. Since % is semidegenerate, this means that IA(X,%)/Zi = 1, thus 2 = P. 2.31. Proposition. Let 23 be semidegenerate and A any simple algebra of 23. If (u,, . . ., uk) is a k-tuple of a %-extension of A which is a solution of a maximalalgebraic system ( S )over ( A , %), then the algebra A(u,, . . . ,u k ) is simple. Proof. Let 0 be the congruence on A(X, %) generated by the equations of (S). By the proof of Th. 1.23, A(X, 23) 10 yields an @)-root extension A(X,, . . . , X k ) of A which, by Th. 2.1 1 c), is isomorphic to A(u,, . . ., uk). Hence A(X, %) 10 s A(u,, . . .,uk). Since 0 is maximal in %, there is no congruence on A(X, 23) between 0 and P, by Lemma 2.3. Hence the algebra A(X, %)I0is simple, bych. I , Th. 1.71., which proves the proposition.
2.32. Proposition. Let % be an arbiirary variety, A an algebra of % with I A I =- I, and A(u,, . . .,Uk) any %-extension of A which is simple. Then there exists a maximal algebraic system ( S ) over ( A , %) which has (ul, . . ., uk) as a solution. Proof. Let A be the congruence on A(X, 23) which has (ul, . .., uk) as a general root, thus pAq if and only if p(u,, . . ., uk) = q(u,, . . .,uk). Let ( S ) be the algebraic system consisting of all equations p = q such that pAq. ( S ) has (ul, . .., uk) as a solution. By the proof of Lemma 1.41, embedding of A into A(X, %)I11 yields a greatest @)-root extension A(%,, . . ., X k ) of A . Clearly, the homomorphism from A(?,, . . ., xk) to A(u,, . . .,uk) fixing A elementwise and mapping Xi to ui, i = 1, . . ., k, is an isomorphism, hence A(u, ,. . . ,uk) is a greatest @)-root extension. If ( S )were not maximal, then, by Th. 2.1 1 e), there would exist an @)-root extension A(b,, . . .,bk)which is not a greatest @)-root extension, and the epimorphism from A(ul, . . .,uk) to A@,, . . ., bk) fixing A elementwise and mapping uito bi would not be an isomorphism. Since A(u,, . . .,U k ) is simple, we conclude that 1 A(b,, . . ., bk)1 = 1 whence 1 A I = 1, a contradiction. Hence ( S ) is maximal.
56
CH. 2
ALGEBRAIC EQUATIONS
2.33. Theorem. Let $‘3 be a semidegenerate variety and A a simple algebra of .% with I A =- 1. Then the k-tuple (ul, . . .,uk)of a %-extensionof A is a solution of a maximal algebraic system over (A, 23) i f and only if A(u,, . . .., uk) is a simple algebra. Proof. This is immediate from Prop. 2.31 and Prop. 2.32. 3. Algebraically closed algebras
3.1. An algebraic system ( S ) over (A, 3)in xl, . . ., xk is called finite if it consists of a finite number of equations. Similarly a mixed algebraic system is called finite if it consists of a finite number of equations and inequalities. Now let % be any variety. An algebra A of 8is called (mixed) algebraically closed if, for every k a 1, every solvable (mixed) algebraic system over (A, 8)in the indeterminates xl, . . ., xk has a solution in A . The algebra A of 93 is called weakly (mixed) algebraically closed if, for every k == I, every solvable finite (mixed) algebraic system over (A, %) in x,, . . ., xk has a solution in A. Thus every mixed algebraically closed algebra is algebraically closed and every weakly mixed algebraically closed algebra is weakly algebraically closed. (Of course all these definitions depend on the variety @. Thus, in fact, we should call A “algebraically closed in %”, but since 8, throughout this section, is being kept fixed, there is no danger of confusion). 3.2. Theorem. Let A be any algebra of 8. Then there always exists some %-extension C of A which is weakly mixed algebraically closed and therefore weakly algebraically closed. The proof of this theorem will depend on 0
3.21. Lemma. Let A be an algebra of %. Then there existssome %-extension &(A)= B of A such that, for every k 1, every solvable mixed algebraic system over (B, %) in the indeterminatesx,, . . .,xk, where the polynomials occurring in the equations and inequalities can be represented by words over A, has a solution in B.
Proof. We may regard every family of equations and inequalities in x,, . . ., xk, k = 1,2, . . . , as a set by neglecting the index sets for the
13
ALGEBRAICALLY CLOSED ALGEBRAS
57
equations and inequalities. Then any two systems, which regarded as sets are equal, have the same solutions. But there is an injective mapping from the class of sets of equations and inequalities to the Cartesian product of two copies of the power set of A ( X , % ) X A ( X ,%), X = {xl, x,, x,, . . .}. Hence the class of sets of equations and inequalities over (A, 23) in x,, . . ., xk,k = 1, 2, . . . , is a set M which can be wellordered, say M = {(S,)]a= 1,2, . . .}, a running through the ordinals p -= 0,o being a fixed ordinal. We put A0 = A. Suppose that a =- 0 is an ordinal such that, for every ordinal 7 7 -= a, a %-extension A, of A has been defined such that A,, is a subalgebra of A,,, for p iY. Then we define A, by A, = U {A, 1 y < 2)-which is a %-extension of A--if (S,), regarded as a mixed algebraic system over U { A , I y < a},is not solvable, A, = (U {A, I y iE } ) (ul, . . .,uk)where (al,. . .,uk) is a solution of (S,) in some %-extension of U { A , I y -== a } if (S,) is solvable. Then, for every p -= a, A , is a subalgebra of A,, thus A,is a %-extension of A . By transfinite induction, A, is then defined to be a %-extension of A , for alla < 0, and A , is a subalgebra of A,, for p < v. We put B = il{A,]cc -= o } which is again a %-extension of A . Now let ( S ) be any solvable mixed algebraic system over (B, 23)which satisfies the hypothesis of the lemma. If we represent the equations and inequalities of ( S )by words over A and take the mixed algebraic system over (A, 8)represented by this representations and regarded as a set, we obtain some set (S,) EM which has the same solutions as (S).Since ( S ) is solvable in some %-extension of B, (S,)-regarded as a system over U ( A , ] y -= a)- is solvable and therefore has a solution in A , S B. Hence ( S )has a solution in B. We are now ready to prove the theorem.
3.22. Proof of Th. 3.2. Let A(') = A and define A("),n == 1, recursively by A(")= &(A("-')). This yields a chain A('), A('), A(2),. . . of algebras of 23, every member of which is a subalgebra of the subsequent one. Hence every member is a %-extension of A . We set C = U (A("'(n 0), then C is also a %-extension of A . Let ( T )be any solvable finite mixed algebraic system over (C, 3) in the indeterminates xl, . . ., x,. Then the polynomials of the equations and inequalities of ( T )can be represented by words of W(CUX ) where just finitely many elements of C occur in ;his representation. Hence these words are words of W(A(")UX),for some n == 0. We consider the system ( T I )over (A("+'),%), represented by these words; it is solvable since it has a solution in some %-extension of C . But
58
ALGEBRAIC EQUATIONS
CH.
2
A("+1)= &(A(")), hence, by Lemma 3.21, (T,) has a solution in A("+1) which is also a solution in C. Thus ( T )has a solution in C , and the theo-
rem is proved.
3.3. Proposition. Let C be any mixed algebraically closed algebra of the variety 23 containing a subalgebra {e} of order 1. Then everyfinite algebra of 23 which contaim a subalgebra of order 1 can be embedded into C . I f C isjust weakly mixed algebraically closed, B has just a finite family of operations and C contains a subalgebra {e}of order 1, then also ever-vfinite algebra of 23 with a subalgebra of order 1 can be embedded into C. Proof. Let E = {ul, . . ., ur} be any finite algebra of 53,X = {x,, . . .,xr} a set of indeterminates, and x :E X a bijection. We construct a mixed algebraic system over (C, B) in X as follows: As equations we take mi = p i if m iis a 0-ary operation of B,wixujl xuj,. . .xujni = xwiujluj,.. .ujnl if o,is an n,-ary operation of 23, ni > 0, and uj, running independently over all elements of E ; as inequalities we take xu, $ xut, for all pairs us, u, of different elements of E. This mixed algebraic system has a solution in the %-extension of C obtained from embedding C into C X E , namely xu, = (e, us),s = 1, . . ., r. Both hypotheses of the proposition imply that this system has a solution in C , say xui = 'u(u,), i = 1, . . . , r. Then V(Wi) = mi, -t
V(OiUj,
. . . u. ) = WiV(Uj,) . . . V(Uja,), J,Ei
'u(us># 'u(u,>.
Hence 'u :E
-+
C is a monomorphism. This completes the proof.
4. Algebraic independence 4.1. Let A be any algebra, B an extension of A, and U = {ul, . . ., u,,) a finite subset of B. If 23 is a variety and 3 E B, then U is calied %algebraically dependent over A if there exist polynomials f , g E A(x,, . . ., s,, 23) such that f 8 g and f ( u , , . . ., u,) = g(u,, . . ., u,) while U is called p-algebraically dependent over A if there exist polynomial functions T,y in the subalgebra . . ., $), of P,,(B) such that Q # y and y(u,, . . ., u,,) = y(ul, . . ., u,,). Clearly this definition does not depend on the order in which the elements of U are written down but just on the .;et U.
04
ALGEBRAIC INDEPENDENCE
59
Subsequently “dependent” shall always mean “algebraically dependent” and & will mean either % or a variety % such that B € %. An infinite subset U of B will be called &-dependent over A if U contains some finite subset, which is &-dependent over A . A subset which is not &-dependent over A is called &-independent over A . In particular, the empty subset of B is &-independent. 4.11. Lemma. Let A be any algebra, B an extension of A , and U = {uzI i E I } a subset of B. Then the following four conditions are equivalent: a) U is &-independent. b) Every finite subset of U is %-independent. c) I f L g C A(x,, . . ., x,, 23) ( y , y C A(&, . . ., En)) such that f(ul, . . . u,) = g(ul, . . .,u,) (y(ul, . . - , u,) = y(u,, . . .,u,)), for a certain subset {ul, . . ., un>of U, then f = g (rp = y ) . d) Zf 0. = 23, then, for all %-extensions C of A , every mapping a :U C can be extended to a homomorphism from A(U) to C jixing every element of A . If & = $‘ 3, then every mapping a : U -+ B can b P extended to a homomorphismfrom A( U ) to B jixing A elementwise. +
Proof. a) + b). If some finite subiet V of U were &-dependent, then U would be finite, say U = { u l , . . ., u,}. Then V = {ujl,. . ., uzs},1 s ij =sn, and there would exist J; g E A(x,, . . ., x,, %) (9,y E A([,, . . ., E,)) such that f # g ( Q #y)andf(u,,, ..., ui,)=g(uil, ..., ui,) (P(’i,> ...,uz,)= y~(u,,, . . ., ui,>).By ch. 1 , Cor. 4.62 (by the fact that A(&, . . ., Ex) can be embedded into A(E,, . . ., 5,) such that the embedding fixes AU {tl, . . ., ts}elementwise), U would be &-dependent, contradiction. Thus a) implies b). b) jc). Obvious. c) j d). Let p E A(U), and p = w(a,, uj) a representation of p as a word. We set pp = w(ai,auj). Then p is a well-defined mapping from A ( U )to C , or B, resp., because of the assumption of c). Clearly p is then a homomorphism extending a and fixing A elementwise. d) a a). Suppose U is not &-independent. Then there exists a &-dependent finite subset {ul, . . ., u,} of U , i.e., there existf, gE A(xl, . . ., x,, 23) (Q,’L’ E & E l , . . ., i n )such ) thatf f- g (9: # y), f@,, * . ., u,) = g@,, . . . > (F(U1. . . ., u,) = y(u1, . . . )I, If & = 23, let C = A(x,, . . ., x,, 2%) and (xui = xi, i = 1, . . ., i i . Then a cannot be extended to a homomorphism fixing A elementwise, - 2
60
ALGEBRAIC EQUATIONS
CH.
2
contradiction. If 0. = p, then there exist b,, . , ., b, E B such that y(b,, ..., b,) # y(b,, . .., bn). Let a : U +.B be the mapping taking ui to b,, i = 1, . . ., n, then again a cannot be extended to a homomorphism k i n g A elementwise, contradiction.
4.2. If B is an extension of A , then every variety 23 such that B is an algebra of 8 yields some notion of %-independence over A in B. Moreover there is also a notion of p-independence over A in B. The relationship between these different notions of independence is described by 4.21. Proposition. Let B be any extension of A . If 2 3 2 2 231 and U is &-independent, then U is also %,-independent, and if U is %-independent for some variety 23, then U is %-independent. Proof. Let %I E 232 and U be &-dependent. Then there exist polynomials J g E A(x,, . . ., x,, 23,) such that f # g and f(u,, . . ., u,) = g(ul, . . .,u,), for some finite subset {ul, . . .,u,} of U. Let y :A(x,, . . ., x,, B2) -+ A(xl, . . ., x,, 23,) be the epimorphism fixing AU{x,, . . ., x,} elementwise, according to ch. 1, Th. 5.22., and f,, g, E A(x,, . . .,x,, &) such that yf, = f, ygl = g . Then fl g,, and &(?A,,. . ., u,) = g,(u,, . . .,u,) whence U is B2-dependent. Suppose now that U is p-dependent. Then there are y , y EA(5,, . . .,5,) such that q~ f y and p(ul, . . ., u,) = y(ul, . . .,u,), for some finite subset {ul, . . ., u,} of U. Let q~ = w(a,, El, . . ., t,), y = v(ai, 5,, . . ., 5,) be any representation of y and y , resp., as words and f,, g, E A(+ . . ., x,, %) such that f, = w(a,, xI, . . . , x,), gl = w(a, x,, . . ., x,). Then f, # g,, and fl(ul,. . . , u,) = g,(u,, . . ., u,), thus U is %-dependent.
+
4.3. Let B be any extension of the algebra A . A subset U of B such that B = A(U) is called an A-generating set of B. An A-generating set U of B is called minimal if no proper subset of U is an A-generating set. 4.31. Lemma. Let A be any algebra of the variety % and X = {xiI i E I } a set of indeterminates. Unless 23 is semidegenerate and 1 A 1 = I, then X is a minimal A-generating set of A(X, 23). Proof. By ch. 1, Prop. 4.1 l , X is an A-generating set of A(X, %). Suppose there is a proper subset of X which is also A-generating. Then, for suitable,
54
61
ALGEBRAIC INDEPENDENCE
pairwise different j , i,, , . ., i, we have xi = w(ai, xil, . . ., xi,>. By hypothesis, there exists a %-extension B of A such that J B J =- 1. Let x : X -+ B be a mapping such that xxi8 = bi,, s = 1, . . ., r, where bil, . . ., bi, are arbitrary elements of B and %xi # w(a,, bil, . . ., bJ, and y2 : F(X, 23) B the extension of x to a homomorphism. B such By ch. 1, Th. 4.31, there exists a homomorphism e : A(X, %) that exi = y2xi,i E I , and pa = a, a E A. Then exj = pw(ai, xjl, . . .,xi,> and hence xxj = w ( a , bil, . . ., bi), contradiction.
-.
-.
4.32. Lemma. Let B be any extension of A and W = {wjI i E I } an infinite minimal A-generating set of B. Then all minimal A-generating sets of B are infinite and of one and the same cardinality. Proof. Let V = {vjjj€ J } be any minimal A-generating set of B. Then, for every vj,there exists some finite subset 5 of W such that wj € A ( 5 ) . Hence = U (Vjl j € J ) is an A-generating set of B. This implies =W and finiteness of J would imply finiteness of W . Hence V is infinite. Moreover, IWI = I U ( V , l j € J ) /5 vjj J j €J ) == z ( ~ ~J I) = j€ IJI N O = / J /= /VI.Similarlyweget]Vi=s/W'/,hence1V1 = jW/.
w
w
-
c(l
4.4. Let B be any extension of the algebra A and IB 1 =- 1. Q shall have the same meaning as before, i.e. & = '$3 or 23, where B is an algebra of 23. An A-generating set U of B is called a &-basis of B over A if U is &-independent over A.
4.41. Proposition. Every &-basis of B over A is a minimal A-generating set of B. Proof. Let U = {uiii E I } be a &-basis of B over A which is not a minimal A-generating set of B. Then uj = w(ai, uil, . . ., uj), for suitable, pairwise distinct indices j , i,, . . ., i,. If 6 = 8, then, by Lemma 4.11 c), xr+, = w(a,, xl, . . ., x,)in A(x,, . . ., x,+,, 23) which contradicts Lemma 4.31. If & = '$3, then again by Lemma 4.1 1 c), [,+,= w(a, 5,, . . .,5,) in A([,, . . ., E,,,). But, if b,, . . ., b, € B, there exists by+,E B such that b,+, f w(ai, b,, . . ., b,) since 1 B j =- 1, contradiction.
4.42. Theorem. I f B has some infinite Q-basis over A , then all Q-bases of B over A are infinite and of one and the same cardinality.
62
CH. 2
AUjEBRAIC EQUATIONS
Proof. By Prop. 4.41 and Lemma 4.32. 4.43. Theorem. Let A be any finite algebra and suppose that, for 0. = %, there is some finite %+extension C of A with 1 C I =. 1 while, for 0: = !$, there is somefinite extension C of A with ICI w 1 being a subalgebra of B . Then all the &-bases of B over A are of one and the same cardinality.
Proof. By Th. 4.42, we can assume that B has just finite &-bases over A . Let U = {u,, . . .,urn}, V = {wl, . . ., vn} be such &bases. If 6 = 23, then %-independence of U over A implies that the epimorphism Q : A(x,, . . . , x ,, 23) A ( U ) as defined in ch. 1, Lemma 4.43, is an isomorphism. Hence, if 91,yi2 are the homomorphisms fromch. 1,s 4.3, we see that B, {eql, @pip} is a free union of the algebras A and F(x,, . . ., x , 23) in 23. If yl : A C is the inclusion map, i.e. yla = a, for all a E A, and y2 : F(x,, . . ., x,, 23) C the homomorphism defined uniquely by y2x, = ci,i = 1, . . ., m, where (cl, . . ., cm) is an arbitrary in-tuple of elements of C , then there is a unique homomorphism z : B C such that yl = zppl and ~2 = zpv2. Clearly za = a, for all a E A while zui = c,, i = 1, . . ., m. Since B = A(U), there is only one such homomorphism from B to C satisfying these conditions. Hence the number of all homomorphisms from B to C fixing A elementwise is 1 C I m . Similarly we obtain I C 1' for this number. Since I C I =- 1, we conclude that m = n. Now let 6 = !$. Let ui-+ ci, i = 1, . . ., m, be an arbitrary mapping from U to C. Since U is !$-independent and C is a subalgebra of B, the mapping w(a,, U J + w(aj,c,) is a well-defined mapping z from B to C . Moreover z is a homomorphism which extends u, ci and fixes A elementwise, and there is just one homomorphism of this kind. Thus the number of all homomorphisms from B to C fixing A elementwise is again ICl". By the same argument, this number is also ICI". Hence again m = n. --t
4
-t
+
+
4.5. Theorem. Let B be any extension of A with 1 B I
=- 1 which has &-bases over A of dlfSerent cardinalities. Then every &-basis of B is finite, and the numbers of elements of all &-bases constitute an arithmetic progression.
Proof. Th. 4.42 implies that every '3-basis of B is finite. Suppose we have ~} already shown that, if {al, . . .,a,}, {bl, . . ., bq}, {c,, . . ., c ~ + are Q-bases, then there exists also a &basis consisting of p f r elements. Let in and m+d be the smallest numbers of elements of &bases, then there
84
63
ALGEBRAIC INDEPENDENCE
exist B-bases with m+2d, m+3d, ~.. elements and if there were a &-basis with m+kd+r elements, 0 < r d, then there would also be a basis with m+r elements, contradiction. Let therefore {al, . . .,up}, (bl, . . . ,bq},{cl, . . . ,c,+~} be &-bases. By Lemma 4.11 d), there exists a homomorphism cp :B B such that p fixes A elementwise and Fbi = ci, i = 1, . . ., q, which is even a monomorphism, since, by Lemma 4.11 b), {cl, . . .,c,} is &independent. Moreover qB = A(cl, . . . ,cq) implies that qa, # c,+,, 1 G i e p , 1 e . j s r . If we can show that {pu1, . . ., yap, c,+~, . . ., c ~ + is ~ }a &basis of B, the proof will be completed. We have +
A(W1,
- . . W p cq+l. . . >
- 3
c,+~) = A(pa1, . . .,P ; ~ J(cq+13 .
.
* 3
. . a,)] ( C , + V . . c,,) = (@I (C,+l, . . C q + J . . ., b,)] (Cq+1, . ., Cq+J . . ., @,)(Cq+l, . . . , c,fr) = A(c1, . . ., cq)(cq+l,. . .,c = A@,, . . .) C q + r ) = B. - [eA(a,,
= [PW,, - A(@,,
.?
-
.?
cq+,.)
.?
*
~ + ~ )
Hence {pal, . . .,yap, c ~ + ~. .,., c ~ + ~is }an A-generating set of B. Let L be an arbitrary %extension of A, for 6 = 8, and L = B if 6 = p, and a an arbitrary mapping from’ {P;u,, . . ., ea,, cq+,, . . . , cq+*} to L, e.g. ayai = di, i = 1, . . .,p , CLC, = di, i = q+ 1, . . ., q f r . The monomorphism p induces an isomorphism 9, : B pB = A(c,, . . ., c,). Also, by Lemma 4.1 1 d), there exists a homomorphism 31 : B L fixing A elementwise such that pi= d,. i = I , . . .,p . We put e = zp-l which is a homomorphism from A(c,, . . .,c,) to L. Again by Lemma 4.1 1 d), there is a homomorphism 6 : B L fixing A elementwise such that tic, = get, i = I, . . . , q , 6c, = ct,, i = q + l , . . ., q + r . Since p fixe\ A elementwise, we see that 6c = ec, for all c E A(c,, . . ., c,); in particular, 6p;a, = @Fai= X U , = di, i = I , . . . , p . Hence 6 is a homomorphism extending a whence, by Lemma 4.1 1 d), {pl,. . ., pup, c ~ + ~. .,., c ~ , + ~is} &independent. +
-
--F
4.51. Proposition. Let B be any extension of A which has a &-basis over A of n elements. Then B has a &-basisover A of m elements if and only if there are words gl(a,, xl, . . .,xn), . . .,g,,(aj, sl,. . .,x n ) ;hl(aj, xl,. . .,x,,), . . ., hn(aj,xl, . . ., x ,) such that, ,for 0 = %. gi(aj,hl(aj, sl,. . . , x,,), . . ., h,(uj, xl,. . ., x n J ) = xi,
i = 1, . . . , in,
(4.51)
64
ALGEBRAIC EQUATIONS
CH.
2
holds in A(x,, . . ., x,, 8)and h,(uj,gl(uj,xl, . . ., x,), . . . , g,(uj, x,, . . .,x,))
=
xi,
i = 1, . .., n, (4.52)
holds in A(x,, . . ., x,, %) while, for 6 = '$3, the same equations hofd in A(t,, . . .),,: and A(E,, . . ., tn)with Ei instead of xi. Proof. Suppose {ul, . . .,u,} and {q,. . ., w,} are &-bases of A. Then vi = gi(aj,u,, . . ., u,), i = 1, . . .,m, and ui= hi(aj,vl, . . . , urn), i = 1, . . .,n, where gi(uj,xl, . . ., x,), h,(uj, xl, . . .,x,) are some words. Hence vi = gi(aj,h,(aj, v,, . . .,wm), . . .,h,(aj, v,, . . .,.),w By Lemma 4.1 1, we obtain (4.51), similarly (4.52). Conversely, suppose that {u,, . . ., u,} is a &-basis of B over A and the conditions of the proposition are satisfied. We set w i = gi(aj,u,, . . ., un), i = 1, . . ., m. Now let ciEL, i = 1, . . .,m, where L is an arbitrary %-extension of A if 0. = % and L = B i f & = $ . Setb i=h ,(aj,c, , . . ., c , ) ~ L , i = l ,..., n, a n d l e t p :B L be the homomorphism fixing A elementwise such that pui = bi, i = 1, . . ., n. Then pvi = gi(uj,pul, . . ., pun) = gi(uj,b,, . . ., b,) = gi(aj,h,(aj, cl, .. . ,c,) . . ., h,(uj, c,, . . ., c,)) = ci,i = 1 , . . ., m. Hence {vl, . . ., v,} is &independent. Moreover, {wl, . . ., urn} is A-generating by (4.52). -f
4.52. Corollary. If an extension B of A has a G-busis over A consisting of one element and a %-basisof n elements, n =- 1, then B has an A-generating set consisting of one Q-dependent element. Proof. Let {ul, . . ., un}and {v} be two &-bases of B, and g(uj,xl, . . ., X ~ ) , h,(aj,x,), i = 1, . . ., n, words satisfying (4.51) and (4.52). By (4.52), h,(uj,g(uj,w, . . ., u ) ) = v, i = 1, . . ., n, hence {g(aj,v, . . ., v)}is A-generating. But h,(aj, xl)= hk(aj,a$ in A(xl, 8) would imply xi = xk in A(x,, . . ., x,, %), hence g(aj,w, . . ., w) is &dependent if LT = 23. A similar argument works for 0. = '$3.
5. Systems of algebraic equations over groups. Algebraically closed groups
5.1. In this section we will specialize some of the results of this chapter to the variety of groups. Throughout this section, 8 will mean the variety of groups while G stands for a group. We will also write, as in ch. 1, G [ X ] for G(X, 23).
§5
SYSTEMS OF ALGEBRAIC EQUATIONS OVER GROUPS
65
We easily see that every algebraic system over (G, 23) is equivalent to an algebraic system pi = 1, i E Z , while every mixed algebraic system over (G, 23) is equivalent to a mixed algebraic system of the form pj = 1,
i€Z,
&#
1, j E J .
Thus it suffices to consider just systems of this special form. We can now restate Th. 1.71 which will also comprise Th. 1.23:
-
5.11. Theorem. The mixed algebraic system p i = 1, i E I , # 1, j € J over (G, 23) in X = {xl, . . ., xk} is solvable i f and only if the normal subgroup N of G [ X ]generated by the subset { p iI i E I} of G [ X ]is such that
N n G = (1) and contains no&.
Proof. Let 0 be the congruence on G [ X ] generated by the subset {(pi,1) j i E Z} of G [ X ]x G [ X ] .If A is any congruence on G [ X ] , then (pi, 1) E A if and only ifp, E ker A, and, by ch. 6, Th. 3.24,we easily see that ker 0 = N. Thus 0 is separating if and only if N n G = {I}, and 6 0 1 if and only if fj EN. Now the theorem follows from Th. 1.71. 5.12. Remark. One sees immediately that all the statements of this subsection are true for any arbitrary variety of Q-multioperator groups if we replace 1 by 0 and “normal subgroup” by “ideal”. 5.2. Not every algebraic system over (G, %), not even every system
consisting of one equation is solvable, e.g. x-lax = 1 has no solution for a # 1. There is, however, a large class of solvable algebraic systems consisting of one equation over (G, 23) as the next theorem will show.
5.21. Proposition. Let p = xb,xb, .. . xb,+, = 1, n == 1, be any equation over G[x],bj E G, i = 0, . . .,n - 1. Then p = 1 is solvable. Proof. Let C be a cyclic group of order n, and G wr C the regular wreath product of G by C. By ch. 6, 0 6.61, this is the set of all pairs (h, q) such that h E C and 9 : C G is a mapping, together with the group operation @vl> ,, (hz,v2)= (h,h2, y ) where yh = (q,h) q,(hh,). If 4 g ) denotes the constant function with value g, then g -+ (1, x ( g ) ) is clearly a monomorphism from G to G wr C , thus embedding G along this monomorphism yields an extension group H of G . Let C = [c] and q~:C .+ G be defined by p)(c’) = b;21-i, i = 0, 1, . . .,n- 1. Weclaim that (c, p) E H -t
66
ALGEBRAIC EQUATIONS
CH.
2
is a solution of our equation. Indeed, ('2
V ) (1,x(bJ)(c,P')( l , x ( b J ) . . . (c,
(1, X(bn-1)) = (cn,VJ)
where, for h E C , v h = (vh) b,p(hc) blp(hc2) . . . p(hc"-l)b,-,. ~ ( c ' )= (b,&lbobi?i-,bl
. . . b,'b,-,-J
Hence
(bi21bn-ibi?2 . . . bi.ibn-l) = 1.
This completes the proof.
. . ., x k } and p € G [ X ] be a polynomial such that, in the normal form p = ao$lalfLT. . . arplXArar of p , we have r =- 0, and moreover, ~ ' A =j (lu, . . ., I,,.), there is t with lfj == 0 for all j or lfj~ O j o rall j and Itj 0 or lfj -=c 0, resp., for one j . Then the algebraic system p = 1 is solvable. 5.22. Theorem. Let X = {xl,
Proof. Since p = 1 and p - l = 1 are equivalent equations, we may assume that ltj 0 and lfj =- 0 for one j . Then p = a,xjlalx:'" . . . a,_,x>a, = I is equivalent to some = x,boxfbl . . . xfb,-l = 1, and hence solvable = 1 , . . ., by Prop. 5.21. But then, x1 = 1 , . . ., x,-' = 1 , x, = c, x, = 1, where c is a solution of p = 1, is a solution of p = 1. 5.23. Corollary. If G is jnite, then every algebraic system of Th. 5.22 has a solution in somejnite extension group of G.
Proof. By Th. 5.22 and the proof of Prop. 5.21. 5.24. Corollary. Every equation x" = g over (G, %), where g E G and n # 0, is solvable and, if G isfinite, has a solution in some finite extension
group of G. 5.25. A very interesting result on the (S)-root extensions of G where ( S ) is any finite algebraic system over (G, 23) has been obtained which, similarly to field theory, ensures the existence of a "primitive element": Let ( S ) be any solvable finite algebraic system over (G, 23) in { x l , . . ., x,} and /GI f 1. Then every (S)-root extension G(al, . . ., a,) of G can be embedded irto some extension group G(a) of G. 5.3. As remarked in $3.1, every weakly mixed algebraically closed algebra is weakly algebraically closed. For groups, we will show the converse. 5.31. Theorem. Every weakly algebraically closed group G with I GI
is weakly mixed algebraically closed.
f
1
§5
SYSTEMS OF ALGEBRAIC EQUATlONS OVER G R O U P S
67
Proof. The proof of this theorem depends on two lemmas we are going to prove first.
5.32. Lemma. Let G be any group and H an extension group of Gsuch that theJinite mixed algebraic system over (G, 23) in x,, . . .,x, ui = 1,
iEZ,
vj # 1, j E J ,
(5.31)
has a solution in H . Moreover, let 1 # gE G. Then there exists some extension group K of H such that the finite mixed algebraic system over (G, 8)in the indeterminates x,, . . ., xk, y , z, sj, jE J , ti, j € J , which we obtain from (5.31) by adding the equations y 2 = 1,
gS1Z2 = 1,
s; = 1,
t,:'sjtjy-'
(sjvj)3= 1, = 1,
(JJZ)2
= 1,
jEI,
j EJ,
(5.32) (5.33) (5.34)
has a solution in K. Proof. Let m be the order of g, and D the dihedral group of order 4m, for m =- 0 ,and the infinite dihedral group, form infinite. Then D = [a, b ] , a2 = b2" = (ab)2= 1. Let L be the free product of H and D with amalgamation b2 = g. Then, in L, the equations (5.32) have a solution y = a, z = b. Let (h,, . . ., h,) be a solution of (5.31) in H , and therefore in L, vj(hl, . . .,h,) = fj E H , j € J , m ( j ) the order of fj, and Cj, j € J , the group generated by {cj,4} defined by the relations ;c = ( ~ ~ =4 d?") = 1 . We claim that cj is of order 2 and 4 is of order m ( j ) . Indeed, if U is the group of unimodular 2 X2-matrices over the integers mod m(j), 2 is the normal subgroup of U consisting of the matrices '1 0 and V the subgroup of U I Z generated by the (\O 1) and -1 1-1 and 4 = (o I)Z, then an easy comelements cj = -1 0 putation shows that cj, 4 satisfy the relations above, cj is of order 2, and 4 of order m(j).Let M I be the free product of L and C1 with amalgamation dl = fi, M2 the free product of M I and CZwith amalgamation d z = f i , etc., then we eventually get some extension group M of L. In M , the equations (5.33) have a solution x, = h,, . .., x, = h,, sj = cj, and a and ciare elements of order 2. By ch. 6, Cor. 6.81, there is an ex-
(-i1' , (
').
)~
68
ALGEBRAIC EQUATIONS
CH.
2
tension group K of M where every cj is conjugate to a whence there exist elements ti in K solving (5.34). This completes the proof. 5.33. Lemma. Let g # 1 be an element of G such that the algebraic system ( T ) consisting of the equations of (5.31), (5.32), (5.33), (5.34) is solvable. Then every solution of the algebraic system ( T ) is also a solution of the mixed algebraic system (5.31). Proof. Let (hl, . . ., h,, y , z, sj, ti) be a solution of the algebraic system ( T ) such that one of the inequalities of (5.31) fails to hold, i.e. zll(hl, .. .,hk) = 1, for some I € J. Then (5.33) implies sl = 1 whence, by (5.34), y = 1. Thus, by (5.32), g = 1, contradiction. 5.34. Proof of Th. 5.31. Let IG/ z 1, with G weakly algebraically closed. Suppose (5.31) has a solution in some extension group H of G . For an arbitrary 1 z g E G , there exists, by Lemma 5.32, an extension group K of H such that the finite algebraic system ( T ) of Lemma 5.33 has a solution in K. Since G is algebraically closed, ( T ) has a solution in G which, by Lemma 5.33, is also a solution of (5.31). This completes the proof. 5.35. Proposition. Every weakly algebraically closed group is simple.
Proof. Since the group of order 1 is simple, we can assume that I G I f 1. Let N be any non-trivial normal subgroup of G and 1 f gl EN. Then the mixed system (5.31) consisting of gl z 1 is solvable, hence, if 1 f g E G , the corresponding system ( T ) of Lemma 5.33 is solvable and has therefore a solution in G . Hence with this solution g = z2 = yz-1yz = (t,'sltl) (z-'t,-'sltlz) =
(~ T ' ( w J ~sltl) ( z - l t ~ ~ ( s s1tlz) ~gJ~
= t(t;l~;lglsltl) ( t l k l t l ) (tT1~hlSlt1)1
[(z-~t;4;~g1s1t1z) ( z - y g l t l z ) (z-lt;1s;1g,s1t1z)]. Hence g EN and therefore N = G . 5.36. Corollary. Every group can be embedded into a simple group.
Proof. This is a consequence of Th. 3.2 and Prop. 5.35.
55
SYSTEMS OF ALGEBRAIC EQUATIONS OVER GROUPS
69
5.4. Remark. We have defined algebraic systems pi = q i over ( A , %) where pi,qi are polynomials of A(X, 93) and X is a finite set of indeterminates. One could also drop the finiteness assumption on X , and it would still be possible to save a considerable part of the theory since the proof of ch. 1, Prop. 6.31 remains valid and thus the value of a polynomial on a place of a %extension of A is well defined. Thus it makes sense to define solutions and solvability of such “unrestricted algebraic systems” as in 0 1. An algebra A such that every solvable unrestricted algebraic system over ( A , 93) has a solution in A is called “absolutely algebraically closed”. Unrestricted algebraic systems have been considered by various authors and our results on algebraic systems are, to some extent, generalizable to unrestricted algebraic systems. A remarkable result on the existence of absolutely algebraically closed algebras in the variety of groups is the following. 5.41. Theorem. There is only one absolutely algebraically closed group, namely the group of order 1. Proof. The theorem will be a consequence of 5.42. Lemma. Let G be a group and A any subgroup of G. Any unrestricted algebraic system over A which has some solution in G, has also a solution in A if and only if A is a semidirect factor (retract) of G . Proof. Let A be a semidirect factor of G . Then G = AN where N a G and A n N = { l}. Let J;(x,) = 1, i E I, be an arbitrary unrestricted system over A in the indeterminates {xI/ I E J} which has a solution { g , / lE J } in G . Then g, = a&,, a, E A , d, EN, thus g,Oa, where 0 is the congruence corresponding to N . Hence A(aJ 01, thus J;:(a,)E A T‘IN = { I } whence {a, I 1E J} is a solution of the system in A . Conversely suppose that the hypothesis of the lemma is satisfied for G and A . Let M = {g,l I E J } be any set of elements of G with G = A(M), and {A(x,)I i E I} the set of all polynomials over A in the indeterminates {x,l I € J } such that J;(g,)= 1. Then the unrestricted system A = 1, i € I, has a solution in G whence it has also a solution {a,/ I € J } in A. Let B = {ay1gll1E J } and N the least normal subgroup of G containing B. Then AN = G. Suppose that a E A n N . Then a = h;’(a;’gt,P hl .. .
70
ALGEBRAIC EQUATIONS
CH.
2
*h,l(aclglk)e*hk where hiE G, ei = & 1. But since G = A(M), we have hi = wi(br,g,), b, E A , thus a-lwl(b,, gs)-l(aclgilY’wl(b,, 8,) . . . = 1. This is an equation over A of the form f;:(g,) = 1. Hence we can replace every g , by a, and obtain a true relation. This shows that a = 1 whence A is a semidirect factor of G. 5.43. Proof of Th. 5.41. Let A be an absolutely algebraically closed group, , FI(A)for IA / Z 1. and H any extension group of A such that /HI =- / A / e.g. By Cor. 5.36, there exists a simple group G containing H as a subgroup. Every unrestricted algebraic system over A which has a solution in G, has also a solution in A , thus, by Lemma 5.42, A is a semidirect factor of G. Hence G has a normal subgroup N such that G = AN and A n N = (1). Since G is simple, N = (1) or G. In the first case we would get 1 GI = I A1 whence ]HI s I A / , contradiction. Hence A = (1).
Remarks and comments
$8 1.2. Systems of algebraic equations have so far been studied mainly for special classes of algebras (see the comments on§ 5). Algebraic inequalities in the sense of our definition have been introduced by W. R. SCOTT [1] for the class of groups. Systems of algebraic equations over arbitrary algebras have been considered by DORGE[l], [2], DORGEand SCHUFF[l], SLOMINSKI [l], HULE[3], and under certain restrictions for the algebras, by SHODA[I], [2]. Our presentation partly follows HULE’Spaper, partly we have tried to generalize some fundamental concepts from the classical theory of algebraic manifolds (see e.g. VAN DER WAERDEN [2]) from fields to more general classes of algebras. Indeed, some of the results of 0 2 seem to indicate that simple algebras in arbitrary semidegenerate varieties behave similarly as fields do in the variety of commutative rings with identity with respect to systems of algebraic equations. Th. 1.23 is, we believe, due to DORGEwho has proved the result for arbitrary algebras even for the case of infinitely many indeterminates and Th. 1.3 has been proved by HULE.Amongst the unsolved problems of this section, we mention: a) Let (5’)be a solvable algebraic system over ( A , %), B a %extension of A , 6: A(X, %) B(X, %) that homomorphism which extends the inclusion monomorphism from A to B, and (65‘)the algebraic system over
-.
REMARKS AND COMMENTS
71
(B, %) which is obtained from applying 6 to each equation of (S). Is (6s)also solvable ? b) Has every solvable algebraic system over ( A , %) which satisfies the first alternative of Th. 1.3 a solution in A?
8 3. Various concepts of being algebraically closed have been used, and most of them are special cases of the general notion of being (m, n)algebraically closed which is defined as follows : Let % be any variety and m,n arbitrary cardinals. An algebra A of % is called (m, n)-algebraically closed in 23 if every (possibly unrestricted, see 9 5.4) solvable algebraic system over ( A , 23) consisting of less than m equations in less than n indeterminates has a solution in A. For example, being weakly algebraically closed in our book means being ( R ~ , R0)algebraically closed. (By slightly modifying the definition of being (m, n)-algebraically closed, one could also cover the classical concept of being algebraically closed for fields, this we leave to the reader). The relationship between the different concepts of being algebraically closed is far from being known completely. Just occasionally there have been investigations on algebraic closedness in its general meaning (e.g. SHODA€31,FUJIWARA [l], BOKUT’ [l]). Special results refer to the variety of groups or sometimes semigroups (see W. R. SCOTT [I], B. H. NEUMANN [I], [2], [3], ERDELYI [l]). Our proof of Th. 3.2 is a generalization of SCOTT’S proof for groups, and Prop. 3.3 is due to B. H. NEUMANN for the special case of groups. In this context we also refer to the concept of an equationally compact algebra which has recently been studied in several papers (see MYCIELSKY and RYLL-NARDZEWSKI [l], WEGLORZ [l], [2], WENZEL [l]). An algebra A of the variety 23 is called equationally compact if every (unrestricted) algebraic system ( S ) over (A: ‘23) has a solution in A provided that every finite subsystem of ( S ) has a solution in A (it is easy to see that this concept is independent of the variety %). 0 4. There are several concepts of independence in classes of algebras (a general discussion of dependence relations in algebras can be found in COHN[l], for groups see LYNDON [l]). One of the most natural and, as we think, also most useful concepts has been introduced and thoroughly investigated by MARCZEWSK~ and his school (there is an exposition of this work in GRATZER [3]). This is MARCZEWSKI’S definition of independence :
72
ALGEBRAIC EQUATIONS
CH.
2
Let B be an algebra and S = {u, I i E I } a subset of B. Then S is called independent if and only if, for any k 3 1, any two functions p, q E Gk(B) (the algebra of GRATZER'S polynomial functions), and any subset { ul, . . .,uk} of S , p(ul, . . .,u k ) = q(u,, . . .,uk) implies p = q . If B is an extension of A , then Lemma 4.11 shows that this definition of independence in B is closely related to our definition of '$-independence over A . Indeed, if the subalgebra of A generated by the 0-ary operations coincides with A , then '$-independence over A in our sense coincides with independence in B in the sense of MARCZEWSKI. Therefore all our results of Q 4 correspond to results on MARCZEWSKI'S independence. Hence the presentation of our 8 4 resembles strongly that in GRATZER [31, 31.
o
3 5. As mentioned, there exists quite a large number of papers on systems of algebraic equations over certain special classes of algebras. Apart from systems of equations over commutative rings, and fields in particular (which are the subject of the classical theory of elimination, and fundamental in algebraic geometry, and have been treated in abundance), there exists also a remarkable literature on algebraic systems over Boolean [2], RUDEANU [l], [3], [5]) and over algebras (e.g. &IAN [l], ANDREOLI groups (e.g. ALLENBY [l], ERDBLYI[I], GERSTENHABER and ROTHAUS [l], HOANGKI 111, ISAACS 111, LEVIN[l], [2], SCHIEK [l],SCHUFF[l], SOLOMON [l]). For algebraic equations over other special classes of algebras, see GOODSTEIN [3], LEVIN[3], RUDEANU [4]. Th. 5.11 (for algebraic systems) is also contained in ERDBLYI[l], Th. 5.22 and its proof is due to LEVIN[l], who has also proved (see [2]) the result which is mentioned in Q 5.25. The results and proofs of 8 5.3 are due to B. H. NEUMANN [I], Lemma 5.42 is a result of ERDBLYI[l].
CHAPTER 3 COMPOSITION OF POLYNOMIALS AND POLYNOMIAL FUNCTIONS
1. Composition algebras 1.1. Let M be a non-empty set, k a positive integer, x a (k+ 1)-ary operation on M , oi an n,-ary operation on M , and xi, y l ,j , I = 0,1, 2, . . ., indeterminates. We define : a) The operation x is called superassociative if x satisfies the law xxxoxl
.. . xky1y2 . . yk = xxoxx1y1 . .
1
yk"x2y1
--
*
-
yk f . xxkyi
- .. Y k .
b) The operation x is called right-superdistributive with respect to wi if x satisfies the law xwiy, . . . y k = m i , for n, = 0, xwixl . . . xniyl . . . yk = wj%x1y1 .'-. ykxxzy] * . . yk . * * x&,y1
for
--
*
yk 9
ni =- 0.
c) The family {sj, . . .,sk} of elements of M is called a selector system for the operation x,if sl, . . .,sk regarded as O-ary operations on M satisfy the laws (1.1.a) i = I, 2, , . ., k, x s i y l . . . yk = y i , (1.1 .b) X X I S l . . . Sk = x l . 1.11. Lemma. There exists at most one selector system for x.
Proof. Let {tl, . . ., tk} be also a selector system for x. Then ti = %sitl. . . tk = si,i = 1, . . ., k , by (1.1.a) and (1.l.b). 1.12. Remark. Let k = 1 and ni = 2. Then superassociativity reduces to ordinary associativity, right-superdistributivity to ordinary right-distributivity, and a selector system for x to an identity for x . 1.2. Let 23 be a class of algebras of type T = {n,1 L -= o } and 0 = {wIIL -=z o} the family of operation symbols of the algebras in 23 where o is a fixed 13
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COMPOSITION OF POLYNOMIALS AND POLYNOMIAL FUNCTIONS
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ordinal. Let A be any algebra of type T with Q as its family of operations, and x a (k+l)-ary operation on A , k =- 0 an integer. If we put w, = x , then Q, = {ac I L < o f l } is a family of operations on A. The algebra A = ( A ; Q,) is called a k-dimensional %-composition algebra if a) ( A ; Q) is an algebra of 23, b) x is superassociative, c) x is right-superdistributive with respect to w,EQ, for all L -= 0. The algebra A = ( A ; @ ) is called a k-dimensional %-composition algebra with selector system if there is a selector system for x in A. Every k-dimensional %-composition algebra ( A ; Q,) with selector system can be regarded as an algebra with the family Q, = {a’ I L < o k + I} is the selector system for x regarded of operations where . . ., as a family of 0-ary operations.
+
1.21. Proposition. r f the class % is a variety with respect to Q, then, for any k, the class of k-dimensional %-composition algebras is a variety with respect to Q1, and the class of k-dimensional composition algebras with selector system is a variety with respect to Q z . This is a straightforward consequence of the definitions of superassociativity, right-superdistributivity, and selector systems since all these concepts are defined by laws.
1.3. A few examples will illustrate the concept of composition algebras : Let % be the variety of sets, i.e. Q = 4, then the k-dimensional 8cornposition algebras are the so-called k-dimensional superassociative systems. In particular, the 1-dimensional superassociative systems are exactly the semigroups. Let be the variety of groups as considered in ch. 1,s 2.4. Then the k-dimensional %-composition algebras are called k-dimensional composition groups. In particular, the 1-dimensional composition groups are better known under the name “near-rings”. If % is the variety of rings as considered in ch. 1,s 2.4, then the k-dimensional %-composition algebras are called k-dimensional composition rings. 1-dimensional composition rings are also known as composition rings or tri-operational (TO-) algebras.
81
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COMPOSITION ALGEBRAS
In case of 8 being the variety of lattices as in ch. 1,s 2.4, the k-dimensional %-composition algebras are called k-dimensional composition lattices. For k = 1 , we simply speak of composition lattices. 1.4. Let A be any algebra, Q its family of operations, k, 1, positive integers, Fk ( A ) and F, ( A ) the full function algebras, p E Fk ( A ) , and (yl, . . .,Yk) E F,(A)k. We define the “composition” p o (yl, . . .,?&)€ F, ( A ) by (31 (yl,
* .?
yk)) (al,. .
.?
=
p(vl(al,
*
,
- - . >yk(al, - a’>), (al, . . .,a‘) E A’. *
-7
1.41. Lemma. For any (yl, . . ., Yk) €F,(A)k, the mapping p o (yl, . . .,Yk) is a homomorphism from Fk ( A ) to F, (A).
p -,
Proof. We have to show that m i o ( y l , . . ., yk) = mi, for ni = 0, and (mipi . . . 9),,> 0 (y1, . . .,yk) = mi[pl (yl, * * Yk)] . . . [pni (yl, . . ‘ vJk)l? for ni =- 0. This is done by showing that these equations hold “pointwise” for any (al, . . .,aJ € A’.
-
7
7
Proof. Again, by showing “pointwise” equality. 1.43. Proposition. Let A be an algebra of the variety %, Q its family of operations, and k a 1 an integer.. On (F,(A); Q), we deJine a (k+ 1)-ary operation x by xpopl . . . p k = yo o (pl, . . .,V k ) . If Ql= {Q, x}, then (Fk( A ) ;Ql) is a k-dimensional %-composition algebra with selector system. Proof.Bych. 1,Prop. 6.11,(Fk(A);Q)isanalgebraof%. For 1 = m = k, Lemma 1.42 yields superassociativity for x while Lemma 1.41 shows the right-superdistributivity with respect to any miE Q. Moreover, the family El, . . .,t k of the projections of Fk(A) is a selector system for x . 1.5. In the preceding proposition, we have shown that every full function algebra Fk(A) over an algebra A of the variety ‘8 is a k-dimensional %-composition algebra if the composition of functions is added to the operations on Fk (A). Since the class of k-dimensional %-composition
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algebras is a variety, every subalgebra of this composition algebra is also a k-dimensional %-composition algebra. The subsequent theorem will show that, up to isomorphism, there are no further k-dimensional %-composition algebras, this means that for every @-compositionalgebra there exists an isomorphic algebra consisting of functions on an algebra of % where the composition operation is just the composition of functions or, as we say, that every %-composition algebra can be faithfully represented by such an algebra of functions. 1.51. Theorem. Let A = ( A ; Q, x) be a k-dimensional %-composition
algebra. Then there exists an algebra D of % such that A is isomorphic to some subalgebra of the k-dimensional %-composition algebra (Fk(D); 52, x) where x is the composition of functions. Proof. The theorem holds for I A I = 1 since then I Fk(A)I = 1 and we may take D = ( A ; 52). Assume now that I A 1 f 1. Let D be any %-extension of ( A ; Q) different from ( A ; 52) such as (F,(A);52). Let 6 : A Fk(D) be defined by -t
8 is certainly injective since there exists (d,, . . ., dk)C Dk -. Ak, hence (8a)(dl, . . ., dk) = a, and (8b)(d,, . . ., d,) = b, a, b E A . We have to show that 8 is a homomorphism, i.e. 8oi = mi, $wial
&a,,
for
ni = 0,
.. . ani = wi6a, . . .6a,,;, . . . ak = x6a, . . . @ak
for
ni =- 0
That these equations hold, can be easily verified, by substituting any (d,, . . ., d,) E D k into either side of the equations, and using right superdistributivity and superassociativity of x for ( A ;Q, x). 1.52. The examples of 1.3 yield some remarkable special cases of Th. 1.51 :
a) Every k-dimensional superassociative system can be faithfully represented by some superassociative system of k-place functions on a set. Every semigroup can be faithfully represented by some semigroup of I-place functions on a set.
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COMPOSITION ALGEBRAS OF POLYNOMIALS AND POLYNOMIAL FUNCTIONS
77
b) Every k-dimensional composition group can be faithfully represented by a composition group of k-place functions on a group. Every nearring can be faithfully represented by some near-ring of 1-place functions on a group. c) Every k-dimensional composition ring can be faithfully represented by some composition ring of k-place functions on a ring. d) Every k-dimensional composition lattice can be faithfully represented by some composition lattice of k-place functions on a lattice. 2. Composition algebras of polynomials and polynomial functions 2.1. (Pk(A); Q ) is, by definition, a subalgebra of (Fk(A);Q). Let no,. . .,n, EPk(A). Since Pk(A)= . . ., E,), we have ni= wi(ai, El, . . ., &), i = 0,1, . . ., k . If x denotes the composition of functions, then xz,, . . . z k = z o o (z3,. . .,n,) = wo(ai, wl(ui, El, . . .,E,), . . ., &(ai, El, . . ., &)) for this equations holds “pointwise” for all elements of Ak. Thus xno . . . z k EPk(A).Hence we obtain from Prop. 1.43 and Prop. 1.21 2.11. Proposition. Let A be an algebra of the variety 8.The subset Pk(A) of the composition algebra (Fx( A ) ;Q, x) is a subalgebra, i.e. (Pk(A);n, x) is a k-dimensional %-compositionalgebra with selector system El, Ez, . . .,
&.
2.12. Remark. In ch. 1, 5 6.2, we have shown that, for any subfamily
@ of 0, the algebra (Pk(A@);@) is a subalgebra of (Pk(A);@).Since Pk (A,) is, by Prop. 2.11, a subalgebra of (Fk( A ) ;@, x), we conclude that
Pk(A@)is closed with respect to the composition of functions, thus (Pk(A,); 0,x) is also a subalgebra of (Pk(A);@, x).
2.2. Let A be an algebra of the variety %,Q its family of operations, X = {xl, . . .,xk}, Y = { y l , . . .,yl}, Z = {zl,. . ., z,} (not necessarily disjoint) sets of indeterminates, and p = p(xl, . . ., X k ) E A(X, %), (ql, . . ., q k ) E A(Y, %)k. We define a “composition” p o (ql, . . ., qk) by po(ql, . . ., q k ) zz p(ql, . . ., q k ) which is, by ch. 1, Prop. 6.31, a Welldefined element of A(Y, %). Restating this proposition for this particular situation, we get 2.21. Lemma. r f ( q l , . . ., qk) is aoy $xed element of A(Y, then the mapping p P O (ql, . . ., qk) is a homomorphism from A ( X , 8) to A(Y, 8).
-
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2.22. By the definition of p(ql, . . ., qk) (see ch. 1, Prop. 6.31), a representation of P O (ql, . . ., qk) as a word can be obtained if we take any representation p = w(aj, x,, . . ., xk) for p and representations qi = vi(aj,yl, . . ., yl ) for qi, i = 1, . . ., k, as words. Then p (41, . . ., qk) = w ( a j , v l ( a j , y1, . . .>Y/>> . ' vk(aj7 y17 . . ' Y l ) ) . Hence .?
7
2.24. Proposition. Let A be an algebra of the variety % and Q its family of operations. For any integer k a 1, dejine a (k+ 1)-ary operation x on (A(x1, . . * xk, xPoP1 . . . Pk ='PO (Pl, * * * P k ) = PO(P1, . . P k ) , ;Q, and let Q, = {Q, x } . Then the algebra (A(xl, . . ., xk, @); Q,) is a k-dimensional @-compositionalgebra with selector system. 9
7
.?
Proof. (A(xl, . . ., xk, 8); Q) is an algebra of 23. Lemma 2.23 applied to X = Y = 2 shows the superassociativity of x, Lemma 2.21 the rightsuperdistributivity of x with respect to-any wi € Q. It is easy to see that the family x,, . . . ,xk is a selector system for x. 3. Composition homomorphisms
Q, x) be k-dimensional %-composition 3.1. Let (C; Q, x) and (D; algebras. A homomorphism from ( C ;Q) to (D; Q) is called a composition Q, x). homomorphism if it is also a homomorphism from ( C ;Q, x ) to (D; If a composition homomorphism is a monomorphism (epimorphism, isomorphism), then it is called a composition monomorphism (epimorphism, isomorphism). Let C = ( C ;52, x) be a k-dimensional @-compositionalgebra. An element c E c is called a constant of C if xcc, . . . ck = c, for all (c,, . . .,ck) E ck. Examples for constants are the elements of A in the @-composition algebras (A(X, 23); Q, x), (Fk(A);52, x), and (Pk(A);Q, x ) . 3.11. Lemma. Let 8 : ( C ;Q, x ) = C -, (D;Q, x) = D be an epimorphism of composition algebras. Then 8 maps any constant of C onto a constant
of D, and a selector system of C onto a selector system of D.
03
19
COMPOSITION HOMOMORPHISMS
Proof. Straightforward. 3.2. We know from Q 2 that, for any algebra A in %, the %-polynomial algebra A(x,, . ..,x,, %) together with the composition x of polynomials is a k-dimensional %-composition algebra. We are now going to determine all composition epimorphisms of A(x,, . . .,x,, %). 3.21. Theorem. Let A(x,, . . ., xk, 8)= A(X, %) be any %-polynomial algebra, and ( C ;9, x) a k-dimensional %-composition algebra such that a) C has a selector system s,, s2, . . .,4, b) ( C ; has a subalgebra B consisting of constants of ( C ;9, x ) such that ( C ;Q) = B(s,, . ., sk). Then every epimorphism q : A B can be uniquely extended to a composition epimorphism e : A ( X , 23) ( C ;9).e maps xi onto si, i = 1 , . . . ,k, and is called the composition extension of q. Every composition epimorphism of A(X, 23) can be obtained in this way.
a)
.
-+
+
Proof. Let q: A -, B be an epimorphism, then q can be considered as a homomorphism from A to (C; Q). Let y : F(X, %) (C; a)be the extension of the mapping xi -,sito a homomorphism of the free algebra F(X, 93). Since A(X, %), {pl,9 2 ) is, by ch. 1, Q 4.3, a free union of the algebras A and F(X, 23) in 8,there exists a homomorphism e : A(X, 93) ( C ;9)such that q = pp1 and y = ep2. By definition of PI,p2, pa = q a , a E A , and exi = si, i = 1 , . . ., k . Since ( C ;Q) = B(s,, . . ., s k ) , e is an epimorphism. Next we show that e is even a composition epimorphism. Let w,,, . . ., wk E A(X, %) and w,, = w,,(aj, x,, . . .,xk) be a representation of w,, as a word. Then xw,, . . . w, = wo(aj, w,, . . .,w,) and hence exwo. . . w k = w,,(qaj, ew,, . . ., ew,). On the other hand, ew,, = wo(qaj7s,, . . ., s,). Since ~tis right-superdistributive, qaj E B, i.e. qaj is . a constant of (C; Q, x), and s,, . . .,s, is a selector system of (C; 9,x), we conclude -+
-
Thus p is a composition epimorphism.
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Let a be any cornposition epimorphism from A(X, %) to ( C ;Q) extending q. By Lemma 3.11, axl, . . ., ax, is a selector system of ( C ;Q, x) and hence, by Lemma 1 . 1 1, axi = sj, i = 1 , . . ., k. Thus r~ and p coincide on A U X which generates A(X, 8). This implies u = p. Finally let a : A(X, %) C be any composition epimorphism of A(X, %). By Lemma 3.1 1, the k-dimensional %-composition algebra ( C ; L?, x) has the selector system axl, . . ., ax,, and the algebra ( C ;Q) has oA as a subalgebra consisting of constants of ( C ; Q , x ) . Since A(X, 58)= [ AU X I , we conclude that ( C ;L?) = (aA)(ox,, . . .,axk). If 7 is the restriction of a to A , then a is just the unique extension of 7 to a composition epimorphism from A ( X , %) to ( C ;SZ). -+
3.22. Three important special cases of Th. 3.21 deserve to be mentioned. Here X = {xl, . . ., xk}. a) Let A be an algebra of the variety 23 and q : A B an epimorphism. Then 7 can be uniquely extended to a composition epimorphism p :A(X, %) .+ B(X, 23)which will be denoted by q(X, a),q [ X ] or q(X). We have exi = xi,and hence e is just the homomorphism of ch. 1, Prop. 4.5. Thus the notation I, = q(X, 23) = q ( X ) is compatible with ch. 1, Remark 4.5 1. be two varieties and A an algebra of 231. Then there b) Let %l C is a unique composition epimorphism from A(X, 232) to A(X, 231)fixing A U X elementwise (see also ch. 1, Th. 5.22). c) Let A be an algebra of the variety 53.Then there exists a unique composition epimorphism from A(X, %) to Pk(A)fixing A elementwise and mapping xi to &, i = 1, .. ., k. This is, by ch. 1, Prop. 6.41, just the canonical epimorphism a. The dependence of a on A, k , and % will, if necessary. be expressed by cr = o(A)= a(A, k) = a(A, k, 93) = a(X, 8)= 4% X , %I. -f
3.3. We are now going to consider the algebras of k-place polynomial functions as k-dimensional composition algebras, in the sense of 9 2. In particular, we shall obtain a result analogous to 5 3.22, a). 3.31. Proposition. Let k w 0 be an integer, A an algebra, and r j : A B an epimorphism. Then r j can be uniquely extended to a composition epimorphism I, :Pk ( A ) Pk (B). e$xes every Ei and is called the composition extension of q. If r j is an isomorphism,so is g. -+
+
03
COMPOSITION HOMOMORPHISMS
81
Proof. Let p EPk(A) and p = w(uj, tl, . . ., t k ) a representation of p as a word. We define pp = w(quj, El, . . .,t k ) then pp is a well defined element of Pk (B). For, if p = v(uj, El, . . .,&) is another representation of p as a word, then w(aj,c,l . . ., ck) = w(uj, cl,. . ., ck), for all (C1, . . .,ck) Ak. Then W ( q U j , qCl, . . .,qCk) = V(?;rUj, YC,, . . ., qck) whence w(quj, tl, . . ., t k ) = v(quj,tl, . . ., Ck) as q is surjective. e is obviously surjective and extends q. Straightforward computation exhibits p as a composition homomorphism. In order to establish the uniqueness of p, we take any composition epimorphism a : Pk ( A ) -. Pk ( B ) extending q. Under a, the selector system tl, . . .,Ek of Pk (A)is mapped onto a selector system of Pk(B) whence, by Lemma 1.11, ati = ti, i = 1, . . ., k. Since Pk(A) = [AU {El, . . ., &}I, c coincides with e. Suppose now that q is an isomorphism. Then 7-l:B -. A is an isomorphism, in particular, qV1is an epimorphism and thus can be extended to a composition epimorphism z : Pk(B) Pk(A). Hence zp and pa are extensions of identity mappings and, by uniqueness, are identity mappings themselves. +
3.32. Remark. Let q : A B be an epimorphism and X = {xl, . . ., xk}. The unique extension p of q to a composition epimorphism from Pk (A) to Pk(B)will be denoted by Pk(q).The diagram fig. 3.1 is commutative. -f
FIG.3.1 3.4. Let R, S be algebras of the variety % such that there is an additional operation x on R, S making k-dimensional %-composition algebras of R, S . The direct product R X S by adding x then also becomes a k-dimensional %-composition algebra, by Prop. 1.21 and ch. 1 , Th. 2.51. In particular, if A , B are algebras of % and X = {xl, . ..,xk}, then A(X, %) >: B(X, 23) and Pk ( A ) XPk ( B ) are k-dimensional %-composition algebras.
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Proposition. Let X = {xl, . . ., xk}, A, B algebras of 8 and A X B . Then there exists a unique composition homomorphism z : U ( X , 8) A(X, 8) X B(X, %) such that zU(X, 23) is a subdirect product of A(X, %) and B(X, %) and zjixes U elementwise. Similarly, there exists a unique composition homomorphism z : Pk (U) Pk( A )X P k (B) such that zPk (U)is a subdirect product of Pk ( A ) and Pk (B) and z jixes U elementwise. In either case, zp = (elp, ezp)where el = n i ( X )or el = Pk(zi), respectively, i = 1,2, and n,are the projections of U. 3.41.
U
-.
=
-+
z will be called the decomposition homomorphism of U(X, 8)or Pk (U),
respectively.
Proof. We will prove just the first assertion, the second assertion can be shown by exactly the same argument. By straight forward calculation we see that z as defined above has all the required properties. Therefore let u be another composition homomorphism with these properties, and p l , i = 1,2, the projections of the composition algebra A(X, %) X B ( X , a). Then ,up, i = 1, 2, is a composition extension of ni,and hence, by Th. 3.21, ,up = n,(X), i = 1,2. Since up = ( p p p , ,u2up),we conclude u = z. 3.42. Remark. Let z1 be the decomposition homomorphism of U(X, %), z2 the decomposition homomorphism of Pk (U), and u(A)x u(B):A(X, 8)X
-
B(X, 8) Pk(A)YPk(B) the epimorphism defined by ( u ( A ) X u(B))( p , q) = (a(A)p,o(B)q). Then diagram fig. 3.2 is commutative. This is an immediate consequence of the commutativity of diagram fig. 3.1. z1
FIG.3.2 3.5. We want to know under what circumstances the decomposition homomorphisms z of U(X, %) and Pk ( U )are epimorphisms or monomorphisms. We start with the following
03
COMPOSITION HOMOMORPHISMS
83
3.51. Proposition. Let X = {x,, . . ., xk}, and U = A X B . The decomposition homomorphism z of U(X, 23) is an epimorphism if and only if, for any ( p , q ) E A(X, 23) X B(X, 23), there exists a word w(y,, . . .,y,, x,, . . .,xk) in the indeterminates yl, . . .,y , and elements a,, . . ., a, E A, b,, . . .,b, E B such that w(al, .. ., a,, x,, . . ., x,) is a representation of p and w(bl, . . ., b,, xl,. . ., x k ) is a representation of q. The decomposition homomorphism z of Pk(U) is an epimorphism if and only if, for any ( p , q)EP,(A)XPk(B), there exists a word w(y,, . . ., y,, l,, . . ., t k ) and elements a,, . . ., a, E A, b,, . . .,b, E B with analogousproperties. Proof. We prove the first assertion, the second one can be shown using exactly the same argument. Let z be an epimorphism, then, for any ( p , q) E A(X, %) X B(X, a), there exists an element u E U(X, 23) such that zu = ( p , 4). Let u = w(u,, . . ., u,, xl, . . ., xk) be any representation of u as a word. By Prop. 3.41, zu = (n,(X)u, z2(X)u),zibeing the i-th projection of U, i = 1,2. Hence p = w(n,u,, . . .,z,u,, x,, . . ., xk) and q = w(z2u1, . . .,n2u,, x,, . . ., xk), is.w(y,, . . ., y,, x,, . . ., xk) is just a word we were looking for. Conversely, if ( p , q ) E A(X, 23)XB(X, %) and w(y,, . . ., y,, x,, . . ., xk) is a word such that w(a,, . . .,a,, x,, . . .,x,) = p and w(b,, . . .,b,, xi, . . .,xk) = q, for some a,, . . .,a, E A , b,, . . ., b, E B, then u = w((a,, b,), . . ., (a,, b,), x,, . . ., x k ) is a polynomial in U(X, 23) such that zu = ( p , q), thus z is an epimorphism.
3.52. Remark. Diagram fig. 3.2 shows that the decomposition homomorphism z2 of P,(U) is an epimorphism if the decomposition homomorphism z1 of U ( X , %) is an epimorphism. 3.53. Proposition. For any algebra U = A,XB, the decomposition homomorphism z of P k ( u ) is a monomorphism. Proof. Let g EPk(U) and ((a,, b,), . . ., (ak,b,)) E uk.Then g((a,, bl), . . ., ( a k , bk)) = ((pk(z1)g)(a,, . . ., ak), (Pk(z2)g)(bl,. . ., bK)).This is established by taking any representation of g as a word in elements of U and the projections ti,i = 1 , . . ., k, or by applying z. Suppose zp = zq, for some p , q E Pk(U). Then Pk(q)P = Pk(ni)q, i = 1,2, whence ~ ( ( a ,bl), , . . ., (ak? b k ) ) = q((a,, b,), . . ., (ak,bk))? for all ((al,b,), . . ., (ak,bk))E Uk. Therefore p = q.
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3.54. Remark. It is not known under what conditions the decomposition homomorphism z of U(X, 23) is a monomorphism. 3.6. As an application of the preceding results we prove: 3.61. Theorem. Let X = {xl,. . ., xk}, % the variety of commuta-
tive rings with identity, and U = A X B an algebra of 23. Then the decomposition homomorphisms 21 : U(X, 8) A ( X , % ) x B ( X ,23) and z2 : Pk( U ) + Pk ( A )X Pk ( B ) are isomorphisms. +
Proof. Let ( p , q ) E A ( X , % ) x B ( X , 23). By ch. 1, Th. 8.21, there are representations of p , q as words of the form
P = C(a,?? I 2 E PI)? 4
=
C(b,$IA
PZ)
Thus x ( y A f I A E PIUP,) is a word in the indeterminates y, satisfying the conditions of Prop. 3.51. Hence z1is an epimorphism, and so is z2, by Remark 3.52. It remains to show that zl is a monomorphism. Let f, g E U ( X , 23) such that zlf = zlg. By ch. 1, Th. 8.21, there are representations off, g as words of the form
f = I(%$11E P),
g = C<.,XL I A E PI
c(
We obtain normal forms for n i ( X ) f and nj(X)g, i = 1,2, if we remove I 1. € P) and (niv,j$ j 1 E P) those summands in the words C((niuA)hA for which niu, = 0 and nivA= 0, respectively. Since zf= zg implies n i ( X ) f = ni(X)g, i = 1,2, we have n,u, = ziv,, for all il E P, hence uL = v,, for all 1 E P. Therefore f = g. In ch. 5, we will show that, if 23 is the variety of groups, the decomposition homomorphisms zl,z2 are, by no means, always isomorphisms. 4. Full congruences 4.1. Let ( A ; Q, x ) be a k-dimensional 23-composition algebra. The congruence @ of ( A ; Q) is called a full congruence if @ is also a congruence of ( A ; Q x ) . Thus @ is a full congruence if and only if pi@qi, i = 0, 1, . . ., k, implies xp,,pl . . . pk@xqoql . . . q k . Ch. 1, Q 1.4 and Prop. 1.21 imply that, for any full congruence @ on
(A;
n),the factor algebra ( A ;Q, x)[@ is a k-dimensional %-composition
04
85
FULL CONGRUENCES
algebra, and the canonical epimorphism from A to A I @ is a composition epirnorphism. Conversely, by ch. 1, Th. 1.51, for any composition epi( B ; Q), the kernel Ker pl = @ is a fullcongruence morphism pl : ( A ; Q) -, on ( A ; Q), and there is a composition isomorphism y : B ..+ A I@ such that y v is the canonical epimorphism from A to A I@. Thus we obtain, up to isomorphism, every homomorphic image of the composition algebra ( A ; Q, x) as a factor algebra of ( A ; Q, x) with respect to a suitable full congruence of ( A ; Q). 4.2. The definition of a full congruence applies, in particular, to the k-dimensional composition algebras Fk(A), Pk(A), and A(X, 23) where X = {xl, . . ., xk}. It has been shown that, for k z 1 and any A, the algebra Fk(A) has just the trivial full congruences. This is also true for a large class of algebras if k = 1, but does not hold with full generality. By 9 3.22 c), the canonical epimorphism B : A(X, 23) P k ( A ) is a composition epimorphism. The kernel @= Ker d is a full congruence on A(X, 23). If y : P k (A) A(X, 23) I @ is the corresponding composition isomorphism, then y-l yields a bijection from the set of full congruences on A(X, 23) I@ to the set of those on Pk(A). Thus the full congruences on Pk(A) are completely determined by the full’congruences on A ( X , %) I @ which, in turn, are determined by the full congruences on A(X, 23) containing @, by ch. 1, Th. 1.71. Therefore we restrict ourselves to the investigation of the full congruences on A(X, %). -t
-t
4.3. Let X = {xl, . . ., xk} and A an algebra of 23. First we give some characterization of the full congruences on A(X, 23).
Proof. The “only if” statement is obvious. Suppose now that @ is a congruence satisfying the hypothesis, and pi@qi, i = 0, 1, .. .,k. By induction on the least minimal rank of the words representing PO, (xpopl * -pk)@(xpOql* * .qk),and by hypothesis, (xPOql * * qk) @(xqOql* * qk)* Thus (xPOP1 * * pk) @(“1Oq1 - . qk).
-
-
-
-
-
4.32. Now let 0 be a congruence on A and (0) the congruence on
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COMPOSITION OF POLYNOMIALS AND POLYNOMIAL FUNCTIONS
CH.
3
A(X, 23) generated by 0, i.e. (0)is the set-theoretical intersection of all congruences on A(X, 23) containing 0.
4.33. Proposition. For any congruence 0 on A, the congruence (0) is a full congruence on A(X, 23). This proposition will be established if we prove the following 4.34. Lemma. p ( 0 ) q holds i f and only if there is a j n i t e chain p = ro, r,, . . ., rt = q of polynomials of A(X, 23) such that, for any two adjacent polynomials ri, ri+,, we have ri = wi(al, . . ., a,, xl, . . ., xk), ri+l = wi(bl, . . .,b,, xl, . . ., x k ) , where wi(yl, . . .,y,, xl, . . . ,xk) is some word in indeterminates yj, j = 1, . . .,m, and where aj, bj E A , ajObj, j = 1, ..., m.
Proof. The existence of such chains between p and q is an equivalence relation @ on A(X, 23). Let w be any ni-ary operation and pk@qk, k = I , . . .,ni. For k = I, . . .,ni, we take chains pk = rko,rkl, . . ., rkfk= qk as described in the lemma. Then cop1 . . . pni = wrlo . . . rnio, cor,,rzo . . . r,,,, cor12rzo. . . rni0, . . ., wqlrzo . . . rnio,wq1r2, . . . rn,07 . . ., wqlqz . . . q,, is again a chain of the type occurring in the lemma. Hence @is a congruence on A(X, 23), and a 0 b implies a@b, thus @ 2 0 whence @ 2 (0).Conversely, if Y is a congruence on A(X, 23) and Y 2 0, then, for any two adjacent polynomials ri, ri+, of a chain as in the lemma, riYri+,, thus Y 2 @. Therefore (0)2 @. 4.35. Proof of Proposition 4.33. We have to show that (0)satisfies the condition of Lemma 4.31. Let po(@)qo and (p,, . . .,pk) E A(X, 23)k. Then there exists a chain po = r,, rl, . . ., r, = qo satisfying the conditions of Lemma 4.34. Moreover, pi = oi(ail, . . ., a,,,, x,, . . ., x k ) , i = 1, . . ., k, for some words vi(yil, . . ., yin, xl, . . ., xk) in the indeterminates yij, and ri = wi(al, . . ., a,, x,, . . ., x k ) , i = 0, . . ., t, for some words wi as in Lemma 4.34. Then wi(yl, . . .,y,, vl(yll, . . . ,yl,, xl, . . ., xk), . ..) is a word such that ri(pl, . . .,pk) = wi(ul, . ..,a,, vl(all,. . . ,a,,, xl, . . ., xk), . . .) and ri+,(p1, . . .,p k ) = W i ( b 1 7 . . ., b,, ~l(all,. . ., a,,, . . x k ) , . . .). Hence po(p1, * * . pk) = rO(p~,. . Pk)7 ri(pi, . . * P k ) , . . .,qo(pl, . . . ,P k ) is a chain satisfying the condition of Lemma 4.34, thus po(pl, . . .,p k ) (0)q,(p,, . . .,pk), i.e. (0)satisfies the condition of Lemma 4.31, and the proposition is proved.
-
7
Y
.?
Y
04
FULL CONGRUENCES
87
4.36. Proposition. Let 0 be a congruence on A . Then the binary relation {O}on A(X, 8)defined by: p{O}q if and only if p(a,, . , .,ak)Oq(a,, . . .,ak) for all (al, . . .,ak)E Ak, is a full congruence on A(X, 8).
-
Proof. Let 17 : A AIO be the canonical epimorphism, and a(A 10): ( A10)( X , 23) Pk( A 10)the canonical epimorphism. Then, by 0 3.22 a) and c), a(A 10)v ( X ) is a composition epimorphism. We have Ker o(A 10)q ( X ) = {O}, hence (0) is a full congruence. -t
4.37. Remark. If {@,I vEZ} is a set of congruences on A and 0 denotes the set-theoretical intersection, then
n
Proof. Since n(0,) 2 O,, the first formula holds. The second formula follows from the definition of { nO,}. 4.4. Theorem. Let @ be any full congruence on A(X, 8). Then there exists exactly one congruence 0 on A such that (0)G @ E (0). 0 will be called the enclosing congruence of @.
Proof. We put 0 = @ n ( A X A ) . Since A is a subalgebra of A(X, 8), 0 is a congruence on A. @ 2 0 implies @ 2 (0). If p@q, then, for any (a,, . . ., ak)E Ak, xpa, . . . a,@xqa, . .. ak whence p(al, .. ., ak) @(al, . . .,ak).Hencep(a,, . . .,ak)Oq(al, . . .,ak),for all (al, . . .,ak)E Ak, thus p{O}q. Therefore @ E (0). Let A be an arbitrary congruence on A, then obviously {A}n( AX A ) = A. Since {A} 2 A, we conclude {A} 2 (A) whence A = { A } n( A X A) 2 ( A ) n( A x A ) 2 A. So we also have ( A ) n( A x A ) = A. If O1 is a congruence on A such that (01)C @ C {@I}, then (&)n( A x A ) C @n ( Ax A ) G { O , } ~ ( A X Awhich ) implies 01 C_ 0 C_ 01,i.e. 0 = O,, and the uniqueness assertion is proved. 4.5. Theorem. The set 8 of all full congruences on A(X, 8)is a complete sublattice of the congruence lattice 2 ( A ( X , 8)). If 2(A) is the congruence ( A X A ) is a complete lattice of A, then a : 3 .€!(A)defined by a@ = @rl lattice epimorphism. -f
88
COMPOSmION OF POLYNOMIALS AND POLYNOMIAL FCKCTIONS
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3
Proof. The set 3 of all full congruences on A(X, 23) coincides with the set of all congruences on the algebra ( A ( X , 23); 0, x ) . If M is any nonempty set of full congruences on A(X, a), then the greatest lower bound and the least upper bound of M in the congruence lattice of (A@, 23); 0 ) and of ( A ( X , 8); Q, x ) coincide, by construction (see ch. 1,§ 1.6). Hence 3 is a complete sublattice of ,Q(A(X,23)). Let A E 2(A), then a(A) = ( A ) n ( A x A )= A, hence (T is surjective. If { Q i I i E Z} is a subset of 8 and @ its greatest lower, Y its least upper bound, then (T@ =on ( A x A ) = n( ~ i E~ z)n 1 ( Ax A ) = n ( @ i n ( A x A ) I i E z )= n(o@ijiEz).
Let 0 be the least upper bound of the set { ~ @ ~ l i € Z in } 2(A), and a(aY)b. Since aY'c Y, we have aYb, therefore there are congruences @,.,,Qj2, . . . , Q j , and elements a = p o , p , , . . ., p , = b of A(X, 23) such that Y = 1,2, . . ., r. Let (a,, . . ., ak) be an arbitrary element of Ak, then
p ~ - l ( ~ l* ,* f
9
ak) @ i v p v ( a ~3
* * *3
ak).
Hence pv-l(al, . . .,ak)(&jv)pv(al, . . .,ak), Y = 1, . . ., Y , whence a0b. Conversely, if aOb, then there are congruences a@i,,. . .,dDi, and elements a = c,,, c,, . . ., c, = b of A such that C + ~ ( O @ J c,, Y = 1, . . ., Y, hence C , - ~ @ ~ ~ C ,and , thus aPb. Therefore a(aP)b, and we conclude (TY= 0. 4.6. Let 0 be any congruence on A . Then, by definition of the enclosing congruence, the set of all full congruences on A(X, 23) with 0 as their enclosing congruence constitutes a complete sublattice &4(X, a), 0 ) of the lattice of all full congruences on A(X, 23). We will now investigate this lattice more closely. 4.61. Proposition. Let 0 be any congruence on A and 8:A
A1 0 the canonical epimorphism. Then 8 ( X ) induces a lattice isomorphism from %(A(-%', %), 0 )to %:((A10)(X,a),0) where 0 denotes the congruence whose classes consist of a single element. -+
Proof. Bych. 1, Th. 1.71, thecanonicalepimorphism C:(A(X, 23); 52, x ) -, (A(X, %); Q, x ) I(@) induces a lattice isomorphism from 2((0))-the lattice of all full congruences Q on A(X, 8)containing (@)-to the
§5
FULL IDEALS OVER MULTIOPERATOR GROUPS
89
congruence lattice of ( A ( X , 23);S, x) I(@). For any 0 E 2((0)), we have C(@n ( A x A ) ) S [@n ( [ A x [A), and conversely let c E [@n ( [ A X CA). Then c = ([a, Cb), a, b E A , and there exist elements p, q E A ( X , 8) such that p@q and [ p = [a, Cq = Cb, whence p ( 0 ) a and q(0)b. Let (a,, . . ., a k ) E Ak, then, since both (0)and @ are full congruences, we havep(a,, . . .,ak)(0)a,q@,, . . .,a,)(@)b,and&, . . .,ak)@q(a,,. . ., a k ) . Thus Z'a = Cp(a,, . . . , ak), Z'b = Z'q(a,, . . ., ak), and (p(al, . . ., a k > & b . . ., a,)) E @ n ( A x A ) .Hence [@bn(CAx[A) G [ ( @ n ( A x A ) and ) thus [@f? ( [ A X CA) = [(@n ( A x A ) ) . Thus C@n ( [ A x ( A ) = 0 if and only if @ n ( A X A )= 0. We conclude that [ induces a lattice isomorphism from 3 ( A ( X , a),@)t o the lattice 2 of all congruences Y / on (A(X, 23);Q, x) (0)satisfying 'Yn( [ A x [ A ) = 0. By Lemma 3.11, the k-dimensional %-composition algebra (A(X, 23); Q, x) I(@) has the selector system [xl = sl, . . ., [ x k = sk, and CA is a subalgebra of (A(X, %); Q) I(@) consisting of constants such that ( A ( X , 23); 0) I(@) = ( [ A )(sl, . . .,sk). Let C(a) be the congruence class of a under 0, then ( O ) n ( A x A )= 0 implies that the mapping C(a) ( a is an isomorphism q from A 0 to [ A . Let o be the composition extension of q, according to Th. 3.21, then a:(A!@)(X,23) (A(X, a);Q) I(@) is a composition epimorphism. Let p, q E ( A I 0 ) ( X , %), and up = oq. Then there are representations p = w(C(a,),xl, . . ., xk), q = v(C(a,),xl,. . ., x k ) of p, q as words, and w((a,, sl,. . .,sk) = v([a,,sl,. . .,sk). Thus w(a,, x l , . . .,xk) (0)E ( Q , xl, . . .,x,). By Lemma 4.34, there exists a finite chain r,,, rl, . . ., r, of polynomials of A(X, 8) such that w(a,, xl, . . ., xk) = ro, v(a,,xl, . . ., xk) = r, and eri = where e = 8 ( X ) . Hence p = pro = er, = q and o is an isomorphism. The composition isomorphism 0-l: (A(X, 8); Q) j(0) ( A 10)( X , 8)induces a lattice isomorphism from the congruence lattice of ( A ( X , 8); Q, x ) I(@) to the lattice of full congruences of ( A 10)(X, 8)mapping 2 onto %:((A10)(X, %), 0). Since 0-15 is a composition epimorphism from A(X, 23) to ( A 10)(X,%) extending 8,we have 0-Y = e and the proposition is proved.
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-t
-t
5. Full ideals over multioperator groups 5.1. Let G = ( G ; +, -, 0, Q) be an Q-multioperator group. Since, by ch. 6, Q 3, the class of S-multioperator groups is a variety, the algebras Fk(G) and P,(G) arealso Q-multioperator groups. If G is an algebra of
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COMPOSITION O F POLYNOMIALS AND POLYNOMIAL FUNCTIONS
CH.
3
the variety 8 which is contained in the variety of Q-multioperator groups, then G(X, 8)is also an Q-multioperator group.
5.11. Proposition. Let 8 be the variety of Q-multioperator groups and G = ( G ; , -, 0, Q, x ) a k-dimensional %-composition algebra. Then G is an {Q, x}-multioperator group.
+
Proof. We have only to show that d o . . . 0 = 0. But xog, . . . g k = x(O+O)g, . . . g , = xOg, . . . gk+xOgl . . . g,, by the right-superdistributivity of x, hence xOg, . . . g , = 0, for any ( g l , . . . , g k )E G". This proves the proposition and moreover shows, that 0 is a constant of G .
5.12. As a special case of Prop. 5.11, we get that, if G is any Q-multioperator group, then Fk(G) and Pk(G) are (52, x}-multioperator groups, and if X = {x,, . . ., x,}, G E 23, and 23 is a variety of Q-multioperator groups, then the composition algebra G(X, 8)is an {Q, x}-multioperator group.
+ +
5.2. Let % be the variety of Q-multioperator groups and ( G ; ,-, 0, 52, x ) any k-dimensional %-composition algebra. Then G = ( G ; , -, 0, a) is an Q-multioperator group and G1 = ( G ; -, 0, Q, x ) is an {G,x>multioperator group. An ideal A of G is called a full ideal if A is also an ideal of GI. Thus, by ch. 6,s 3, an ideal A of G is a full ideal if and only if xg,,g,. . .g,-,(g,,+u)g,+, . . . gk-xgogl . . . g,, . . .g, E A , for all a E A , and for all (go,g,, . . .,gk)E Gk+', v = 0, 1, . . ., k . Let @ be a congruence on G and ker@ the kernel of @. As shown in ch. 6 , 9 3, ker is a lattice isomorphism from the congruence lattice g ( G ) to the ideal lattice a(G). By definition, ker maps the set of all full congruences on G onto the set of all full ideals of G . Thus the ideal A of G is a full ideal if and only if ker-I A is a fullcongruence. This correspondence enables us to derive a theorem on full ideals from every theorem on full congruences. Hence the results of§ 4 can be stated in terms of full ideals of G(X, 8).
+,
5.3. Let X = {x,, . . ., xk}, G any algebra of 8,a variety of Q-multioperator groups. The full ideals of G(X, 8)will be investigated. 5.31. Lemma. An ideal A of G(X, 23) is a full ideal if and only implies xap, . . . p k E A , for any ( p l , . . .,p k ) E G(X, 8jk.
if a E A
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FULL IDEALS OVER MULTIOPERATOR GROUPS
91
Proof. Put @ = ker-l A , then a E Ais equivalent to saying a@O. Let A be a full ideal, then di is a full congruence, hence, by Lemma 4.31, a E A implies zap, . . . pk@Xopl . . . pk = 0 since 0 is a constant of G(X, 23). Hence zap, . . . pk E A. Conversely, let a E A imply xap,. . .pk E A , for any (p,, . . .,P k ) E G(X, @lk. Let podiqo, then (po-qo)diO. We conclude z(po-qo)pl . . . P k E A , and right-superdistributivity of ~t implies xpopl . . . pk@-qopl . . .P k . Hence di is, by Lemma 4.31, a full congruence and A is a full ideal.
5.32. Let D be any ideal of G, and (D) the ideal of G(X, 23) generated by D, i.e. (D) is the set-theoretical intersection of all ideals A of G(X, 8) containing D. 5.33. Proposition. (D) = ker (ker-I D), thus (D) is a full ideal. Proof. Set B = ker(ker-lD). If a E D, then a(ker-l D)O, i.e. a E B. Hence ( D ) C B. Let A be any ideal of G(X, %) such that D C A. If a ker-l D b, then a ker-l A b whence ker-1 A is a congruence on G(X,@) containing ker-l D. Therefore ker-l A 2 (ker-1 D), and we conclude A 2 B and so (D) 2 B. By Prop. 433, (ker-10) i s a full congruence on G ( X , %I.
5.34. Let D be any ideal of G . Then {D} will denote the set of all p E G(X, %) such that p(g,, . . .,gk) E D , for all (g1, . . .,gk) E Gk. 5.35. Proposition. {D} = ker{ker-l D}, in particular, {D} is a full ideal. Proof. Since 0 is a constant of G(X, @), we have p E ker {ker-l D } if and only if p(g,, . . ., gk) E D, for all (gl, .. ., gk) E Gk, and, by Prop. 4.36, {ker-I D } is a full congruence.
5.36. The full ideal {D} is called the residue polynomial ideal of G(X, %) generated by D. By Remark 4.37, Prop. 5.33, and Prop. 5.35, we get
for any set (Dv11’ E I } of ideals of G.
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COMPOSITION OF POLYNOMIALS AND POLYNOMIAL FUNCTIONS
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3
5.4. Theorem. Let A be any full ideal of G(X, 23). Then there exists a unique ideal D of G, the so-called enclosing ideal of A, such that (D) C A C {D). W e have D = A n G . Proof. Let 0 be the enclosing congruence of ker-l A and D = her 0. Then (0)C ker-1 A C {O}, hence ker (ker-10) C A C ker {ker-l D } . This implies (0) C A E {D}. Let C be any ideal of G such that ( C ) C A C {C},’ then ker-l (C) C ker-l A E ker-l{C} whence (ker-l C ) G ker-lA C {ker-lC}. By Th. 4.4, ker-1C = ker-lD, thus C = D. Moreover, 0 = (ker-1 A)n (G X G) implies D = ker 0 = A 17 G.
5.5. Theorem. The set of all full ideals of G(X, 23) constitutes a complete sublattice 6 of the ideal lattice R(G(X, 23)) of G(X, 23). The mapping z which assigns to each full ideal its enclosing ideal is a complete lattice epimorphismfrom 6 to the ideal lattice R(G) of G. Proof. This boils down to rewriting Th. 4.5 in terms of ideals, in the sense of 9 5.2.
5.51. Corollary. The intersection and the s u h of any set of ideals of $?(G(X, 23)) where each ideal is a full ideal is again a full ideal. 5.6. Next we are going to state Prop. 4.61 in terms of ideals. We remark that, by definition of enclosing ideals, the set of all full ideals of G(X, 23) with the enclosing ideal D constitutes a complete sublattice &(G(X,%), D) of the lattice C? of all full ideals of G(X, 23).
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5.61. Proposition. Let D be any ideal of G and 8 : G G I D the canonical epimorphism. Then the composition extension 8 ( X ) of 8 induces a lattice isomorphism from G(G(X, %), D) to @((G ID) ( X , B), 0), 0 meaning the zero-ideal of G ID. Proof. It suffices to prove that, for every A E 6 ( G ( X , %), D), 6 ( X ) A = (ker) 8 ( X )(ker-1)A. Then Prop. 4.61 will yield the proposition. But b E (ker) $(X)(ker-l)A implies (b, 0) E 8 ( X ) (ker-l)A, hence there exist elements c, d E G(X, 23) such that 8(X)c = b, 6(X)d = 0, and c(ker-l A)d. Therefore c-d E A , and we have b = 6 ( X )(c-d) E @(X)A. Conversely, if b E 8 ( X ) A , then we easily get b E (ker) 8 ( X ) (ker-I)A.
06
93
FULL IDEALS OVER COMMUTATIVE RINGS WITH IDENTITY
6. Full ideals over commutative rings with identity 6.1. Let % be the variety of Commutative rings with identity regarded as algebras with the family { + , -, 0, ., 1) of operations. 8 is a variety of Q-multioperator groups where 8 = { ., l}. Thus, for any R E %, we may consider the full ideals of the Q-muhioperator group R(xl, . . .,xk, 8)= R[x,, . . ., xk], and we are ready to apply the results of $0 5.3 to 5.6 to this particular case. So the intersection and the sum of ideals of R[xl, . . .,xk]which are full ideals are again full ideals, and the mapping z which assigns to each full ideal its enclosing ideal is an epimorphism with respect to intersection and sum. There is a third operation on the ideals of a ring, namely the product of two ideals for which we prove the following 6.11. Theorem. Let U , V be full ideals of R[xl, . . ., xk]. Then the ideal product U V is also a full ideal, and the mapping z under which every full ideal is mapped onto its enclosing ideal, is an epimorphism with respect to forming ideal products.
Proof. Let W E U V , then w =
'n i=l
uivi, uiE U, viE V , i = 1,
for some n. Right-superdistributivity of for
. . ., n,
and Lemma 5.31 imply, that,
~t
P I , . . ., P k E R[x1, . . ., x k ] , xwp1 . . . Pk
n
=
fiici, i= 1
ci E u,
E
@;i
v,
i = 1, . . ., n. Hence xwpI . . . P k E UV whence, by Lemma 5.31, UV is a full ideal. Since zU S U , zV G V , we have ( z U )( z V )C UV and therefore ( ( z U ) ( z V ) )C UV. If w t UV, then w =
n
i=l
uivi, ui E U, vi E V ,
i = 1 , . . ., n. Hence w(rI, . . ., rk) E ( z U )(zV), for all (rl, . . .,rk)E Rk implying that UV C {(zU)(zV)}.Therefore z ( U V ) = (zU)(zV). 6.12. Remark. If A, B are ideals of R, then (AB) = ( A ) (B) and {AB} 2 { A } { B } .This will follow from 6.13. Lemma. I f C is any ideal of R , then the ideal ( C ) of R[x,, . . ., xk] consists of all polynomials of R[x,, . , ., x,, which have a normal form c(aAx.'11 E P ) as in ch. 1, Th. 8.21, such that a, E C , for all 1 E P.
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3
Proof. Let C , = (C(aA$ I L E P)I a, E C, for all L E P}. Certainly C , G (c). Conversely, since C G C , and C , is an ideal of R[x,, . . ., xk], we also have ( C )E C1. 6.14. Lemma 6.13 implies that ( A )(B) E (AB), moreover AB C (A)(B), hence (AB) C_ ( A ) (B). By definition of { A } , {B}, and the ideal product, we get immediately { A ) { B }E {AB}, and Remark 6.12 is proved. A partial converse of Th. 6.11 can be proved : 6.15. Proposition. Let W be any full ideal of R[x,, . . .,xk]such that the enclosing ideal C of W is the ideal product of some comaximal ideals A and B, i.e. A + B = R. Then W = UV where U is a full ideal with enclosing ideal A and V is a full ideal with enclosing ideal B. Proof. We put U = W + ( A ) and V = W + ( B ) . Then, by Cor. 5.51, U and V are full ideals, and, by Th. 5.5, U has the enclosing ideal AB+ A = A whereas V has the enclosing ideal AB+B = B. Moreover, UV = WW+W(A)-t-W(B)+(A)(B) G W + ( A ) ( B ) = W+(C) = W , by Remark 6.12. On the other hand, U f V 2 A + B = R whence U+ V = R[x,, . . . , xk].Hence UV = U n V 2 - W (see ch. 6,§ 4.3).
6.2. Rings of polynomials over a commutative ring with identity possess some further operations which crop up in a fairly natural way, namely the partial derivations a/axt = Eli, i = 1, . . ., k, of R[x,, . . ., xk].They can be used for constructing new full ideals from given full ideals. 6.21. Theorem. Let V be any full ideal of R[x,, . . ., xk]and A its enclosing ideal. Then V’ = {fCR[x,, . . ., X k ] l f € V , aifE V , i = 1, . . ., k } is also a full ideal of R[x,, . . ., xk], A its enclosing ideal, and V‘ G V. V’ is called the derivative of V.
Proof. V is not empty since 0 E Y‘. Let f , g E V’, then f , g E V and aif, aig E V , i = 1, . . .,k. Hence ,f-g E V , 3,tf-g) = aif-aig E V implying that f - g E V’. I f h E R[x,, . . .,X k ] , then f h E V and a,(fh) = (a,,f)h+f(a,h) E V whence f h E V’. Thus V’ is an ideal of R[x,, . . . ,xk]. Let pi c R[xl, . . ., xk],j = 1, . . ., k, then, by Lemma 5.31, ~t fpl . . .pk E V. Applying the chain rule for partial derivations, we get aiIcfpl . . . Pk = k
1 [St(a,.f)pl . . . &]
V = l
aipyE V , by Lemma 5.31. Thus Icfp,
. . . Pk E V’
06
95
FULL IDEALS OVER COMMUTATIVE RINGS WITH IDENTITY
so that, again by Lemma 5.31, V' is a full ideal. Clearly V' C V , in particular, V' C { A ) . Also A C V', hence ( A ) C_ V' since V' is an ideal. This proves the proposition.
6.22. Lemma. Let U, V be full ideals of R[x,, . . ., xk] such that U G V. Then U' C V'. If {VyI u E I } is a set of full ideals and D = n(V, I u E I ) , then D' = n(Vv'I Y E I).
n
Proof. The first assertion is obvious. As a consequence, D' C (V,' I u E I). Conversely, let f E n(P'*' I Y E I), then f E V,, and a,.fE V,,, u E I. Therefore f € D,aif E D whence f E D'. 6.23. For any full ideal V of R[x,, . . .,x,], we can now define V(O)= V , V(")= (V("-'))', n = 1, 2, . . .. By Th. 6.21, every V(")is a full ideal. 6.24. Proposition. V(")consists of all polynomials f E R[x,, . . ., xk] such that the partial derivatives of the orders k = 0, 1, . . ., n off are contained in V where a partial derivative o f f of order 0 means f itself. Proof. By induction on n, using the definition of V(").
6.25. Corollary. Let V" be the ideal product of n copies of V where V is a full ideal and n a 1 an integer. Then V" C_ V("-'). Proof. Let f E V".Then f is a sum of elements v1v2 . . . vn,vvE V. Applying sum and product rule for partial derivatives, we find that the partial derivatives of the orders k =sn - 1 of .f are contained in V. Thus V" S y(n-1).
6.3. Definition. A full ideal V of R[x,, Y, = V.
. . ., xk] is called a D-full ideal
if
6.31. We want to characterize the D-full ideals of R[x,, . . ., xk] = S . For this purpose, we consider the algebra S = ( S ; f , -, 0, -,1, x, a,, . . ., a,). Setting Q, = { ., 1, x , a,, . . ., a,}, S becomes an S,-multioperator group. Because of 5 5.12, this is a consequence of aiO = 0, i = 1 , . . ., k.
96
COMPOSITION OF POLYNOMIALS A N D POLYKOMIAL FUh’CTICKS
CH.
3
6.32. Theorem. The ideals of the Ql-multioperator group S coincide with the D-jiull ideals of R[x,, . . ., xk].
Proof. By ch. 6, Lemma 3.4, a subset V of S is an ideal of S if and only if it is a full ideal of R[x,, . . .,xk] and ai(g+a)-aig E V , for all a E V , g E R [ x l , . . ., xk], 1 e i == k. This is equivalent to saying that V is a full ideal and aia E V ,for all a E V , i.e. V is a D-full ideal of R[x,, . . .,x k ] . 6.33. Corollary. The intersection and the sum of a set of ideals of R[xl, . . ., xk] which are D-jiull ideals are also D-jiull ideals. The ideal product of any two D-jiull ideals is also a D-jiill ideal.
Proof. The first assertion follows from Th. 6.32, and ch. 6,§ 3.2. If U , V are D-full ideals and p E U V , then p =
aip
2 n
=
n
v=l
uvv,, u, c U, v, E V. Hence
aivs,)E UV, thus (UV)‘ = UV.
( ( ~ i ~ , ) ~ , , + ~ ,
v=l
6.34. Corollary. Let W be a D-jiull ideal of R[x,, . . ., xk] such that the enclosing ideal C of W is the ideal product of two comaximal ideals A and B. Then W can be represented as the ideal product of a D-jiull ideal U and a DlfuIl ideal V having A and B, resp., as their enclosing ideal.
The corollary will follow from 6.35. Lemma. For any ideal C of R, the ideal (C) of R [ x l , . . ., xk] is a D:full ideal.
Proof. By Lemma 6.13, and the elementary properties of partial derivations. 6.36. Proof of Corollary 6.34. As the proof of Prop. 6.1 5 and by taking in account Cor. 6.33 and Lemma 6.35. 6.4. For any full ideal V of R[x,, . . ., xk], we define the D-core 6V of V as the sum of all D-full ideals of R[x,, . . ., xk] contained in V . By Cor. 6.33, 6V itself is a D-full ideal contained in V.
47
FULL IDEALS OVER FIELDS
6.41. Proposition. SV = n 6V coincide.
97
n 3 0), and the enclosing ideals of V and
Proof. We set U = n(Vtn)In a 0), then U is a full ideal of R[x,, . . . ,x k ] . By Lemma 6.22, U' = n ( V ( " + ' ) / n a O )= n ( V ( " ) I n a l )whence U C U'. Thus U is a D-full ideal and U C V . We conclude U G 6V. Conversely, let W be any D-full ideal such that W C_ V , then W E V("),n z=0, again by Lemma 6.22, thus W G U. In particular, 6V C U showing that 6V = U . Th. 6.21 and Th. 5.5 imply that V and 6V have one and the same enclosing ideal. 6.42. Corollary. If {VyIv C I } is a set of full ideals and D = fl(V,l v E Z), then 6D = n (W,l v c I ) . Proof. By Prop. 6.41 and Lemma 6.22,
7. Full ideals over fields 7.1. Let Q be any field, then Q is in the variety 23 of commutative rings with identity and we may apply all our results of Q 6 to Q[xl. . . .,xk]. The information that Q is a field yields, however, some more specific results on full ideals. 7.11. Theorem. If Q is any infinite field, then Q[x,, ideals apart from the trivial ones,
. . .,xk] has no full
Proof. Let U be a full ideal of Q[x,, . . .,x k ] . Since Q has just the trivial ideals, the enclosing ideal of U is either Q or the zero ideal 0 of Q. The first case yields U 2 ( Q ) = Q[x,, . . ., x,], and the second case U E (0). But then f E U implies f (rl, . . .,rk) = 0, for all (rl, . . ., rk) E Qk, whence f = 0. For k = 1, this follows from a well-known theorem on polynomials over fields, and induction on k proves the result. Therefore U = 0.
7.2. If (2 is a finite field, then Th. 7.1 1 does not hold. For xiQ'-xl E (0)
as we know from a well-known theorem on finite fields, but- 1 E (0). Thus {0}is a non-trivial full ideal of Q[q, . . ., x k ] . A complete classi-
98
COMPOSITION OF POLYNOMIALS AND POLYNOMIAL FUNCTIONS
fication of the full ideals of Q[xl, k = 1. This is achieved by
.. ., xk], Q finite, is
CH.
3
known just for
1.21. Theorem. Let Q be a jinite jield of characteristic p and order q. Then every non-trivialfull ideal V of Q[x] can be uniquely represented as V = ( X ~ ” - X ) ~ ~ ~ ( X ~ ~ ’ .-.X. f) l~ (*~~ @ ~ - where x ) ~ ~ r =- 0, el =- e2 =. . . e, =- 0, a, z 0, v = 1, . . ., r, and aj =- ai i f ej is aproper divisor of ei. Any such V is a non-trivial full ideal. Proof. Clearly, any such V is a non-trivial ideal. The principal ideal (x4”-x) consists of those polynomials of Q[x] which vanish for all elements of the extension field of order q‘ of Q, hence, by Lemma 5.31, (x4“-x) is a full ideal of Q[x]. By Th. 6.1 1 and Cor. 5.51, Vis afull ideal. Next we show that every non-trivial full ideal V can be written as in the theorem. Since Q[x] is a principal ideal domain, V = (f),for some monic non-constant polynomial f~Q[x]. Let C be the algebraic closure of Q and c1 C C any root off. If g E Q[x], then xfg = f o g E V whence f o g = rf, for some rEQ[x]. Hence f(g(cl)) = r(cl)f(cl)= 0. This implies that every element of the subfield Q(cl) of C is a root off. Let c1 f c2 E Q(cl), and ai, i = 1, 2, be the multiplicities of cias roots off. Then we have the factorization in C[x]
f = (X-CCl)al(X-C2)a” . . . (x-cc,)a.
where s =- 1 , ci ti cj, for i + j , a, that c2 = g(cl) and set
=- 0,
Y =
for
1,
(7.21)
.. ., s. Let gEQ[x] such
g’(c1) # 0
where el is the degree of Q(cl) over Q. Then h(c1) = cz, and h‘(c1) Z 0. By (7.21), we have
foh
=
(h-cl)”’(h-c2)”z.. . (h-c,)a,.
h E V and thus is a multiple of A hence a2 == a l . Since c2 was chosen arbitrarily in Q(cl), we conclude that (xqel-xX)al/f.But c1 was an arbitrary root offin C , hence we can use any cifor this argument. Thus c1 is then a root of fo h with multiplicity a2. But fo
(f)c q=l(x4c~-x)nc
(7.22)
57
FULL IDEALS OVER FIELDS
99
where e, denotes the degree of Q(c,) over Q . If this inclusion were not an equality, then f = gv where v is the monk least common multiple of the polynomials (XQ"-x)ai and g is a non-constant polynomial of Q [ x ] .Then g has a root in C which is also a root ci,say, off. But x - c , divides xqei-x whence ( X - C , ~ + ' divides f, a contradiction. We may order the right-hand side of (7.22) by decreasing e,. Of all those ideals with the same ej,we need just that one with the greatest a,. In the representation of Y thus obtained there may be an ideal (X""'--x)"j such that ej is a proper divisor of ei and aj =s a,. But then (xqej-xx)a, divides ( x q C i - - x ~and , we may drop (xqE'-x)q in our representation. After completing this procedure whenever possible, we get a representation of V as in the theorem. It remains to establish uniqueness. Suppose there are two representations of V as in the theorem, e.g.
where h is a monic, non-constant polynomial of Q [ x ] .Without loss of generality, we may assume 0 ir'=s s. Every root of the polynomial xqe"- x is also a root of h, and conversely every root of h is a root of some xqev--x since h divides the product of the (xqev-x>"".Thus the sets of roots of h and of roots of all xq6"-x coincide. From the theory of finite fields we know that e, is the greatest degree over Q occurring amongst all the roots of xq""-x. Hence el = fl since both equal the greatest degree over (2 ocurring amongst the roots of h. If c is a root of h of degree el over Q , then x - c divides xqe'-x, but ( x - c ) ~does not, nor does x - c divide any other xq"-x. But (x@- x>"' divides k which divides the product of all (xqeU--x)"". Thus al is the multiplicity of c as a root of h, and arguing in the same manner for the second representation, we conclude a1 = bl. Suppose now that e, =A,, a, = b,, for 1 4 v =s t, and
where g is a monic polynomial of Q [ x ] .Then h = gk, for some k C Q [ x ] Let
100
COMPOSITION O F POLYNOMIALS AND POLYNOMIAL FUNCTIONS
CH.
3
for some monk 1 C Q[x], then (h) = (g)n(1). If d is the monic greatest common divisor of g and I, then h = glld whence k = I/d. Thus every root of k is a root of I, and hence a root of xqeV-x,for some v =- t. Therefore the maximal possible degree over Q of any root of k is e f + , , and if c is a root of ~ ~ " f + of ~ -degree x e,,, over Q then arguing as before we see that c, as a root of I, has the multiplicity a,+,. By a wellknown theorem on finite fields, x-c divides xqe"-x where v =s t if and only if e,+, divides e,. Since g is a least common multiple of the polynomials (xq""-xX)lly, v s t, the multiplicity of c, as a root of g , equals a = max a, where v runs through those indices which satisfy v =st and e,+,/eV.By hypothesis, a < a,,, whence c, as a root of d has just multiplicity a thus, as a root of k, c has multiplicity 2, = a,+,-a =- 0. Therefore e,, is characterized by being the greatest of all degrees over Q of the roots of k , and so is .ft+l, by the same argument. If c is a root of k of the greatest possible degree, then c has multiplicity a,+,-max a, where v runs through all indices such that v == t and e,+,/e,,,and, for the same reason c has multiplicity b,+,-max b, where v 4 t andf,+,/f,. We summarize and obtain e,+, = f,+,and, by induction, a,,, = b,+l. Hence e, = f,, av = b,, for 1 G v == Y. If s >. Y, then we could use the same argument as before. But now k = 1 while, on the other hand, k would have roots, a contradiction. Hence Y = s, and the uniqueness is established.
,
7.3. Again let Q be a finite field of characteristicp and order q, With the information of Th. 7.21, it is now easy to determine all the D-full ideals of Q [ x ] . All to be done is to compute the D-core of every non-trivial full ideal of Q[x]. We do this first for the full ideal W = (xqC-xX)a.I f f € W , then f = (xqe-xX)ag, for some g E Q[x], thus
&f=f'
= (X~-xX)(Ig'-u(Xq'-xX)n--lg.
Hence f ' E W if and only if a(~~~--x)"-~gE W. Therefore W' = W if pla while W' = (xq'-xX)a+' if p f a . By Prop. 6.41, SW = (xqe-x)d where 5 is the least integer m > a being divisible by p . By Cor. 6.42, if V = fl;=l (xq"'--x)OYaccording to Th. 7.21, we have SV = flin1(x@-x)li. is the least integer m a u , being divisible by p . Cancelling where redundant ideals in this representation of SV,we get a representation as
§8
RESIDUE POLYNOMIAL IDEALS OF D E D E K I N D DOMAINS
101
in Th. 7.21. Here all the a, are divisible by p . Conversely, any ideal having such representation is a D-full ideal of Q [ x ] .We may state our results as 7.31. Theorem. Let Q be a finite field of characteristicp and order q, and V a full ideal of Q [ x ] represented as in Th. 7.21. Then V is a D-full ideal i f and only i f every exponent a, is divisible by p .
8. Residue polynomial ideals of Dedekind domains
8.1. Let % be a variety of Q-multioperator groups, A any algebra of 8, X = {xl, . . ., xk},and D any ideal of A . Then by§ 5.34, the residue polynomial ideal { D } of A(X, 8)consists of all p E A(X, %) such that p(gl, . . ., gk)E D, for all (gl, . . .,gk)E Ak. By Prop. 5.35, { D } is a full ideal of A(X, %). The elements of { D } will be called “residue polynomials mod D”. What is the significance of considering residue polynomial ideals? Let q : A B by any epimorphism of %-algebras. The diagram fig. 3.1 shows that /I= Pk(q)o(A) = a(B) q ( X ) is a composition epimorphism from A(X, 8) to Pk(B), and p(Ker p ) q if and only if Ag,, . . .,gk)(Ker q) d g , , . . .,gk), for-all (gl,. . .,gk)E Ak.Thus Ker B = {Kerq}. Using the notation of ch. 6,§3.2 and 65.3, we get kerB= ker Ker /?= ker {Ker q} = ker {ker-l ker Ker q} = {ker Ker q} = {kerq}. Thus the residue polynomial ideals of A(X, %) are just the kernels of the composition epimorphisms B. Now let % be the variety of commutative rings with identity and R a ring of 93. Again R[x,, . . ., xk]will stand for R (X , 23) and will be called the ring of polynomials in xl, . . ., xk (over R). For any ideal D of R , we wish to get more information about {D}. If R is noetherian, then R[xl, . . ., xk] is also noetherian, thus { D } could be characterized in this case by writing down some ideal basis for {D}. This will actually be done for the case where R is a Dedekind domain, and RID is finite. In particular, we may take R to be the ring of rational integers and D # 0. Let S be any ring and B any ideal of S . ker-I B is then the correspondingcongruence on s. As usual we will write a = b mod B for a(ker-lB)b. +
8.2. Having spelled out our next aim, we start by taking a Dedekind domain R and an ideal D of R such that RID is finite. We are looking for an ideal basis of { D } . If D = R then { D } = R[xl, . . ., xk] = (l),
102
cn. 3
COMPOSITION OF POLYNOMIALS AND POLYNOMIAL FUNCTIONS
and if D = 0, then R RIO is finite and hence a field. In this case I R I = q, for some prime power q, and certainly (0) 2 (x;-xl, . . ., x;f-xk) as follows from a well-known theorem on finite fields. On the other hand, we also know that, for k = 1, (0) E (x:-xl). Let f E {0}, then, by ch. 1, Th. 8.21, f = ( x l - x k ) g + h where g E R[x,, . . .,xk] and q--l
C pi(xl, . . ., x , - , ) x ~ . Clearly h E (0). Therefore, for any (rl, . . ., 1 - ~c )Rk-l, we have C pi(r1, . . . , Y k - 1 ) ~ ; E (0) whence pi(xl, . . .,
h=
4-
i= 0
Y ~
i=O
x ~ - ~ ) E { O } 0, ~ i ~ q - By l . induction, hE(x4,--xl, ..., X E - ~ - X ~ - J , hence f € ( x ; - x 1 , . . ., XI-x,). We conclude that (0) = ( x ; - x l , . . ., x;f-xJ. Now let D be any non-trivial ideal of R such that R ID is finite. Since R
is Dedekind, D has, up to ordering, a unique factorization D =
n r
Pf"
i=l
where the Pi are pairwise distinct non trivial prime ideals of R and . . ., r. It is also known that the ideals Pi" are pairwise comaximal. Since P,'i E {P,"'},we see that also the ideals {P?} are pairwise comaximal. The product of comaximal ideals coincides with their intersection, thus by 5.36, ei> 0, i = 1,
{D} =
{
i=l
P?}
=
{n;,lp}
=
n;==, {P?} =
i=l
{P,'i}.
Hence it suffices to determine ideal bases for the {q}. Since D G all the prime ideals Pi have finite factor rings R IPi.
e25 Pi,
8.3. Let P be a non trivial prime ideal of R such that RIP is finite. Then RIP is a finite field of order q where q = p", for some prime p . We introduce a number-theoretical function cp which depends on P . Let ;Ibe the function from the set of positive integers to the set of non-negative integers defined by A(k) = max (A I qRdivides k). Then cP shall be the function from the set of non-negative integers into itself defined by k
+(O) = 0,
cp(k) =
C A@),
i=l
for
k z 0.
(8.31)
A well-known number-theoretical formula (see ch. 6, 5 9.3) tells us that
08
RESIDUE POLYNOMIAL IDEALS OF DEDEKIND DOMAINS
103
where sq(k) is the sum of the digits in the q-adic expansion of k . sP actually depends just on j R j PI,not on P itself. Whenever we keep P fixed, we will write E for F ~ For . every integer n == 0 and every integer 1 s a =sq- 1, we define elements s(aq") as follows: s(qn),s(2qn), . . ., s((q- 1)q") is an arbitrarily chosen full set of coset representatives for the non-zero elements in P" I P n f l- by ch. 6, Lemma 4.52, P" I Pn+l consists, indeed, of q elements. Additionally we put s(0) = 0. If k is any non-negative integer and k = amqm+ . .. +a,q+ao is the q-adic expansion of k , we define
. .. +s(a,q)+s(a,).
r, = s(a,q'n)+
In particular, rfqn= s(tq")E P", for 0 e t -= q. 8.31. Lemma. Let n if i = k mod q".
2
0 be an integer. Then ri
= rk
mod P" i f and only
Proof. The lemma holds obviously for n = 0, hence we may assume n =- 0. Let i = xa,qt, k = Cb,q' be the q-adic expansion of i and k , resp., then ri = Cs(a,q'), r, ==Cs(b,q'). If i s k mod q", then, for 0 e t -= n, we have a, = b,, hence s(a,q') = s(b,q'), 0 == t < n. As a s(b,q') E r, mod P". Conversely, if consequence, ri = s(a,q') =
1
f-=n
t-=n
ri z r, mod P", then
C s(a,q') 5 C s(b,q') mod P"whence s(a,,qo)= r
t
s(boqO) mod P. Therefore s(aoqO)= s(boqo). By induction, s(a,q') = s(b,q'), 0 ==t -= n, hence afq' = b,q', for 0 =st -= n. We conclude that i = C atqt = C b,q' = k mod q". 8.4. We define a sequence of polynomials tn E R [ x ] ,n = 0, 1, 2, . . . by
n
n-1
to = 1,
tn =
i=O
(x-r,),
for n
> 0.
8.41. Lemma. For every n 2 0, we have t, E {P"(")}, but tn(rn)B. PECn)+'. In particular, t,, # {P'(")+l}. Proof. We first show that t, C {P"'")}. For E(n) = 0, this is obviously true. So we may assume that &(n)= m =- 0. Let r E R be arbitrarily chosen. Then there exist integers di, 0 =S di -= q, i = 0, 1, ..., m-1, such
104
that r
COMPOSITION OF POLYNOMIALS A N D POLYNOMIAL FLh CTIOhS
= s(do) mod
P , r-s(do)
s(dn,-lq'"-l)mod Pm,i.e. r
m- 1
= s(d,q)
CE.
3
m-2
1 s(d,q')
mod P2, . . ., r -
i=O
G
m-1
= 1 s(d,q') mod Pm. Then, for I = C d,q*, t=O and hence t,(r) = t,(rl) mod P". For 1 -= n, I=O
we have r = rI mod P" obviously t,(rl) = 0. If 1 2 n, then, by Lemma 8.31, if we set S =
C l(l-i),
n-1
i=O
n-I
we get t,,(r,) = Jl
1=0
(r,-ri) 6Ps, but tn(rl)6 Ps+'. By (8.31) and
(8.32), S = &(l)-&(l-n) = (/-s4(/))/(q- 1)-[(/-n)-s4(1-n)]/(q-1). But, since sq(a+b) G s,(a)+sq(b), we have s4(/)--s,(/-n) s s,(n). Hence S 3 (n-s,(n))/(q- 1) = ~ ( n= ) m whence t,,(r) = tn(rl)= 0 mod P". Similarly we show that tn(r,) 6 For tn(rn)6 Ps+l, where S = ~ ( n-)~ ( 0=) &(n).This completes the proof.
8.5. Lemma. Let e
0 be an arbitrary integer. Every polynomial f C R [ x ]
such that f E {P") and [ j ]e m can be written as f
a, C P"-"(') if E ( i ) belongs to {P'}.
<
e, and ai C R
if
E(i)
2
m
=
1 aiti
where
i=O
e. Every such polynomial
Proof. The latter statement is a consequence of Lemma 8.41. Suppose m
fc {P"},[ j ]s m. Then clearly f = 1 aiti, for some a; C R. We have i=O f ( 0 ) = a0 whence a0 c P'-'('). By induction and Lemma 8.41, we conm dude that a,t, E {P"}, for I -= j , hence g = .C. aiti C {P'}. Therefore l=J
g(ri) = aj$(rj)E P', hence aj C Pe-'(j), again by Lemma 8.41.
8.51. Theorem. Let 0 # p E P, bi c R , i = 0, 1 , . . ., such that Pi = (pi,bi), s the least integer such that &(qs)3 e, and T I = {t,,}, T2 = {pe-f(qi)tqi/ 0 e i -=c s}, T3 = {be-eE(qi)tqiI 0 ==i s}. Then TlUT2UT3 is a basis for the ideal {P'}. in particular, P = ( p ) , then T1U T2 is a basis .for { P"}.
v,
Proof. By Lemma 8.5, T 1 U T 2 U T 3C {P"}.Letfc {P'}, then, by Lemma 8.5, f
m
=
aili, a; t P'-'('),
i=O
Obviously, for n
2
for &(i)< e, and aiE R, for ~ ( i==) e.
0, tn/t,+l, and e(k+ 1) = ~ ( k if) q I" k f 1. Hence,
$8
RESIDUE POLYNOMIAL IDEALS OF DEDEKIND DOMAINS
105
for I -= s, there exist elements ci, di E R such that
- pe-&Ot
qrg + be-,(,,,t,ih>
for some g, h E R[x].This proves the first assertion. If P = ( p ) , we may set bi = 0, for all i. This completes the proof. 8.6. We want to generalize Lemma 8.5 and Th. 8.51 to R[xl, . . .,4. For this purpose, we introduce some new notations. Let N be the additive semigroup of non-negative integers and Mk the direct product of k copies of N . For any r E N , and tc E M k , rtc shall mean the element of M k whose components arer-times the corresponding component of tc. Apartial order =son h f k will be defined by (inl, . . .,mk) ==(nl, . . .,nk)i f and only if mi =S ni, i = 1, . . .,k. Let Epk : Mk N be defined by &pk(m1,. . .,mk)= -+
k
C +(mi). In
i=l
particular, we have cP1 =
E~
and again we abbreviate
&pk by writing &k instead. Finally, for all L = (il, . . ., ik)E Mk, we define t, E R [ x l , * . xkl by t, = . tik(xk). .?
8.61. Lemma. Let e == 0 be an arbitrary integer, p = (ml, . . .,mk) E Mk, and f E R[xl, . . ., x k ] such that f E {P'} and, regarded as a polynomial in xj over R[x,, .. ., .. ., xk],has degree == mi, j = 1, - .-,k. Then f = z ( a l t kI L e p) where a, E Pe-'r(l),for & k ( i ) < e, and aLE R, for &k(L) a e. Any such polynomial belongs to { P e }and has degree e mi as a polynomial in xi, j = 1, . . ., k.
Proof. Let f be a polynomial as in the hypothesis of the lemma. By Lemma 8.41, tiJxj)E {P"('j)},j = 1, . . ., k, hence t, E {P"(')}whence f E {P"}.Since [ti,(xj)]= i,, we see that, for L = (il, . . ., ik), t, is of degree $ in xi, and thus f is of degree =S mj in xi. Conversely, let f C {P'} be of degree =smi in xi, j = 1 , . . .,k. For k = 1, Lemma 8.5 yields the result. Let k
=- 1, then f = C mk
i,=O
gi,(x,, . . . , x ~ - tik(xk), ~ ) whichis obtained
from the normal form f o r f a s in ch. I, Th. 8.21, by collecting all those summands containing equal powers of xk and applying the division algorithm using tm,(xk), tm,-1(xk),. . ., to(+). We observe then that every g,,(x,, . . ., x k P l ) is of degree s mi in xj, j = I , . . ., k-1. For any
106 (I17
COMPOSITION OF POLYNOMIALS AND POLYNOMIAL FUNCTIONS
* * -7
rk-l)
CH.
3
E Rk-',
f ( r l r . . .)rk-l,
mk
xk) = ik=O
gi,(.l,
..
- 7
r k - l ) tik(xk)E
(p')
and, by the same argument as in the proof of Lemma 8.5, E {PE-e(ik)} if E ( i k ) we conclude that gjk(xl, . . ., e. By inducgi,(xl, . . x k - - l ) = xa(il, ..., ik-,, i k ) t i l ( x l ) . * t i k - l ( X k - l ) where tion, '(il, . . ., i k - 3 , i k ) E {p'-E(ik)-(€(il)+ . . . + E ( i k - - l ) ) } = {~ '- "k ( ')if } &k(L) i e, and the sum extends over all (il, . . .,ik--l)== (ml, . . .,mk-l). Substitution of this expression for the gikinto our expression for f completes the proof of the lemma. .?
a
8.62. Theorem. Let p , b,, s be dejned as in Th. 8.5 1, a = (s, s, . . .,s) E Mk, qL1 1 a, Fk(qL)-== 4, and Tl = {tq" L =s a , Ek(qi) e}, T2'= {pe-Ek(qc)t T3 = {be-a(qL+q,lL s a, E k ( q L ) e}. Then TlUT2UT, is a basis for the ideal {P'} of R[x,, . . ., xk]. In particular, if P = ( p ) , then TIU T, is a basis for {P'}.
Proof. Using Lemma 8.61, the proof runs along the same lines as that of Th. 8.51 and is thus left to the reader. 8.63. Remark. If R is the ring of rational integers, then P = ( p ) ,for some prime p . Moreover q = p and we can take s(1q') = Iq', hence rl = I, for 1 = 0, 1; 2 , . . . 9. Residue polynomial ideals over groups 9.1. In the preceding section, a description of the residue polynomial ideals of some special types of polynomial rings could be achieved comparatively easily. We know much less in the case of groups, however, the Kurosh subgroup theorem will at least tell us a lot about the group structure of the residue polynomial ideals of G(X, %) = G [ X ]where 23 is the variety of groups, G any group with group operation -,and X = {xl, . . .,xk} a set of indeterminates. First we prove a qualitative result : 9.11. Theorem. Let N be any normal subgroup of G, and { N } the residue polynomial ideal of G [ X ]generated by N. Then the group V = ({N}, ., - l, 1 ) is isomorphic to a ,freeproduct F +KK of a free group F and a group K which is the free product of some copies of N.
Proof. According to ch. 1, Cor. 9.22, G [ X ] ,{ q l ,p2} is a free product of G and the free group F(X, %) = F(X). By a well-known group-theoretical
39
RESIDUE POLYNOMIAL IDEALS OVER G R O W S
-.
107
result, there exist monomorphisms yi:F(xJ F(X), i = 1, . . .,k , such that F(X), {yl, . . ., y k } is a free product of the groups F(xi). Thus G[X] = F(xJ ++c . . . fF(xk)++Gwhere ++ denotes the free product of groups as usual. Let 7 :G GIN be the canonical epimorphism, then, by 5 5.3, we have V = { N } = { p Ip c G [ X ]and p ( g , , . . .,gk)C N , for all (gl, . . .,gk) E Gk}, and it is easy to see that V = ker a(G I N ) q ( X ) . Since V is a subgroup of a free product, we are ready for applying the Kurosh subgroup theorem (see ch. 6 , s 6.7) to V and obtain since V is normal in G[X]) +
where F I is a free group and 4, j = 0, 1, . . ., k , are suitable index sets. Since VnF(xj) is free as a subgroup of a free group, the free product of the first and second factor in (9.1) is a free group F. Moreover, V n G = {N)nG = N , by Th. 5.4, whence the third factor in (9.1) is a free product of some copies of N and the theorem is proved. 9.2. The Kurosh subgroup theorem also yields some quantitative information which, for finite G, enables us to determine the rank of F in V z F.1Ac K as well as the number of free factors isomorphic to N in K. In the following, we keep the notation of Th. 9.11. 9.21. Proposition. If G is a finite group, then K is the free product of Pk(GI N ) ] IN I / 1 G I copies of N while the rank of F equals
i
I
1+ jP,(G IN) ( k - l NlllG I).
Proof. By the Kurosh subgroup theorem, 1101 is the number of cosets of G[X] modulo GV. Since V = ker a(G IN)q(X), we see that G[X] 1 V s Pk(GIN)whileGVIV= G ~ V ~ G = G ~ N . H e n c=e IP,(GIN)IINI/IGI ~Io~ which proves the first statement of the proposition. Similarly 141 is the number of cosets of G [ X ]modulo F(xj)V,j = 1, . . .,k . But F(xj)VI V s F(xj)IVnF(xj) and V n F ( x j )= {$I g k c N , for all g c G } = [x;] where e is the exponent of GIN. Thus F(xj)VIV 2 C,, the cyclic group If r(F) and r(F1)are the of order e which shows that I 4 I = Pk(G1 N ) rank of F and F I , respectively, then r(F) = r(Fl) k IPk(GIN) ] / e ,by definition of F since r(VnF(xj))= 1. But, again by the Kurosh subgroup r(F1) = klP,(GlN)I-lP,(GIN)l INI/IGI-kIPk(GIN)!/e+l. theorem, Hence r(F) = 1+IPk(GIN)/( k - l N l / l G l ) which proves the proposition.
1
I/.
+
108
COMPOSITION OF POLYNOMIALS A N D POLYliOMIAL F C h C 7 1 C b S
CH.
3
9.22. Corollary. (ker a(G); -I, 1) is a free group. If G isfinite, then this free group has rank 1 I P,(G) I ( k - 1/I G I).
+
a,
Proof. By Th. 9.11 and Prop. 9.21 if we put N
=
(1).
9.23. Remark. The actual description of the residue polynomials of { N } is, in general, unknown, but a few results into this direction have been obtained. The reader is referred to ch. 5, 1.
10. Derivation families with chain rule 10.1. Let R be a commutative ring with identity, R [ x l , . . .,x,] the polynomial ring over R in the indeterminates x l , . . ., x,, and a, = a/axi, i = 1, . . ., k, the partial derivations of R[x,, . . ., x,]. This family (a,, . . .,ak) is a well-known example of a family of mappings (a, . . .,a,), 6 , : R [ x l ,. . ., x,] R[x,, . . ., x,], i = 1, . . ., k, such that, for any f, g , g,, . . .,g , C R[x,, . . ., x,] and I = 1, . . .,k, the following equations are valid: 6 , ( f + d = S l f + 6/g, (10.11) -+
(10.12)
S,(fg) = ( s / ” f f g + f ( S / g ) ,
‘IXfg, . . . g , =
k
C [x(’,f)g,
r=l
. . gkl 6,gi. *
( 10.13)
The problem arises of determining all the families (Sl, . . ., 6,) of such mappings satisfying (10.1 l), (10.12), (10.13). More generally, for any ring S , a mapping 6 : S S is called a derivation of S if 6 satisfies (10.1 I), (10.12) in the place of S,, for all f, g E S . Substituting ai for S,, i = 1, . . ., k, in (10.13), we can read off the well-known chain rule. Thus we will call R[x,, ..,,xk], any family (al, . . ., 6,) of mappings tii: R[x,, . . ., x,] i = 1, . . ., k, a derivation family with chain rule, or, in brief, a “Cderivation family” if this family satisfies (lO.ll), (10.12), (10.13). It will be useful to adopt the following notation : Let F = (At)be any matrix with entries At c R[x,, . . ., x,] and g = (g,, . . .,g,) a k-tuple of elements of R[x,, . . .,x,], then the matrix (&g, . . . g,) shall be denoted by F o g. -f
-
10.2. Let (al, . . ., 6,) be an arbitrary C-derivation family and f,g,, . . .,gk arbitrary elements of R[x,, . . .,x,]. If xfg, . . . g, is represented by
§ 10
DERIVATION FAMILIES WITH CHAIN RULE
109
substituting g,, . . ., gk for x, . . ., xk into the normal form of f as in ch. 1, tj 8.2, and if we apply (10.11) and (10.12) to this representation, we obtain from a straightforward computation
. . . gk But xfx, . . . xk = L hence
S,f=
k i=l
[x(aif)gl . . . gkl 61gi'
k
C (6lxi)( a i f ) ,
(10.21 1)
(10.2 12)
i=l
and tying (I 0.13) and (10.21 1) together and using the right-superdistributivity of x, we end up with k
1 (6/gi)[.(Sif-aif)gl . . . gk] = 0,
i=l
I = 1, . . ., k. (10.213)
We now introduce a matrix notation where s is used for the row index while t means the column index: G = (6,g,), P = (a,g,), D = (S,x,), E = k x k-identity matrix, 0 = k X k-zero matrix, thus G, P, D, 0 and E are k x k-matrices. Moreover we introduce k x 1-matrices c = (6,f- a , f ) , b = (a,f), o = (0), and 1 >: k-matrices g = (8,) and h = ( x t ) . Then (10.213) becomes (10.214) G(co 9) = 0. From (10.212) we get (10.215) G = DP and c = Db-b = (D-E)b. ( 10.2 16) (10.216) and right-superdistributivity of x imply
and using (10.214) and (10.215), we get DP(Dog-E)(bog)
= D.
(10.218)
Since (10.218) holds for all f and g, we may take, in particular f = xj, j = 1, . . ., k, whence DP(Dog-E) = 0 ,
forall
g c R [ x , , ..., xklk.
(10.219)
110
COMPOSITION OF POLYNOMIALS AND POLYNOMIAL FUNCTIONS
CH.
3
On the other hand, if D = (6,x,) is any k X k-matrix over R[xl, . . ., xk] satisfying (10.219), an easy computation shows that the mappings a,, 1 = 1 , 2, . . ., k, defined by k
are derivations of R[xr, . . ., x,], i.e. (10.11) and (10.12) is satisfied (one could have taken any matrix D for this purpose). Thus (10.21l), (10.212), (10.215), (10.216), and (10.217) hold, while the condition (10.219) yields (10.214) and also (10.213). The right-superdistributivity of x and (10.211) yield (10.13), hence (al, . . ., 6,) is a C-derivation family. Thus our problem boils down to determining the matrices D satisfying (10.219). If D is such a matrix, let
D
= Ar+ArPl+
. ..
+A1+Ao
(10.220)
be the representation of D according to ch. 1 , 6 8.3, that means Ai,i = 0,1, . . .,r, is a k x k-matrix whose entries are forms of degree i. We first show that A , is a scalar matrix, i.e.
A , = dE,
for some d E R .
(10.221)
For k = 1, (10.221) is obvious, hence assume that k z 1. Choose g such that g , = x1 . . . xk and gj = 0, f o r j # I, then P = (a,g,) = P,-l is a matrix whose entries are forms of degree k - 1. Right-superdistributivity of x and (10.220) imply D o g = (A,og)+(A,-,og)+
. .. +(Alog)+A,.
If qi is a form of degree i, and qj, a form of degree j , 1 = 1, . . ., k, then xqiqjl . . . qjk is a form of degree ij. Hence substituting our particularly chosen g into (10.219) yields ( A r + A r - l + . . . + A ~ + A o ) P ~ P ~ ( B r ~ + B , .r.-.~+B,+(Ao-E)) ~,+ =0
(10.222) where B,, i = 1, . . ., r, is a k x k-matrix whose entries are forms of degree ik. After performing the matrix multiplication, we obtain a sum where the only summand with entrks of degree k - 1 is A O P k - l ( A-E), ~ hence AOPk-l(AO-E)= 0. (10.223)
0 10
Let A,,
DERIVATION FAMILIES WITH CHAIN RULE
=
111
(a,,), (xl . . . xk)/x,= u,, t = 1 , . . ., k, and E = (ast), then
AoPk-,(Ao-E) = (c,,) where c,, = (ar,--6J
k
C aspi.Hence, by (10.223),
i=l
(a,,-8,JA0 = 0 , t, I = 1, . . ., k .
(10.224)
For I ti t, we get arraN= 0 and (arr- l)a, = 0 whence a,, = 0. Furthermore, (au - l)at, = (a,, - l)a, = 0 whence a,, = all. Thus a,, = d, t = 1, . . ., k, for some d c R, and (10.221) is proved. Now we substitute (x,, . . ., x k ) for g in (10.219), then P = E and (10.225)
D2 = D.
Substituting (10.221) into (10.220) and (10.220) into (10.225) and considering the terms of degree 0 of the entries, we see that d 2 = d. Suppose that D = A,+A,-,+ . . . +A,+dE, A, # 0, I =- 0. (10.226) Then, again substitution into (10.225) and comparison of the terms of degree I in the entries, gives 2 dA, = A,. (10.227) Since d 2 = d, we have dA, = 0 and therefore A, = 0, a contradiction. Thus D = dE where d E R and d 2 = d. Conversely, if D = dE where d c R and d 2 = d, then, for all g, DP(D0g-E) = (dE)P(dE-E) = (d2-d)P = 0. We summarize our results as 10.21. Theorem. The C-derivationfamilies of R[x,, . . .,x k ]are exactly the families (d a,, . . ., d a,) where d C R is an idempotent. 10.3. It is well-known that for idempotents d of commutative rings R with identity, d R is a direct summand of R, and r dr is the projection from R to dR. Conversely, every projection of R onto a direct summand of R is a mapping r -,dr where d is an idempotent. Thus we may state Theorem 10.21 also as -f
10.31. Proposition. The C-derivation families of R[x,, . . ., xk] are exactly the families (al, . . ., 6,) of mappingsfrom R[x,, . . .,xk] into itself such that di = n( X ) ai where n is a projection from R to a direct summand of R.
112
COMPOSITION OF POLYNOMIALS AND POLYNOMIAL FUNCTIONS
CH.
3
10.32. Corollary. R is directly indecomposable if and only if the only C derivation families of R[x,, . , .,xk]are the family of the partial derivations and the family consisting of a k-tuple of zero mappings. 10.4. Remark. The investigations of 9 10.2 also show that any family (al, . . ., 6), of mappings from R[x,, . . .,xk] into itself satisfying (10.11) and (10.12) also satisfies (10.13) if and only if 1 RI = 1 . Generally, we may ask for what rings R two of these three conditions ensure that the third condition is satisfied. This problem is by no means trivial and no complete answer is known as yet. W. MULLER[l] could prove: a) If the additive group of R is torsionfree, then (10.11) and (10.13) imply (10.12). That torsionfreeness cannot be dropped, is shown by taking the field of order 2 for R as a counterexample. b ) If k 3 2, then (10.12) and (10.13) imply (10.11). If k = 1 and R is an integral domain, then also (10.12) and (10.13) imply (10.11). 11. Polynomial vectors and polynomial function vectors 11.1. Let % be any variety,
its system of operations, k z= 1 an integer,
?ISk the variety of k-dimensional %-composition algebras, A = ( A ; s2, x) an algebra of Bk,and Ak = ( A k ;Q) the direct product of k copies of the
algebra ( A ; a). We define a binary operation infix notation, in Ak by
(al, . . ., ak)o (bl, . . ., bk) = (xa,b,
0 ,
written by means of
. . . bk, xu&, . . . bk, . . .)
. . . bk),
and consider the algebra V ( A ) = ( A k ;Q, 0 ) . Let p : A .+ B be any homomorphism of algebras of B k . Then we define the mapping G(p): tT(A) V ( B ) by V(@) (a1, . . . ak> = (@a,, . . eak).
-
'3
11.11. Theorem. For any A E ?ISk, the algebra V ( A ) is an algebra of the variety of 1-dimensional %-composition algebras. P ( A ) has a selector system if and only i f A has. For any homomorphism Q : A B the mapping V(p):V ( A ) + V ( B ) is a homomorphism. r f E : A A is the identity automorphism of A , then @ ( E ) :Q(A) -. P ( A ) is the identity automorphism of (?(A). Moreover, if p : A B, and 0 :B -. C are homomorphisms, +
+
+
0 11
POLYNOMIAL VECTORS A N D POLYNOMIAL FUNCTION VECTORS
113
then i;T(ae) = v(u)i;F(e). p ( p ) is a monomorphism (epimorphism, isomorphism) if and only i f e is a monomorphism (epimorphisrn,isomorphism).
Proof. As a direct product of k copies of ( A ; Q), the algebra ( q ( A ) ;Q) belongs to 23. Superassociativity of x implies associativity of o . Similarly, the right-superdistributivityof o follows from the right-superdistributivity of 7t. A selector system for o means an identity, and (slr. . .,sk) actually is an identity if and only if {sl,. . .,s,} is a selector system for x . Straightforward computation shows that is a homomorphism and all the remaining statements of the theorem are obvious.
(ace)
11.12. Remark. In the language of category theory, Theorem 11.1 I tells us that @ is a covariant functor from the category of k-dimensional %-composition algebras to the category of 1 -dimensional !&composition algebras.
11.13. Lemma. Let A be any algebra of T8, and B a subalgebra o j A, then iT(B) is a subalgebra of G ( A ) . IJ' A , B are algebras of B,, then the mapping y : p ( A x B ) v ( A )x V ( B )dejined by
-
?!'((a,, b J ,
. . . > (a,, b,))
=
..
((a19
(b1, . .
.7
- 9
bk))
is an isomorphism.
Proof. The first statement is obvious, the second one follows from straightforward computation. 11.2. Let 23 be any variety, L?its system of operations, k a 1 an integer, and X = {XI, . . ., xk}. Since A(X, %), Fk(A), and Pk(A) are k-dimensional to these algebras : %-composition algebras, we can apply the functor a) The elements of @ ( A ( X , 2?)) are the k-tuples (pl, . . .,p,) of polynomials of A(X, 8),being called polynomial vectors and denoted by small Gothic letters. b) The elements of q ( F , ( A ) ) are the k-tuples (yl, . . .,y k )of functions from Ak to A . 11.21. Lemma. The mapping
v: p(Fk(A))
+
Fl(Ak) dejined
by
P(Y~,. . . Y)k) (al3 . . ., = (yl(al, . . ., a,), . . .,vk(al, . . ., a,)), for all (al, . . .,a,) E A,, is a composition isomorphism.
114
COMPOSITION OF POLYNOMIALS AND POLYNOMIAL FUNCTIONS
CH.
3
Proof. Obvious. c) The elements of @(P,(A)) are the k-tuples (yl, . . ., y k )of polynomial functions from Ak to A . These elements will be called polynomial function vectors and denoted by small Gothic letters. By Lemma 11.13, V ( P k ( A ) )is a subalgebra of @(Fk(A)),hence p@(P,(A)) is a subalgebra of F ~A( ~ ) . 11.22. Remark. Pl(Ak) is a subalgebra of cpq(Pk(A))since q@(P,(A)) is a subalgebra of F1(Ak)containing all constant functions of F1(Ak)and the projection E, E Fl(Ak); but, in general, Pl(Ak) c p V ( P k ( A ) ) .For example, let A be polynomially complete, k I> 1, and I A / =- 1. Then Pk(A) = Fk(A), hence y(;F(P,(A))= Fl(Ak). Since Ak is not simple, Ak cannot be 1-polynomially complete, thus Pl(Ak) c p@(Pk(A)).
-
11.3. Let cs : A(X, 3) Pk(A)be the canonical epimorphism. cs is, as we know, a composition epimorphism. Hence (;F(o):@(A(X, '%))-, @(P,(A)) i 5 an epimorphism and therefore = ~ @ ( c:s@(A(X, ) 8)) F1(Ak) is a homomorphism such that P(Pl, . . * 3 P k ) (a,,
-
. . 4 = (Pl(a,, . 3
* *
.> 4,. .
.?
Pk(% @I,
. . ., a,)), - . ak) E Ak. f
3
By Th. 11.1I , @(o)and j3are monomorphisms if and only if cs is. Clearly,
p is an epimorphism if and only if A is k-polynomially complete.
-
Let A , B be algebras of %? and q : A B an epimorphism. Then (7(rW,a)): G ( A ( X ,W) V ( B ( X ,%)) and @(Pk(q)):@(Pk(A)) -, (F(P,(B)) are epimorphisms. If q is an isomorphism, so are q(X, '%) and P,(q), hence @(r(X, 23)) and @(Pk(q)) are isomorphisms too. The diagram fig. 3.1 remains commutative when @ is applied to all the algebras and all the mappings there. Let U = A X B , t, the decomposition homomorphism of U(X, 8) and z2 the decomposition homomorphism of Pk(U).Then -t
-
tF(zi) :V(u(x, W)
~FGz) :(7( f ' k ( U ) )
*
@(A(X, 8)X B ( X , %)),
G?(Pk(A)XPk(B))
are composition homomorphisms. By Lemma 11.13, there are isomorphisms "1 1/)2
:(S(A(X, % ) X B ( X ,W)* G ( A ( X , % ) ) X @ ( B ( X ,%)), (F(Pk(A) XPk(B))
-
@(Pk(A)) X@(f',(B)),
0 11 hence
POLYNOMIAL VECTORS AND POLYNOMIAL FUNCTION VECTORS
yl@(zl) :@( U(X, 23)) y2q(z2)
:@ ( P k ( u ) )
+
-+
V ( A ( X ,%)) X@(B(X,
@(Pk(A))
115
%I),
x@(Pk(B))
are homomorphisms. yl@(zl) and y2@(z2) are monomorphisms (epimorphisms, isomorphisms) if and only if z1 and ZZ, resp., are monomorphisms (epimorphisms, isomorphisms). Thus Prop. 3.53 and Th. 3.61 imply
-
11.31. Proposition. The homomorphism y2@(z2) :@(pk(A X B ) ) G ( P k ( A ) ) x @ ( P k ( B ) )is always a monomorphism. If % is the variety of commutative rings with identity, then both the homomorphism YlV(Z1)
:@((Ax@( X , 29)* G(4X %I) X G - ( B ( X %))
and y2@(z2) :@(P,(A X B ) )
-+
@(Pk(A))x@(Pk(B)) are isomorphisms.
11.32. The diagram in fig. 3.2 yields another commutative diagram (fig. 3.3) if we apply @ and define @(a(A)) X@(o(B)) in the same way as a(A) x a(B) in 0 3.42.
(71U(X,'%)1
Y G (71)
FIG.3.3 11.4. Let H = ( H ; 0 ) be a semigroup with identity, &(H) the subsemigroup of H consisting of the units of H-which is a group- and @(H) the subsemigroup of H consisting of the regular elements of H. Then &(H) c @ ( H ) while d ( H ) = @ ( H ) if and only if @(H) is a group. This is, in particular, the case if H is finite. Let A be an algebra of '% and X = {xl. . . ., xk}. If o is the operation introduced in $11.1, we observe that (@(Fk(A));o), (@(Pk(A));o), and ( ~ ( A ( x%I); , 0 ) are semigroups. Since Fk(A), Pk(A), and A(X, 3)have a selector system each, these semigroups have identities, by Th. 11.11. In order to simplify our notation, we will write &(. . .) for &(('3(. . .)) and @(. . .) for @(G(. . .)).
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COMPOSITION OF POLYNOMIALS A N D POLYNOMIAL FUNCTIONS
CH.
3
11.41. Lemma. Let pl be the isomorphism of Lemma 11.21. Then q&(Fk(A)) = p@(Fk(A))= Sym (Ak),the symmetric group over the elements of Ah. In particular, &(F'(A)) = @(Fk(A)).
Proof. Every element of Sym ( A k )has an inverse in F1(Ak).Furthermore, every element of p(R(F,(A)) is a regular element of Fl(Ak) and thus a permutation of Ak. Hence Sym ( A k )E pl&(Fk(A))c pl@(F,(A)) E Sym ( A k ) which proves the lemma. 11.42. We set P ( P k ( A ) )n 9-l Sym ( A k )= 8 ( P k ( A ) )This . is a subsemigroup of @(Pk(A))as follows from 11.43. Lemma. &(Pk&4))G d(P,(A)) E @(Pk(A)).Zf A is jnite, then equality holds.
Proof. We have &(Pk(A))G @(Pk(A)) and &(Pk(A))L 8 ( F k ( A ) )= y-lSym (A'). Furthermore, 8(Pk(A))= pl-' Sym ( A k )f?G ( P k ( A ) )= @(Fk(A))n @ ( P k ( A ) ) E @(Pk(A)).If A is finite, then also G ( P k ( A ) ) is finite, thus &(Pk(A))= @ ( P k ( A ) ) . 11.44. Remark. In general, the inclusions in Lemma 11.43 are proper. We give examples. Let k = 1, and A the field of real numbers regarded as a commutative ring with identity. Then the element 6; E P,(A) is not in &(Pl(A))but certainly in J(P1(A)).For the second proper inclusion we get an example by taking k = 1 and, for A , the infinite cyclic group. Then E: 6 8(P@)), yet t: E @(PI@)). 11.45. Let o :A(X, 23) -+ Pk(A)be the canonical epimorphism and Ma subset of V ( P k ( A ) )As . usual V ( o ) - ' M denotes the set of all g € p ( A ( X , %)) such that @(o)g E M . Lemma 11.43 implies &(A(X
3))E @(a)-1 &(P,(A)) c G(4-l d(P,(A)) c @(a)-1 Q ( P ' ( 4 ) .
If A is finite, then both the second and third inclusion becomes an equality. By definition, d( Pk(A)) consists of all polynomial function vectors f €(?(Pk(A)) such that yf is a permutation of Ak. These vectors will therefore be called polynomial permutations while the elements of &(Pk(A))are named permutation polynomial vectors. For the sake of convenience, we abbreviate (@(Pk(A)); o> = Vk(A) and (J(m0); 0 ) = Uk(A).
0 11
POLYNOMIAL VECTORS AND POLYNOMIAL FUNCTION VECTORS
117
11.5. Let A, B be algebras of % and q :A B an epimorphism. As stated in 5 11.3, @(Pk(q)): Vk(A) Vk(B)is an epimorphism which we will also denote by Vk(q).The question arises whether, for 2, = 8,8, @, there is any relation between the subsemigroups Vk(q)2,(Pk(A))and G ( P k ( B ) )of Vk(B). A preliminary answer is given by -+
-+
c 8(Pk(B)). If B is jinite, then 11.51. Proposition. Vk(q)4(Pk(A)) Vk(q)Uk(A)G Uk(B). If q is an isomorphism, then Vk(q) maps Z ( P k ( A ) ) isomorphically onto G(Pk(B)),for 2, = 8,8, @. Proof. The first assertion is evident. Also, if q is an isomorphism, then Let f E 8(Pk(A))= Uk(A), the third assertion is true for 2, = & and then qf c Sym (Ak). Thus, for any (u,, .. .,uk)E Ak, there exists (zl,. . ., z k ) E Ak such that f o (zl,. . .,zk) = (ul, . . .,uk). Hence, by definition of Vk(q), ( Vk(r>f) 0 (wl, . . .,v k ) = (w,,. . .,vuk) whence qvk(q)f: Bk Bk is surjective and is also injective if B is finite or q is an isomorphism. This proves the second assertion as well as the third assertion for 2, = 8.
a.
-f
11.52. Remark. We shall see later on that, for various classes of algebras, Vk(q)Uk(A)= Uk(B) always holds, if A is a finite algebra of such a class. 11.53. Remark. We have already seen that diagram fig. 3.1 remains commutative after applying @ to it. As a consequence, we get that, for Z = &, 8, 42, the inclusion G ( q ( X , %))@(u(A))-l%(P,(A)) c @ ( u ( B ) ) - ~ % ( P ~ ( B ) ) holds if and only if Vk(q)'iS(P,(A)) E %(Pk(B)). Hence if q is an isomorphism, then @(q(X, 8))maps p(o(A))-' Z(Pk(A)) isomorphically onto G ( u ( B ) ) - l G ( P k ( B ) ) .Moreover, @(q(X, %)) @(u(A))-l %(Pk(A)) = T(a(B))-l %(Pk(B))implies Vk(q)%(Pk(A)) = %(P,(B)j-however, the converse does not hold in general.
-.
11.6. Let A , B be algebras of 2 ' 3, U = A X B , and z2 the decomposition homomorphism of Pk(U). Then, by Prop. 11.31, y2p(z2):Vk(u) Vk(A)X Vk(B)is a monomorphism. Again we may ask how 53(Pk(U)), % = 8, 8, 02,behaves under yz(a(zz).
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COMPOSITION OF POLYNOMIALS AND POLYNOMIAL FUNCTIONS
CH.
3
11.61. Proposition. For 5 = 8, 8 and arbitrary U, the monomorphism y2q(z2) maps z ( P , ( U ) ) into 'Zi(P,(A))X %(P,(B)).
Since y2GF(cJ (El, . . ., = yZ(z2t1,. . ., z2tk) = ((El, . . ., E,), i.e. y2@(z2) maps the identity of Vk(U)onto the identity of V , ( A ) x V,(B), the assertion is proved for 5 = 8.For 55 = 8, we require the following
Proof.
(El, . . ., E J ) ,
11.62. Lemma. For any f C V,(U) and any ((al, bl), . . ., (a,, 6 ,)) E Uk, we have f o ((al, bl), . . ., (u,, bk)) = ((ul, wl), . . ., (u,, 21,) if and only if V k ( 4 f 0 (a1,. . ,a,) = . . .,24,) and V,(zJf 0 . . .,b,) = (w1, . . .,O k ) where z1,z z are the projections from U to A and B, resp. f
Proof. Let f = (pl, . . ., F,). then
f 0 ((q,bJ, . . .
Y
(a,,
bk))
'
=
= [(Pk(zl)Plo (a13 (Pk(nl)pk
(a13
* * .7 *
. .>
a,), Pk(n,)p, Pk(n2)
0
pk
@I, (b19
f
. ., b k ) ) ? . * -, ' * *9
bk))l
which proves the lemma.
By definition, for any algebra C , we have f C 8 ( P k ( C ) if ) and only if, for any (wl,. . .,w,) E Ck, the equation f o (cl, . . ., c,) = (wl, . . ., w,) has exactly one solution (cl, . . ., c,). By Lemma 1 I .62, f C S(P,(U)) if and only if Vk(zl)f 8(P,(A)) and Vk(z2)f E 8(P,(B)). Since yzq(z2)f = ( Vk(zl)f, V,(zz)f), the proof of the proposition is completed. 11.63. Corollary. The monomorphism y2@(z2) maps U,(U) onto
( Uk(A) x U k ( B ) )
n
Y)Zq('2)
Vk(u).
Proof. This is an immediate consequence of the preceding proof.
11.64. Proposition. If the monomorphism y2@(z2) is an isomorphism, then y&F(z2) maps Iz;(P,(U)) onto Zi(P,(A)) x z ( P k ( B ) ) ,fbr 2, = 8,8, Q. Proof. For 77 = 8, this follows from Cor. 11.63. If 77 = 8 or Q, then we use the general fact that for a direct product of semigroups with identity H = L x M we always have &(H) = 8 ( L ) x d ( M ) and Q(H) = Q(L) X @ ( M ) .
11.65. Remark. Proposition 11.61 and diagram fig. 3.3 show that, for 2,= 8, 8, the homomorphism yl@(zl) maps @ ( U ( U ) ) - ~ ~ ( P ~into (U>)
§ 12
PERMUTATION POLYNOMIALS AND POLYNOMIAL PERMUTATIONS
119
P(u(A))-l%(Pk(A))x(F(o(B))-l‘zi(P,(B)). Together with Prop. 11.64, diagram fig. 3.3 also shows that if y&F(z1) is an epimorphism, then, for ‘i5 = &,8, it maps c;f((u(L’))-lZ;(Pk(U)) onto @(o(A))-l/Si(Pk(A))x P ( u ( B ) ) - l S ( P , ( B > )Prop. . 11.31 and Prop. 11.64 then imply
a,
11.66. Proposition. Zf % is the variety of commutative rings with identity, then for ‘77 = 8, the homomorphism y1q(z1) maps G?(u(U))-l/Z;(P,(U)) isomorphically onto @(a(A))-lCZ;(P,(A)) x @ ( u ( B ) ) - %(P,(B)) ~ and y z P ( z z ) maps %(Pk(U)) isomorphically onto 5 ( P k ( A ) )x-(Pk(B)).
&,a,
12. Permutation polynomials and polynomial permutations 12.1. Let C be any algebra of the variety 23, of k-dimensional 23-composition algebras and S a subsemigroup of the semigroup (@(C); 0). An element f C is called a “part of S” if there is an element (L, ...,fk) E S such that f=&, for some i = 1, .. ., k. The set of all parts of S will be denoted by Q(S).Thus Q(S) is a subset of C . 12.11. Lemma. Q has the following properties: a) ZfS E T, then Q(S) E P(T). b) Zf p :C D is any homomorphism, then @(S) = Q(@(e)S). c) Zf e :B -+ C is any epimorphism, then e - q ( S ) = Q ( @ ( e ) - ] S ) . -+
Proof. a) is obvious. Let f E QQ(S), then there exists an element (tl, . . ., fk) E S such that f = eti, for some i = 1 , . . .,k, hence f E Q(@(p)S). This argument also works vice versa. Let f E e - v ( S ) , i.e. e f € Q ( S ) , hence there exists an element of the form (tl, . . ., e f , . . ., tk)E S and thus some (ul, . . ., f, . . ., uk)E @(e)-lS whence fc Q(@(e)-l S). Again the argument can be reversed.
12.12. Lemma. If C contains a selector system {sl, . . .,sk) and if, for every permutation n of (sl,. . .,sk),the k-tuple (ns,,. . .,risk) is in s, then f is a part of S if and only if there exist elements f,, .. .,fk E C such that (Lf2,
. .. , f k )
ES.
Proof. Straightforward.
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COMPOSITION OF POLYNOMIALS AND POLYNOMIAL FUNCTIONS
CH.
3
12.2. We are going to apply the results of the preceding subsections to the case where C = Fk(A),Pk(A),and A(X, %), X = {xl, . ..,xk}. First of all let C = Fk(A). The subset (p(&(F,(A))) of Fk(A) is called the set of permutation functions. Since, in this case, the hypothesis of Lemma 12.12 is satisfied, we conclude that e E F,(A) is a permutation function if and only if there exist functions @,, . . ., @k E Fk(A) such that, for f = (e, e2, . . ., @ k ) , the mapping rpf is a permutation of Ak. There is also another characterization of permutation functions, namely 12.21. Proposition. A function e E Fk(A) is a permutation function if and only if, for any u E A , the set of all solutions in A of the equation e(xl, . . ., xk) = u has cardinality 1 A I k - ’ . Proof. Let e E Q(&(Fk(A))).Then there exist functions ez, . . ., @k E F,(A) such that, for any (u, b,, . . ., bk)E Ak, the system of equations e(xl, . . ,x k ) = u, . . .,x k ) = b,, . . .,e,(x,, . . .,xk) = 6, has one and only one solution. Hence we can map the set of solutions of @(XI, . . .,xk) = u bijectively onto the set of all elements (u, b,, . . .,bk) E Ak which has cardinality I Conversely let M(u) be the set of all solutions of e(xl, . . .,xk) = u, for any u E A and suppose that 1 M(u)1 = I A Ik-l. Then we can choose bijections yu :M(u) Ak-l, for each u E A , e.g. y,(al, . . .,a k ) = (e2(a,, . . .,a k ) , . . .,&(al, . . .,a,)), where eiE F,(A), i = 2, . . ., k. Taking f = (e, Qz, . . ., ek), we find that pf is a permutation of Ak whence p is a permutation function.
.
+
12.22. The elements of Sk(A) = P , ( A ) n Q ( d ( F , ( A ) ) ) , i.e. the set of polynomial functions which are also permutation functions, will be called permutation polynomial functions while, if 0 :A(X, 23) -+ Pk(A) is the canonical epimorphism, the elements of o-’,!Sk(A)are called permutation polynomials. In other words, a permutation polynomial is a polynomial which “induces” a permutation function. The next case we consider is where C = Pk(A).Applying C p to &(P,(A)), 8(Pk(A)),and @(Pk(A)), we can again use Lemma 12.12 for testing elements ofPk(A) whether or not they belong to one of these semigroups. As a consequence of Lemma 12.11, we obtain Q(&(P,(A))) c (p(J(Pk(A)))c P ( @ ( P k ( A ) ) )The . elements of Q(8(P,(A))) = Ssk(A) will be called strict permutation polynomial functions while the elements of u-?SSk(A) will be called strict permutation polynomials. This means that a poly-
9 12
PERMUTATION POLYNOMIALS AND POLYNOMIAL PERMUTATIONS
121
nomial function y is a strict permutation polynomial function if and only if there exist functions y2, . . .,y k E P,(A) such that, for f = ( y , y 2 ,. . .,y,), the mapping pf is a permutation of A” while a strict permutation polynomial is a polynomial which “induces” a strict permutation polynomial function. 12.23. Proposition. Any strict permutation polynomial function is also a permutation polynomial function and any strict permutation polynomial is also a permutationpolynomial. I f k = 1 or the algebra A is polynomially complete, then also the converse holds. Proof. Evident.
12.3. Proposition. Let A, B be czlgebras of % and r j : A + B an epimorphism. If v,(rj)U,(A) E Uk(B), then Then Pd.7) p ( e ( P d A ) ) ) p(&(p,@))). p k ( V ) S S d A ) c SS,(B), and if V,(rj) U,(A) = U,(B), then P,(rj) SS,(A) = SS,(B). If r j is an isomorphism, then P,(q) maps SS,(A) bijectively onto SSk(B).
Proof. This is a straightforward consequence of Prop. 11.51 in connection with Lemma 12.11. 12.31. Proposition. Let A , B, r j be as in Prop. 12.3. If A is$nite and every congruence class of Ker r j has one and the same order, then Pk(rj)S,(A) C S,(B). If r j is an isomorphism, then P,(q) maps, in any case, S,(A) bijectively onto Sk(B). Proof. If 9 is an isomorphism, then, by Prop. 3.31, Pk(q)is also an isomorphism. eES,(A) means that eEP,(A) and the equation @(xl, . . ., x,) = u has a solution set of cardinality I A I,-’ for all u E A , by Prop. 12.21. Hence the equation Pk(y)@(XI, . . ., x,) = yu has a solution set of cardinality I A I,-’ = I B I,-’ whence P,(y)e E S,(B), by Prop. 12.21 and the second assertion is proved. Assume now that the hypothesis of the first assertion holds. Let e E S,(A), then again ,o E P,(A) and e(xl, .. .,x,) = u has a solution set in A of cardinality for all u E A . Let v E B and M the solution set for this equation where u runs through all elements of A such that yu = v. Let N be the set of solutions in B of the equation
122
COMPOSfl'lON OF POLYNOMIALS AND POLYNOMIAL FUNCTIONS
CH.
3
(Pk(q)e)(xl, . . .,x k ) = v, then the mapping qk : Ak Bk, qk(z,, . . ., z,) = (7zl, . . ., qz,) maps M onto N . For any two elements (zl,. . ., z,), (tl, . . ., t k ) E A ~ we , have (yz,, . . ., 72,) = (qtl, . . ., qtk) if and only if zi(Ker q) ti, i = 1, . . .,k. By hypothesis, every congruence class of Ker 7 has cardinality I A I// B 1, hence every element of N is the image of exactly (I A I/iBl)k elements of Ak under qk, and all these elements are in M . But IMI = ( l A ] / ~ B ~ ) ~ A ~hence k - - l ,IN1 (IAI/IB()k= / A l k / l B I . Therefore IN1 = I Blk-' and, by Prop. 12.21, P,(q)e E S,(B). -f
12.32. Remark. Proposition 12.31 applies, in particular, to finite multioperator groups and hence to finite rings and finite groups. 12.33. Remark. Let A , B, 7 be as in Prop. 12.3. Diagram fig. 3.1 shows that, for T = SS, and T = Sk, we have q(X, 23) o(A)-l T ( A )G o(B)-l T(B) if and only ifP,(q) T(A)C T(B),and that, for q being an isomorphism, q(X, '$3) maps o(A)-l T ( A ) bijectively onto o(B)-l T(B). 12.4. Proposition.Let A , B be algebras of 23, U = A X B, and z2 the decomposition homomorphism of Pk(U).Then z2O(&(Pk(U ) ) )C_ Q(&(P, ( A ) ) )x ' p ( e ( P k ( B ) ) ) and z2ssk(u) G SS,(A)XSS,(B). ~fz2 is an isomorphism, then z2 maps 'p(&(Pk( U ) ) ) bijectively onto $9( &(P,(A)))x 'p(&(P,(B))) and also SS,(U) bijectively onto SS,(A) xSS,(B).
Proof. Let 5 = & or d. In Lemma 12.11 b), put e = z2, C = P,(u), and D = Pk(A)xP,(B).Then, by Lemma 12.11 a) and Prop. 11.61, we get . , P ( 5 ( P k < W ) )= P(GYz2) 5 ( m U ) ) ) C $_ 9 ( Y L 1 ( p ( P k ( A )x) %(P,(B)))).
In the case that z2 is an isomorphism, by Prop. 11.64,equality holds. But 'p(%'(z;(Pk ( A ) )x z;(Pk ( B ) ) ) )= 'P(%(Pk ( A ) ) )x /p(p( Pk ( B ) ) ) , by definition of y 2 and the applicability of Lemma 12.12 to %(P,(A)) and 2r(Pk(B)). 12.41. Proposition. Let A , B be jinite and U = A x B. Then z2S,(U) C Sk(A)XSk(B),and i f z2 is an isomorphism, then z2 maps S,(U) bijectively onto S,(A)XS,(B).
Proof. Let ni, i = 1, 2, be the projections of ( P k ( z l ) e ,Pk(n,)e), e E Pk(U). Both U and xi, i
U. Then z2p = 1, 2, satisfy the
=
o 13
123
SUBSEMIGROUPS DEFINED B Y PARAMETRIC W O R D S
hypothesis of Prop. 12.31, hence z,S,(U) E Sk(A)XSk(B). If z2 is an isomorphism and (e, y) E S,(A) XS,(B), then there exists an element x E Pk(U) such that z2x = (e, y). Let (ai, bi) E U, then x((al, '19 ' b k ) ) = (p,(zl) X(a,, - . *, Pk(z& kfb1, . - 3 b k ) ) = (@(al,. . ,a,), ~ ( b ,., . .,b,)). Hence, for any (u, v) t U , the set of solutions in U of the equation ~(x,, . . ., xk) = (u, w) has order I A I B -',I = I Ulk-l whence x E S,(U), by Prop. 12.21, and the proof is completed.
.
.I
-
12.42. Remark. Diagram fig. 3.2 implies that, for T = SS, and T = S,, we have zlo(U)-'T(U) L a(A)-lT(A)Xo(B)-lT(B) if and only if zzT(U) L T ( A ) X T ( B ) .If the latter condition holds and z1is an epimorphism, then the first inclusion becomes an equality. This consideration together with Th. 3.61, Prop. 12.4, and Prop. 12.41 yields 12.43. Proposition. If 23 is the variety of commutative ritzgs with identity and U = A y B in %, then 2 2 maps SS,( U )bijectively onto SS,(A) x SS,(B) while 21 maps o( U)-lSS,(U) bijectively onto o(A)-l SS,(A)Xo(B)-lSSk(B). g,moreover, U isJinite, an analogous result holds for Sk instead of ssk.
13. Subsemigroups defined by parametric words 13.1. Let X = {xl, . . ., x,} and A an algebra of the variety 23. Then (@(A(x,23)); 0 ) = Ck(A) is a semigroup of polynomial vectors. In order to stick to consistent notation, we will write C,(q) for @(q(X, 8)) where q : A -+ B is a homomorphism. In C,(A) we want to single out various special types of subsemigroups such as the set of all polynomial vectors x' in C1(A) where A is a group and 1 runs through the integers, or the set of all polynomial vectors over a commutative ring A with identity where all the components are of degree 1. This section is devoted to the investigation of such subsemigroups. In order to be able to state our problems we give a few rather general definitions. Let 20be a mapping which assigns, to each algebra A of 23, a subsemigroup oPo(A)of C,(A). Then dl= @(u) 20,o : A(X, 23) -,P,(A) being the canonical epimorphism, assigns a subsemigroup .Jl(A) of V,(A) to each A E 23. We define a,(A)= -@,(A)r?, @(o)-' Uk(A)which is either empty or a subsemigroup of @ ( c T )U,(A), -~ then d 3= p(a).& assigns a subsemigroup of Uk(A)or the empty set to each A of 8.We get immediately J 3 ( A )= J l ( A ) f!U,(A). We cannot expect many results under these general conditions. Thus we
124
COMPOSITION OF POLYNOMIALS AND POLYNOMIAL FUNCTIONS
make an
-
assumption.
We
CH.
3
say that -Po has property HI if
c k ( q ) -@,(A) = d2,(B), for any two algebras A , B of 8 and any epimorphism q : A B. Methods for the construction of such mappings 20 with
property H I will be elaborated in $0 13.2 and 13.3.
Proof. a) follows from the definitions and diagram fig. 3.1, b) follows from J 3 ( A )= -’@,(A)f’i U,(A) and a). c) The “only if” part follows from diagram fig. 3.1. Conversely let Vk(q)dP3(A)C B3(B), then @(o(B))C,(q)B2(A)!Z &,(B) n Uk(B), by diagram fig. 3.1, hence C!f(r)-@,(A) E =-%(B)n Uk(B) = -P2(B).
ca(@>)rl
13.12. Lemma. Let U = A x B an algebra of 8 and z1, z2 the decomposition homomorphisms of U(X, 8)and Pk(U),respectively. If LOhas property HI, then y l ~ ( z l )maps J i ( U ) into .-@,(A)xpPi(B),for i = 0, 2 , and y2@(z2)maps J i ( U ) into L i ( A )XBi(B),for i = 1,3. Proof. Let zi,i = 1, 2, be the projections of U. Then lylP(z,)f = (Ck(zl)f,C,(n~>f) whence the lemma follows for i = 0. Similarly, with the aid of Lemma 13.11 a), we obtain the result for i = 1. By Prop. 11.61, y 2 ~ ( z 2maps ) Uk(U)into Uk(A)X U,(B), hence, by Lemma 13.11 b), the lemma follows for i = 3, and finally for i = 2, by Lemma 13.11 c).
13.13. We define another property for 20: We say Bo has property H 2 if y ~ ( ? ( z ~ maps ) J 0 ( U ) onto Bo(A>XBo(B).Again we postpone the construction of such mappings 20to $0 13.2 and 13.3. 13.14. Lemma. rf20hasproperty H2, then y&?-(z1) induces an epimorphism, ,for i = 0,2, and y2@(z2) induces an isomorphism, for i = 1,3, from 2 , ( U ) to J , ( A ) X d i ( B ) .
Proof. For i = 0, this is the hypothesis while, for i = 1, the lemma follows from diagram fig. 3.3 since y 2 q ( z 2 ) is injective, by Prop. 11.31. By Prop. 11.61, y 2 V ( z 2 maps ) -P3(U) into -@3(A)XJ3(B) and, by Cnr.
I13
SUBSEMIGROUPS DEFINED BY PARAMETRIC WORDS
125
11.63, this mapping is onto. For i = 2, the result is then again a consequence of diagram fig. 3.3. 13.2. We are now going to construct mappings Y Owith property H I , or properties H1and H 2 . Some definitions will go ahead. Let 52 be the family of operations of the variety '23 and r a non-negative integer. A parametric word vector '$3 of order r is a mapping from the Cartesian product of r copies of the set of non-negative integers to the set of k-tuples of words over 52 in the indeterminates pi,lwI, . . ., pi, i = 1,2, . . . and xl, . . ., xk, for r =- 0, and a fixed k-tuple of this kind, for r = 0. A specialization vector of !$ in the algebra A E % is a k-tuple of words in A U {xl,. . ., xk} we obtain from substituting an arbitrary word in A U {x,, . . ., x,} for each ,wioccurring in the k-tuple of words assigned by '$3 to each r-tuple of non-negative integers. In particular, we have to substitute an element of A for each ,,wI,and into a fixed k-tuple, if r = 0. The specialization '$(A) of '$3 in the algebra A is the set of all polynomial vectors of @(A(xl, . . .,xk,'23)) which are represented by the specialization vectors of '$3 in A . Thus '$3(A)is a subset of Ck(A), for any A in 23. The parametric word vector '$3 is called semigroup generating if '$3(A)is a subsemigroup ofCk(A), for every A in %. Three examples will illustrate the concept of semigroup generating parametric word vectors. The first and second vector are of order 0 while the third one is of order k2. It is easy to see that these parametric word vectors are, indeed, semigroup generating. a) % an arbitrary variety, Sp = (lw,, 2w1,. .., k w , ) . b) % the variety of commutative rings with identity,
'$3
=
(oW1Xlfow2, lwlX2+lwg,
. . .) k-1W1Xk+k-1w2)*
c) '23 the variety of commutative rings with identity, and '$3 mapping the k2-tuple of non-negative integers 1 = (n,,, . . ., nlk, n21, . . ., nkk) to the k-tuple of words
'$3(1) = (owlx;"x:z
. . . XE'b, owzx;p' . . . XZZ, . . ., OwkX;lL'Xp. . . *).
13.3. Theorem. If is a semigroup generating parametric word vector of arbitrary order, then the mapping 20defined by J?o( A ) = '$3(A) has property H I .
-.
Proof. Let 7 :A B be any epimorphism andf E !$(A),then f is represented by some specialization vector of !$ in A . We obtain C,(q)f by replacing
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COMPOSITION OF POLYNOMIALS AND POLYNOMIAL FUNCTIONS
CH.
3
each a E A in this specialization vector by qa E B. Thus Ck(q)f is represented by a specialization vector of '$ in B, i.e. an element of '$(B) whence ck(q) B0(A)& ao(B). Conversely, if g E '$3(B),then g is represented by some specialization vector of '$ in B. If we replace each b E B in this vector by any inverse image a E A of b under q, we get a specialization vector of '$ in A . This vector represents some f E '$(A) and Ck(q)f = g whence B o ( B )C_ C,(q) =&(A). This completes the proof. 13.31. Theorem. Suppose '$3 is a semigroup generating parametric word vector of order 0. Let U = A X B be a direct product of algebras of % and assume that the decomposition homomorphism z1of U(x,, . . ., x,,%) is an epimorphism, for v = 1, . . ., k . Then 20= '$ has property Ha. Proof. By Th. 13.3 and Lemma 13.12, ?pJ@(zl) maps p ( U ) into '$(A)X '$(B).Conversely, if (f, g) E '$(A) X '$(B),then f and g are represented by specialization vectors of '$ in A and B, resp. If *wjoccurs in '$3, let u(aj, xl, . . ., xv), v(bj, xl, . . ., x,) be the words which are substituted for vwi in these specialization vectors. We choose f C U(x,, . . ., x,,%) such that zlf E A(xl, . . .,x,,, 8)xB(x,, . . .,.x,,23) is just represented by the pair of these words and take a word ,wj(uj,xl,. . .,xu)in U LJ {xl, . . . , xy} representing f. If we subject every vwi in '$ to this procedure, we get a specialization vector of $ in U which represents some 4 E '$(U) such that YJ@(ZJ)$ = (f, 9). This proves the theorem.
13.32. From Th. 3.61 we conclude that Bo = '$3 has property H 2 in the example a), b) of Q 13.2 for % being the variety of commutative rings with identity. Hence, in these cases, &!;(A>< B) Bi(A)x d i ( B ) , i = 0, 1,2, 3. 13.4. Let % be any variety. We want to construct new parametric word vectors from given ones. Let '$1, Y2 be parametric word vectors of the orders r and s, resp. We define a new parametric word vector of order r f s by means of the following procedure : Let (ml, . . ., m,, n,, . . ., n,) be an r+s-tuple of non-negative integers and suppose that the images under Plcontain the indeterminates owi, lwi, . . .,kwiand xl,. . ., xk while the images under '$2 contain the indeterminates ovj, lvi, . . p j and y,, y2, . . ., yI. Suppose that '$J(ml,. . ., m,) = a and q2(n1, . . ., n,) = b. In b we replace each yi by the new indeterminate xk+;, each ovjby the new indeterminate kw-j, I,
a 13
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SUBSEMIGROUPS DEFINED B Y PARAMETRIC W O R D S
and each ,pi,v =- 0, by the new indeterminate k f p j (the negative index in ,w-~ serves just to distinguish the new indeterminate from ,wj which occurs in a). Thus we have obtained a new l-tuple 6 from the old I-tuple b. We define a mapping El = '$2 wr Pl by (V2 wr (ml, . ., m,., n,, . . ., n,) = (a, 6), a kfl-tuple of words in the indeterminates owi, lwi, . . . , k W i , k+lwi, . . .,k+,wj,kw-;?,x l , . . ., X k + [ . Thus $3 is a parametric word vector. For r = 0 or s = 0, it is clear how we have to modify our construction. 0 = p2wr '$1 is called the wreath product of p2by '$1. As a consequence of the definition we get that the wreath product is associative, i.e. '$3 wr (VZwr '$1) = (!& wr '$2) wr '$1. Thus we may omit brackets.
vl)
~
13.41. Lemma. Let Q = '$2 wr '$1. Then the specialization 0 ( A )of $31 in A is the set M of all polynomial vectors f = (f,),) of C k + , ( A )such that f1 E pl(A)and f 2 E p2( A(xl, . . ., x,, 23)) where Y2 is the parametric word vector we get from replacing each yi by x , + ~in the I-tuples of p2.
Proof. Every word in A U { x l , . . ., x ~ + is ~ }a word in W( A U { x l , . . ., xk}) U {xk+l, . .., xkiy) and vice.versa. Thus the polynomial vectors of a ( A ) are exactly the polynomial vectors of M . 13.42. Theorem. I f the parametric word vectors '$1, '$2 are both semigroup generating, then the parametric word vector Q = '$2 wr p1is also semigroup generating.
Proof. We have to show that a ( A ) is a semigroup of C,+[(A),for all A in '23. Let f = (fl, fd, g = (91,9 2 ) E D ( 4 and suppose f o g = (01, $ 2 ) . By Lemma 13.41, fl, g1 E pl(A),hence fl contains just xl.. . ., xk and we have therefore ljl = f l o g1 E '$1(A) by hypothesis. Again by Lemma 13.41, f2, g2 E p 2 ( A ( x l , . . ., x,, 23)). In order to obtain an expression for lj2, we have to replace each xi in f2 by the i-th component of g. Since g1 E !&(A), the elements of A(x,, . . .,xk, 23) occurring in the components of f2 are taken into elements of A(xl, . . ., xk, a), and we obtain some element fz E @,(A(x,, . . ., xk, %)) after replacing x i , for i = I , . . ., k. Continuing by replacing the x ~ + for ~ ,i = 1, . . ., I , we have to substitute now the i-th component of gz and finally end up with some vector $2 which belongs to PS2(A(x1,. . ., x,, 23)) since qP2 is semigroup generating. By Lemma 13.41, f o g E D(A).
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COMPOSITION OF POLYNOMIALS AND POLYNOMIAL FUNCTIONS
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13.43. It is easy to see that, for any variety 8 and k = 1, if '$3 is the parametric word vector lwl of order 0, then the wreath product $5 wr '$3 wr . . . wr '$3 of k copies of '$3 is the parametric word vector of 5 13.2, example a). If 8 is the variety of commutative rings with identity, k = I , and '$3 is the parametric word vector owlxlf ow2 of order 0, then the wreath product of k copies of '$3 is the parametric word vector of 5 13.2, example b) as one can easily see. 13.5. Let and p2be semigroup generating parametric word vectors of the orders r and s, resp., and El = pp2 wr Fl. Then do= Q, d?: = PI, 2:= '$32 are mappings as dealt with in 9 13.1. Let A be any algebra of 23, then &?,(A), "P3(A) (unless empty) are subsemigroups of Vk+[(A), 2 : ( A ) and &:(A) (unless empty) are subsemigroups of V,(A), and 2&4) and .&:(A) (unless empty) are subsemigroups of V,(A). Let 0 : A(Xl, . . ., X k + l , 8) Pk+l(A), C1 : A(X,, . . ., xk, 8) Pk(A), and o2: A(x,, . . , ,x,, 23) P,(A) be the canonical epimorphisms, and v, pl,pz the isomorphisms of Lemma 11.21 for k+I, k , and I, resp., instead of k . Let [ : Ak+' AkXA' be the bijection defined by :(al, . . ., ak, ak+ls. . ., ar) = ((al, . . ., a,), (ak+l,. . ., a,)). Then the mapping z :f'l(Ak+') P1(Akx A') (z :Sym (Ak+') Sym (AkX A')) defined by zf = cf5-l is a semigroup (group) isomorphism. Suppose now that f E A?o(A)= &4). By Lemma 13.41, f = (fl, fa) where fl E Q1(A) and f2 E g2(A(x1,. . ., x,'. 8)).Therefore, for any (a, b) E Ak x A', we have
-
--
-+
-+
-f
c(vV(df)C-l (a, b) = (fl 0 a, f2(a) 0 b)
Hence
= ((pl@(0dfl)a, (dmz)fz(a>)b).
T(pGY4f)C-1(a, b) = (%a, W b ) (13.5) where x E p12:(A), A(a) E ~ I ~ A ? ~Thus ( A ) . zcp maps J?,(A) monomorphically into the wreath product y 2 2 : ( A )wr p12t(A), this time the wreath product is to be understood as the usual wreath product for semigroups. 13.51. Proposition. The mapping zq~maps &,(A) monomorphically into the semigroup wreath product p2&T(A)wr plA?:(A). r f A is jinite, then also J),(A) is mapped monomorphically into p,B:(A) wr yr12i(A) under zp, and &,(A) is mapped onto this wreath product if zp maps &,(A) onto Y , J w ) wr PlJ&4. Proof. The first assertion has already been proved. Let g E &,(A), then 9 E Uk,,(A),thus yg E Sym (Ak+').Therefore [ q ~ g C - E~ Sym (AkXA'). By
0 13
129
SUBSEMIGROUPS DEFINED BY PARAMETRIC WORDS
(13.5), [(pg) T-l(a, b) = (xu,A(a)b) where x E F1(Ak) and A(a) E FI(A'). Hence x is surjective and A(a) is injective, for all a E Ak. Since A is finite, x and A(a) are permutations which proves the second assertion. The third assertion can be proved by using (13.5) and the fact that if A(a) and x are permutations, so is the mapping on the left-hand side.
13.6. We ask for conditions under which zp maps J l ( A ) isomorphically onto pz2;(A)wr @:(A). We first prove
-
13.61. Lemma. The equation z p J l ( A ) = p,&;(A) wr plB:(A) holds i f and only if, for any mapping 8:Ak J:(A), there exists a polynomial Vector fz(x1, ., xk, X k + 1 , . . ., X k + / ) E !&(A(x,, . . ., xk, 8))such that
..
@(a,, * a k ) = @(02) for all (al, . . ., ak)E Ak.
fZ(al,
9
..
* 3 ak7
xk+]?
* *
-
3
xk+/)?
Proof. J I ( A ) consists of all elements @(o)f such that f E &!,(A). By Lemma 13.41, ~ o ( A is) the set of all polynomial vectors f = (fl, f2) where fl E !&(A) and f2 E FZ(A(xl,. . .,xk, 8)).In 0 13.5 we have seen that (Zp;G(o)f) (a, b) = ((p1GYm)f1)a, (pZCF(o2) f2(a))b). Since zp maps monomorphically from J l ( A ) to; p&?;(A) wr p,B:(A), it maps J l ( A ) isomorphically onto this wreath product if and only if, for any x E &:(A) and any mapping 8: Ak k f ( A ) , there exists some fl E $$(A) and fz E pZ(A(Xl, . . .,xk, 8)) such that @(o,)fl = x and -+
G(O2) f2(a1,
*
.
xk+ 1,
.?
for all (al, .. .,a k ) E A ~ . 13.62. Theorem. Let El p,&?(A) wr p&(A) i f
=
* * *
,X k + / )
= 8(a]3
* *
.?
ak),
p2wr Pl.zp maps J l ( A ) isomorphically onto
vzis of order 0 and A is polynomially complete.
Proof. We have to show that the hypothesis of Lemma 13.61. is satisfied. Let pZ = (gl, . . .,gJ where g, = gj(oui.p i , . - ., pi,yl, . . ., Y J , j = 1,2, . . ., I, and 8: Ak -+ &!:(A) be any mapping. Then, for any a = (al, . . ., ak) E Ak, we have 8a = (h17. . ., h,) where hi = gj(owi(a), Ivi(a, El), 2ui(a, 51, Ez), . . ., /vi(a, 61, * . ., t/),51%* . E/), j = 1,2, . . ., Z. Since A is polynomially complete, for any ,v, occurring in gj there exists a polynomial r ~ s ( ~ l ,. ., xk, x ~ +. .~.,,& f r ) E A(x;, . . .,Xk+,., 23) such that, for any a E Ak we have .?
.
rWs(al,
.-
9
51,
9
., 6,) = ,v,(a,
51,
- - ., tr).
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COMPOSITION O F POLYNOMIALS AND POLYNOMIAL FUNCTIONS
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3
Substituting these polynomials into !& of Lemma 13.41, we get a polynomial vector f,(x,, . . .,xk+J E p 2 ( A ( x l , . . ., x,, 23)). The j-th (T(02)f2(al,. . .,a,, x,+~, . . ., xr) is hj, thus component of q ( 0 2 )f2(al,. . . , ak, x , + ~ ,. . . , xr)= 8(al, . . ., ak) and the hypothesis of Lemma 13.61 is satisfied.
13.63. Let us now apply Theorem 13.62 to our examples a) and b) of
3 13.2 which, as stated in 3 13.43 are wreath products of parametric
word vectors of order 0. Th. 13.62 and Prop. 13.51 show that, if A is polynomially complete, then, in the case of example a), the semigroup J 1 ( A ) is isomorphic to the wreath product of k copies of the symmetric semigroup of A and B3(A)is isomorphic to the wreath product of k copies of the symmetric group of A. We also see that, if A is a finite field, then, in the case of example b), -!?,(A) is isomorphic to the wreath product of k copies of the one-dimensional linear inhomogeneous semigroup of A and &(A) is isomorphic to the wreath product of k copies of the one-dimensional linear inhomogeneous group of A . Remarks and comments
0 1. MENGER[3] was the first person who fully realized the significance of the concept of superassociativity, and it was he who introduced selector [l] and SKALA[ l] on k-dimensystems. There are also papers by DICKER sional superassociative systems. MENGER [I] also introduced 1-dimensiona1 composition rings and thus axiomatized the composition of functions from a ring into itself. MENGER and his school have written several papers on 1-dimensional composition rings which then bore the name of trioperational algebras; we refer to a survey on this work by MENGER[2], but also to ADLER[l]. Near-rings were introduced by ZASSENHAUS about 1935 (for a special case) and have been the subject of quite a lot of papers ever since whereas composition lattices have hardly been investigated (see MITSCH[l]). HION[I], [2], [3] considered %-composition algebras in general and related algebras for the first time. Composition of functions on sets has been treated in many papers, mostly for the needs of formal logic, but sometimes also from a purely algebraic viewpoint (MENGER and WHITLOCK [l], WHITLOCK [l], SCHWEIZER and SKLAR[l]). The cornposition algebra FI(R)where R is a commutative ring with identity has been treated by NOBAUER[16]. BERMAN
REMARKS AND COMMENTS
131
and SILVERMAN 121 proved Th. 1.51 for the case that f2contains just 2-ary operations, NOBAUER [20] gave a proof in the general case.
5 2. HULE[l], [2], was the first one to study the composition of polynomials and polynomial functions over arbitrary algebras. For commutative rings with identity and fields, in particular, the theory of composition of polynomials has been a tool for numerous branches of mathematics ever since the beginnings of algebra. but there was not a systematic approach until this century (a survey of papers on this subject will be given in our remarks and comments on 5s 6, 7 and on Ch. 4). 5 3. Our definition of a constant was inspired by MENGER who defined constants for composition rings. Our condition of Prop. 3.51 is closely related to a concept of independence for algebras having the same type, which was introduced by FOSTER [3]. 5 4. The theory of full congruences in arbitrary polynomial algebras has been developed by HULE[l], [2]. For the results on full congruences of F,(A) which are quoted in 0 4.2, we refer to BERMAN and SILVERMAN [l], NOBAUER and PHILIPP [l], [2], and PHILIPP [l]. 5 6. MANNOS[I] has studied ideals of composition rings in general while a systematic theory on full ideals of polynomial rings over commutative rings with identity was subsequently developed by NOBAUER [lo]-[ 141. A computation and discussion of the Jacobson radical (i.e. the intersection of all maximal full ideals) of an arbitrary polynomial ring R[x,, . . ., xk] with composition over a commutative ring R with identity is due to MLITZ[I]. It has been an outstanding problem for a long time to find all full ideals of the polynomial ring Z[x,, . . ., xk], Z being the ring of rational integers. This problem has not even been solved for k = 1. 5 7. A part of Th. 7.21 is due to MILGRAM [l], in its present form it was proved by NOBAUER [23], Th. 7.31 has its origin in BURKE[I]. The problem of determining all full ideals of Q[x,, . . ., xk] for a finite field Q and k =- 1 has not yet been solved. A first attack on this problem for k = 2 was launched by STUEBEN [l]. Recently CLAYand Do1 [l] have computed the Jacobson radical and all maximal ideals of the near-ring (K[x]; +, x ) where x stands for the composition and Kis a field, IKl # 2.
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COMPOSITION OF POLYNOMIALS AND POLYNOMIAL FUNCTIONS
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3
8 8. DICKSON[ 5 ] called the polynomials of {D}“residue polynomials mod D’. Papers on residue polynomials over the rational integers are KEMPNER [l], [2], LITZINGER [l], NOBAUER [ 2 ] , [4], NIVEN and WARREN [ l ] . LEWIS[l] started with the investigation of residue polynomials for the case of arbitrary Dedekind domains. The approach of our book, however, has been used by LAUSCH[l] for k = 1 and generalized by AIGNER [ I ] to k 5 1. There is a connection between the residue polynomials of the ring Z of rational integers and the so-called integral-valued polynomials whose definition is: Let K be the field of rational numbers, then a polynomial fcK[.x,, . . ., xk] is called an integral-valued polynomial if f(a,, . . .,ak)E Z , for all (al, . ., ak)E Zk. If (n) is an ideal of Z , then evidently f C Z[x,, . . .,xk] is a residue polynomial mod (n) if and only if the polynomial (I / n ) f is integral-valued. For details of integral-valued polynomials we refer to CARLITZ [ 2 ] ,STRAUS [l], but there are also many other papers on this subject.
.
0 10. The presentation of this section follows NOBAUER [19].
0 11. It
is still an open problem to characterize all those 1-dimensional %-composition algebras which are isomorphic to some algebra ,@(A), for a suitable k-dimensional composition algebra A. We refer to SAIN[ l ] for the case where 23 is the variety of sets. The considerations of this section were so far treated only for special varieties, mainly for commutative rings with identity (for a survey on relevant papers we refer to the remarks and comments on Ch. 4, § 4).
5 12. Polynomial permutations, permutation polynomial vectors and permutation polynomials of A were investigated up to 1950 only for the case where % is the variety of commutative rings with identity, A is a finite field and k = 1, but for this case an extensive literature had piled up (see remarks and comments on Ch. 4, @ 8,9). Later on, other commutative rings with identity were studied in this respect, then also k = 1 was dropped (see remarks and comments on Ch. 4, Q 4). Furthermore in recent years also the variety of groups was object of investigations in this direc[2] tion (see remarks and comments on Ch. 5) and more recently MITSCH and SCHWEIGERT [I] treated permutation polynomials in the variety of lattices. Our Prop. 12.21 is due to P. GRUBER.
REMARKS AND COMMENTS
133
0 13. Several examples of subsemigroups of the semigroup C,(A) which are defined by parametric words are known for the case where 23 is the variety of commutative rings with identity. The best known such example is the subsemigroup of polynomial vectors with linear forms as their components. This subsemigroup and the group of its permutation polynomial vectors have been studied in numerous papers. For other examples [l], NOBAUER [3], [6], [8], [Ill, [17], [18]. we refer to KALOUJNINE
CHAPTER 4
COMPOSITION OF POLYNOMIALS AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
1. Prime factor decomposition with respect to composition 1.1. Let 23 be any variety with 2 ! as its family of operations, A an algebra of 23, X = {xl, . . .,xk} a set of indeterminates, and Q(A(X, %)) = ( A ( X , % ) k ; 9, o ) the algebra introduced in ch. 3, 5 11. By ch. 3, Th. 11.11, this is a l-dimensional %-composition algebra, thus the algebra S = (A(X, 0 ) is a semigroup with identity. There is little known about the structure of S . Only for the case where 23 is the variety of commutative rings with identity and A is a field (or sometimes, more generally, an integral domain) and k = 1, there exist some satisfactory results. The most interesting and important results will be derived in this and the subsequent sections. 1.2. Let 23 be the variety of commutative rings with identity, D any integral domain of 23, and x an indeterminate. We consider the semigroup ( D [ x ] ;o ) = S which has x as identity. The degree [ f ]of any polynomial f # 0 of D [ x ] has been defined in ch. 1, S; 8.3. It will be useful for our further considerations to set [O] = 0; then every polynomial will have some well-defined degree. 1.21. Proposition. For any two polynomials f , g E S , we have [ f o g ] =
[f1 [gl.
Proof. By ch. 1 , Th. 8.11, f , g have normal forms that can be written as a,x"+un-,x"-l+ . . . +a,x+a,, g = b,x"+ . . . +b,x+b,. Then f o g = f ( g ) = anf+a,-lg"-l+ . . . +a,g+a,. If [g] = 0, we have [ f o g ]= 0 = [ . f ][ g ] ,and if g f0, bych. 1, Prop. 8.31, we have [gj]= J [ g ] for j 3 0 whence [ f o g ] = n[g] = [f][g].
f=
1.22. Remark. The proof of Prop. I .21 also shows that, for an arbitrary ring D of 23, we have [ f o g ] 6 [ f ] [ g ] . 134
91
PRIME FACTOR DECOMPOSITION WITH RESPECT TO COMPOSITION
-
135
1.23. Corollary. The mapping f [ f ] from S into the multiplicative semigroup of integers is a homomorphism 8. Proof. This is a restatement of Prop. 1.21. 1.24. Corollary. I f f E S has a right or a left inverse, then [f ] = 1.
Proof. By Cor. 1.23, since 1 is the only non-negative integer which has a multiplicative inverse.
1.25. Proposition. Every f E S with [ f ] # 0 is a right-regular element of S . Proof. Suppose that g of = h of, then the right-superdistributivity of o implies ( g - h ) o f = 0. We apply 8 of Cor. 1.23 to this equation, then g - h = a E D . Hence 0 = ( g - - ) o f = a o f = a and therefore g = h.
1.26. A polynomial f E S is called indecomposable if [f ] =. 1 and there < [f]and is no representation o f f of the form f = f i o f 2 where [fi] [f2]< [f 1. E.g., every polynomial of prime degree is indecomposable. 1.27. Proposition. Every f E S such that [f ] =- 1 has a representation of the form f = of, o . . . of, where every J;: is indecomposable.
f,
Proof. By Prop. 1.21.
1.28. There arises now the question of in how many different ways f can be decomposed this way. Only for the case where D is a field of characteristic zero (for partial results for prime characteristic see FRIEDand MACRAE [l]), a complete answer is known which we will elaborate in the remainder of this section and the subsequent one, leaning heavily on the definitions and results of ch. 6 , s 5. 1.3. Let K be any field of characteristic zero and S the semigroup ( K [ x ] ;0 ) . Since K is an integral domain, all the results of 5 1.2 can be applied to K . An extension of Cor. 1.24 can be obtained for fields : 1.31. Proposition. f S has a right (left) inverse i f and only In this case f has a unique inverse.
if 1f J
=
1.
136
COMPOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
CH.
4
Proof. “Only if” follows from Cor. 1.24. If f = ax+b, then g = ( l / a ) .(x-b) is an inverse ofjwhich is unique since o is associative. 1.32. A polynomial f E K [ x ]is called normed if its normal form according to ch. 1 , Th. 8.11, is f = xn+a,_,x”-l+ . . . +alx. Thus every normed polynomial is monk. Iff, g are normed, then also fg and f o g are normed. 1.33. Lemma. Let U be any intermediate4eld of K ( x ) such that U contains
a polynomial of positive degree. Then there exists one and only one normed polynomial h E K [ x ]such that U = K(h). Proof. By ch. 6, Lemma 5.8, U = K(r) where rE K [ x ] and [r] z 0. But clearly r = ah+b where a # 0 and b are suitable chosen elements of K and h is a normed polynomial, hence K(r) = K(h). I f k is any normed polynomial of K [ x ] such that U = K(k), then, by ch. 6, Lemma 5.81, k = l o h and h = m o k whence [I] = 1. Thus k = ch+d, c, d E K, c # 0. Since h, k are normed, we concludec = 1, d = 0, hence k = h. 1.34. Theorem. a) Every polynomial f ES with [f]=- 1 has a ‘>rime factor decomposition” of the form
f
=:
Iofio
...
(1.31)
of,
where I is a linear polynomial and every f;. is a normed indecomposoble polynomial. b) If (1.32) f = mog,o . . . cgs is any other prime factor decomposition o f f , then I = m, r = s, and the degrees [gi]can be paired off with the degrees c) The decomposition (1.31 ) can be transformed into the decomposition (1.32) by a finite number of steps of the following kind: From a decomposition f = n oh, o . . . oh,, select a “partial product” hi o hi+l and replace it by a partial product ki o ki+, of normed indecomposable polynomials ki, ki+l such that hi o hi+l = ki o ki+,. d ) There is only a3nite number of dflerent prime factor decompositions off.
[A].
01
PRIME FACTOR DECOMPOSITION WITH RESPECT TO COMPOSITION
137
Proof. Let .f.= 1of, o . . . of, be any prime factor decomposition, and f,+, = x. In the chain K(f)= K ( f i of,o . . . of,,J C K ( f ,o . . . of,+l) C . . . C K(f,oj;+,) C_ K(f,+,) of subfields of K(x), every inclusion is proper, by ch. 6, Th. 5.71,since [ A ] =- 1, for i G r. Let U be a field such thatK(f,of,+,o . . . of,,,) C U C K(f,+,o . . . of,+,), for 1 G v -c r. By ch. 6, Lemma 5.8, U = K(t), t E K [ x ] ,and by ch. 6, Lemma 5.81,there are p , q E K [ x ] such that.f,oL.+,o . . . of,,, = p o t , t = qofY+,o . . . hencef, of,+,o . . . o f , + l = p o q of,,, o . . . of,,,. By Prop. 1.25, f, = p o q , thus [ p ] = 1 or [q]= 1 whence the chain above is a maximal
chain. Thus every decomposition (1.31)off yields some maximal chain K ( f ) = So c S, c . . . c S , = K ( x ) from K(f)to K ( x ) which we will call the chain belonging to this decomposition. The length of this chain equals the number of factors in the decomposition while, by ch. 6, Th. 5.71, [ S i : S i - , ]= [A], i = 1, . . ., r. Clearly, if (I .32) yields the same maximal chain from K ( f ) to K(x), then s = Y and K(f,cf,+,o.. . of,) = K(gyogv+lo.. . og,), 1 == Y =z r. Since J;,gi, i = 1, . . ., r, are normed, Lemma 1.33 implies that f,of,+,o ... of,=g,,ogV+,o . . . og,, l = s v s r , whence, by Prop. 1.25,ji = gi, i = Y, r- 1, . . ., 1 and. 1 = m. Hence every maximal chain from K ( f )to K ( x ) belongs to at mos; one prime factor decomposition off. Next we will show that every such chain belongs to at least one prime factor decomposition offso that we may consider maximal chains in the place of prime factor decompositions. Thus let K(f)= SOc S1 c . . . c S, = K ( x ) be any maximal chain from K(J’) to K(x). By Lemma 1.33, there are normed polynomials ujr i = 0,1, . . ., r, such that Si= K(u,), in particular, u, = x. By ch. 6, Lemma 5.81,we have f = louo, [1] = 1, and ui =A+, O U ~ + ~[A,,] , =- 1, i = 0, 1, . . ., r-1. Since ui, ui+, are normed, A,, is normed, too. Suppose that A+, is not indecomposable, say ,h+,= p o q where [PI < [A+ll. k 1 [L+1l9then ui = P 0 ( 4 o u i + l ) , and since [ p ] =- 1, [q] > 1, we would have Sic K(qoui+,) c Si+,, contradiction. Hence ui = f ; + , ~ f ; . + ~.o. . of*, i = r-1, r-2, . . ., 0, therefore f = I o f i o f,o . . . oj,, which is a prime factor decomposition of.6 having So c S, c . . . c S, as its maximal chain. We need some further property of the correspondence between maximal chains and prime factor decompositions. Let So c S, c . . . c T, c . . . c S, be a maximal chain which differs from SO c S1 c . .. c S, c . . . c S, only by the m-th member. Thenfhas a certain prime factor decomposition (1.32) corresponding to this maximal chain. Let
138
COMPOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
CH. 4
Trn= K(vrn)where urnis normed, then, as before, we get .f = l o u o ,
u.I = f. , +
l ~ ~ i + l ,
i = 0, 1, . . ., m-2,
urn = grn+1Oum+19 u., =fr + l o ~ i + l , i = m + l , ..., r - 1 . By Prop. 1.25, grnogmfl =frnofrn+l. Hence (1.32) becomes in this case urn-1
= grnovrn,
f= 10fi0 - - . ~
..
f r n - ~ ~ ~ m ~ ~ n ~* +of, ~ ~ f r n + ~ ~
where grnOgrn+, =zfrnofrn,,. The proof of the theorem now consists of translating the relevant results of ch. 6, § 5, into the language of prime factor decompositions. Hence a) holds since there exist maximal chains from K ( f ) to K(x). Since allJ;., g i , are normed, b) is a consequence of ch. 6, Th. 5 . 8 6 ; ~ )also follows from part c) of this theorem while d) results from ch. 6, Remark 5.87. 2. Standard solutions of p o q = r o s 2.1. Theorem 1.34 shows that, in a finite number of steps, all the different prime factor decompositions of a polynomial f can be obtained from a given decomposition as soon as we know all the solutions of the equation p o q = ros (2.1) where p , q, r, s are normed indecomposable polynomials. This type of solution will be called astandard solution of (2.1). Indeed, let D be the given decomposition off. Then we can determine the set D Iof all decompositions of f , which are obtained from D by at most one replacement of two subseetc. Finally, by Th. quent factors. In the same way, we can derive a set D2, 1.34 c), d), we endup with aset D,of decompositions such that Dk+l = D,. Hence D, consists of all the different prime factor decompositions off. There seems to exist, so far, no explicit investigation of the problem to decide, whether or not a polynomial f is indecomposable, and of the problem, to determine a prime factor decomposition of a given polynohas pointed out, the first problem can besolved mial. As A. SCHINZEL by considering all possibilities for a decomposition off'into two factors and deciding in every case whether the resulting system of equations for the unknown coefficients of these factors is solvable (which, of course, is rather tedious). Clearly then the second problem can also be solved.
2.2. We are now going to reduce the problem of finding the standard solutions of (2.1) in K [ x ] where K is any field of characteristic zero.
02
STANDARD SOLUTIONS OF p o q =
ros
139
2.21. Lemma. Let f E K [ x ]be indecomposable over K [ x ]and L any extensionjeld of K. Then f is also indecomposable over L[x].
Proof. Suppose that f = p o q, p , q E L [ x ] ,then, for any 0 # c E K and for any linear polynomial I t L [ x ] ,we have (l/c)f = ((1/c)p o I-]) o ( I o q) where I-1 is the inverse of I under o . For suitably chosen I and c, q* = I o q is normed and f * = (1:c)f is monk whence p* = (1ic)po1-l is also monic. Let p* = xrn+clx"-l+ . . ., q* = x"+a1x"-l+ . . . +a,-,x, f * = xnrn+blxnm-l + . . ., then f* = p* o q* yields bj = maj+ wj-,(al, . . ., aj-,),
j
=
1, 2, . . ., n - 1,
where w,, = 0 and wi(xl, . . ., xi), i = 1,2, . . ., n-2, is some polynomial over the ring of rational integers. Thus bj E K implies ajE K, j = 1,2, . . ., n- 1, whence q* EK[x]. Assume that p* 4K [ x ] , then p* = u+v where [v]-= [u], u E K [ x ] ,v 6 K [ x ] , and the first coefficient of v is not in K. Hence f * - u o q* = v o q* and the left-hand side is in K [ x ]while the right-hand side is not, contradiction. Thus p* f K[x] and f = cf* = (cp*)o q* whence [cp*]= [ f ] or [q*]= [ f ] .Therefore [ p ]= [ f ] or [q]= [ f ]and f is indecomposable over L [ x ] . 2.22. Corollary. Let L be the algebraic closure of K. Then the standard solutions p , q, r, s of (2.1) in K [ x ]are just those standard solutions of (2.1) in L [ x ]which are polynomials of K [ x ] . 2.23. Remark. Corollary 2.22 shows that we can restrict ourselves to algebraically closed fields K when solving (2.1). 2.3. A further reduction of the problem of solving (2.1) can be reached by classifying the solutions in the following way: Letp, q, r, s be a standard solution of (2.1) and I EK[x] an arbitrary linear polynomial. Then p o (qo I ) = r o ( s o I). There exist unique linear polynomials ml, m2 E K [ x ] such that m l o q o I and m2oso I are normed. Let g = (pom;')o (ml o q o I ) = (r o mL1) o (m, o s o I). Then there exists a unique linear polynomial u E K [ x ]such that u o g is normed, hence u o p om;', m, o q o I, u o r o m;', m2 o s o I is also a standard solution of (2.1). Any solution of that kind will be called conjugate to p , q, Y, s. "Being conjugate" is an equivalence relation on the set S of all standard solutions of (2.1) and thus yields a partition of S. Our problem will be solved as soon as we know a full system of representatives of this partition.
140
COMPOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
CH.
4
If q = s, for a standard solution p , q, Y, s, then, by Prop. 1.25, p = r. Any solution of the form p , q, p , q will be called a trivial solution. Each class of pairwise conjugate standard solutions consists either of trivial solutions or it contains no trivial solution. Of course, only the non-trivial solutions are of interest to us. 2.4. Our next aim is to derive two special classes of standard solutions of (2.1). 2.41. Let zbe any prime, ,p an integer, 0 i e i z,and t E K [ x ] monic. Then xn o ( x g t ( 9 ) )= (xet(x)n)o 9. Thus if xnt(xq)is indecomposable, is a standard solution of then p = 9, q = xet(xn),r = xet(x)", s = (2.l)-the indecomposability of x?tt(x):follows from Th. 1.34 b). A standard solution of this type is called a power solution. For t = I , we get the power solution p = 9, q = xe, r = xe, s = xn, zbeing a prime. 2.42. Remark. So far no handy method could be developed to decide whether or not a polynomial x e t ( 9 )is indecomposable. Clearly if e + z [ t ] is a prime, x e t ( Y ) is indecomposable, and the equation x2((x3)4+(x3)>') = (x(x6 x')) o x2shows that otherwise the polynomial can be decomposable.
+
2.43. Let m be an integer and p any real number. Then de Moivre's equation (cos pi+i sin p)m = cos mp+i sin mp implies, for m 0,
(2
cos mcp = (cos ($I)"
+ (7)(cos
(cos p)"-2 ( 1 -cos2 p)+
94"-4
( 1 -cosZp)'-
. . . = t,(cos
p)
where t, is a well-defined polynomial of degree m over ther ational integers, the so-called Cebyshev polynomial of the first kind of degree m. We first sum up some well-known results on cebyshev polynomials which we will need later on : 2.44. Lemma. (i) Zf p ,P a 0, then t,, = t,, o t , . (ii) u p a 0, then t , o ( - x ) = (- 1)''tt,. (G) If p == 0, then t, satisfies the d@erential equation p2(1- t:) = = (1-x");.
Proof. (i) t,,(cos p) = cos pvp = cos p(vp) = t,(cos ~ 9 ; = ) t,(t,(cos p)). Since cos ranges over an infinite set, we conclude t,,(x) = t,(t,(x)).
02
STANDARD SOLUTIONS O F p o q =
(ii) t,(-cosy) = t,(cos (n-p))
( - l), t,(cos sj). Hence t,(
-x)
= cos
ros
p(n-p) = (-1)”
141
cos pp =
= ( - l), t,(x).
(iii) Differentiation of cospup; = t,(cos p) yields psinpp = tL(cos p) sin up;. Hence p2( 1 - tE(cos p;)) = (1 -cos2 p) t:(cos p) whence pZ( 1 - t;) = (1 -X2)t;.
2.45. Remark. Part (i) of Lemma 2.44 shows that t, is indecomposable[if and only if p is a prime. Moreover, t, G t, = t,.o t,. Thus if p, v are primes and I,, l,, u are suitably chosen linear polynomials, then p = u o t , o l;l, q = 1, G c,, r = u o t, o 1;l. s = o t , is a standard solution of (2.1). Such solution is called a Cebyshev solution. 2.46. Theorem. Every non-trivial standard solution of the equation p o q = r o s is conjugate to some power solution or to some cebyshev solution. Proof. Theorem 2.46 will result from a whole bunch of lemmas, the proof of which will fill the remainder of this section. .5. Proposition. Let p , q, r, s be any non-trivial standard solution of (2.1). Then [q] = [ r ] = p, [ p ] = [s]= v where (p,v) = 1. Proof. By hypothesis, we have p c q = r o s = f. Hence, by Prop. 1.25, q f s while Lemma 1.33 implies K(q) f K(s). Then the proof of Th. 1.34 shows that K ( f ) c K(q) c K ( x ) and K ( f ) c K(s) c K ( x ) are maximal chains of fields from K ( f ) to K(x).Thus K ( f ) = K(q) f’ K(s) and K ( x ) = K(q,s). By ch. 6, Th. 5.84, [K(x):K(q)] = [ K ( s ) : K ( f ) ] ,hence, by ch. 6, Th. 5.71, [q] = [r]whence [ p ] = [s].Furthermore ch. 6, Th. 5.84 b), implies ( [ q ] ,[s])= 1 since K(q, s) = K(x). 2.51. Lemma. If p o q = r o s = f, then the minimal polynomials in the indeterminate y of the elements x over K( f ), K(q), K(s), q over K( f ) , s over K( f ) are the polynomials f ( y )-A q(y)-q, s(y)-s, p(y)--f, and r ( y )-“L respectively. The polynomial p ( y )-f remains irreducible over K(s), and r ( y )-f remains irreducible over K(q). Proof. The statement for the first three polynomials follows from ch. 6, Th. 5.71, while [K(q): K ( f ) ]= [ p ] , [K(s): K ( f ) ] = [r]shows that p(y)-f and r ( y ) -f are the minimal polynomials for q and s, respectively. Since
142
COMPOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
CH.
4
K(s) (q) = K(x) and [K(x): K(s)] = [s] = [ p ]by Prop. 2.5, p(y)-f must be irreducible over K(s). Similarly r(y)-fis irreducible over K(q).
-
-
2.52. We define a binary relation on K(x) by: u v means u = ev where e = 1 or - 1 . Clearly, is a congruence on the multiplicative semigroup of K(x).If K ( f ) is an extension field of K ( g ) and h E K ( f ) , then, as in ch. 6 , § 5.88, MfI,(h) will denote the norm of h with respect to this extension.
+
2.53. Lemma. I f c E K , then
rnxlf(X-c)
-
laz,,,(x-c>
f-f(c>,
m, , f(4 c) m, s(4 -c) -
-
-
f-p(c), f-p(c),
-
q-q(c),
-
faz,~,(x-c)
-
as 1 f(S -c> J -rG-37 mx,(s c) f- @).
,
-
s-s(c),
(2.51) (2.52) (2.53)
Proof. Since K(x-c) = K(x), we have [K(x-c) : K ( f ) ] = [ f ] , hence the minimal polynomial of x -c over K(f)is f ( y + c) -5 and the constant term of this polynomial is f(c) -f. Thus 6‘ZXif(x-c)- f - f ( c ) . Similarly all the other relations of (2.51)and (2.52)are obtained. Since K(s) ( 4 - c ) = K(s) (q), we have [K(s)( q - c ) : K(s)] = [ p ] , by Prop. 2.5, hence the minimal polynomial of q -c over K(s) is p ( y c) -f. Thus 02, s(q-c) f-p(c). Similarly the last relation follows.
,
+
-
Proof. Applying the chain rule to f = p(q) = ~ ( s ) ,we get f’= p’(q)q’ = r‘(s)s’. Hence mxI &f’) I ,(p’(q)) 02,I ,(q’) p’(q)’ 02, I &’) and mx1 ,(f’)= ax,(r’(s)) (s’) (72, lf(r’(s))laz, ,(s’). The last relation holds since K is algebraically closed whence we may write r’(s) = p n ( s - c i ) , c, E K. By Lemma 2.53, CaZ,,,(r’(s)) p p ~ ( a z x l , < s - c i ) p.”nmsIf(s-ci) caZSlf(r’(s))whence the last relation follows. (2.54) is then an immediate consequence; (2.55) is obtained in a similar way.
mx
mx,
-
-
-
-
-
2.55. Lemma. The linear polynomial x-ai &ides q-a
-
-
if and only if
6Yxl,(x-ai) q-a, and x - a , divides s-a if and only if caZ,,,(x-ai)
STANDARD SOLUTIONS OF p o q =
02
143
ros
s -a. Furthermore q -ai divides f -a E K [q]i f and only if OZq, (q-a,) f - a , ands-a, divides f - a E K [ s ] ifandonly ifOZ,,,(s-ui) .- f - a .
-
-
Proof. If (x-a,)/(q-a), then q(ai) = a, therefore q-a = q-q(ai) M,, &x-u,), by Lemma 2.53. Conversely OZ, ,(x-a,) q --a implies q-q(a,) = q-a, by Lemma 2.53, hence ( ~ - a , ) / ( q - a ) . In all the other cases, a similar argument can be used. 2.56. By Lemma 2.53 and Lemma 2.55, there is, for each linear factor q -a, of p‘(q), a unique linear polynomial f-kiE K [ f ]which is divisible
by q-a,. This polynomial f-ki is called the linear factor corresponding to q-a,.
2.57. Proposition.Let p =- Y. Then the set of linear jactors couresponding to the linear factors of p ‘ ( q ) consists of at most two dgerent polynomials.
,
Proof. Let q-a be any linear factor of p’(q), w its multiplicity in p’(q), and q’ = p n ( x - b i ) . Then Oil,, ,(q‘) ppnOZx &X -bi). By Lemma 2.55, the multiplicity of q-a in caZ,,,(q‘) equals the number ZI of factors x - b , of q’ such that (x-b,)/(q-a). Hence the multiplicity of q-a on the lefthand side of (2.54) equals pw+u..-Similarly the multiplicity of q-a in OZ, ,(d) equals the number t of factors x-ci of s’such that (x -ci)/(q -a). The unique linear polynomial f -k E K [ f ] which is divisible by q-a is f-p(a). Since (dldq)(f-p(a)) = p‘(q), the multiplicity of q-a in f - p ( a ) is w + l . If r‘(s) = p n ( s - d , ) , th en MSlf(r’(s)) ppnM,,,(s-di), hence, by Lemma 2.55, the multiplicity off -p(a) in OZ, f(r’(s))equals the number u of factors s-d, of r‘ such that (s-d,)/(f-p(u)). Therefore the multiplicity of q-a on the right-hand side of (2.54) equals u(w+ 1)+t. Hence pw+v = u(w+ 1)+ t. Moreover [r’]= p- 1 implies u == p- 1 < p, hence p-2u-t = pw+v-p(w - 1) -22.4 --21 -t -2
,
-,
=
u(w+1)-p(w-1)-2u-v
= (u-p)(w-l)-ZI
SZ
0,
therefore p == 2u+ t . Suppose now that (q-ai)/(f-ki), i = 1 , 2, 3, where k, f k, # k3# k,. Then we obtain the inequalities p 6 2ui+ ti, i = 1, 2, 3, where ui, ti stand for u, t if we substitute ai for a. Hence 3p e 2(u,+u2+u,)+ f t,+t,+ t,. But by definition of ui, ti, we have u,+u,+u, =s p- 1 , tl+t,+t3 =sY-1 -= p-1, thus3p -= 3 ( p - l ) , contradiction.Thisproves the proposition.
144
COMPOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
CH.
4
2.58. Proposition. If p =- v and the linear factors corresponding to the linear factors of p’(q) are all equal, then the solution p , q, r, s of (2.1) is conjugate to some power solution. Proof. Letf-k be the common linear factor corresponding to the linear If w i2 0 is the mulfactors of p’(q) and f - k = (q-al)‘I . . . (q-a,)‘v. tiplicity of q -a, in p’(q), then, since f ’ = p’(q), we have ti = wi+ l .
Thus v = and
V
V
i=l
i= 1
C ti = v+ C w, = v+(v-
1) hence v
=
1, therefore t , = v,
p ( q ) = f = (q-a)”+k.
(2.56)
If v were not a prime, say v = m, then f = ((q-a)“)“+k whence K(f)c K((q-a)”) c K(q), contradiction. K(x) = K(s)(q-a) implies that K ( x ) is normal over K(s) since, by (2.56), q - a is a root of y’-(r(s)-k) E K(s) [ y ] and K contains the v-th roots of unity whence K(s)(q-a) is a splitting field of y”-(r(s)-k) over K(s). Furthermore, by a well-known theorem of Galois theory, the Galois group of K(s) (q-a) over K(s) is cyclic, thus K ( x )is a cyclic extension of K(s). By ch. 6, Th. 5.71, the polynomial s(y)-s E K(s) [ y ]is irreducible over K(s) and has the root x in K(x), thus s(y)-s splits completely into linear factors over K(x).Therefore s(y) - s has v different roots vi, i = I , 2, . . .,v, in K(x).Since K(v,) = K(x),i = 1, . . .,v, a well-known theorem on transcendental extensions implies that vi = (bix+ci)/(dix+e,),bi, ci,di, e, E K. But s(vi)=s(x),hence if we substitute for v,into this equation and multiply by a suitable power of dix+ei, then we see that (dix+ei)/(bix+ci)”, and we conclude that vi is a linear polynomial of K [ x ] , i = 1 , . . ., v. Now let w be a v-th root of unity, G a generator of the Galois group of K ( x ) over K(s), and
I = x+o)(ax)+W2(o2X)+ . . .
+CO-1(O”--lX)
the Lagrange resolvent. Since the o’x are just the vi and since there exists w such that K(s) ( 1 ) = K(x), we have 1 = bx+ c where b, c E K, b # 0, for such to. Furthermore I” E K(s) implies I” = gos, for some g E K [ x ] , by ch. 6, Lemma 5.81. Comparing the degrees in the last equation, we see that [ g ] = 1 . Thus s = m o x” o 1 where m, I E K [ x ] are linear, and (2.56) implies that p = ml o x”oI I where ml, l1 E K [ x ] are linear.
82
STANDARD SOLUTIONS OF p 0 4 =
145
r 0S
By substituting into (2.l ) , we obtain ml o x”o 11 o q = r o m o xv o I, hence, i f 0 # c E K , then ( m ; l o r o m ) o x ” = ( x v o c x ) o ( ( l / c ) x o l l o q o I - l )thus , ( ( l / c ” ) x om c l o r o m ) ox” = xuo ((1Ic)xoIl o q o Z-l). We choose c in such a way that the polynomial in the bracket on the right-hand side and therefore also the polynomial in the bracket on the left-hand side becomes monic. Then the last equation becomes x ’ o q = FoxU (2.57) where 4, F E K [ x ] are monic. Let q = xfi alx”-l . . ., F = x” blx’“-l + . . ., then (xfi+alxfi’-l+. . .)” = x””+blx/“-”+ . . .. Coefficientwise comparison yields a, = 0, hence a, = 0, . . ., hence a = 0, similarly a,,,, = = 0, a”+, = 0, . . ., u , , + ~= - ~0, etc. Thus q = xp+avX-u+a,,x’‘-2‘+ + . . . +akyxp-kv where 0 -= p - k v < v. If we set y - k v = p, then q = xQf(x”), t E K [ x ] .If we substitute into (2.57),then X’ o xef(x”)= Fox’ whence Fox” = xet(x)”o x”, thus F = xef(x)”,by Prop. 1.25. Going back to the definitions of q, F, we now obtain
+
+
+
$,-,
s = mox”o1, q = (l;10cx)0401,
r = (m, o cvx)o T o m-’, p = m , o ~ ” o /=, ( m l o c Y x ) o x v o ( ( l / c ) x o / l )
and the proposition now follows from 5 2.3. 2.6. By Prop. 2.57, we have now to investigate the case where the set of linear factors corresponding to the linear factors of p’(q) consists of two different polynomials. We state Hypothesis (H) : f -k , f -k l are the linear factors corresponding to the linear factors of p‘(q). The assumption p =- v will not be needed.
2.61. Lemma. Zf every linear factor of f - k E K [ q ] is a divisor of p‘(q), then f - k l has at least three dgerent linear factors that are not divisors o f P’(4). Proof. By hypothesis, f - k = [q-al)”l+l . . . (q-am)”m+l where a, # a,, for i # j , and c, =- 0, i = I , . . .,m, since (f -k)’ = p’(q). Thus the greatest common divisor d of f - k and p‘(q) is of degree [ d ] =
nt
1 ul.
i=l
If
v, = 1, for i = 1 , . . ., m, then f - k would be the square of some polynomial g E K [ q ]whence K(f)= K ( f - k ) = K[g2),therefore K ( f ) c K [ g ) c K(q), thus K ( g ) = K(q) and [ g ] = m = 1 . This would imply that f - k = (q-ul),, hence [ p ] = 2, [ p ‘ ] = 1 . But the only linear factor ofp’
146
COMPOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
cannot divide f -k and f furthermore m+
k l simultaneously, by 52.56. Hence
-
m
m
i=l
i=l
C vi= [ p ] = v, therefore
vi
CH.
4
m
C vi z m,
i=l
=- (1/2)v. Let dl be
the greatest common divisor off-kl and p‘(q). Then d and dl are relatively prime since f -k and f-kl are, hence (ddl)/p’(q),thus [ddl] =S v- 1. This implies [dl] =sv- 1 - [ d ] vj2- 1. Let f - k , = (q-cJW1+l . . . (q-cn)w*+lwhere ci f cj, for i i j , then n
[d,] =
i=l
wi, therefore
C wi n
v/2- 1. Hence n = v-
n
i
i=l
i=l
If I is the number of indices i such that wi = 0, then v whence 1 z=2n-v =- 2, Q. E. D.
w i =- v/2+ 1. I+2(n-l)
2.62. Lemma. I f the polynomial f - k E K[s] contains at least one linear factor s-b which is not a divisor of r‘(s),and the polynomial f - k l E K [ s ] contains at least one such linear factor s-bl then each of these polynomials contains exactly one such linear factor, the polynomials satisfy the equation pz(f-k) (f-ki) = (5-b) (s-bi) ~‘(s)’,
and every linear factor of r’(s) is either a divisor of f-k or of ,f-kl, but not all the linear factors of r’(s) divide one and the same of these polynomials.
-
Proof. By hypothesis (H) we see that, b%n,l,(p‘(q)) = l%,I,(vn(q-ai)) v”(f-k)U (f-kl)”l where v + v l = v- I and w1 f.0. Hence, by (2.55),
mx, LS’)
r’(s>y
-
v”(f-k)U (f-k,)”’
mx,(4’). I
(2.61)
Now let g be the product of all linear factors off-k, and gl be the product of all linear factors of f-kl, which do not divide r’(s), then, ’ ) . by (2.51), laz,,,(s‘) is a polynomial in s of by (2.61), g ’ g ~ / ~ x l s ( sBut, degree v - 1 whence [g] = [gl] = 1 , and the first assertion of the lemma is proved. Letf-k = (s-b) (s-c1)e1 . . . ( S - C ~ ) ~ ” ’be the decomposition into linear factors of f - k C K[s], then ei a 2, i = 1, . . ., m, hence p =1
+ C ei z=1 +2m, thus m ==( p - 1)/2. Since C (ei- I)+m
we have
m
m
i=l
i=l
m
1=i
= p-1,
(ei- 1) 3 ( p - 1)/2, therefore the greatest common divisor of
STANDARD SOLUTIONS OF p o q =
02
r os
147
r’(s) and f -k is of degree 2 ( p - 1)/2. Similarly, the greatest common divisor of r’(s) and f - k l is of degree (p-1)/2. But since f-k and f - k l are relatively prime, so are those greatest common divisors, thus their product divides r’(s).But [r’] = p - 1 whence each of these greatest
common divisors is of degree (p- l)/2. Hence
m
i=l
(ei- 1) = ( p - l)/2,
implying that rn = (p- 1)/2, ei = 2, i = 1, . . ., m. Thus f - k = (s-b) (s-cl)’ f - k l = (s-b,)
Since(s-c,)
(S-C~J’
... (S-C~)’, . . . (s-cmJ2.
(2.62)
. . . (s-cc,,) (s-cll) . . . (s-cml) is a divisor of r’(s)of degree
p- 1, we have
r’(s) = p(s-cl)
. . . (s-cm)(s-cll) . . . (s-cml)
(2.63)
which proves the second assertion of the lemma. The third assertion is a straightforward consequence of (2.62) and (2.63). 2.63. Remark. For proving (2.62) and (2.63), we required only the first statement of Lemma 2.62, but not Hypothesis (H). 2.64. Lemma. If f-k, f - k l E K[s] satisfy the hypothesis of Lemma 2.62, then each of the polynomials f-k E K[q] and , f-kl E K [ q ] contains at least one linear factor that is not a divisor o f p’(q). Proof. We evaluate 02,f ( ~ ’ ( s )using ) , (2.621, (2.63), and Lemma 2.55, and then substitute into (2.54):
Suppose that every linear factor of f-k divides p’(q), then, by Lemma 2.61,f-kl has at least three different linear factors that are not divisors of p’(q). Let g be the product of these linear factors, then, by (2.64), g(p-l)iz/OZx,,(q’). But this is a contradiction since [ g ]= 3 whereas [OZx ,(q’)] = p - 1. A similar argument works forf- k,. 2.65. Lemma. I f the polynomials f - k E K [ s ] , f - k l E K[s] satisfv the hypothesis of Lemma 2.62, then the polynomial f -k E K [q] contains exactly one linear factor q -a that does not divide p’(q), and f -kl E K [ q ]
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COMPOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
CH.
4
contains exactly one linearfactor q -al that does not dividep'(q). Moreover there are elements d # dl E K such that p2(q-u)(q-al)
= (x-d)(x-d1)q'(x)'.
Proof. By the last statement of Lemma 2.62 and Lemma 2.64, we see that hypothesis (H) and the hypothesis of Lemma 2.62 is also satisfied for the solution r, s, p , q of equation (2.1), i.e. the statements of Lemma 2.62 hold if r , p and s, q change their places in the hypothesis. Hence Lemma 2.62 implies thatf-k contains exactly one linear factor q -a that does not divide p'(q), and f -kl contains exactly one linear factor q -al that does not divide p'(q). By (2.64), (q-a)(p-l)'' (q -a1)(J+l)/2/(azx l q(4 ') whence OZx,q(qr) p"(q (q -a1)(p-')''. (2.65)
-
If q-a = (x-tl)'"l'' . . . (x-tJwn+l is the decomposition of q-a into linear factors of K [ x ] , ti z tj, for i # j , then Lemma 2.55 and Lemma 2.53 imply f-k
-
OZqlf(q-a)
-
(azxl,(q-a)
-
(s-s(tl))wl+l . . . (s-s(t,,))wiL+l.
But then (2.62) implies wi =s1, i = €, . . ., n. By (2.65), q'(x) has exactly ( p - 1)/2 = m linear factors which divide q -a and MI,s 1 implies that all these linear factors are different. Hence wi = 1, for exactly ( p - 1)/2
indices i. Thus
q-a = (x-d)(x-el)2 4-a'
= (x-dl)(x-ell)2
. . . (x-em)', . . . (x-e,,J
(2.66)
A similar argument as in Lemma 2.62 completes the proof. 2.7. Proposition. If hypothesis ( H ) and the hypothesis of Lemma 2.62 are satisjied, then the solution p , q, r, s of (2.1) is conjugate to some debyshev solution. Proof. We require two lemmas which will be proved first. 2.71. Lemma. Let
(T
be any positive integer. Then the dflerential equation
*
a2( 1 -z') =
(1 -x2)z"
(2.71)
has the solutions z = t,(x), and these are the only solutions which are polynomials over K of degree (T.
STANDARD SOLUTIONS OF p o q =
52
r os
149
Proof. Suppose that z = aoxu+alx“-l+ . . . +a, is a solution of (2.71). From substituting z into (2.71), we get for i = 1, 2, . . .,Q,
-a~(2aoai+w(a,, . .., q-,)) = -(20ao(o-i)ai+v(uo,
a,,
. . ., ui-,))
where w(xl, . . ., xi-,) and v(xo, . . ., xi-,) are quadratic forms over the rational integers. Hence 2oiaoai=f (ao,. . .,ai-J wheref(xo, xl, . . .,xi-,) is some quadratic form over the integers which is independent of z. Therefore
2oi(ai/a0)= f(1, a,/ao, . . .,ai-,/ao),
i = 1, 2,
. ..,Q.
Hence ai/ao= I.;, i = 1, . . .,Q, is a rational which does not depend on z. Thus, if t is any polynomial of degree Q which is a solution of (2.71), then every polynomial z of degree Q which is a solution of (2.71) satisfies z = ct where c E K . We substitute z = ct into (2.71) and obtain 02(1-cC2t2)= (1 -xX2)c2t’2= c202(1- t2),hence c = & 1. Conversely z = t is a solution of (2.71). Moreover, z = t,(x) is a solution of (2.71) by Lemma 2.44 (iii). 2.72. Lemma. Let Q be anypositive integer, u, u1, v,v1,u f U I , v # v1, any elements of K and y = y(x) a polynomial of degree c over K which satisfies the dgerential equation o y y -u) ( y - u1) = (x-v) (x -v1)y12.
(2.72)
Then there exist linear polynomials I = ax+b, m = cx+d such that z = l o y o m is a solution of (2.71). Proof. By (2.72), oz(ysrn-u)(yom-ul) = (m-v)(m-v1)(y’orn)2. Since z‘ = a(y‘om)c, we have a2c%2(1-1 o z-u)
(1-1 o z-u,)
= (m-v) (rn -v1)zf2.
Now we choose m, 1suitably, namely
m = $(v1-v)x++(v1+v),
1-1
= +(ul-u)x+$(ul+u).
Then the last equation becomes Q2[(Vl
-vI2/(u1 =
-421
1 &1--2))
gu1-u) (z+ l).+(ul-u) (z- 1) =
( x + 1).+(211--2))(x-l)2’2
whence z is a solution of (2.71).
(2.73)
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COMPOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
CH.
4
2.73. Proof of Proposition 2.7. By Lemma 2.62 and Lemma 2.65, pY f -k) ( f -kd = (s - b ) (s - b d r'(sI2, (2.74) p2(q-a) (q -ad = ( x -d>( x -dd q'(x)* The proof of Lemma 2.65 shows that hypothesis (H) and the hypothesis of Lemma 2.62 are also satisfied for the solution Y, s , p , q of (2.1), thus
(f-kl) = (4 -a>(4-a11P W 2 9 (2.75) v2(S-b)(S-bl) = (X-d)(X-d1)s'(X)2. That a, a, and b, bl actually change their roles in (2.74), (2.75), follows from Lemma 2.62 and Lemma 2.65. That d, d1 in (2.74), (2.75) are the same follows since x - d , x-dl are the only simple linear factors of q-a and q-a,, resp., by (2.66), and q-a, q-al are the only simple linear factors of f - k , f - k l , resp., by Lemma 2.65 whence, in both cases, x - d , x - d , are the only simple linear factors in K [ x ] off- k , f- k,, resp. Now (2.74), (2.75) are differential equations of the type as in (2.72), hence, by Lemma 2.72 and Lemma 2.71, r(s) = Ic'o Elt, o my', q(x) = I;lo E2t, o m;', y 2 ( f -k )
p(q) = I;,
s(x) =';I o c4tvo m; 1, i = l ) ..., 4. (2.73) implies m;' = mT1 = n, I;, = m3, 1;' = m,. Since too E , X = = ( E ? X ) o t, by Lemma 2.44 (ii) we obtain s = mlo&qxotyon, q = m30E2xoEpon, r = 1;' o clxo cix o t, o E ~ oXm;', p = Ii1o ~ ~ ~ xi x t,o oo F ~ oX m;', o e3tvom i l ,
Ei=&l,
and the proposition follows.
2.8. The proof of Th. 2.46 has now boiled down to showing that hypothesis (H) implies the hypothesis of Lemma 2.62. Throughout this subsection, we will assume that p > Y and hypothesis (H) holds. 2.81. Proposition. Hypothesis (H) and p r v imply that the hypothesis of Lemma 2.62 is satisfied. Proof. We will again require a few lemmas to prove the proposition. 2.82. Lemma. I f c E K , ( q - a ) l ( f - c ) , ( s - b ) / ( f - c ) , then the polynomials q -a, s - b c K [ x ] have at least one linear factor in common. Proof. Letf-c = (s-b)mfl wheref, is not divisible by s - b . Coprimality of s - b and fi then implies u(s-b)+vfl = 1, for some polynomials
STANDARD SOLUTIONS OF p o q =
02
r os
151
u, v E K [ s ] ,hence s ( x ) - b and f i ( s ( x ) ) are also relatively prime in K [ x ] . If q-a and s - b were relatively prime, we would have (q-a)/fi(s(x) whence f i ( s ( x ) ) = (q(x)-a)h(x), for some h(x) E K [ x ] . Hence ~ x ! , ( f l ( s ( x )=) )~ ~ , ~ ( q ( x ) - a ) ~ ~ , Lemma ~ ( h ( x2.53andLemma2.55 )). imply fi(s)y (f-c)g(s), for some g(s) f K[s] whence (s-b)lfi(s), contradiction.
-
2.83. Lemma. Let f - k =
rn i=l
(q-a,)”( =
IT (s-bj>”j where n
the a, and
j=1
bj, resp., are pairwise distinct elements of K . If u, =- 1, then q-a, and s‘(x) have some divisor x - c , in common. r f vj =- 1, then s - b j and q‘(x) have some divisor x -cj in common. Proof. Let x-c, be a linear factor of q-ai of multiplicity gi, then x-ci is a linear factor of f - k of multiplicity g,ui. There is one and only one index j such that (x-ci)/(s--bj). If eij is the multiplicity of x - c , as a factor of s-bj, then x-c, has the multiplicity eGvjas a factor of f - k . Suppose that q-a, and s’(x> are relatively prime, then eii = 1 , hence vj = g,ui. By Lemma 2.82, every s - b j has some linear factor in common with q-ai whence every vj is divisible by ui. Therefore f - k = h(s)”(,for some h(s) E K [ s ] . This implies K ( f ) = K(h”*)c K(h) E K(s). Hence K(h) = K(s), [h(s)]= 1. Therefore ui = [r] = p. But f = k + n ( q - a i ) ” , and [q] = p imply pv = [ f ]3 pz which contradicts p =- Y. Hence q-a, and s’(x)have some linear factor in comaon. T o prove the second assertion, we proceed in an analogous way to before and get f - k = h ( q p , for some h(q)EK[q]. Again [h(q)]= 1 and vj = [ p ] = Y. Hence f = k f (q-aa)”, thusp’(q) = v(q -a)’-’ which contradicts hypothesis (H). Hence s - b j and q’(x) have some linear factor in common if vj =- 1. 2.84. For the remainder of this subsection, we now assume that every linear factor of f - k f K[s] is a divisor of r‘(s).This will finally lead to a contradiction and thus will prove Lemma 2.62 so far asf -k is concerned. But also the statement about f - k l will follow for reasons of symmetry. 2.85. Lemma. Let f - k E K[s]have n dgerent linear factors, then n -= p12. Proof. By hypothesis, every linear factor off- k E K [ s ] is a divisor of r‘(s), thus every such linear factor has multiplicity vj =- 1 in f - k . Since
152
COMPOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
2=1 vj = p, we have n ==p/2. If n = 1.12, then vj n
j
=
CH.
4
2, for j = 1, . . ., n,
whencef - k = h(s)2for some h(s)E K [ s ] .As in the proof of Lemma 2.83, we would have p = 2, hence v = 1, contradiction. Hence n < p/2.
2.86. Lemma. Let m be the number of all linearfactors of r’(s) which divide f - k l . Then m -= p/4.
Proof. The number of all linear factors of r’(s) dividing f-k n
(vj-1) = p-n.
equals
By Lemma 2.85, p - n = (p/2)+j where j
0.
j=1
Hence, by Lemma 2.55,
-
M S ,r’(s)) (f- k ) ( p I 2 ) + j (f-kJ“ f(
h(f)
(2.81)
where h ( f ) and ( f - k ) ( f - k l ) are relatively prime. By (2.54), p’(q)”mxI ,(q’)
N
(f-k)‘”’2’+i(f-kl)mh(f)faZx1 ,(s’)-
(2.82)
Hence, if (q-a)/(f-k) but q-a does not divide p’(q), then (q - L Z ) ( ~ ’ ~ ) + ~ / ,(q’). ( ~ Z ~ , Since [q’] = p -.l, there is at most one such q -a. Suppose that e different factors q -bi o f f -k, are not divisors ofp‘(q). By (2.82), each of these factors occurs in MXI ,(q’) with multiplicity m. If t is the multiplicity of q-bi in Mxlq(q‘), then exactly t of the linear factors of q‘(x) are divisors of q -bi. If x-c is such a linear factor and v its multiplicity in q’(x), then (x-c)/(f(x) - k I ) . Since ( f ( x ) - k l ) ’ = p’(q(x))q’(x), we see that x-c has multiplicity in the greatestcommon divisor d off(x)-kl and q’(x). Thus d is divisible by some product of linear factors of q-bi which has degree t a m. Hence [ d ]a em. On the other hand, (2.81) implies that p-n+m = (p/2)+j+m s p-1, thus n a m+ 1. Then, by Lemma 2.83, there are at least m+ 1 different linear factors of q’(x) which are divisors of f ( x ) -k. Hence [d]==p-m-2, thus em == p-m-2. Therefore m(e+ 1) -= p. By way of contradiction, assume now that m ==p/4. Then (p/4)(e+1) -= p whence e s 2. Suppose that f-k has no linear factor q-a which is not a divisor of p‘(q), then, by Lemma 2.61, e* 3. Hence there is exactly one q--a dividing f - k but not dividing p’(q). Then (2.82) implies that (q -a)(p’2)+j/OZx,(q’) whence (p/2)+j linear factors of q’(x) divide q--a and therefore also .f(x)-k. Hence [d]=sp- 1 -(p/2)-j < p/2. This implies em -= p/2, thus e(p/4) -= p12 and e & 1 . If e = 0, then, by Lemma 2.61,f-k would
02
STANDARD SOLUTlONS OF p o q =
r os
153
have at least three different linear factors q -a not dividing p’(q), contradiction. Thus e = 1. We summarize: There is exactly one linear factor q-a of f - k that is not a divisor of p’(q) and exactly one linear factor q-iil off-kl that is not a divisor of p’(q). Remark 2.63 implies that f-k
f-k,
=
( q - ~ ) ( q - a ~. .) .~(q-a,)’,
- .-
= (4 - a,)(q
P’(4) =
y(q
-4
* * *
(2.83)
(4 -a,J2,
(4 -a&q - a 3
.
* *
(4 - a d
where 1 = (v - 1)/2.By Lemma 2.83, each q -ai has some divisor x -ciand each q-ail has some divisor x-cil, which is also a divisor of s’(x). Since all these divisors are different we see that s’(x) = v(x--cl) ... (x -cl)(x -ell) . . . (x -cn) and therefore
-
6-2x,q(s’(x))
”%-a,)
- -
k-a,)(q-a,J
.. -
(q-a,J
(2.84)
Substitution of (2.83) and (2.84) into (2.82) yields, after computing the exponents of q-all on either side of (2.82), the inequality 2m+ 1 2 p whence m a (p-1)/2. On the other hand, (2.81) implies that (p/2)+j+m e p- 1 whence m < ( p - 1)/2, contradiction. Hence m -= p/4. 2.87. Proof of Proposition 2.81. It remains to lead the assumption made
in Q 2.84 to a contradiction. By hypothesis (H), there is at least one linear factor q -a off- k which divides p’(q). Therefore q -a has multiplicity u =- 1 in f-k. By Lemma 2.83, there is a linear factor x-c dividing q - a and s’(x) whence the degree of the greatest common divisor off-kl and s’(x) is at most v-2. If v = 2, then [p’] = 1 which contradicts hypothesis (H). Hence v =- 2. Thus we can find an integer g which is maximal with respect to g(v-2) =sp, and p > Y implies that g a 1. Let d 2 1 be the number of different factors q-bi of f-k, which are divisors of p’(q) and wi their multiplicities in p’(q). Then (2.82) implies that
pwi+ui = m ( w i f l)+ti
(2.85)
where ui,t i are suitable non-negative integers. Hence (p-m)w, = m+ti-ui. By Lemma 2.86, (3/4)pwi (p/4)+ti-ui. Since w i a 1, we have ti =- 0. Suppose that ti =- plgd, for every index i, then C ti w p/g. But ti is just the number of linear factors of s’(x) dividing q-bi. If e, is the multiplicity of suchalinearfactorx- c in s’(x), then (x -c)/Cf(x) -kl)
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COMPOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
CH.
4
and (,f(x)-kl)’ = r’(s(x))s’(x) implies that x-c has multiplicity v in the greatest common divisor off(x)-kl and s’(x). Hence this greatest common divisor is divisible by the product of these ti linear factors of s’(x), and thus has degree 2 ti. Therefore
c ti
=sv - 2 .
(2.86)
This implies p/g < v-2, contradicting the definition of g . Hence there is at least one indexjsuch that ti s p/gd. By (2.85), pw, =sm(wj+ 1)+ ti e m(wj+ 1)+ plgdwhencebyLemma2.86, p/4=-m~(p/g(wj+l))(wj- l/gd). Hence wj+ 1 4wj-4/gd, thus 3wj < 1 +4/gd & 5. Therefore wj = 1. Thus 3 < 1 4/gd whence gd < 2. This can only be the case if g = d = 1. By definition of g , this means that 2(v-2) > p while the definition of d shows that f - k l has exactly one factor q-b which divides p’(q). Since wj = 1, this factor has multiplicity 1 inp’(q), hence multiplicity 2 in f - k , . Hencef-kl has v-2 linear factors which are not divisors ofp’(q). Then, by (2.82), mx,,(q’)is divisible by at least m(v-2) linear factors whence p- 1 3 m(v-2). Therefore m(v-2) < p, hence m =s1. Substitution of wi = 1 into (2.85) yields p+ui = 2m+ti s Z i t i . But 2+ti =sv by (2.86). Hence p 4 v which is the contradiction we needed, and the proposition is proved.
+
2.9. Proof of Theorem 2.46. By Prop. 2.5, every non-trivial standard solution of (2.1) is either of the form p , q, Y, s or of the form r, s, p , q where p , q, Y, s is a standard solution of (2.1) such that [q] = [ r ] = p, [ p ] = [s] = v, and p z v. By Prop. 2.57, Prop. 2.58, Prop. 2.81, and Prop. 2.7, this solution is conjugate to some power solution or to some Cebyshev solution. 3. Permutable chains over fields and integral domains 3.1. Let R be an r commutative ring with identity and ( R [ x ] ;0 ) = S. If I R I =- 1, then S is a non-commutative semigroup, for ax o (x+ 1) # (x+ 1)oax if a # 1. The problem of determining all the commutative subsemigroups of S seems to be rather difficult. If R is an integral domain and C any commutative subsemigroup of S , then Cor. 1.23 implies that the set of the degrees of the polynomials in C constitutes a subsemigroup of the multiplicative semigroup of non-negative integers. C is called a permutable chain, or a P-chain (over R) if every polynomial in C is of
43
155
PERMUTABLE CHAINS OVER FIELDS AND INTEGRAL DOMAINS
degree =- 0, and for any k =- 0, there exists a polynomial in C of degree k. This section is devoted to determining all the P-chains over fields and various other types of integral domains. 3.2. Let K be any field, C = {g,ji E I} a P-chain and 1 E K[x] a linear polynomial. Evidently C , = {Iw1 ogi o Z j i E Z} is a P-chain, too. C , is called a conjugate (over K ) of C. Also I - l o g o I and g will be called conjugate. Clearly conjugacy is an equivalence relation on the set of all P-chains over K and therefore induces some partition of this set. Thus all the P-chains over K will be known as soon as we are able to pick one representative for each class of this partition. 3.3. Two special P-chains are already known to us: a) {x,x2,x3, . . .} = S is a P-chain over K being called the P-chain of powers. b) By Lemma 2.44 (i), {tl, t2, t3, . . .}, ti the i-th Cebyshev polynomial, is a P-chain over the field of rational numbers, thus { g, = (2x) o t, o (+x) I n = 1,2, . . .} is also a P-chain over the field of rational numbers. Let usputs, = eino,+e-inp,n= 0, 1,2,. . .,thens,s, = ( e i n p + e - i n ~ ) ( e i p + e - i p i ) = S,,+~+S,,-,, n = 1,2, . . . . Hence s , , + ~= S , , S ~ - - S ~ - ~n, = 1,2, . . ., therefore s2 = s2,-2, s3 = +3s1 (3.3) and, by induction on n, we easily see that s, = fn(s,) where.f,, is some monic polynomial of degree n over the rational integers. But the definition of s, and Euler’s formula imply that s, = 2 cos np whence 2t,(cos p) = 2 cos np = f,(s,) = ,f,(2 cos p). Thus 2t,,(x) = f,(2x), hence f , = (2x) o t,,o (+)x = g,,,therefore g, is monk. 3.31. Remark. Induction on n shows that the coefficient of normal form of f,= g,, is zero, for n 2.
9-lin
the
3.32. Let Z be the domain of rational integers, L the prime field of K, q :Z L the unique ring homomorphism, and q[x] : Z[x] 7 Z[x] the unique extension of q to a composition epimorphism (see ch. 3,s 3.22). Then {q[x]gl, q[x]gz, . . .} = T is a P-chain over Z. Since, by ch. 1, 9 8.1, 7 Z [XIcan be embedded into K[x], we see that T is also a P-chain over K being called the P-chain of Cebyshev polynomials. -f
--f
156
CH.
COMPOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
4
3.33. Theorem. Every P-chain over K is some conjugate of either the P-chain of powers or the P-chain of cebyshev polynomials. Thus there are exactly two dgerent classes of P-chains over K.
Proof. We require several lemmas which will fill the remainder of the subsequent subsection before we prove the theorem. 3.4. The second statement of the theorem on the non-conjugacy of the P-chains of powers and of Cebyshev polynomials can easily be seen: Let I = a x f b , then I - l o x " o l = (l/a)(ax+b)"-(b/a) where, for some n a=2, the coefficient of x'+' differs from zero unless b = 0. Thus if I-' G x" o I = g,, for all n, then b = 0 by Remark 3.31 whence a"-' = I , for all n. Therefore a = 1, but S # T by (3.3), contradiction. Let us now assume that K is algebraically closed. 3.41. Lemma. If char K Z 2 and g E K [ x ] is any polynomial of degree 2, then, for any n a 1 , there exists at most one polynomial of degree n that is permutable with g. If char K = 2 and g E K [ x ] is any polynomial of degree 3, then,for any n 3 1, there exists at most one polynomial of degree n that is permutable with g.
Proof. Let char K # 2, g = ax2-tbx+c, a # 0, and f =
n
a$-'
of
1=0
degree n and permutable with g, i.e. g of = f o g . Then equating the coefficients of x~~we obtain aat = aOanwhence a,, = a"-'. The coeffici = 1,2, . . .,n - 1, on either side yield the equations ients of gnPi,
a(2a,ai+q(a,, a2, . . ., ai-,)) = k,, a,, . . .,a+,) where q is a quadratic and I is a linear form over K. Thus a,, a,, . . ., a,-, determine ai uniquely, for i = 1, . . ., n - I . Moreover, the coefficient of x n on either side yields the equation
a(2aoa,+ q(a,, a2, . . .,a,,-,))
+ ba, = K ~ oa,,,
. . ., a,-J
where again q is a quadratic and I is a linear form over K. Thus a, is uniquely determined by a,, a,, . . ., a,-l. IfcharK = 2,thenlet g = ax3+bx2+cx+d, a
#
Oand f =
n
C a,x"-'
i=O
a polynomial of degree n which is permutable with g. Again g of = f o g
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PERMUTABLE CHAINS OVER FIELDS AND INTEGRAL DOMAINS
157
yields some conditions for the coefficients of x3" and x3"-', i = 1, . . .,n. In particular, the coefficients of x3" give us aai = aOan whence a: = a"-' and since char K = 2, we see that a, is uniquely determined by a. The coefficients of x3"-', i = 1 , . . ., n - 1 yield the equations
a(3a:aff C(a0,
* *
-
9
ai-1)) = I(a0, a17
* * *
> ai-l)
where c is a cubic and I is a linear form over K. Similarly, a( 34an+ c(a,,, . . .,an-,))
+ bai = Z(ao,a,, . . .,an-l)
where c is a cubic and 1is a linear form over K. Thus again ai, i = 1, is uniquely determined by a,, a,, . . ., aiVl.
. ..,n,
3.42. Corollary. If P is any P-chain over K , then P contains exactly one polynomial of degree n, for every n 3 1.
Proof. Obvious. 3.43. Corollary. If char K # 2 and gl E K [ x ] ,[gl] = 2, or if char K = 2 and g2E K [ x ] , [g2] = 3, then gl and gz, resp., belong to at most one P-chain.
Proof. Obvious. 3.44. Lemma. If char K # 2, g c K [ x ] , and [g] = 2, then there is one and only one polynomial m = x2+d which is a conjugate of g. r f char K = 2, g E K [ x ] , and [ g ] = 3, then there is one and only one polynomial m = x3+dx+e which is a conjugate of g.
Proof. Let char K # 2, g = ax2+bx+c, 1-1 = ( l / u )( x - v ) whence, for any K,
a # 0, 1 = ux+w. Then
+
1-1 o g o I = (I lu) (a(ux+ w)2+ b(ux+ w) c -v) = aux2+ (2av
+b)x+ ( 1 12.4)(g(v)
-
v).
Hencel-logoI,is oftheformxz+difandonlyifu = a-l.v = -(2a)-l b. Thus there is exactly one I such that I-1 og G I is of the form as required. If char K = 2, and g = a1x3+ blx2 clx+ dl, al # 0, I = ux v, then, for any K,
+
+
+
1-l o g o I = ( I /u>(al(ux =
v)3+
+
+
bl(ux v ) ~ +cdux v) -t 4 -v) =
a1u2x3+(3alv+ bi)U~2+(3alw2+2 b 1 ~ +c ~ ) x + ( 1 /u) (g(v)-v).
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COMPOSI~ONAND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
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Hence l - l o g o l is of the form x3+dx+e if and only if u2 = a;', v = -(3a1)-I bl. Since char K = 2 and K is algebraically closed, such u, v E K do exist and are uniquely determined by al, bl. Thus m = x3+ dx+ e ic uniquely determined by g. 3.45. Corollary. If char K # 2, tt.tn every P-chain over K is a conjugate
of one and only one P-chain which contains some quadratic polynomial x2+a. If char K = 2, then every P-chain over K is a conjugate of one and only one P-chain which contains some cubic polynomial x3+ax+ b. Proof. By Cor. 3.42, Cor. 3.43, and Lemma 3.44. 3.46. Proposition.' I char K # 2, theh the only P-chains which contain somequadratic polynomial x2+a are the P-chain of powers and the P-chain of cebyshev polynomials. These are also the only P-chains containing some cubic polynomial x3 ax b i f char K = 2.
+ +
Proof. The P-chain S has the property as in the proposition and T also has, by (3.3). We have to show that no. other P-chains have this property. Let char K f 2, and P any P-chain containing some polynomial x2+ a. Case 1: char K # 3. Letf = a ~ x ~ + a ~ x ~ + aE~P,x then + a ~f ( x 2 + a ) = f * + a whence f(-x)"a = f ( x,+a) =f(x),+a. Thus f ( - x ) = f ( x ) or.f( -x) = -f(x). Since a0 # 0, we can only have f ( - x ) = -f(x), thus a1 = a3 = 0. Then a ~ ( x ~ + a ) ~ + a ~ ( x= ~ (+aao) ~ ~ + a ~ x ) ~whence +a 3 a, = a:, i.e. a, = 1. Moreover, 3a,a = 2a0a2, hence a, = za, and 3a,a2+a, = a; whence 3a2+,a3 = +a2. Thus a = 0 or a = -2. Case 2: charK = 3. Let f = b a x 5 + b , x 4 + b 2 x 3 + b , x 2 + b ~ x + b ~ E P . As before, we conclude f = box5+b2x3+b4x. Hence b o ( ~ 2 + a ) 5 + b , ( ~ ~ + a ) ~ f b ~ ( x=~( + b ~a x) ~ + b ~ x ~ + b ~ xWecompare )~+a. thecoefficients on either side of this equation and obrain b, = bi, thus b, = 1 ; furthermore 5ab, = 2b0b,, therefore b, = a, and 10b,a2+b2 = bi+2b,b4 implies b4 = -a. Finally 10boa3= 2b2b4 yields a3 = a2 whence a = 0 or a = 1 = -2. Thus in either case we have a = 0 or a = -2. Since x2 E S while x2-2 C T by (3.3), the P-chain Peither is equal to S or to T, by Cor. 3.43. Now let char K = 2, P any P-chain containing some polynomial x3++x+b, and f = b 0 x ~ + b ~ x ~ + b ~ x 3 + b ~ ~ ~ + b ~Then ~+b~EP.
+
03
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PERMUTABLE CHAINS OVER FIELDS A N D INTEGRAL DOMAINS
,f(X3+ax+b) =.f3++af+b. Making abundant use of char K = 2, thisequation reads as
(bo(x3+ ux + b)+ bl)(xl*+ a4x4+ b4)+ (bZ(X3+ux + b )+ ~
+
~ ) ( X ~ + U b2) ~ X ~ +
+ . . . = (bix" +b:xs +b:x6 +b;x4+ bix2+ b:) (box5+b,x4 + + b z ~ ~ + b 3 ~ ~ + b 4 x + b s. .) .+
We compare the coefficients on either side of this equation. Then b, = b:, thus bo = 1. Furthermore 0 = bib, whence bl = 0, then boa = b:b,+b:b, whence b, = a. Moreover bob+b, = b3,+bib3 whence b, = b, and 0 = bibo+ b:b,+ bib, whence b, = a2. Furthermore 0 = bib,+ b:b3+ bib5 whence b j = 0. We continue with comparing the coefficients and obtain b, = bib,+ bi+ b:b, whence a = b2+ a3.Furthermore 0 = bib,+ bib,+ b:b, whence a2b = 0. Thus a2b2 = 0 and therefore a3 = a2b2+as = a5, i.e. ~ ~ ( ~ 2 -= 1 )0. Hence a = 0 or a = 1 = -1 and a = b2+a3 implies b = 0. Since x3 E S while x3-x E T, by (3.3), again P = S or P = T follows. 3.47. Cor. 3.45 and Prop. 3.46 imply that Th. 3.33 is true if K is algebraically closed. 3.48. Proof of Theorem 3.33. Let K be an arbitrary field, and P any P-chain over K. By the preceding result, P is conjugate to U = S or T in the algebraic closure L of K . Let I = ux+v a linear polynomial over L that transforms P into U , then, if p 2 is the quadratic polynomial in P, we have l-lop,c I = x 2 + d while, for the cubic polynomial p 3 E P, we have 1-13p301 = x3+ex where d = 0 or -2 and e = 0 or -3. Let p2 = ax2+bx+c, p 3 = a1x3+b1x2+c1x+dl, then au = 1, 2aw+b = 0, 3alw+ bl = 0. Thus, whatever the characteristic of K may be, u, C K , whence P i s a conjugate of U over K which proves Th. 3.33. 17
3.5. Let J be any integral domain. We obtain all the P-chains over J if we determine all the P-chains of the quotient field K of J and pick those which consist of polynomials over J . By Th. 3.33, the P-chains over K are just I - l o S o 1, l - l o T o I where S is the P-chain of powers; T the P-chain of cebyshev polynomials, and 1 an arbitrary linear polynomial of K [ s ] .
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COMPOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
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Thus we have to check which of these P-chains consist of polynomials over J . 3.51. Proposition. Let J be any integrally closed domain, K the quotient jield of J , and 1 = ux+v a linear polynomial of K [ x ] .Then I-1 O So 1 is a P-chain over J if and only i f u, v E J and v2-v E J u while I- o T o 1 is a P-chain over J if and only if u, v E J and v -2 E Ju. Proof. a) Let I - l o S o l be any P-chain over J , then both 1-10x201 and 1 - l o x 3 o 1 belong to J [ x ] . Computing the normal forms of these polynomials, we see that u E J , 2v E J, (l/u) (v2-v) E J , 3v2 E J whence 4v2-3v2 E J . Since J is integrally closed, we therefore have u, v E J , v 2 -v E Ju. Conversely if 1 = ux+ v and u, v E J , v2 -v E Ju, then I - l o xi o I E J [ x ] ,i = 1,2,3, . . . which follows from a straightforward computation. b) Let 1-l o T o I be any P-chain over J , then both I-1 o (x2-2) o I and 1-l o (x3-3x) o 1 belong to J [ x ] .Computing the normal forms of these polynomials, we see that u E J , 2v E J , 3v2-3 E J which, as in a), shows that u, v E J . Moreover, for all gi E T , we have I F 1 ogi o I E J [ x ] .Computing the constant terms of these polynomials, we obtain g,(v)-v E Ju, i = 1 , 2, 3, . . . The case i = 2 yields v2-v-2E Ju. By§ 3.3, g,,,, = g,x-g,-, whence gn+l(v) = g,(v)v-g,-l(v). Therefore v2-2v E J u and so v-2 E Ju. Conversely if 1 = u x f v satisfies the condition of the proposition, then g2(v)-v = v2-v-2 E Ju. Suppose gi(v)-v E Ju, for i =s n. Then g,+,(v)-v = g,(v)v-g,-,(v)-v = (g,(v)-v)v(g,,-,(v)-v)+(v2-2v)E Ju. Thus g,(v)-vE Ju, i = I , 2, . . ., and a straightforward computation shows that l - l o g , o l E J [ x ] ,i= 1,2, 3, . . .
3.52. A similar argument as in the proof of Prop. 1.31 can be used to show that, in the semigroup ( J [ x ] ;o), the elements which have an inverse are just the polynomials I = ux+v where u is a unit of J . These elements form a group L with respect to the operation 0 . Two P-chains C and C1 over J are called conjugate (over J ) if there exists 1EL such that C I = I F 1 o C c 1. Clearly conjugacy is an equivalence relation on the set of all P-chains over J . 3.53. Theorem. Let J be any integrally closed domain. Then a full system of representatives for the classes of conjugate P-chains over J is given by the P-chains ( ~ x + v ) - ~ o S o ( u x + vand ) ( ~ x + 2 ) - ~ o T o ( u x + 2where ) u
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PERMUTATION POLYNOMIAL VECTORS AND POLYNOMIALS OVER RINGS
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runs through a full system U of representatives for the non-zero classes of associate elements of J and, for each u, the elements v run through a full system V(u)of representatives for the idempotents of J 1 Ju. Proof. By Prop. 3.51, these P-chains are, indeed, P-chains over J . Furthermore, S and Tpre not conjugate over the quotient field of J , thus are not conjugate over J and, by considoring the polynomials of degree 2 or of degree 3, one can easily check that, in either type of P-chains, any two distinct P-chains are not conjugate over J . On the other hand, if ul, vl E J , v;-q E Ju,, then there exists a unit e E J and an element u E U such that u1 = eu. Hence $-vl E Ju, therefore we can find an element v E V(u) such that v~ = v+ku. But ulx+wl = (ux+v)o(ex+k) whence (ulx+vl)-loSo(ulx+vl)is aconjugate over J o f ( u x + ~ ) - ~ o S o ( u x f v ) . If u1, v 1 E J and vl-2 E Jul, then again u1 = eu, u E U, 01 = 2+ ku whence ulx+ v1 = (ux+2) o (ex+ k ) and we can proceed as before. Hence every P-chain is represented by one of the theorem. This completes the proof. 4. Permutation polynomial vectors and permutation polynomials over rings 4.1. Let R be any commutative ring with identity, D an ideal of R, q(D):R -+ R 1 D the canonical epimorphism, q(D)(X):R[xl, . . .,xk] (RID) [x,, . . ., xk] the unique extension of q(D) to a composition epimorphism and C,(q(D)):C,(R) -+ Ck(RID) the corresponding epimorphism as described in ch. 3, § 13.1. --f
4.11. Definition. A polynomial vector f E Ck(R)is called a permutation polynomial vector mod D if the polynomial vector C,(q(D))f i s a permutation polynomial vector over R ] D . A polynomial p E Rlx,, . . ., xk] is called a (strict) permutation polynomial mod D if the polynomial q ( D ) ( X ) p is a (strict) permutation polynomial over R [ D. 4.12. Lemma. a) A polynomial vector f = (pl, . . . , p k ) of Ck(R) is a permutation polynomial vector mod D if and only if, for every (rl, . . ., rk) E Rk, the system of congruences
pi(xl, . . .,xk) _= ri mod D, has a unique solution mod D .
i = 1, . . .,k,
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COMPOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
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4
b) A polynomial p t R[x,, . . .,xk] is a permutation polynomial mod D i f and only i f , for every r f R, the set of incongruent solutions of the congruence p(x,, . . ., xk) = r mod D
has cardinality I R I D I k p l . c) The polynomial p f R[x,, . . ., xk] is a strict permutation polynomial mod D if and only if there exist polynomials p2, . . .,P k E R[x,, . . ., xk] such that the poIynomia1 vector ( p ,p2, . . .,Pk) is a permutalion polynomial vector mod D. d) Every strict permutation polynomial mod D is a permutation polynomial mod D. Proof. a) follows immediately from the definition of a permutation polynomial vector over R ID (see ch. 3 , s 11.45), b) is a consequence of ch. 3, Prop. 12.21,~)stemsfrom ch. 3 , s 12.22, and d) from ch. 3, Prop. 12.23.
5Z C . r f R 1 C is finite, then every permutation polynomial vector mod D is also a permutation polynomial vector mod C , and every strict permutation polynomial mod D is also a strict permutation polynomial mod C . If R I D is finite, then every permutation polynomial mod D is also a permutation polynomial mod C. 4.2. Proposition. Let C , D be ideals of R , D
Proof. a) Let f be a permutation polynomial vector mod D, then Ck(q(D))fis a permutation polynomial vector over R I D, thus - with the notation of ch. 3 , s 3.22 - we have T ( a ( R D ) ) )Ck(v(D))f t Uk(RID). Let 8: R I D R IC be the unique epimorphism such that 8q(D) = q(C). Since R j C is finite, ch. 3, Prop. 11.51 implies p ( p k ( 8 )a(R j D ) q(D)( X ) )f € Uk(RI C ) . But by diagram fig. 3.1 of ch. 3, we havePk(8)a ( R I D ) q(D)( X ) = = a(R 1 c, q(c) (x)? thus @(a ( R1 c>) ck(v(c)) fE a(RI c) q(D) Uk(RIC)whence f is a permutation polynomial vector modC. The statement on strict permutation polynomials now follows from Lemma 4.11 c). b) Let p be a permutation polynomial mod D, then q(D)( X ) p is a permutation polynomial over R \ D, thus a(R ID) q(D)( X ) p€ Sk(RI D). By ch. 3, Prop. 12.31, pk(8)5(R 1 D ) q(D)( X ) p f Sk(R1 C). As in part a) of the proof, we conclude that p is a permutation polynomial mod C .
-
$4
PERMUTATION POLYNOMIAL VECTORS AND POLYNOMIALS OVER RINGS
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4.21. Proposition. Let C , D be comaximal ideals of R and E = CD. Then a polynomial vector f f Ck(R) is a permutation polynomial vector mod E if and only if f is a pynutation polynomial vector mod C as well as mod D. A polynomial p E R[x,, . . .,xk] is a strict permutation polynomial mod E if and only i f p is a strict permutation polynomial mod C as well as mod D. If moreover R I C and R j D are finite, then p is a permutation polynomial mod E if andonly i f p is a permutation polynomial mod C as well as mod D.
-.
Proof. y : RIE R l C X R l D defined by yq(E)r = (q(C)r, q(D)r) is an isomorphism, by the Chinese remainder theorem. Thus by ch. 3, Remark 11.53, f is a permutation polynomial vector mod E if and only if c&)C,(q(E))f is a permutation polynomial vector over R I C X R 1 D which is, by ch. 3, Prop. 11.31 and Prop. 115 6 , the case if and only if ylp(zl) Ck(y)C,(q(E))f consists of a permutation polynomial vector over R 1 C and a permutation polynomial vector over RID. But by definition of y1 and z1 (see ch. 3, 9 11.3), we have ylp(zly(X)q(E)(X))f = (Ck(q(C))f,Ck(q(D))f)whence the first statement fo~lows. Similarly the other two statements are proved: By ch. 3, Remark 12.33, p is a (strict) permutation polynomial mod E if and only if y ( X ) q ( E )( X ) p is a (strict) permutation polynomial over R ( Cx R 1 D.By ch. 3, Prop. 12.43, this is the case if and only if z l y ( X ) q(E)( X ) p is a pair consisting of (strict) permutation polynomials over R I C and R ID,respectively. But z l y ( X ) q ( E )( X j p = (q(C)( X ) p , v(D)( X ) p ) which completes the proof. 4.3. Let f = (f,,. . . , f k ) be any polynomial vector in x,, . ., x, over R, and Of = (a,f,) which is a k Xk-matrix over R[x,, . ., x,] where a,f, is the entry in the s-th row and t-th column, and a, = ajax, is the t-th partial derivation. af denotes the determinant of Of, i.e. the Jacobian off. Polynomial vectors over R will be regarded as k X 1-matrices over R [x,, . . . , xk]. If (tdik), ( o ~ )are both m X n-matrices over a ring S and P is an ideal of S , then (Ui& = (2)ik) mod P shall mean that Uik = Ojk mod P, for all pairs (i,k ) ~
4.31. Proposition. Let Q be any primary ideal of R with associatedprime ideal P such that R 1 Q is finite and Q f P. Then a polynomial vector f = (f,,. . .. . f k ) in x,, . . ., xk over R is a permutation polynomial vector mod Q if and only if a) f is a permutation polynomial vector mod P, and b) the congruence af(xl, .. ., x k ) E 0 modP has no solution in R.
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COMPOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
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4
Proof. The proposition will follow from the following 4.32. Lemma. Let Q be any primary ideal which is dgerent from its associatedprime ideal P. Then q E {Q}implies aiq E (P}, .for any q E R[x,, . . ., xk], i = 1,2,..., k. Proof. We havz to show that (aiq)(rl, . . .,rk) E P, for all (rl, . . ., rk) E Rk. It will suffice to prove that p E { Q } implies p' E {P}, for all p E R [ x ] , since then q(x,) = q(rl, r2, . . .,x,, . . ., rk) E { Q } implies q'(x,) E {P}, i = 1, 2, . . ., k. Let a E P, a 4 Q, then, for any rE R , Taylor's formula implies that p(r+u) = p(r)+p'(r)a+p2a2+ . . . +pman'where p z , p3, . . ., p,E R . Since p ( r + a ) E Q , p(r)EQ, we have a(p'(r)+p,a+ . . . pmam-') E Q . But a 4 Q implies p'(r) E P since Q is primary and P its associated prime ideal.
+
4.33. Proof of Prop. 4.31. Let f be any polynomial vector satisfying the conditions a) and b). Since there exists some integer 1 =- 0 such that P' E Q, it suffices to show that f is a permutation polynomial vector mod P", n = 1, 2, . . ., by Prop. 4.2. This will be done by induction on n. By a), f is a permutation polynomial vector mod P,thus let us assume that f is a permutation polynomial vector mod P-',for some y1 =- 1, and that f E Rk is arbitrary. We have to show that
. . .,xk) = f mod P"
fo(x,,
(4.31)
has a unique solution mod P". Then, by Lemma 4.12 a), f is a permutation polynomial vector mod P". Let u = (ul, . . ., uk) E Rk be any solution of the system fo(xl, . . ., xk) = f mod P"-' (4.32) which exists because of Lemma 4.12 a) and the induction hypothesis. Suppose b = (al, . . .,vk) is a solution of (4.31). By induction hypothesis, b = u+I where IE(P"-l)k, the k-th Cartesian power of Pn-'. Since 2(n- 1) a n, Taylor's formula yields
fo(u+l) where A =
= (fou)+AlmodP"
(4.33)
((a&) (ul, . . .,uk)), whence AI
= ?-(fox)
mod P".
(4.34)
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PERMUTATION POLYNOMIAL VECTORS AND POLYNOMIALS OVER RINGS
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*
By b), Det A $ 0 mod P whence there exists some k x k-matrix B over R such that AB = BA 3 E mod P where E is the k x k-identity matrix. Hence (4.34) implies that
I z B(f-(fou)) modP".
(4.35)
Thus there is at most one solution b mod P" of (4.31). The existence of such a solution now follows easily: Let b = u + l where 1 is picked as above. Then fob=
fo(u+I)
= (fou)+AB(f-(fou))
E
fmodP",
by (4.33). Hence b is a solution of (4.31). Conversely let f be any permutation polynomial vector m o d e . By Prop. 4.2, f is also a permutation polynomial vector mod P . Moreover fl = [@(a(R 1 Q)) C,(q(Q)) f E Uk(RjQ), and since R 1 Q is finite, Uk(RIQ) is a group by ch. 3, Lemma 11.43. Thus f l has a inverse g1 E Uk(RIQ). Let g be any polynomial vector of C,(R) such that P(@IQ)) Ck(g(Q)) g = gl. Then, if F = (XI7 . ., xk), we have G(~RIQ>)~,
) = P ( ~ $ I Q > ) C ~ ( V ( Q ) ) Fthus (gof) , 9.f = ~ + q , l'j = (hl, . . .,hk)E C,(R), hi E {Q}, i = 1 , . ..,k. The chain rule for partial derivatives yields O(g o f ) = (Og of) Of. If 6, denotes the Kronecker symbol, we may write d(gof) = d(x+l'j) = (at(xs+hs))= (a,+ a,h,), and ath, E {P}, by Lemma 4.32. Hence (dg o f) Of = (a,,+ a,h,) and switching over to determinants, we get (ago f) af = I a,,+ ath, I E 1 mod {P}.Therefore (ago f) af = 1 + p , p E {P}, whence af(ul, . . .,uk) g 0 mod P , for all (ul, . . ., uk)E Rk.
.
4.34. Proposition. Let Q be any primary ideal of R with associated prime ideal P, Q # P, and R I Q jinite. Then a polynomial f E R[x,, . . ., x k ] is a strict permutation polynomial mod Q i f and only i f a) f is a strict permutation polynomial mod P , b) the system of congruences aif(xl, . . .,x k ) = 0 mod P,i = 1, . . ., k , has no solution in R .
Proof. Let f be any polynomial satisfying a) and b), then q(P)(X)f = gl is a strict permutation polynomial over RIP. Thus we can find polynomials g,, . . ., gk in x l , . . ., xk over R I P such that g = (gl, . . ., gk) is a permutation polynomial vector over R 1 P. If I R I PI = n , then in R 1 P
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{g-xi}E {0}, i = 1, . . ., k . Therefore, if h, are arbitrary polynomials over RIP, the polynomial vector I
k
k
\
is also a permutation polynomial vector over RIP. If we put hlj = 0, . . ., k, then (4.37) 49 = (i3,g,-hS,) mod (0).
j = 1,
If (u1, . . .,u k ) = U E (RI p)k, then by b), ( a l g i ( u ) , a,g,(U), . a&I(u))= (dl1(u),. . ., d,k(u)) = b,(u) is not the zero vector. Since R I P is a field, we may regard (R 1 P)k as an R I P-vectorspace, thus we can find vectors b,(u) = (dll(u), . . ., dik(u)), i = 2, . . ., k , in ( R ~ P such ) ~ that b,(u), b,(u), . . ., bk(u) are R I P-linearly independent. The finiteness of R 1 P moreover implies that R I P is polynomially complete. Hence we can choose h,, E (RIP) [x,, . . .,x k ] , 2 =S s G k, 1 =s t e k , such that h,,(u) = a,g,(u) -dat(u), for all u E (R I P)k. When substituting these polynomials Iz,, into (4.36), the congruence (4.37) yields n.9
(49)0 u
= (dStW.
Hence (a$) (ul, . . .,uk)= I dJu) 1 f 0, for all (ul,. . . ,uk)E (R I P)k because of the linear independence of b,(u), . . ., bk(u). Let J; E R[xl, . . . , x k ] , i = 2, . . ., k , such that q(P)(X)x = g j + c ; = , h,(x:-xj) then f = (f,f,, . . .,f k ) is a permutation polynomial vector mod P, and q(P) af = a9 implies that a f ( x l , . . ., x k ) = 0 mod P has no solution in R. By Prop. 4.31, f is a permutation polynomial vector mod Q whencefis a strict permutation polynomial mod Q. Conversely let f be any strict permutation polynomial mod Q then, by Prop. 4.2, f i s also a strict permutation polynomial mod P. Moreover there exists f = ( f , f,, . . .,fk) which is a permutation polynomial vector mod Q. Hence, by Prop. 4.31, af(ul, . . ., uk) $ 0 mod P, for all (ul, . . ., uk) E Rk, therefore the system a,f(x,, . . ., xk) E 0 mod P, i = 1 , . . ., k , has no solution in R.
(x)
4.35. Corollary. I f f (f)is apermutationpolynomial vector (strictpermutation polynomial) modulo some primary ideal Q of R with associated prime deal P # Q, R I Q 1 jinite, then f (f)is a permutation polynomial vector
1
§5
.
SEMIGROUPS OF POLYNOMIAL FUNCTION VECTORS
167
(strict permutation polynomial) modulo any primary ideal Q of R d@erent from P , having P as the associated prime ideal and R I Q 1 finite. ~
4.4. Let R be any Dedekind domain and M an ideal of R such that R I M is finite. The results of§ 4.2 and§ 4.3 yield a practical method of deciding whether or not a given polynomial vector f in xl, . . ., xk over R is a permutation vector mod M or a given polynomial f E R[x,, . . ., x],
is a strict permutation polynomial mod M. For if M = Py . . .Pp where the prime ideals P,are pairwise distinct, then P:k and P,"7are comaximal primary ideals, for i # j , and R I Pf" is finite, for i = 1 , . . ., r. Thus we have just to investigate the behavior of f or A respectively, and of af or a,f, . . ., a,f, resp. modulo the prime ideals P,.In particular, this method applies to the domains of the algebraic integers in algebraic number fields where the rational integers are a special case. It is, however, not known yet how to decide whether or not a polynomial f is a permutation polynomial mod M . A result similar to Prop. 4.34 has not yet been derived. Nor do we know whether or not there are rings R, in particular finite residue class rings of Dedekind domains, such that not every permutation polynomial over R is a strict permutation polynomial. 5. Semigroups of polynomial function vectors and polynomial permutations over finite factor rings of Dedekind domains 5.1. Let R be any Dedekind domain, M an ideal of R such that RIM is finite, and V,(R I M ) , U,(R I M ) as in ch. 3, Q 11.45. We recall that V,(R I M ) is the semigroup of all polynomial function vectors, U,(R I M ) consists of all polynomial permutations over R I M , and Uk(RI M)is asubsemigroup of V,(R j M ) . Since R 1 M is finite, Y,(R I M ) and U,(R [ M ) are also finite, and U,(RIM) is a group by ch. 3, Lemma 11.43. We observe that all results of this section will hold, in particular, for the case where R is the domain of rational integers and M # (0).
5.2. In ch. 3, Prop. 11.51, we have proved the result: Let B be any B an epimorphism, finite, A any arbitrary algebra of a variety '23, q :A and P',(v) : vk(A) V,(B) the extension of 7 to a composition epimorphism, then V,(q)U,(A) C Uk(B).This result can be sharpened for 8 being the variety of commutative rings with identity, and A , B finite residue class rings of a Dedekind domain R as follows: -2-
+
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COMPOSITiON AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
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4
5.21. Theorem. Let R be any Dedekind domain, A = R I M and B = RI N finite residue class rings of R and “/I A B an epimorphism. Then vk(q) U k ( A ) = Uk(B)* --f
/ \
RI L
Rl M
V I V
RIN
B,
(RIM11 ( L I M ) FIG.4.1
-
Proof. By ch. 6, Prop. 3.3, ker Ker rj = LI M where L is some ideal of R such that M c L. Let % : R I M R I L be defined by x(a+M) = a f L , for all a E R , then ker Ker x = L I M . Thus if v : R I M ( R I M ) I ( L j M ) is the canonical epimorphism, there are isomorphisms 61, 6 2 , by ch. 1, Th. 1.51, such that the diagram fig. 4.1 is commutative. Hence V,(q) = Vk(Scl)Vk(62)Vk(x).Since 8i, i = 1,2, are isomorphisms, it is sufficient to show, by ch. 3, Prop. 11.51, that V,(x) U,(RIM) = U,(RIL). This, however, is a consequence of +
5.22.Lemma. If R j M isfinite, L 2 M , and f is any permutation polynomial vector mod L, then there exists a permutation polynomial vector g mod M such that g = f mod {L}. Proof. Let M = PT . . . P: be the primary decomposition of M , then L=P? wheredi=sei,i= 1, ..., r. Step 1. We show that there exist permutation polynomial vectors gi mod i = 1, . . ., r, such that gi G f mod {Pp}. If di = 0, we may take gi = F = (xl,. . ., x,). If di ==2, then f is a permutation polynomial vector mod p?, by Prop. 4.21 whence, by Cor. 4.35, f is also a permutation polynomial vector mod PF, thus we may take gi = f. Now let di = 1, then f is a permutation polynomial vector mod Pi.
...e
e,
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SEMIGROUPS OF POLYNOMIAL FUNCTION VECTORS
169
Let j R 1 Pi = n and f = (f,,. . ., f k ) . Then, if hsj E R[xl, . . ., xk] is arbitrary, the polynomial vector k gi = (f1+
h,j(xy
-xj),
j=1
satisfies gi
G
..
k *9
fk f j=1
hkj(xy- x j ) )
f mod { P i } .Moreover, agi
I (a,f,-hst) I mod {Pi}
Since R 1 Piis a finite field, thus polynomially complete, we can choose h,, such that h,, = a,f,-8,, mod { P i }where ?is, is the Kronecker symbol. Then ag, G 1 mod {Pi} whence, by Prop. 4.31, gi is a permutation polynomial vector mod P,'i. Step 2. Since P,'i G {P,'"}, i = 1, . . ., r, the ideals {Pp}are pairwise comaximal, hence the Chinese remainder theorem implies that there exists some polynomial vector g such that g = gi mod {PF},i = 1, .. .,r. By Prop. 4.21, g is a permutation polynomial vector mod M . Since g = f mod {P?},i = 1, . . ., r, wehave g = f mod { L }by ch. 3, 0 5.36.
5.3. The finiteness of A in Th. 5.21 i s indispensable. The following lemma will yield a &nstruction of such a counterexample. 5.31. Lemma. Let R be any injinite Dedekind domain which has just jnitely many units, and such that every ,factor ring modulo a non-trivial prime ideal is finite. Then af is a unit of R, for every permutation polynomial vector f over R . Proof. By hypothesis, f is a permutation polynomial vector mod (0), therefore, by Prop. 4.2 and the finiteness of R I P2 which follows from ch. 6, Lemma 4.52, f is also a permutation polynomial vector mod P2,for every prime ideal P # (0) in R. Hence, by Prop. 4.31, af(rl, . . ., rk) g 0 mod P, for any (rl, . . ., rk)E Rk.Therefore R af(rl, . . ., rk) = R, i.e. af(rl, . . . , rk) is a unit of R,for any (rl, . . ., rk)E Rk.We now show by induction on k , that if g(xl, . . ., xk) E R[xl, . . ., xk]such that g(rl, . . .,rk) is a unit of R for all (rl, . . ., rk) E Rk,then g is a unit of R. This is true for k = 1 because of a well-known theorem on the number of roots of a polynomial. Suppose the assertion holds for k - 1 instead of k, and let g =
C ai(xl, . . .,xk-l)xL such that, for arbitrary (rl, . . .,rk-J E R~-', r
i=n
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COMPOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
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As stated, g(rl, . . ., rk-+ xk) is then a unit of R whence ai(xl, . . . ,xk-l) = 0,for i == 1, and a,(x,, . . ., X k - 1 ) assumes only units as values. By induction, aO(x,, . . .,&-I) is a unit of R, thus g is a unit of R. In particular, af(xl, . . ., x k ) is a unit of R. 5.32. Remark. Since every element of Ck(R)which has an inverse in Ck(R),is a permutation polynomial vector over R by ch. 3, 9 11.45, the following proposition would be a partial converse of Lemma 5.31 : If R is any integral domain of characteristic zero and f E Ck(R) is a polynomial vector such that af is a unit of R, then f has an inverse in Ck(R).But no correct proof of this proposition for k 3 is yet known 5.33. We now construct the counterexample as announced at the begin-
ning of this subsection. Let R be the ring of rational integers, k = 1, A = R, B = R I ( p ) , p =- 3 a prime, and 7 : R + R / ( p ) the canonical epimorphism. Since R satisfies the conditions of Lemma 5.31, every permutation polynomial vectorfover R satisfies eitherf' = 1 orf' = - 1 whence {f= x+c, -x+cJcER} is the totality of permutation polynomial vectors over R. Therefore Ul(R)= {et+cl e = 1, c E R } whence 1 V I ( $ Ul(R)I = 2p. But B is a finite field and therefore polynomially complete, thus IU1(B)1 = p ! . Hence Vk(7)Ul(R) # Ul(B). 5.4. Let E be any ideal of R such that E=CD where C and D are comaximal ideals of R , and y : R I E .+ RI C x RI D the isomorphism as in the
proof of Prop. 4.21. Then, by the results of ch. 3, 9 11.3, Vk(RIE)2 v k ( R I C X R I D ) = vk(RIC)XVk(RID),and by ch. 3, §11,6, U,(RIE)= Uk(RICX R [D) Y U k ( R j C ) xUk(RID). Therefore, if M = P:' . . . P: is the primary decomposition of M , we have Vk(R1 M ) Y V,(R I q) X . X V,(R 1 f',".) and Uk(RI M ) 2 Uk(R1 f':) X . . . X uk(R 1 P:). Thus for R I Mfinite the structure of Vk(RI M ) and Uk(R1 M ) will be determined as soon as we know the structure of Vk(R1 P"),Uk(RIP),for any prime ideal P of R, e 3 1, where RIP is finite. Since RIP is also a field and therefore polynomially complete, V,(R IP) is isomorphic to the symmetric semigroup and Uk(RIP)to the symmetric group of (RIP)k,by ch. 3, Remark 11.22. Thus it remains to investigate the case e > 1.
..
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SEMIGROUPS OF POLYNOMIAL FUNCTlON VECTORS
171
5.5. We first introduce some new notation : As in ch. 3, 0 8.6, let N be the additive semigroup of non-negative integers and Mk the direct product of k copies of N . On Mk, we introduce a partial order =s by (m,, . . ., mk) G (n,, . . ., n k ) if and only if mi s n i , i = 1, . . ., k. Let epK = &k:Mk-,N
be defined by Fk(m,, . . ., mk) =
C &(mi)where k
E~
=
E
is defined as in
i=l
ch. 3, 0 8.3. Moreover, let a :Mk -, N be defined by u(ml, . . .,mk) = k
C mi, F = (xl,. . .,xk), zL= xk . . . xf,for L = (i,, . . .,ik) E M~ (see ch. I,
i=l
0 8.2). Since every finite subset of M k has an upper bound in Mk, we can write f = 1(a,$ I A 6 a), for every f E R[xl, . . .,xk] where a, E R
can be possibly zero, for some I.. If we regard the polynomial vectors over R as k X 1-matrices, then every polynomial vector f E Ck(R)can be written as f = x(a,$l A == u), a, regarded as k X 1-matrices over R. As in ch. 3,§ 8.3, we set 1 R I PI = q. We deviate from the usual convention by setting P" = R, for n =s 0, and every ideal P of R and d" = 1, for all a E R and n == 0. A subset W of Rk for any e is called a vector system mod P' if the mapping 5 :Rk -+ ( R 1 Pe)k defined by [(a,, a2, . . ., ak) = (al+ P", . . ., ak+P') is such that the restriction of 5 to W is a bijection. If a E P, a 6P2, W 1a vector system mod P and We-l a vector system mod Pe-', then, by ch. 6, Lemma 4.53, both the sets W = {UfabI u E W,, b E We-l} and W = {b+ae-'uI u E W,, b E We-,} are vector systems mod P'. 5.51. Lemma. Let a E P, a @ p2, i = (il, . . ., ik) E Mk, and d, E ~ + ~ ( + & k ( ~ ) . Then there exists a polynomial s,E (P"}such that s, = d'd'"~'+C(c,a'(y
I 1E < 1)
(5.51)
and a polynomial u, E {P'-'} such that L
= da"(')-1~'+ b C(b,a"(")-lh1l
A < l )
(5.52)
Proof. Let s, = d,a"(')t,where t, has been defined in ch. 3 , § 8.6. By the proof of ch. 3, Lemma 8.61, t, E {P"")}, hence s, E {P'}. If we expand s, into power products of the indeterminates xi, then we see that s, is as in (5.51). Similarly, u, = d,&')-lt, E {P-'} and is as in (5.52).
z g E {P'} and a$ be the leading term oj its normal form according to ch. 1, Th. 8.21. Then a, E Pe-'Ap).
5.52. Lemma. L.et 0
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COMPOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
CH.
4
Proof. By induction on k . For k = 1, the lemma has been proved in ch. 3, Lemma 8.5. Suppose the lemma holds fo rk - 1, then let g E R [ x l , . . .,xk], g E {P'} and a$,
p = (ml, . . .,mk), its leading term. Let g =
0
C
xihi,
j=ml
where h j € R [ x z ,. . ., xk], then h,, # 0, and the normal form of hm, has the leading term a,FI where = (xz, . . ., xk), p = (m2, . . ., mk). For any (u2, . . ., vk) E R k - l , the polynomial g(xl, v2, . . ., wk-l) = 0
hj(v2, . . ., t)k-,)x{ is in {P"}. Since the lemma holds for k = 1,
j=ml
we conclude that hml(v2,. . ., 2 ) k - l ) E Pe-s(ml) whence hmlE {Pe-c(ml)}. By induction, a, E p e - & ( m d - & k - l ( a = p"-En(P)
5.53. Lemma. Let r0 = (wl, . . ., wk) E Rk and 1 = (il, . . ., ik) E Mk. Then there exists a polynomial t,, F_ R[xl, . . ., xk] such that, for any 1: = (rl, . . ., rk) E Rk,
= f modP', tmL(rO+r)= OmodP', t,,(bi++)
for
jor
~ + =r ~JI modP, Wfr $ mmodP.
(5.53)
Proof. Let E E R be any full set of representatives for the units of R I P". Then c1 = ( n 1 n E E ) also represents some unit of RIP", thus there exists some c E R such that cc1 = 1 mod P". Let too = c (x+nl n EE) and toi = (xtooy, i = 1, 2, . . . . For w c R, we set tWi= toi(x-w), i = 1, 2, . . . . Then we claim
n
fl
too(r) E 1 mod P',
for
r
= 0 mod P ,
too(r) z OmodP",
for
r
f-
OmodP.
(5.54)
For, too(r) = c n ( r + n I n E E ) . Any e c R representing a unit of RIP also represents a unit of RIP" by ch. 6, Lemma 4.51. Hence if r = 0 mod P, then { r f n l n E E } is again a full system of representatives for the units of RIP" whence the first congruence of (5.54) follows. Furthermore if r f 0 mod P, then - r represents some unit of RIP and therefore of RIP" whence n = -r mod P", for some n E E, and the second congruence follows. (5.54) implies that toi(r) = ri mod P',
for
toi(r)= 0 mod P", for
r
= 0 mod P,
r $ 0 mod P.
I5
SEMIGROUPS OF POLYNOMIAL FUNCTION VECTORS
Thus t,,(w+r)
= ri mod P",
t,,(w+r)
E
for
0 mod P", for
w+r
173
= w mod P,
w f r g w mod P.
Let t,, = twlil(xl) tWzi,(x2) . . . twkjl;(Xk). Then t,,, satisfies (5.53). 5.54. Lemma. Let a E P , a 4P2, and g = xc,a'(')xA such that g(b) = 0 mod P" for every b of some vector system Wepl mod PP-l.Then g E {P"}.
w
Proof. Let W , be a vector system mod P and = {b+ae-luI u E W,, b E We-,} as in the beginning of this subsection, then g(b+ae-lU) = Cc,a"(')(b + u"-'u), f xc,a"(')b' E g(b) mod P'.
-
5.6. Let C : Rk (RIP")kbe the mapping as defined in Q 5.5, and W any vector system mod P". Then 5 maps W onto (RIP"), bijectively. If p : @(Fk(RlP')) f'l((RIP")k)denotes the composition isomorphism of ch. 3, Lemma 11.21, then let 15 : Vk(R1 P") Map ( W , ( R 1 P"),) (where Map ( M , N ) means the set of all mappings from a set M to a set N ) be defined by xf = (pf)l. If (9f)T = (pfI)C, then pf = pfl, whence f = fl. Thus x is injective for any e. In'order to determine the mappings of V k ( RI P")= V,, it suffices to know the mappings of xVk. But these are given by
-
-
5.61. Proposition. Let a E P, a 4 P2, W = {U+ aq I f E 2 1 , 9 E Ze-,} a vector system mod P" where 2, is a vector system mod P and Zepl a vector system mod Pe-', and, for every integer r, W , a vector system mod P' containing the "zero vector" o = (0, 0, . . ., 0). Then the mappings
f+aq
-
x(afLau(')9'lE M k ) modPe,
a, E We-o(L)-Fq(L), (5.61)
are mappings of xV,, every mapping of x v k has the form (5.61), and the mappings of (5.61) are pairwise distinct. Proof. a) Let 8 E xVk,then there exists fl E V,(R I P') such that 8 = (pf& thus there is a polynomial vector f C,(R) such that 8 is the mapping
U+aq If 7
=
-
f o(U+aq) mod P'.
( e - 1 , . . ., e - 1) E M,, then Taylor's formula yields
f o (f+aq)
= C(bfca"(oq' I L -= 7) mod P".
(5.62)
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COMPOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
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For any f E Z,, there is some a,, E We--o(,)-Ek(,) such that b,, = a,, mod Pe-u(q)-Ek(v), thus b€7 - a,,+b, where b, E (Pe-"(q)-Ek(q))k. By Lemma 5.51, there exists a polynomial vector 5 , E C,(R) such that 3, E {P"}"and 5,, = ~ , u " ( ~ ~ ~ ~ + C2( -= C ,q). U Since " ( ~ )5,~ E~{P"}k, I (5.62) implies that
f o (€+at)) = uf,uu(%)~+ C(bi:)a"(')t)L I L < q) mod P". Let No = { L C M k /L =sq}, Nl = No-. {q}, ql be a maximal element of Nl and apply the same procedure to ql instead of q, then take a maximal element of N2 = N1-- {q,} etc. In each step, is replaced by afai E We+u(,d)-s+,c)while the other vectors f$: are changed only for z 4 qi. Thus after a finite number of steps we arrive at f o (f+at)) = C (a,,a'-"')$'~L G q) mod P",for every I, E Ze-,, where a,' E We-u(L)--Ek(L), for every 1. Hence every 8 E xVk is of the form (5.61). b) We now show that every mapping of the form (5.61) belongs to xVk. Let 8 be such a mapping, then let t,, be as in Lemma 5.53, for tu E 2, and L E Mk, and g = C(C(amrtmr 1 L E M k )I ku E ZJ, then this is a finite sum, indeed, since We-o(r)-s(r) = ( 0 ) unless t s q. By Lemma 5.53, we obtain g o ( f + a g > = C(C(a,'t,,(f+at)) I l E M " ) ltu EZI)
= C(a&'(f + at))I 1 E Mk) = C(a,,a"(')t)' I L E M k ) mod P".
Hence 8(f+at)) = go(€+at)) mod P",for all f+al) E W . c) We have to show that the mappings of (5.61) are pairwise distinct. Suppose that C(arp"(')t)'It E Mk) = C(b,p"("'$'lL t M k )mod P", for all €+at) E W ,
and let € E 2,. Then
C((a,,
-
b,,)a"("'$' I L E M k ) = o mod P", for all g E 2e-l .
By Lemma 5.54, ~ ( ( a f , - b & f " ' ) ~ ' ~EtM k )E {P"}".If (a,,-b,p)uu(p)zp is the leading term of this polynomial vector according to ch. 1, Th. 8.21, and qp# b, then, by Lemma 5.52, (q,-bfp)u@) E (Pe-Ek(p))k, hence at,, = b,, mod Pe-u(p)-ek(p)since u P2.By hypothesis, at, = bf,, a contradiction. Hence a,, = br,, for all L E Mk.
§5
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SEMIGROUPS OF POLYNOMIAL FUNCTION VECTORS
5.62. Corollary. If N(k, e) = { L E Mk/e-c$L)-&k(i) =- 0 } and T C(e--4L)--k(t)/ L E ~ ( ke)), , then
I Vk(R I P")
=
=
(5.63)
4 Tkqx.
Proof. This is an immediate consequence of counting the mappings of (5.61) and applying ch. 6, Lemma 4.52. 5.63. As a special case of Cor. 5.62, we get ' V,(R I P2) = q(k+2)kqk. ~
9 11.4, Uk(RIP") = uk is a subsemigroup of Vk-actually is a group. We are now going to characterize those mappings of (5.61) that are elements of XU,. Let (0, 0, . . ., 0) = OEM,, aj = (0, . ..,1, . . ., 0) E Mk with 1 as the i-th component and 0 elsewhere, and Det (ul, . . ., uk) the determinant of the column vectors u,, . . ., u k E Rk which sometimes is also denoted by D(u,, . . . ,uk). 5.7. By ch. 3, u k
5.71. Proposition. A mapping of theform (5.61) belongs to X U , ifand only i f
a) The family {afoI f E 2,) is a vector system mod P, and b) Det (af8,, . . . , af8J f 0 mod P,;for every f E 2,.
Proof. Let f + at) C(af,a4')t)'lL E M k ) mod P' be a mapping of (5.61), then, by Prop. 5.61, there exists a polynomial vector g E C k ( R )such that +
go(f+at))
= Ca,,a'(')t)'
mod P",
for every f+at) E W .
(5.71)
Let t) = (y,, .. ., yk), and aif = .. ., i3,fk), for any polynomial vector f = (f,, . . .,f k ) . Then Taylor's formula implies go(f+at)) = g o € + a where 8, E Ck(R) and
Sto 0 = 0.
c [(a,g)otlyj+a28fot) k
i=l
(5.72)
Substitution of (5.72) into (5.71) yields
k
for every t) EZ,-,. There is exactly one 9 EZ,-,, such that t) = o mod P'-l. If we substitute this particular g into (5.73) and observe that a E P, we get g o f = a,, mod P '. (5.74)
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COMPOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
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4
Since e z=2, the congruence (5.73) remains valid if we replace P" by P2, thus (5.73) and (5.74) together with a E P, a Q P2 imply that k
k
i=l
i=l
1 [ ( a i g ) o f I ~ i C afs,Yi mod P ,
for all t, EZ,-,, hence also for all t, E Rk. Thus taking y i = 1 and y j = 0, for i # j , we obtain aMi = (i3,g)of mod P whence
= (@$of mod P . Caf,a5(k)t,6 mod P" belongs to xu,, then g is a
-.
Det (afa,, . . .,Ja,
(5.75)
permutaIf f+aq tion polynomial vector mod P'. Therefore by Prop. 4.31, g is a permutation polynomial vector mod P and (i3g)of g 0 mod P, for every f E Z1. Hence (5.74) and (5.75) show that a) and b) are satisfied. Conversely if these conditions are satisfied, then (5.74) implies that g i s a permutation vector mod P while (5.75) shows that (ag) o f $ 0 mod P, for all f E Rk. Again by Prop. 4.31, g is a permutation vector mod P",hence the given mapping is in XU,. 5.72. Corollary. If @k(P)denotes the number of nonsingular k X k-matrices over the field RIP and T has the meaning of Cor. 5.62, then I ~
Uk(R1 P'), = qk!@k(P)qk q(T-k-l)kqk.
(5.76)
Proof. By Prop. 5.61 and Prop. 5.71, we have lUkl = IVkl ( r / s )where = qekqkq(e-l)k2qk and r = qk!q('-1)kqk[@k(P)q('-2)kP]qb. Then (5.63) implies (5.76).
5.73. As a special case of Cor. 5.72, we get IUk(RIP2)) = qk!@k(p)qkqkqk. This follows from Q 5.63. 5.74. Remark. It is well-known (see ch. 6, .(qk-q) . . . ( q k - q k - 1 ) .
3
7.2) that @ k ( P )= (qk-l)
5.8. Proposition. Let a, W , and W,, r any integer, have the meaning of Prop. 5.61. Then the mappings
f+ag -&+a
E
( 2 1 3
Z&,
i
bf0+
C b,,a 5(L)-1t,L)mod P', LZO
b,o E We-1,
b,, E We-o(L)-ek(c)>
(5.81)
§5
SEMIGROUPS OF POLYNOMIAL FUNCTION VECTORS
177
are mappings of x v k , every mapping of xVk has the form (5.81), and the mappings of (5.81) arepairwisedistinct. Amapping of theform (5.81) belongs to XU, if and only i f u is a permutation of Z1 and Det (brsl, . . ., bfak)f 0 mod P,f o r every f E 21.
Proof. Let 8 E xVk, then 8 is of the form (5.61), by Prop. 5.61. By ch. 6, Lemma 4.53, a,, = af+abf, mod P" where af E 2, and b,, E W e - l . Hence 19 is a mapping of (5.81). Conversely, every mapping 8 of (5.81) is a mapping of (5.61) whence 8 E xVk. If any two mappings of (5.81) are equal, say and
then, by Prop. 5.61, b,, = cfL, for any L =- 0, and f € 2 ,while af+abp, = ,6f+acr,, for any f E 2, whence a = @ and b,, = c,,. Suppose now that a mapping 8 of (5.81) belongs to xu,.Rewriting 8 in the form of (5.61) and applying Prop. 5.71, we see that Det (bfB1, . . ., bf8J g 0 mod P, for every f €2,and that the family {af+ab,~fEZ,) is a vector system mod P. Hence {aflf€2,)is a vector system mod P whence a is a permutation of 2,. Conversely if a mapping 8 of (5.81) is such that a is a permutation of 2, and Det (bral, . . ., bra,) 9 0 mod P, for all f, then rewriting of 8 in the form (5.61) and applying Prop. 5.71 shows that 8 E
xu,.
5.9. Lemma. Let Q be any commutative ring with identity, A an ideal of Q and H ( A ) the set of all polynomial vectors f E C,(Q) of the form f = xa,xa where a, f D mod AUot)-', for every a., Then H ( A ) is a subsemigroup of C,(Q) containing the identity F of C,(Q).
Proof. Clearly the sum of any two polynomials Ca,F', a, = 0 mod A"(')-' for every a,, is again a polynomial of this form. Thus it suffices to prove: Let ,u = (ml, . . ., mk) M,, a E Q, such that a = 0 mod and g = (g,, . . .,gk) E H ( A ) then if ag;"' . . . g;l'k = CuazL,we have u, = 0 mod AUQ-', for every un. This will follow as soon we have shown: For any integer r a 0, a E Ar-', and polynomials hi = CvjLj~'j, vjtj E AU('j)-'
178
COMPOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
CH. 4
for every 5, j = 1, . . .,r, from ah, . . . h, = z u , follows ~ ~ u, c A"@-', for every u,. But ah, . . . h, = Cavul,,. . . vJ1+ ...+'r and avlLl. . . vrL,E Ag whereg=r-l+o(i,)-l+ ... +u(i,)-l =o(c,+ ... + ~ , ) - l . 5.91. As before, let R be any Dedekind domain, P a prime ideal of R, and RIP finite. In Lemma 5.9, take Q = RIP",A =PIP', then we obtain a subsemigroup H(PI P") = Hk(Pe,P ) of Ck(RI P'). Therefore iF'(o(RI P')) Hk(P", P ) = sk(P', P ) is a subsemigroup of v,(R [ P")while s k ( p e ,P ) U k ( RP") / = T,(P",P ) is a subsemigroup of U,(RIP") containing (El, . . ., 5,) and is even a group. Let e =- 1, ZeF1 = W a vector system mod Pep', and x1 : Vk(R1 P"-l> Map ( W , ( R I F-'),)the injection being defined in an anaiogops manner as ;5 in Q 5.6.
n
+
5.92. Proposition. Let a E P , a ( P 2 , Z e - , a vector system mod Pe-', and W , a vector system mod P', containing D, for every integer r. Then
t,
+
C bta0(')--lqkmod Pew',
b,+
k>O
t, E Ze-'
(5.91)
where b, E We-,, b, E We-a(r)-Ek(k), .for t =- 0,is an element of x~S,'(P'-~, P), every element of XlSk(p"-', P) is of the form (5.91), and these mappings (5.91) are pairwise distinct. A mapping (5.91) belongs to XITk(pe-', P ) i f and only i f Det (bdl, . . ., bdt) $ 0 mod P . Proof. We proceed as in the proof of Prop. 5.61. a) Let z E X ~ S , ( P " -P~) , then z can be written as
t,
-t
mod Pe-'
where aL= D mod for all i occurring in this sum. Since Ra G P, but Ra S P2, we have Ra"+P"-' = (Ra)"+P"-l = Pmin(n,e-l) , for any n. Hence, for any p E P", we have p E uamin(n, mod Pe-l, for some u E R . Taking q as in § 5.61, we conclude that z is of the form
t,
+
C
bkaO(L)-'t,k mod P e p , .
1sq
Now we apply the same procedure as in the fist part of the proof of Prop. 5.61, using Lemma 5.51, to show that z can be written in the form (5.91).
§5
SEMIGROUPS OF POLYNOMIAL FUNCTION VECTORS
179
b) By definition of sk(p", P), every mapping (5.91) belongs to X1sk(Pe-',
c) Suppose two mappings with coefficients b,, c, of (5.91) are equal, then (b, - co)+ C (b, - cL)u0(')-l$E {Pe-l}k.Again we may argue as in l>O
the third part of the proof of Prop. 5.61 to show that b, = c,, for all t . d) Suppose that the mapping in (5.91) belongs to X1Tk(Pe-', P). Then b,+ C ~ , u " ( ' ) - ~isF ~a permutation polynomial vector mod Pew', L>O
whence, by Prop. 4.31, this is also apermutation polynomial vector mod P. Hence bo+
k
b,$'
is also a permutation polynomial vector mod P.
i=l
Therefore the system
C b,$* = o k
mod P of linear congruences has
+
i= 1
just the trivial solution whence Det (bdl, . . .,b,J 0 mod P.Conversely if Det (bsl, . . ., Pdn) $ 0 mod P, then bo+ C bLuu(L)-l~' satisfies both L>O
the conditions of Prop. 4.31. Hence such a mapping as (5.91) belongs to XITk(pe-',
5.93. We set (R = K and ( R1 Pe.71)k= L. Let 2, be any vector system m o d P , Z , - l a v e c t o r s y s t e m m o d P " - f an d ~ , :Z ~ Ce-l:Ze-l be definedas Cing5.5.Then W = {f+at)/fEZl, gEZ,-,} is a vector system mod P" and Q : W K X L defined by p(f+at)) = ([lf,[e-lLj) is a bijection. Let [ be the mapping as defined in 9 5.5. If 8'1(KXL) denotes the symmetric semigroup of K X L , then 6 : v k F l ( K X L ) which is defined by 8f = p C - l ( q f ) [ p - l , is a monomorphism. Thus v k S @[-lvk[p-l = rkand uk @[-luk[@-l = By Prop. 5.8, every mapping of Fk S F , ( K x L ) is of the form -+
+
+
+
where the bars mean the cosets mod P and mod PE-', resp. of which the element under the bar is a member. Hence every mapping of rkcan be written as 9) -+If, B(t)6) (5.92)
(t
where x1 = [,LX[;~E Fl(K) and B(f) E gSk(Pe-', P).Conversely by Prop. 5.8 and Prop. 5.92, every mapping (5.92) belongs to We also conclude that a mapping (5.92) belongs to Vk if and only if tcl is a permutation of K and B(f) E gTk(P'-l, P),for all f E Z,.
rk.
180
COMPOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
CH.
4
We recall the definition of the wreath product of two permutation groups and consider also its straightforward generalization to semigroups of mappings. Then we may summarize our results as 5.94. Theorem. The semigroup V k ( RI P") is isomorphic to the wreath product of the semigroup qjSk(Pe-', P ) by the symmetric semigroup Fl(( RI P)')), and the group Uk(R1 P') is isomorphic to the wreath product of the group qjT,(P"', P ) by the symmetric group S y m ( RI P)k. 6. Ideal power semigroups 6.1. Let Q be any commutative ring with identity, A an ideal of Q, and H ( A ) the subsemigroup of C,(Q) defined in 5 5.9. Clearly H(Q) = Ck(Q) while H ( ( 0 ) ) = {a,+CaatsSL}which will be denoted by L(Q).Moreover if A1 is an ideal of Q such that A1 C A , then H ( A I )C H(A). We put J(A) = H(A)f' @(0)-1 Uk(Q)which is the set of all permutation polynomial vectors of H ( A ) . J ( A ) contains the polynomial vector 5, hence J(A) is non-empty and therefore is a subsemigroup of H ( A ) . Furthermore we put p ( a ) H ( A )= S(A) which is a subsemigroup of Vk(Q) and @(a) J ( A ) = T(A)is a subsemigroup of U,(Q). S(A) and T(A) are related by T ( A) = S ( A ) n U,(Q). 6.2. Proposition. Let 7 : Q Ql be any epimorphism of rings with identity, Ck(7):c k ( Q ) -,C,(Q,) and V,(q) : Vk(Q)+ Vk(Ql)the epimorphismsdefined in ch. 3 , s 11.3. Then --t
a) C,(r) H ( 4 = H ( r 4 , Vk(rl)s(4 = S(rA). b) V,(q) U,(Q) G U,(Ql) impliev Vk(7)T ( A ) C T(qA).If Ql is finite, the last inclusion always holds. c) C,($ J(A) C J(7A) if and only if Vk(q)T ( A ) C T(7A). d) If q is an isomorphism, then Ck(q)maps H ( A ) isomorphically onto H ( q A ) and J ( A ) isomorphically onto J(qA), Vk(q)maps S(A) isomorphically onto S(qA) and T ( A ) isomorphically onto T(qA). Proof. a) follows from the definition of H(,4) and ch. 3, diagram fig. 3. I . b) follows from
u w ( c~v)k wS(A)n
Vk(d
u,(Q)c s(qA)nVJQJ
= T(VO.
The hypothesis of b) is satisfied if Ql is finite, by ch. 3, Prop. 11.51.
06
IDEAL POWER SEMIGROUPS
181
c) C,(q) J ( A ) E J(qA) implies V,(q) T ( A ) c T(qA), by ch. 3, diagram fig. 3.1. Conversely if V,(q)T(A)E T(qA),then, again by ch. 3, diagram fig. 3.12 @(o>C,(q)J(A) s (?(a> J(7A) C Uk(Ql>,hence c k ( q )J(A) E H(qA)nGYo)-' Uk(Q1) = J(qA)d) If q is an isomorphism, then C,(q):C,(Q) -. C,(Q,) is an isomorphism, by ch. 1, Prop. 4.5 and ch. 3, Th. 11.11, and C,(q) maps @(a)-, U,(Q) isomorphically onto @(o)-l U,(Q,), by ch. 3, Remark 11.53 whence the first part of d) follows. Similarly the second statement can be proved.
-
6.3. Proposition. Let Q = Q I X QZbe adirectproduct of commutative rings with identity, A,, A, ideals of Q,, Q,, respectively,yl@(zl) :C,(Q1 X Q,) c,@I) X C , (Q2). Y~?(Z,) : v, X QJ -. V,(Q,) x V, (Q,) the composition isomorphisms of ch. 3, Prop. 11.31. Then yl@(zl) induces isomorphisms from H(A1 x Az) to H(A1)x H ( A 2 ) and from J(A1 X Az) to J(A1) >( J(A2). Similarly yzT(z2)induces isomorphismsfrom S(A1x Az) to S ( A I ) X S ( A ~and ) from T(A1x-42)to T ( A l ) XT( A 2 ) .
<el
Proof. By Prop. 6.2, yl@(z,) H(A, X A,) G H(A,) X H(A,). Since A; X A; = (A,XA,)', this inclusion is an' equality. By ch. 3, Prop. 11.66, y,(F(z,) maps tT(o)-l Uk(Q) isomorphically onto T(o)-l U,(Q,) X @(o)-l U,(Q2), hence y1Q(zl) maps J(A) into J(A,)XJ(A,) and this mapping is onto since ylV(zl) is a composition isomorphism from C,(Q) to C,(Q,)xC,(Q,). The second assertion follows from ch. 3, diagram fig. 3.3,
6.31. Remark. By ch. 6, 4.6, every ideal of Q, XQ, is of the form A , X A,, where Ai is an ideal of Qi, i = 1,2. 6.4.Theorem. Let Q be any commutative ring with identity and A a nilpotenf ideal of Q. Then the subsemigroup J ( A ) of H(A) is equal to &(H(A)), the group of units of H(A), and therefore is a group. A polynomial vector f = Caa? of H ( A ) belongs to J(A) if and only if the determinant D(a8,, . . ., ad,) is a unit of Q. Proof. Let I be the least integer such that A' = (0). If f E &(H(A)), then f E &(c,(Q))Y thus W o l f E d(GYo)Ck(Q))= d(v,(Q)) E u,(Q), by ch. 3, Lemma 11.43, whence t E J(A). Let now f E J(A) and f = Since 17(o)iE T ( A ) C UJQ), the equation f o g = u has a solution z,
182
COMPOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
for every uE Qk, hence f o g k
= u mod A has
C aa,gd' E u mod
Hence the system a,+
CH.
a solution g, for every
4
u.
A of linear congruences also
i=l
has a solution, for every u E Qk, thus the matrix (aal,. . .,ask) of column vectors a#, has a right inverse mod A . Hence the determinant D(aal, . . .,ask) = d is a unit mod A . Therefore there exist elements u E Q, a E A such that du = 1 -a whence du(l+a+a2+ . . . +a'-') = 1. Thus d is a unit of Q. Hence we have to show that if f E H ( A ) and D(aa,, . . ., ask) is a unit of Q, then f E &(H(A)).Let @ ( H ( A ) ) = @ the set of those elements f satisfying our hypothesis. LZ is not empty since g E LE. Let f , g E H ( A ) , then using matrix notation for the linear coefficients o f f , g, and f o g, we can write
C
f = a,+Fx+
uAxA, g = b,+G$+
where F, G , C are suitable k x k-matrices over Q. Then f o g = a+ , Fb,+ FGg mod ( A ) whence C ICI
C
u(A) B 2
E
bA$,
FG mod A , thus
IFIiGIm o d A
(6.4) (6.4) shows that @ is a subsemigroup of H ( A )since every unit mod A is also a unit of Q , by a preceding result. It now suffices to show that every element g of @ has a left inverse in &. But if we show that g has a left inverse in H(A), we are done, by (6.4). Let g = a+, C aAgA, then g =
(a,+$)o
C
it is of the form
o(A) s 1
C
o(a)
=I
Since a , + ~ has a left inverse in H(A),
a&'.
we can assume that g =
3
C
If g has a left inverse, then clearly
a$.
u(A)
1
bAgA.We proceed by induction on I. If I = 1,
o(A) 3 1
then A = (0),thus g = Gg where G is a matrix and ]GI is a unit of Q. In this case, g has a left inverse in H(A), by a well-known theorem on matrices. Let 1 =- 1 and g = @+ C bAgA,G a matrix, and [GI a .(A)
ZZ
2
unit of Q. We put Ql = Q I A'-' and let 17 : Q .+ Q I A'-' be the canonical epimorphism. Then (qA)'-' = 7 k - l = (0). Let yF be the matrix being obtained from applying 7 to each entry of F, F any matrix over Q, then c k ( q ) g = (qG)g+ C (176,)~' E @(H(qA)). By induction hypo.(a)
thesis, there exists f = &+
Z=Z
C
o(a)
32
aAgAE H(qA)such that f o C,(r])g = g. Let
B7
IDEAL POWER SEMIGROUPS OVER FACTOR RINGS OF DEDEKIND DOMAINS
f = Fx
+ C z a,xa E H ( A ) such that C,(q)f o(a)
= f, then f o g =
183
x +Ij where
Ij = o mod (A'-'). Let GI be a matrix over Q such that GIG = E, the identity matrix, and 91 = Glx. Then g l o g = x + I where I = o mod ( A ) . If we
set f l = f-Ij.91, then f l o g = fog-(Ijog1)og = x + I j - I j o ( ~ + I ) . By Taylor's formula Ijo(x+l) = lJox = lJ whence f l o g = But H ( A ) is closed With respect to , -, as remarked in the proof of Lemma 5.9, and therefore f l E H(A).
+
x.
6.41. Corollary. If A is a nilpotent ideal of Q, then T(A) = &(S(A)), therefore T(A)is a group.
Proof. J ( A ) = &(H(A))implies T(A) E d(S(A)).Conversely &(S(A))C s(A)n u,(A)= T(A). 6.42. Corollary. If A is a nilpotent ideal of Q and q :Q Qlis an epimorphism such that ker Ker q is nilpotent, then C,(q) J ( A ) = J(qA) and V,(r)T(A)= T ( W --f
Proof. By Th. 6.4, J ( A ) = { z a , x ' € H ( A ) I D(aal, . . ., a,J is a unit of Q}. Since qA is also nilpotent, we also have J(qA) = {xb,zA€ H(qA)I D(bal, . . . , baJ E &(QI)}. Suppose we have already shown that q-l&(Ql)= &(Q), then Prop. 6.2 a) implies C,(q) J ( A ) = J(qA) while the second assertion follows from the first one and ch. 3, diagram fig. 3.1. Thus it remains to show that q-lJ(Ql)= &(Q). Let eE &(Q), then qe E Conversely if e € q-l&(Ql) then there exists f1 E Q l such that (qe)f1 = 1. Let f E Q such that f l = qf, then q ( e f ) = 1, hence ef = 1 - a where a E ker Ker q. By hypothesis ef ( 1 +a+ . . . +am-') = 1 for some m whence e E &(Q).
&(el).
7. Ideal power semigroups over factor rings of Dedekind domains 7.1. Let R be any Dedekind domain, U a non-trivial ideal of R,q(U) :R + R I U the canonical epimorphism, and W any ideal of R. Then q(U)W = (W+ U )1 U is an ideal of R 1 U . Let G stand for any of the letters H , J , S, T of the preceding section. Our aim is to get information on the subsemigroups G(q(U)W)= G,(U, W ) of C,(R I U ) , V,(R 1 U ) ,respectively, using the results of 0 6.
-
7.2. Suppose that U = C D where C , D are comaximal ideals of R.Then y :R I U R 1 C X R 1 D, where yq(U)r = (q(C)r,q(D)r) is an isomorphism
184
COMPOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
CH. 4
(see the proof of Prop. 4.21). Also yq(U)W = q(C)Wxq(D)W whence Gk(U, W ) % G(q(C)WXQ(D)W)2 Gk(C,W )X Gk(D,W ) ,by Prop. 6.2 d) and Prop. 6.3. Hence if U = P p . . . P: is the primary decomposition Of u, then G , ( u , w)% Gk(P,"',W ) X . . . xGk(Pf., w). Let P be any non-trivial prime ideal, W # (0) an ideal of R such that W = P f Q where Q, P are comaximal ideals in R, and f 3 0. 'Then q(P")W = (q(P)pf) (q(P")Q)= q(P')Pf since Q(P')Q = RIP". Thus
G,(P", W ) = G(q(Pe)W)= G(r(Pe)Pf)= Gk(P', P f ) .
(7.2)
We summarize the results so far obtained: 7.21. Proposition. Let G = H , J , S , .or T , Pi, i = 1, 2, trivial prime ideals, e; =- 0, .(. 3 0, i = 1, . . ., r, then
.. ., r f s non-
7.22. This proposition reduces the investigation of the semigroups Gk(U,W ) , U non-trivial, W arbitrary ideal of R, to considering semigroups Gk(P', (0)) and G,(P", P f ) where P is a non-trivial prime ideal, e z 0, f a 0. I f f a e, then Gk(Pe,P f ) = G((0))= Gk(P', (0)).
7.3. By 5 6.1, H( (0)) = L(R 1 P"),the subsemigroup of C,(R I P")consisting
of all polynomial vectors with linear polynomials as their components. Clearly @(a) maps H( (0)) isomorphically onto S( (0)) and therefore J((0)) isomorphically onto T((0)).By Th. 6.4, J ( ( 0 ) ) is a group consisting of all those linear polynomial vectors whose determinant is a unit. The semigroup H((O)),and the group J ( ( 0 ) ) in particular, have been studied very thoroughly in a great number of papers. Furthermore G,(Pe, P") = G(RI P") which is one of the semigroups ck(RIPe),@(@>-'v,(R/P"),y,(RIP'), Uk(R[P").By Th. 5.94, the last two of these semigroups will be completely investigated-if R I P is finite-as soon as the structure of Sk(Pe-l,P ) , T,(Pe-', P ) is known while little is known about the first two semigroups. 7.4. The remainder of this section will be devoted to the semigroups G k ( p ,P f ) where G = H , J , S , or T, P is a non-trivial prime ideal, R 1 P finite, and e w O,.f z 0.
07
IDEAL POWER SEMIGROUPS OVER FACTOR RINGS OF DEDEKIND DOMAINS
,185
7.41. Proposition. Let r =se and 8 : RIP' RIP' be the unique epimorphism such that 8q(Pe)= q(Pr).Then G(6) Gk(P', P f ) = Gk(Pr,P f ) where G(8) = c k ( 8 ) i f G = H , J and G(8) = Vk(8)i f G = S, T. -+
Proof. By Prop. 6.2, Ck(8)Hk(Pe,P f ) = ck('8)H(q(P')Pf)= H(V(Pr)P)= Hk(Pr,P f ) whence, by ch. 3, diagram fig. 3.1, the proposition also holds for G = S . Since q(P')Pf is a nilpotent ideal of RIP', and ker Ker 8 = P'I P', r > 0, is also a nilpotent ideal of RIP", Cor 6.42 implies that
Ck(6)Jk(P", P f ) = ck(8) J(V(Pe)Pf)= J(q(P')Pf} = Jk(Pr, P f ) which is also true for r = 0 since in this case, I Jk(P', P f )I = 1. Again ch. 3, diagram fig. 3.1 proves the proposition for G = T. 7.42. Proposition. J,(P', P f ) and T,(P', P f ) are groups. Proof. Since Gk(Pe,P f ) = G(q(P')Pf)and q(Pe)Pfis a nilpotent ideal of RIP', Th. 6.4 and Cor. 6.41 imply the statement of the proposition. 7.43. Proposition. Let rj = q(Pe),a E P, a cf P2, and W, a vector system mod P' containing the zero vector 6, for any integer r. Then every polynomial vector
where b,f W e , b,E We-f(u(L)-l), for L f 0, belongs to Hk(P",P f ) , every element of Hk(Pe,P f ) is of the form (7.4) and the vectors (7.4) are pairwise distinct. A vector (7.4) belongs to Jk(P", P f ) if and only if Det (b6,, . . ., b,J $ 0 mod P.
I ~ J ' a,
Proof. Let E Hk(Pe,P f ) ,then f = where = o mod (qPfY(')-'. But Ra+P' = P implies qP = ( R j P') (qa) whence (qPf)"("-' = q(Raf(u(L)-l)). Thus, for suitable 6, f Rk,we have f = C,(q) (6,
+ C b,(afy(L)-lE'. Since 1x0
)
b, s b, mod Pe-f(ucL)-l) if and only if (af)"(')'-'b, = (af)u(L)-lbL mod P', we conclude that is of the form (7.4). The other two statements on Hk(P', pf) are clear. Furthermore, by Th. 6.4, a vector (7.4) belongs to Jk(P', P)if and only if D(ybal, . . .,qbs,) = q ~ ( b * ,.,. ., bOk)is a unit of RIP". This is the case if and only if D(bsl, . . ., b6J $ 0 mod P .
186
COMPOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
CH.
4
7.44. Remark. The greatest amongst the degrees of the polynomials in the vectors (7.4)is called the length of Hk(Pe,Pf)and Jk(P",Pf).Thus the length is the greatest integer 1 such that e-f(I- 1 ) =- 0, hence it is the least integer greater than or equal to elf. 7.5. Proposition. Let a E P, a 8. P2,Z , a vector system m o d P , W , a vector system mod P' containing the zero vector 0,for all integers r, and x the mapping defined in 0 5.6. Then the mappings
9
-+
b,+
C L W O
bC(af)u(L)-19L mod P e ,
9E Z e ,
(7.5)
from 2, to (R where boE We, b, E We-f~v~+l)-tk(t), L -= 0, belong to XSk(Pe,Pf),every mapping of xSk(Pe,Pf)has the form (7.5) and the mappings of (7.5) are pairwise distinct. A mapping (7.5) belongs to xTk(Pe,pf) i f and only if Det (bdl, . . ., b,J 3 0 mod P. 7.51. Remark. Prop. 5.92 is a special case of Prop. 7.5.
Proof. Using the same argument as in the proof of Lemma 5.51, we may take u, = d,af("(')-l)t,.
7.53. Proof of Prop. 7.5. We proceed along the same lines as in the proof of Prop. 5.92., now using Lemma 7.52 and also considering Prop. 7.43. The detailed proof is left to the reader. 7.54. Corollary. If
07
IDEAL POWER SEMIGROUPS OVER FACTOR RINGS OF DEDEKIND DOMAINS
187
7.55. Theorem. Let R (P be finite, e =- 0, f w 0 . Then the groups Jk(Pe,P f ) and Tk(Pe,P f ) are soluble if and only if the linear group GL(k, R 1 P ) is soluble. The group Uk(RIPe), e =- 1, is soluble if and only if GL(k, R I P ) and the symmetric group Sym ( R IP)k are soluble.
Proof. By Prop. 7.41, there exists an epimorphism from Gk(Pe,P f ) to Gk(P,P f ) = Gk(P,(0)), for G = J and T . But Gk(P,(0))is isomorphic to the group of all polynomial vectors over R I P of the form Ax 6, I A 1 # 0, by 9 7.3. The mapping 6 defined by d ( A ~ f 6 )= A is obviously an epimorphism whose image is GL(k, R I P). Hence GL(k, RIP) is a homomorphic image of G,(P', P f ) under some epimorphism 6. By Cor. 7.54, 1 ker Ker 6 I is some power of q whence ker Ker 8 is a p-group and therefore is soluble. Thus the first assertion of the theorem is true while the second statement follows from Th. 5.94 and well-known properties of wreath products.
+
7.56. Remark. From group theory we know that GL(k, RIP) is soluble
if and only if k = 1 or if k = 2 and / R I P / = 2 , 3 while Sym (RIP)k is soluble if and only if k = 1, IRIPI = 2,3 or k = 2, and iRIP/ = 2. 7.6. By definition of the semigroups Gk(Pe,pf), the mapping @(o) maps
Hk(pe,P f ) Onto Sk(pe,P f ) and Jk(Pe,P f ) onto Tk(pe,P f ) . The proof Of Lemma 5.9 shows that (Hk(P",P f ) ; f, -, 0 , 0 ) is an !&multioperator group where si! = ( o ) -Hk(Pe,P f )is, in fact, a near-ring-and Jk(P', P f ) is a group by Th. 6.4. Hence we see that Kl = ker Ker q ( a ) I Hk(P", pf) is an ideal of Hk(Pe,P f ) and Kz = ker Ker p(o)I Jk(Pe,P f ) is a normal subgroup of Jk(Pe,P f ) .We are going to derive some results on Kl and Kz. 7.61. Lemma. Let q = q(Pe), a € P , a 4P2, and W , a vector system
mod P' containing the zero vector 0,for every integer r. Then the polynomial vectors
(7.61)
belong to Hk(Pe,~ f ) every , vector of H k ( P ,P*) is of the form (7.61), and the vectors of (7.61) are pairwise distinct if c, E We, c, E We,(,(,,-,), L =- 0. A polynomial vector (7.61) belongs to Jk(P",P f ) if and only if Det (ca,, . . ., ca,) g! 0 mod P .
188
COMPOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
c
CH.
4
Proof. Let f be a vector (7.61). By definition oft, (see ch. 3,§ 8.6), we have t, = a,,$ whence f = C,(q) C b,za where b, = C ~,a,,(af)"(~)-~ = a s
a=o
,=a
bA(af)u(A)-l,for some b,. Hence f E Hk(Pe, Pf).Suppose that some polynomial vector (7.61) equals 0, but c,(u')"'~)-' 9 o mod p", for some A. We take 2 maximal in hfk with respect to this property, thus b, E cA (af)"(")-l D mod P', a contradiction. Hence the polynomial vectors (7.61) are pairwise distinct. Since the number of polynomial vectors (7.61) equals the number of polynomial vectors (7.4), by Prop. 7.43 every elePf)is of the form (7.61). If ment of Hk(Pe,
+
then this equation also holds if q = q(P). Hence caj E bajmod P, .. . ,k, thus D ( c , ~., . ., ca,) = D(bal, . . .,bSk)mod P. Prop. 7.43 implies the last statement of the proposition.
j = 1,
7.62. Theorem. The vectors
belong to K1,every element of K1 is of the form (7.62), and the vectors (7.62) are pairwise distinct, if u, E Wmin F k ( ~ , The elements of ( 1 -f(m--l)). KB are just the polynomial vectors F+4 (7.63) where 9 is a vector of (7.62), and the vectors (7.63) are pairwise distinct.
Proof. Suppose we have already proved the statement on K I . Then every polynomial vector (7.63) belongs to @(u)-'U,(R/P')nH,(P",Pf)= Jk(Pe,Pf) and therefore to K2. Conversely if f E K2, then f-X C K1, thus f is a vector of (7.63). That the vectors of (7.63) are pairwise distinct is obvious. We now prove the first assertion: Let g E K I , then, by Lemma 7.61, g is of the form (7.61), and Q E Klimplies that c,+ c,(af)u6)-1t,E{Pe}k. We claim that c,(af)"(')-' z D mod Pe--Ek(L), for every 6. Suppose by way of contradiction, that L is maximal in the lexicographic order of N kwith respect to ~ , ( a ~ ) " ( '3) - D~ mod P"-',('). But then we have a contradiction to Lemma 5.52. Since, by the proof of Prop. 7.43, qP = (RIP")(qa), thus qp'-Ek(L)= (R1 P")(qd-ek")), we have C,(U~)"(~)-' = b,a'-"k(')mod P '.
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IDEAL POWER SEMIGROUPS OVER FACTOR RINGS OF DEDEKIND DOMAINS
189
for some b,E Rk and every L. Hence we may replace ck(af)@(')-'in (7.61) by b,d-Ek('),for every L which satisfies e - F k ( i ) >f(c(~)-l)and so we obtain a representation for g which is of the form (7.62) but without the condition on u, being taken in account. But ~ ~ ( =6 s) e - f ( o ( ~ ) - l )if and only if $(a([)-1) =se-Ek(L), thus min
(&k(L),
e-f(o(L)-I))+max
Hence if we replace an element -
u, = u, mod P
min (E ~ ( )L, e-
u,, i
(e-Ek(L),
=-
~ ( U ( L ) - I )= ) e (7.64)
o,in (7.62) by some U, such that
' ( r ( L ) - l ) ) , the vector g remains unchanged. Thus g may be taken as in the theorem. Conversely, every element of (7.62) is of the form (7.61) and therefore belongs to Hk(P', Pf>. Moreover such an element belongs to Ck($ {Pe}k,by ch. 3, Lemma 8.61, and therefore to K I .That the elements (7.62) are pairwise distinct follows from (7.64).
7.63. Corollary. @(a) maps Hk(Pe, ~ f isomorphically ) onto sk(pe,~ f j and Jk(P",Pf)isomorphically onto Tk(P', P f ) i f and only if e ,rf (4- 1) whereq = IRIP;.
Proof. By Th. 7.62, I K z ( = IK11, hence we see that the restriction of @(o) to f f k ( p e , P f ) ,Jk(P', Pf),resp., is an isomorphism if and only if [ K l l = 1. By Th. 7.62, this is the case if and only if min ( F ~ ( L ) , e - f ( a ( i ) - 1)) -c 0 for every L > 0. This is true if and only if E ~ ( L )=- 0 implies e-f(u(L)- 1 ) G 0. Since L = (4, 0, . ..,0) implies E ~ ( L )= 1, by ch. 3, (8.32) and also a(r) = q, we have e ==f(q--1). Conversely, since Ek(L) =- 0 implies a(&)>- q, e G$(q - 1 ) yields the implication in question. 7.64. Lemma.
If (il, . . ., ik) = L,
then max (e-Ek(L), f ( o ( ~ ) - l ) ) >- e/2.
Proof. Suppose that the lemma fails to hold, then there exists some L z o such that f(o(i)-l)
<
e/2 -= E ~ ( L ) . But
c &(ir),and by ch. 3, k
&k(L)
=
r=l
(8.32), we have F ( i r ) = (ir-sq(ir))/(q-l). By a well-known formula
2 i r ) 2sq(ir),hence
, k
of elementary arithmetic, sq(
k
1
=z
r=l
r=l
190
COMPOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
-
CH.
4
7.65. Theorem. Let x + g be as in (7.63), then x+g g deJines a mapping z : K2 + (K1; +, -, 0 ) and z is an isomorphism. (K1; +, -, 0) is isomorphic to the direct product of k copies of the direct product
n((R,
pmin (QW, e-f(U(+-l));
p-group.
+ , -, 0) I L E Mk) and therefore is an abelian
Proof. By Th. 7.62, z : KO -+ K1 is a bijection. Let n be the least integer greater than or equal to e/2, and f l , f 2 E K2. By Th. 7.62, f l = x+g1, f 2 = x+g2 where giEKl, i = 1,2, hence f 1 0 f 2 = x+g2+g10(x+g2). By Th. 7.62 and Lemma 7.64, gi = C,(q)a"&, i = 1,2, for some ljiECk(R). Thus, since 2n a e, Taylor's formula implies that g1 o (x+ g2) = ck(q)(a"Bo(~+a"fj2)) = c k ( q > d ) l = 91. Hence f1.72 = ~ + g 1 + 9 2 , therefore r(flof2) = g l + g 2 = rfl+zf2. This means that z is an isomorphism. The mapping which assigns to each g of (7.62) the family
{uLmod P
(K1;
min
E~(L
(
+, -,O)to
'2
e
1 6 GMk)}, is obviously an isomorphism from
-f(w(+l))
min ( E ~ ( L ) . e-.f(U(+l)).
rI((RlP
, +,
-,O)kIGw.
7.7. By the homomorphism theorem, Sk(pe, Pf)2 Hk(Pe, Pf)lKl, T k ( p e pf) , Y Jk(Pe,P f )I K2. We have-just investigated Kl and K2, so it is sufficient to discuss Hk(P",P f ) and Jk(Pe,Pf)in order to get some information on S k ( p e , Pf) and Tk(Pe,Pf). The near-ring H,(P", pf) has not been considered yet. There is something known about the group Jk(Pe,P f ) ,in particular, for R being the ring of rational integers and k = 1. The farthest-reaching results for this particular case have been obtained by KOWOL[l]. All these investigations have turned out to be quite elaborate.
8. Characterization of permutation polynomials over finite fields 8.1. Permutation polynomials over finite fields in one indeterminate have already been considered as early as a century ago by HERMITE. Since then plenty of papers have been written on this topic whereas permutation polynomials over finite fields in more than one indeterminate had not been investigated until a few years ago with rather few re&. This and the subsequent section are to develop the most remarkable aspects of the theory of permutation polynomials over a finite field K of order q in one indeterminate x .
88
191
CHARACTERIZATION OF PERMUTATION POLYNOMIALS OVER FINITE FIELDS
8.2. The first and most fundamental problem is finding a method of deciding in the simplest possible way whether or not a given polynomial f €K [ x ]is a permutation polynomial. Basically there is always an answer, namely by computing the values off for all elements of K , but this method does not give any deeper insight into the nature of permutation polynomials. In order to state and to prove a theorem due to HERMITE and DICKSON which shall illustrate what we mean by “deeper insight”, we require the following definition : The reduction of a polynomial f E K [ x ] is the (well-defined) polynomial g E K [ x ] such that [g]-= q and g = f mod (xq-x). Since ( 9 - x ) = {0}, we conclude that g is just the polynomial of least possible degree such that ag = a$ We can now state 8.21. Theorem. A polynomialf over thefinitefield K of order q and characteristic p is a permutation polynomial if and only if a) f has exactly one root in K, b) The reduction off I , 0 -= t < q - 1 , t g 0 mod p , has a degree less than or equal to q -2. The proof will depend on two lemmas which we are going to prove first. 8.22. Lemma. Let K be a finite field of order q and s K. Then, if
n,= 1 -
5j , we have n,s = 1 and n,t
sq-l-j
= 0, for t # s.
j=q-l
0
8.23. Corollary. Let p :K -,Kbe any mapping. Then p =
C
cj5j
where
j=q-1
c0 =C((es)(l- s ’ - ~ ) / s € K ) .
Proof. n,s
=
1-
c 0
sq-’ = 1. Furthermore
j=q-1
tns = s - s q + t - [ q -
C 0
Sq--1--i5i+l
= s-
C 0
ssq--l--i
5 i =sn,
i=q-1
j=q--2
whence t ( n J )= s(n,t). Hence nst = 0, for t # s. Furthermore p = s E K ) implies the corollary.
c((ps)z,I
8.24. Lemma. Let a,, . . .,aq be any family of elements of a finite field K of order q. Then these elements are pairwise distinct if and only if 0, i= 1
-1,
for t = 1,2,
for t = q - I .
.. . , q - 2
192
COMPOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
CH.
4
Proof. By Lemma 8.22, the elements a,, . . ., u4 are pairwise distinct if and only if the polynomial function
equals the constant function with value 1 . Since (0) = (xq-x), this is true if and only if the condition of the lemma is satisfied. 8.25. Proof of Th. 8.21. By Cor. 8.23. the reduction o f f ' is some 0
polynomial
C
j=q-1
cjxj where cq-l = -C(f(s)'isE K). If f is a per-
mutation polynomial, then a) holds obviously and by Lemma 8.24, C(f(s)'IsE K) = 0, t = I , . . ., q - 2 , whence b) follows. Conversely let s E K ) = - 1 while a) and b) be satisfied. Then a) implies that C(f(~)~-ll b) implies C(f(s)'js E K) = 0, for 0 i t < q - 1, f g 0 mod p . Since char K = p , we have C ( ~ ( SIs)E' K) ~ = (C(f(s)'I s E K))" whence we can drop the hypothesis t $ 0 mod p . Hence Lemma 8.24 implies that f is a permutation polynomial.
8.26. Corollary. If d > 1 is a divisor of q - 1, then there is no permutation polynomial over K of degree d . 8.3. Next we will characterize permutation polynomials from a quite different point of view. If K is any field and f € K [ x ] , then @(f) = [ f ( x )- f ( y ) ] / [ x-y ] is a polynomial of K[x, y ] which is a unique factorization domain. Let C be the algebraic closure of K. An irreducible polynomial u E K [ x ,y ] is called absolutely irreducible if u is irreducible regarded as a polynomial of C [ x , y ] .This definition can be extended in the obvious way to polynomials u of K[x,, . . ., x,,]. A polynomial f E K [ x ] is called exceptional over K if no absolutely irreducible factor occurs in the prime factorization of @(f)E K[x, y ] . 8.31. Theorem. Let K be any jield, f E K [ x ] exceptional over K , and char K = 0 or char K w [f]. Then of E P,(K) is an injective polynomial .function. thus f is a permutation polynomial if K isjnite. (Recently S. D. COHEN [l], by means of algebraic number theory, has proved that, if K is finite, every exceptional polynomial over K-without any restriction -is a permutation polynomial).
08
CHARACTERIZATION OF PERMUTATION POLYNOMIALS OVER FINITE FIELDS
193
The proof of this theorem depends on some lemmas we are going to prove first. C wiIl denote any algebraic closure of K we keep fixed. 8.4. Every non-zero polynomial u E K [ x , y ] has a representation u = pox"+plx"-l+
. . . +p,
(8.41)
where p i E K [ y ] ,i = 0, . . .,n, po # 0. If po E K , then u is called semimonic in x while u is called monk in x if p o = 1. Let u E K [ x ,y ] be semimonic in x. A splitting field of u over K is a subfield S of C which is a finite normal extension of K such that u, being regarded as a polynomial of S [ x ,y ] , splits into absolutely irreducible factors. A splitting field of u over K is called minimal if it is contained in every splitting field of u over K. 8.41. Lemma. Let u E K [ x ,y ] be semimonic in x . Then u has one and only one minimal splitting field S,(u) over K.
Proof. p;'u E C[x,y ] has a unique decomposition into irreducible polynomials, monic in x, up to the order, say p;lu = v1v2 . . . v,
(8.42)
If we adjoin all the coefficients of the polynomials wi and their conjugates to K , we obtain some finite normal extension field S of K which is a subfield of C . Since (8.42) also holds in S , this field is a splitting field of u over K. If T is any splittingfieldof u over K, then (8.42) is also the decomposition of p;lu into irreducible polynomials, monic in x , in T [ x ,y ] . Thus, since T is normal over K, we have T 2 S. Hence S is a minimal splitting field of u over K. 8.42. Let S , be a normal extension field of K and vl, v,ES,[x, y ] . ol, v2 are called conjugate over K if there is an automorphism a of S , fixing K elementwise such that a [ x ,y]v1= VZ. "Being conjugate" is obviously an equivalence relation in S J x , y ] . 8.43. Lemma. Let w E K [ x ,y ] be monic in x and irreducible over K .
Then any two divisors of w in S,(w) that are monic in x and irreducible, are conjugate over K .
194
COMPOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
CH.
4
Proof. Let Z = { z E S,(w) 1 z separable over K}, then 2 is a subfield of S,(w). Since S,(w) is a finite normal extension of K, we conclude that Z is also a finite normal extension of K. Let
w = v1v2 . . . vr
(8.43)
be the decomposition of w into polynomials monic in x and irreducible over 2. If a E Gal Z IK, the Galois group of Z over K, the automorphism a[x,y ] of Z [ x ,y ] permutes the polynomials vi. If (Gal 21K ) [x, y ] = {a[x,y] la E Gal 21K}, then the product of all the different elements of the orbit of vl under (Gal ZI K ) [x,y ] is a divisor of w.This product is, however a polynomial over K and monk in x,hence it equals w.Therefore the orbit of v1 consists of vl, . . ., vr,and the lemma holds for Z = S,(w). If Z is a proper subfield of SK(w),then Z is not a splitting field of w, therefore, WLOG, we may assume that vl splits into at least two irreducible factors in SK(w),say (8.44) v1 = v11v12 . . . VIS where vli is monic in x and irreducible over S,(w). Let char K = p , and e the exponent of the extension S,(w) IK. Then vfjE Z [ x , y ] , f o r j = 1, . . ., s. On the other hand, (8.44) implies that 1
= vPevPe 11 12
. . . vz.
(8.45)
Hence v$ is a divisor of vlpe in Z [ x , y ] . Since v1 is irreducible over 2 and Z[x, y ] is a unique factorization domain, v$ = v?, for some rj, j = 1, . . .,s. Hence (8.46) v;; = vav?2 . . . v z ; hence, by the uniqueness of the prime factor decomposition in S,(w), all the elements vlj are equal, therefore
v1 = v"l
(8.47)
is the prime factor decomposition of v1 in S,(w). If a E Gal Z I K such , = vi and B is the extension of a to an automorphism of that ~ [ xylv, S,(w), then (8.47) implies v, = [Btx,Ylvll]" (8.48) which is the prime factor decomposition of ct in S,(w). Substitution of (8.48) into (8.43) yields the decomposition of w in S,(w) into factors which are monic in x and irreducible. Hence any two of these factors are conjugate to uI1 over K.
88
CHARACTERIZATION OF PERMUTATION POLYNOMIALS OVER FINITE FIELDS
195
8.5. Lemma. Let u E K [ x ,y ] be semimonic in x , a = a(x, y ) the form consisting of all terms of maximal degree in the normalform of u, a(x, 0) # 0, and c E K such that the polynomial a(x, c ) 6K [XI has no multiple roots. Then SK(u)is a subfield of the splitting field of a(x, c ) over K in C .
Proof. Let D be the splitting field of a(x, c) over K i n C , and pol24 = UIUZ . . . u,
(8.51)
the decomposition of p;lu into polynomials monic in x and irreducible over D. If U is a finite generating set of SK(u)over K and T is the set of all elements of C conjugate over D to elements of U , then L = D(T)is a finite normal extension of D containing SK(u)and is the least extension w.r.t. being a normal extension of D in C and containing SK(u).Clearly L is a splitting field of u over D, hence L is also a splitting field for every uj over D. If every uj is absolutely irreducible, then D 2 S,(U), and we are done. Hence we may assume, WLOG, that u1 splits into a t least two factors in L. Let u1 = u;ulz
.. . UIS
(8.52)
be the decomposition of u1 into polynomials monic in x and irreducible over L. Since L 2 S,(u,), the elements ulj are polynomials over S,(ul). By Lemma 8.43, there is an automorphism a of S,(u,) fixing D elementwise such that a[x, y]uli = ulj. Let a, = a l ( ~y, ) and alj = alj(x,y ) , j = 1, . . ., s, be the forms consisting of the terms of maximal degree in the normal form of ul,ulj, resp. Then a, = al1alZ. . . a,, whence
By (8.51), a1 divides a whence nl(x, c ) divides a(x, c). Since D is the splitting field of a(x, c ) in C , a,(x,c) splits into linear factors in D and therefore in L. By (8.53), every a,(x, c) splits into linear factors of D, thus alj(x,c) E D[x], j = 1, . . ., s. But a f x ,y]uli = ulj implies a[x, y]ali = arj whence a [ x , y ] a,,(x, c ) = ali(x,c)- But a fixes D elementwise, therefore al,(x, c ) = alj(x,c), for all pairs (i, j ) . Since a,(x, c) divides a(x, c) which, by hypothesis, has no multiple roots, al(x, c) € D. Hence al(x, y ) is a multiple of y and a(x, 0) = 0, a contradiction. Thus every uj is absolutely irreducible and the proof is complete.
196
COMPOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
CH.
4
8.6. Lemma. Let m =- 0 be an odd integer and a E K. Then xm-a is either irreducible over K, or there exists 0 < d < m, dim such that xd-b divides x"-a for some b E K.
Proof. Let m
=p
Then x p - a =
be a prime and x p - a = f g , f, gEK[xl, Tfl, [gl < P.
n(x-5"~)
P-1
"=O
where c is in the splitting field of xp-a,
and 5, runs trough the p-th roots of unity. Hence b = f ( 0 ) = & 5c' where 5 is some p-th root of unity and 0 < r -= p. Therefore ( & b)P = - crp = a'. There exist integers u, v such that wfvp = 1 whence a = a u r + v ~ - (+b y pavp. Therefore a is the p-th power of some element a, E K whence x -a, divides x p-a. Now we use induction on m and assume that m is not a prime, and xm-a is not irreducible. Then every root of P - a is of degree less than m over K. If p is a prime dividing m, k = m[p, and we assume that xk-a is not irreducible, then, by induction, xd-b divides xk -a for dlk, d -=z k and b E K whence xmlP-a = (xJ-b)g(x), for some g ( x ) E K [ x ] Hence . P - a = (xdp-b)g(xp),dp/m, dp < m. Therefore we may assume that xk-a is irreducible over K. Let c be a root of 2 - a . By the first part of the proof, xp-c is either irreducible over K(c) or has some linear factor in K(c). in the first case e is a root of xp-c, then [K(e):K ] = [K(e):K(c)][K(c): K] = pk = m, but em = ck = a implies [K(e):K ] -= m, a contradiction. Therefore x p-c has some root e E K(c). If %(s) denotes the norm of s w.r.t. K(c)IK, then k odd implies that /az(c) = (( - a ) = a while, on the other hand m ( c ) = /aZ(eP)= /aZ(e).. Thus a is the p-th power of some a, E K, therefore xm-a = (xk)"-a:, hence xm-a has the factor xk-a, in K[x]. 8.61. Lemma. Let char K f n , x"-a be an irreducible polynomial in K [ x ] ,c a root of ?-a, and suppose 1 is the only n-th root of unity in K. Then K is the only subfield of K(c) which is normal over K.
Proof. We choose a subfield L of K(c) maximal w.r.t. being normal over K, and assume, by way of contradiction, that L contains K properly. If [K(c):L] = d, then din. Let cl, . . ., c d be the roots of the minimal polynomial for c over L. Since every ci is a root of x" -a, we have ci = cTi where 5, is an n-th root of unity. Hence clc2 . . . c, = cdT where 5 is some n-th root of unity, on the other hand, clc2 . . cd E L, thus 5 E K(c).
.
08
CHARACTERIZATION OF PERMUTATION POLYNOMIALS OVER FINITE FIELDS
197
Hence L(C) is a subfield of K(c). Since L(C)is a splitting field for some suitable polynomial over K , L(5) is normal over K , and the maximality of L implies that C EL. Therefore cd E L. Let n = md, and P - a = fg. Then x"-a = f ( X d ) g ( x d ) whence P - a is also irreducible over K with cd as a root. Therefore [K(cd):K ] = m -- [ L :K ] . Since K(cd) C L, we have L = K(cd). Since L is normal over K , the polynomial x"-a splits into linear factors in L [ x ] ,thus L contains every m-th root of unity. Let p be the least prime dividing m and E a primitive p-th root of unity. Then E E L and [K(&):K ] divides m. Hence [K(E):K ] = 1 or [K(E): K ] p . But [ K ( E )K: ] / ( p - 1 ) whence E E K , and char K z p implies E f 1. This is a contradiction since E is also an n-th root of unity.
8.62. Lemma. Let char K f n and suppose 1 is the only n-th root of unity in K. r f a E K, then there is some root c E C of x"-a such that K(c)n K(C) = K, fur every root of unity [ E C . Proof. If n were even, then char K would be odd and - 1 E K would be an n-th root of unity different from 1. Thus n is odd. Let d be the least divisor of n such that x"-a has some divisor x d - b E K[xx],then Lemma 8.6. implies that x d - b is irreducible in K [ x ] . Let c be a root of xd- b. Since K contains no d-th root of unity apart from 1, Lemma 8.61 implies that K is the only subfield of K(c) which is normal over K . If C E C is a root of unity, then K ( [ ) is a normal, separable, abelian extension of K , thus K(c)nK(C) is normal over K since Gal K(5)(Kis abelian. Hence ~ ( c ) n ~ (=r K ). 8.7. Lemma. Let f E K [ x ] be exceptional over K and a, b E K, a # b, = f ' ( b ) = 0.
f(a) = f(b). Thenf'(a)
Proof. Let @(f) = p(x, y ) as in 9 8.3, then V(a, b ) = 0 . Since p(x, y ) is semimonic in x , this polynomial is the product of an element of K and factors that are monic in x and irreducible in K [ x ,y ] . Sincef is exceptional none of these factors is absolutely irreducible, therefore each such factor g splits into at least two absolutely irreducible factors in S,(p), thus g splits into these factors in S,(g). By Lemma 8.43, these factors are conjugate over K. Hence there are at least two factors pl, pz amongst the irreducible factors of g? in S,(p) and therefore also amongst the irreducible factors of u = f ( x ) - f ( y ) in S,(p) such that rpl(a, b) =
198
COMPOSITION AND POLYNOMIAL FUNCTIONSOVER RINGS AND FIELDS
= p2(a, b) = 0. We conclude that f ' ( a ) = (&)/(ax)
CH.
(a, b) = 0 andf'(b)
4
=
-@ U ) / @ Y ) (a,b) = 0. 8.71. Lemma. Let f E K [ x ]be exceptional over K andf = anx"+an-,x"-l+ + . .. +a,x', char K { r, r > 1, a, f 0. Then 1 is the only r-th root of unity in K.
Proof. Let
@(f) = anuluZ . . . u,
(8.71)
be the decomposition of @(f) = p into polynomials which are monic in x and irreducible over K, and w = w(x, y ) , wj = wj(x,y ) , j = 1, . . ., t, the forms consisting of the terms of minimal degrees in the normal forms of 9,uj, resp. Then w = anwlwZ . .. w,. (8.72) On the other hand,
where Tiruns through all the roots of ( xr- l ) / ( x - 1 ) in C. Suppose there is an r-th root C # 1 of unity in K. Then C is one of the elements Ci whence x-Cy divides w. By (8.72), x-5y divides some wi. WLOG, we may assume x-Cy divides wl. Since f is exceptional, the polynomial U I splits into k z 1 polynomials, monk in x and absolutely irreducible in S,(p), thus u1 also splits into these factors in S,(u,). Then Lemma 8.43 implies that these polynomials are conjugate over K. Let vl, . . ., wk be the forms consisting of the terms of minimal degree of these polynomials, then these vj are also conjugate over K, and
w1 = v1v2
. . . Vk.
(8.74)
Since x-Cy divides wl, one of the polynomials t; is divisible by x-[y. But C E K and the polynomials vj are conjugate, hence x-Ty divides every vj. Therefore w has a multiple linear factor. But since char K f r, this contradicts (8.73).
8.72. Lemma. Let f E KIx] be exceptional over K of degree n, char K { n, C a primitive n-th root of unity in C, and L Z! C a jinite extension of K such that L n K(C) = K. Then f is also exceptional over L.
08
CHARACTERIZATION OF PERMUTATION POLYNOMIALS OVER FINITE FIELDS
199
Proof. Suppose f is not exceptional over L, then the polynomial @(f) = has some factor g E L [ x ,y ] which is monic in x and absolutely irreducible, hence g is an irreducible divisor of p in S,(y). If a = a(x, y ) is the form consisting of the terms of maximal degree in the normal form of cp, then a = a,,(x"-y")/(x-y) whence a(x, 0) # 0 and a(x, 1) has no multiple roots. By Lemma 8.5, S,(p) is a subfield of K(Q. Hence the coefficients of g are contained in L n K(C) = K , and p is not exceptional over K , contradiction.
8.73. Proof of Th. 8.31. Let f E K [ x ] be exceptional over K of degree n, char K = 0 or char K =- n, and L an extension field of K in C which is maximal w.r.t. f being exceptional over L. The existence of such a field L follows from ZORN'SLemma. If uf€ P1(L) is injective, uf€ P 1 ( K ) is injective a fortiori, hence we can assume that f is not exceptional over any proper finite extension field of K in C . Suppose now that uf E P l ( K ) is not injective. Then there exist a, b E K such that a # b, f ( a ) = f(b). Let g = f ( (b-a)x a) -f(a). Then if 0 f d € K , the polynomial h = dg is exceptional over K , but not exceptional over any proper finite extension field of K in C , and h(0) = h(1) = 0. Let
+
h(x) = a,x"+a,-lx"-~+ . . . +a,x', h(x+ 1) = b,~"+b,-,x"-~+ . . . +b$,
a, # 0,
b, # 0.
For a suitable choice of d, we have b, = 1. Since h(0) = h(1) = 0 implies h'(0) = h'(1) = 0, by Lemma 8.7, we have r =- 1 and s =- 1, and by hypothesis, r , s are not divisible by char K . Hence Lemma 8.71 shows that K contains no r-th and no s-th root of unity except 1. Let 5 be a primitive n-th root of unity in C . By Lemma 8.62, there exists some root c of x'-u, such that K ( c ) n K ( [ )= K . By Lemma 8.72, f is also exceptional over K(c) whence c E K . Similarly, there exists some root d of x s - c in K . Therefore xrS-a, has the root d in K . We put @(h) = y ( x , y ) , then h(x)-h(y) = ( x - y ) p ( x , y ) whence h(x")-h(y'+l) = (a"-yr- 1) p(xs, y'+ 1). Let p(x, y ) be the form consisting of the terms of minimal degree in the normal form of p(xs,y'+ l), then
Hence --p(x/d, 1) = xrs- 1 = q. Since char K
7 rs, 1 is a simple root of
200
COMeOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
q. Let a, = a,,u,u,
. .. u,
CH.
4
(8.76)
be the decomposition into polynomials which are monic in x and irreducible over K. Then
~ ( y2' f, I) = a,p,(2, y ' f 1)
. . . u,(2, yr+ l),
thus p(x, y ) is the product of the forms consisting of the terms of minimal degree of the polynomials uj(xb,y ' f 1). Since h is exceptional over K, every uj splits into at least two irreducible factors in S,(a;) which are conjugate over K, by Lemma 8.43. Hence also every uj(xs,y'+ 1) splits into at least two factors which are conjugate over K, and so do the forms consisting of the terms of minimal degree in uj(x', y't- 1). Hence also p(x, y ) splits in S,(q) such that there is at least one other factor to each factor which is conjugate. Thus q = -p(x/d, 1 ) splits in the same way in S,(y). Therefore every root of q in K is a multiple root which is a contradiction since 1 is a simple root of q. 8.8. Under what conditions does the converse of Th. 8.31 hold? That means we want to know under what conditions on K and f E K [ x ]it is true that of injective implies that f is exceptional over K. 8.81. Theorem. There exists a sequence c1, c2, . . . of integers such that for any finite field K of order q =- c,, and (n, char K ) = 1 the following statement is true: Zf f E K [ x ] is a permutation polynomial and [f]= n =- 1 , then f is exceptional over K.
Proof. We require a lemma for the proof of the theorem. 8.82. Lemma. There exists a monotonically increasing sequence c1, C Z , . . . of positive integers with the following property: Zf K is any finite
c3,
jield such that I K I =- cd, then, for every absolutely irreducible polynomial p E K [ x ,y ] of degree d which is not of the form a(y -x), the equation p(x, y ) = 0 has a solution (x, y ) in K such that y # x.
Proof. We put co = 0 andconstructthe sequence C O , ~ 1 , 1 2 2 ,. . . recursively: Wechoosec,suchthatcdzcd-,andq-(d-1)(d-2) d(q)-k(d)>2d+2, for every q w c,, where k(d) is the constant of ch. 6, Th. 9.41. This is a
08
CHARACTERIZATION OF PERMUTATION POLYNOMIALS OVER FINITE FIELDS
201
sequence we require. In order to show this we argue as follows. Let I K I = q =- cd, and p E K[x, y ] an absolutely irreducible polynomial of degree d which is not of the form a(y-x). Then z“p(x/z, y / z ) = u(x, JI, z ) is an absolutely irreducible form of degree d of K[x, y , 23 since u(x, y , z ) = = glg2implies p(x, y ) = gl(x, y , 1) g,(x, y , 1) whence one of the gi(x, y , 1) would be a constant, thus gi = bz‘ and gi itself is a constant. By ch. 6, Th. 9.41, if n is the number of non-equivalent solutions in K of u(x, y , z ) = 0, then n a q - (d- 1) (d-2) d ( q ) - k ( d ) , thus n == 2d+3. The number of non-equivalent solutions in K, where z = 0, is at most d+2, and the number of non-equivalent solutions where z # 0 and y = x equals the number of different solutions in x of u(x, x, 1) = p(x, x) = 0. Since p(x, y ) is irreducible and not of the form a(y -x), p(x, y ) is not divisible by y-x, hencep(x, x) E K [ x ]is not thezero polynomial whencep(x, x) = 0 has at most d solutions in K. Hence there is at least one solution of u(x, y , z ) = 0 such that z f 0 and x f y , therefore p(x, y ) = 0 has a solution in K as stated in the lemma. 8.83. Proof of Th. 8.81. Let f E K[x] be a permutation polynomial such that the hypothesis of the theorem is satisfied for the constant c, of Lemma 8.82. If p: = @(f) E K[x, x], then the equation q(x, y ) = 0 has no solution (x, y ) in K such that y # x. Let q~ = a,p, . . . p , be the decomposition of p: into polynomials which are monic in x and irreducible over K. Suppose, by way of contradiction, that f is not exceptional, then, WLOG, pl is absolutely irreducible. We put [pl] = d. If pl = a(y-x), then f ( x ) - f ( y ) = (x -y)2g(x, y) , for some g(x, y ) E K[x, y l , hence f’(y) = 2(x -y) g(x, y ) -(x - Y ) ~ (Elg(x, y ) lay), thus f’(x) = 0, a contradiction since (n, char K) = 1. Therefore pl is not of the form a(y -x), and I K 1 z c,, =- cd. Hence pl(x, y ) = 0 has a solution in K such that y # x and so has ~ ( xy,) = 0, by Lemma 8.82, a contradiction. 8.84. Remark. The hypothesis (n, char K ) = 1 is indispensable, for xp is a permutation polynomial if char K = p , but xp-yp = (x-y)” shows that x p is not exceptional over K. 8.85. Lemma. Let f E K[x] be exceptional over K of degree n char K { n. Then K contains no n-th root of unity except 1.
=- 1 and
Proof. We proceed almost in the same way as in the proof of Lemma 8.71, but replace “form consisting of the terms of minimal degree” by “form
202
COMPOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
CH.
4
consisting of the terms of maximal degree". The details of the proof are left to the reader.
8.86. Theorem. If n =- 1 is an integer und K is a finite field of order q such that (4,n) = 1, (q- 1, n) w 1, and q > c,,, then there is no permutation polynomial of degree n in K[x].
Proof. Let t be any generator of the multiplicative group of K, then if ( q - 1, n) = d =- 1, we see that t'Z-lld is an n-th root of unity different from 1. By Lemma 8.85, there is no exceptional polynomial of degree n in K[x]. By Th. 8.81, there is no permutation polynomial of degree n in Kbl. 8.87. Corollary. Let n r 0 be even and K afinitefield of order q such that (q, n) = 1 and q =- c,. Then there is no permutation polynomial of degree n in K[x].
Proof. Clear. 8.88. Remark. The hypothesis (qL1, n) =- 1 is indispensable, for if (q - 1, n) = 1, then x"is a permutation polynomial in K[x]. 8.89. Remark. CARLITZ has conjectured that, for any even number n =- 0, there exists a constant b, such that, for any finite field K of odd order q > b,, there is no permutation polynomial of degree n in K . Cor. 8.87 shows that this conjecture is true for n = 2". In all the other cases, the conjecture has been verified only for n = 6 (DICKSON [2]) and n = 10 (HAYES[l]) so far.
8.9. Th. 8.31 and Th. 8.81 together show: For every n r 1, there is a constant 1, such that, for any finite field K such that char K =- I,,, a polynomial f E K [ x ] of degree n is a permutation polynomial if and only i f f i s exceptional over K. We may take I, = max (cn,n). 8.91. Remark. Th 8.86 and Cor. 8.87 tell us something about the nonexistence of permutation polynomials of given degree n in certain finite fields which is ultimately a consequence of the Riemann hypothesis for algebraic function fields over finite fields. This hypothesis, however, has
58
203
CHARACTERIZATION OF PERMUTATION POLYNOMIALS OVER FINITE FIELDS
also some consequences for the existence of certain permutation polynomials over certain fields, such as 8.92. Theorem. Let e z 1 be any integer. Then there exists some constant me such that, in any finite field of order q, q E 1 mode, q z me, there is an element a # 0 with the property: Every polynomial f = xc(x(q-l)'e+a)k, (c, q - 1 ) = 1, k 3 1, is a permutation polynomial.
Proof. Let K be a finite field of order q and q of multiplicative order e and
= 1 mod
e. Let w E K be
a system of equations in the unknowns x , y2, .. ., ye-1. (8.91) satisfies the conditions of ch. 6, Th. 9.42. since w-w" # 0, ws-1 # 0, and (w-wws/w-l) (w'--l/w--l) = (w-ww'/w-l)(ws-l/w-l) implies w(w'-w') = w'- w" whence w = 1 or w' = w', thus t = s. Hence ch. 6, Th. 9.42 implies that, for q C , where C is some positive constant, the number n of solutions of (8.91) in K satisfies n > q/2. But the number of those solutions of (8.91) for which 2 = 0 or 1, or w, or (ws--w/ws-l) is at most (e+l)e'-'. Hence if q =- me, for some suitable constant me, the system (8.91) has always a solution x, y 2 , . . ., ye-1 such that xe is different from 0, 1 , w, and (w"-w/w"-1), s = 2, 3, . . ., e-1. Let us choose such a solution x , y2, . . ., ye-1 and put a = (xe-w/l -xe) # 0. Then, for 2 =ss =se-1, we have
wS+a = w"+Since ws + a
f
xe- w - ( 1 - w")Xe+ ws- w - -___ ( l - w ) ~ ' = y:(l+a). 1 -xe 1 -xe 1 -xe 0, 1 s s =s e, we obtain
Let f be as in the theorem and suppose that for u, v E K, f(u) = f(w). Thenf(u)(q-l)'e = f ( ~ ) ( ~ - whence l)/~ Uc(q-ll/e
(u(q -1Ye + a ) N q - l ) / e =
vc(q-l)lc(w(q-l)/e+
a)Nq-l)le
.
(8.93)
If u or v equals zero, then also v or u, resp., equals zero, otherwise ws+a = 0, for some s. If both u and w are different from zero, then d4-')le and d 4 - ' ) l e are e-th roots of unity, hence ~ ( q - - l ) / ~= wr
204
and
COMPOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
CH.
4
= w', for some r and t . Hence (8.93) can be written as wrywr
+a ) 4 z -
We
Then (8.92) implies W y l+ a ) k ( q - W
=
,,,yW' +
=
W y +l a ) k ( q - - l ) . e .
1)le
.
(8.94) (8.95)
Since a # -1, we have wrc = wtCwhence wr = w' since (c, 4-1) = 1. Therefore u(4-1)/e = V(4-1)le . 3ut .f(u) =f(u) implies uc(u(4-l)le+a)"=
wC(w(4-1)'e+a)k. Since $+a z 0, for all s, we have uc = wc and therefore u = v since (c, q - 1) = 1. We conclude that J' is a permutation polynomial. 9. Semigroups of permutation polynomials and groups of polynomial permutations over finite fields 9.1. Let K be a finite field of order 4 and ( K [ x ] ; o )= S the semigroup of polynomials over K with polynomial composition o as operation as in 5 1.2. A polynomial f € K [ x ] is called regular if f ' ( a ) # 0, for all a E K. 9.11. Lemma. The set S1 of all regular polynomials of K [ x ] is a subsemigroup of the semigroup S .
Proof. The chain rule implies ( f o g ) ' = (f'oglg' whence f , g E S 1 implies f o g E S1, and SI is not empty since x E SI. 9.12. By ch. 3,§ 11.45, the subset T of Sconsisting of all permutation polynomials over K is a subsemigroup of S . If U is any subsemigroup of T , then a U is a subsemigroup of oT = U 1 ( K )= Sym K , the latter equality holds since K is polynomially complete by ch. 1, Th. 12.21. Hence a U is a subgroup of Sym K . Conversely if W is a subgroup of Sym K , then clearly u-lW is a subsemigroup of T .
-
9.13. Lemma. The canonical epimorphism a :K [ x ] P l ( K ) induces a mapping from the set of all semigroups of permutation polynomials onto the set of all subgroups of Syrn K . Furthermore c also induces a surjective mapping from the set of all semigroups of regular permutation polynomials to the set of all subgroups of Sym K .
89
205
SEMIGROUPS OF PERMUTATION POLYNOMIALS AND GROUPS
Proof. The first assertion has just been proved above. If U is a subsemigroup of T and U I = Uf'SIis not empty, then U1is also a subsemigroup of T. We have to show that, for any function zE P1(K),there is a regular polynomial g EK[x] such that ag = z. Let f € S such that uf = z. Since ker Ker o = (x"-x), we have ak = z,for any k = f + h ( x 4 - x ) . Since k' = f'-h mod ker Ker a, and K is polynomially complete, k E K [ x ]is regular, for some suitable h E K [ x ] .
9.14. Remark. The importance of regular permutation polynomials stems from the fact that if Q is a primary ideal of the ring R with associated prime ideal P such that R [ Q is finite and Q # P , then f E R [ x ] is a permutation polynomial mod Q if and only if q ( P ) ( x ) f i s a regular permutation polynomial of the finite field RIP, by Prop. 4.31. 9.2. We are now going to consider some special classes of semigroups of permutation polynomials as recently investigated in various papers (e.g. WELLS[ l ] ) . Throughout this subsection, K will denote a finite field of order q, Q the multiplicative group of K, A the subgroup of Q of order m, and Bj, j = I , 2, . . ., ( 4 - l)/m = k the cosets of A . In each Bj, we select an elemefit bj. 9.21. Lemma. The polynomial (9.21)
is the unique polynomial in K [ x ]such that [pi] < q, pj(0) = -m,pj(y) = 1 for y E Bj, and pj(y) = 0 otherwise. Proof. Since (0) = ( x 4 - x ) , there is at most one such polynomial. Furthermore pj(0) = -mb;-l = -m. If y E B j , then y = bja, a E A , hence y m = by whence Bj is just the set of solutions of xm = by in K. Hence pj(y) = -mkbyk = -(q-1) = 1, for y E B j . Moreover (9.21) implies
(
xxmpj = -mx by
k-1 t=o
b:X-f)mxfm+by(xq-l-~)j,
hence xxmpj = xb,"p,mod (0). Therefore pj(y) = 0 if y
#
0 and y
6Bj.
206
CH. 4
COMPOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
9.22. Lemma. Let G be a group of permutations of A , H a group of permutations of the set B = {bl, . . ., bk}, and U the set of all polynomials f E K [ x ] of the form
f=
c ieb,) 4K14P f k
(9.22)
f=l
where e t H and I,, t = 1, . . ., k, is a polynomial of K [ x ] such that I,(0) = 0 and I,(y) = zty, for all y E A , z, E G . Then every polynomial of U is a permutation polynomial, and oU is a subgroup of Sym K isomorphic to the wreath product of G by H .
Proof. Clearly f(0) = 0, and if a E A , then, by Lemma 9.21, f ( b i a ) = (pb,) (z,a). Hence f is a permutation polynomial. If x:B X A Q is defined by x(bi, a) = b,a, then x is a bijection and x-l(uf)x(bi, a) = (ebi, -cia). Thus x-'(uf)x is an element of the wreath product W of G by H. Clearly uf x-'(uf)x is a bijection from oU to W and the inverse of this bijection is an isomorphism from W to oU. -F
--t
9.23. Theorem. Let m be a divisor of q - 1, k = (q - l)/m, A the subgroup of Q of order m and B = {bl, . . ., bk} a system of representatives for the cosets Bj of A . Then: a) The set U1 of all permutation polynomials of the form xg(xm)where g E K [ x ] is a subsemigroup of T and aU1 is isomorphic to the wreath product of the regular representation 2, of A by the symmetric group Sym B of B. b) The set U 2 of all permutation polynomials of the form xg(xm)kis a subsemigroup of U1, and oU2 is isomorphic to the direct product of k copies of Z,. c) The set Us of all permutation polynomials of the form xrg(xm)where I' > 0, is a subsemigroup of T, and oU3 is a group extension of oU1 by the group of prime residue classes mod m. Proof. a) U1 is obviously a subsemigroup of T . In Lemma 9.22, set G = 2, and H = Sym B. Then I, = a,x+g,, where a, E A and g, E K [ x ] buch that gf(y)= 0, for y E (0}UA . Hence U consists of all polynomials
f
=
c ieb,) a,b,lxp, + c (eb,)g W x )Pt k
k
t=1
f=l
3
hence a U = a V where V C U is the set of all polynomials h
c ieb,)a,b,lxPt. k
=
f=l
(9.23)
09
207
SEMIGROUPS OF PERMUTATION POLYNOMIALS AND GROUPS
Every polynomial (9.23) is a permutation polynomial of the form h=x
k
C c,P,,
C,
EQ.
(9.24)
t=l
Conversely such a polynomial (9.24) can be written as h
k
= t=l
c,b,b;'xpt.
Since h maps Bj = bjA onto cjbjA by Lemma 9.21, the cjbis form a full set of representatives for the cosets Bj whence c,b, = (eb,)a,, for every t, where e is a permutation of B. Hence V is the set of all permutation polynomials of the form (9.24). By (9.21), every such permutation polynomial belongs to U, whence oV G UU,, and if conversely f = xg(x") € U,, then f ( b p ) = bjag(bj"),for every a E A , whence
Hence oU1 G aV, and we conclude that oU1 = oU. Lemma 9.22 completes the proof of part a). b) U z is obviously a subsemigroup of U1. Let x : aU1 W , where W is the wreath product of Z,,, by Sym B, be an isomorphism and I. the epimorphism which maps every element of W onto the corresponding permutation of Sym B. We put L = ker Ker Ax. The proof of Lemma 9.22 shows that if h is of the form (9.23) then oh EL if and only if e is the identical permutation. Since oU1 = oV, we have L = OR where --t
Since
we have L = OR, where R1 is the set of all permutation polynomials of k
k
the form xi C d,p) , and every such permutation polynomial belongs \t=1
73 .
1
'
to U z . Conversely if f = xg(x")" E U z , then f(bia) = b,ag(by)k implies " -
. Hence oU2 = L which proves b). c) U3 is obviously a subsemigroup of T. If f = xrg(xnl) E Us, and (r, m) = d =- 1, then there exists 1 z z t R such that zd = 1 whence zr = zrn= 1 and f ( z ) = f(1). Hence (Y, m) = 1 . If xrg(x'"),xsh(x"')c U3 and oxrg(xm)= oxsh(xm),then, for all a E A, we have a'g(1) = a'h(1).
208
COMPOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
CH.
4
Since g(1) # 0, a'+ = 1, for all a E A, whence r E s mod m. Therefore we can define a mapping 8 :aUs group of prime residues mod rn by Ba(x'g(x")) = r mod m, which is obviously a homomorphism. Let (r, m ) = 1, n = JJ(primes p I p .Y m and p 14- 1). I f b =- 0 is a solution of b E r mod m, b = 1 mod n, then (b, 4-1) = 1 whence x6€U3 and B(oxb)= r mod m. Hence 8 is an epimorphism whose kernel is just aU1. +
9.3. Proposition. a) The set L of all linear polynomials of K [ x ]is a subsemigroup of S which is even a group. Every polynomial of L is a regular permutation polynomial. b) L is isomorphic to the semidirect product of the additive group A of K by the multiplicative group Q of K. The canonical epimorphism o :K [ x ] P l ( K )maps L injectively into PI(K), hence aL z L.
-
Proof. Obvious. 9.31. A permutation polynomial of the form X" is called a power permutation polynomial.
9.32. Proposition. Let K be aJinitejeid of order q and Z the ring of rational integers. Then a) N = {x" € K [ x ]I n 2 l } is a commutative subsemigroup of S. A polynomial of N is a permutation polynomial if and only if (n, 9-1) = 1 and is regular if and only if n = 1. b) Let P be the semigroup of permutation polynomials of N. Then ON is isomorphic to the multiplicative semigroup of ZI(q-1) and oP z &(ZI (q - 1)). Hence I aP I = y(q - 1) where p denotes the Euler y-function. Proof. Obvious. 9.4. Lemma. Let Z be the ring of rational integers, y l , yz indeterminates
and k =- 0 an integer. Then ( k / ( k - t ) )
(k - t ) is an integer, for
0 == t
and t Z, and
holds in Z [ y l , yz] where [k/2]denotes the greatest integer t
=sk/2.
< k/2
§9
SEMIGROUPS OF PERMUTATION POLYNOMIALS AND GROUPS
209
Proof. This is an immediate consequence of Waring's formula (ch. 6 ,
5 9.2).
9.41. Let R be a commutative ring with identity. A polynomial gk = gk(a, x ) over R of the form
(9.42) where k > 1 is an integer and a E R, is called a Dickson polynomial over R. 9.42. Remark. Dickson polynomials are closely related to Cebyshev polynomials of the first kind: If we substitute y1 = eY, y2 = e-'9 in (9.41), then 2 cos k v = g k ( l , 2 cos 9) whence, if tk denotes the k-th 2 cos y), i.e. g k ( ] ,2x1 = 2tk(x). Cebyshev polynomial, 2tk(cos v) = If R = K is a field and char K + 2, a # 0, and d ( a ) is a square root of a in an extension field of K, and i f the image of any polynomial f C Z [ x ] under the extension of the epimorphism from Z to the prime field of K to a homomorphism from Z [ x ]to K [ x ]which fixes x is again denoted by f, then
hence
= ( l / d a ) k g k ( a ,x,
gk(',
=
2(da)k t k ( x / 2 da>.
(9.43)
Since &(o, x ) = xk, we will consider only Dickson polynomials gk(a, x ) where a f 0. 9.43. Theorem. If K is a finite field of order q and characteristic p and 0 # a E K , then a Dickson polynomial gk(a, x) E K [ x ] is a permutation polynomial of K if and only if ( k , q2- 1 ) = 1 , and gk(a, x) is a regular permutation polynomial of K if and only if ( k ,p(q2- 1)) = 1.
Proof. Let z # 0 be any element of an arbitrary extension field of K , then the substitution y l = z, yz = a/z in (9.41) yields gk(a, z+(a/z)) = p+(a/z)k*
(9.44)
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COMPOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
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4
Each of the q quadratic equations in K , X'-~X+U
= 0,
(9.45)
Y EK,
has two solutions (which may possibly coincide) in some extension field H of K of order q2. Let M(a) be the subset of H consisting of all those elements of H which are solutions of an equation (9.45). If 0 f u E K , then u+(u/u)=rwhenceuEM(a). Suppose uEM(a)--Kandx2-~x+a=0 is an equation (9.45) which is satisfied by u. Since 8 z = zq defines an automorphism 8 € Gal HI K, also u4 is a solution of this equation. Hence uq+l = a. Conversely if u E H a n d uq+' = a, then u+(a/u) = u+u4 = r E K whence u E M(a). Therefore
M(u) = {uE HI
u4-'
=1
or
uq+l = a).
(9.46)
Suppose now that (k, q2- 1) = 1, and gk(a,s) = gE(a, t), for some elements s, t E K . Let u, v E H such that u+(a/u) = s and v+(a/w) = t, then (9.44) implies uk+ (a/u)" = uk+ ( a / ~whence ) ~ uk = vk or uk = ( a / ~ ) ~ . Therefore u = v or u = a,%, thus s = t, and gk is a permutation polynomial. Now suppose that (k, q2- 1) = d =- 1. If d is even, then q is odd and k is even whence gk(a, x) contains only even powers of x, by (9.42). If 0 f c E K , then c f -c, but gk(a, c ) = gk(a, -c), hence g, is not a permutation polynomial. If d is odd, then there exists an odd prime p such that p/d, hence plk and p/(q - l), or p / k and p/(q 1). In the first case, there are p elements w E K with up = 1, and these elements also satisfy vk = I whence gk(a, v+a/w) = 1+ak, for all these elements w, but vl+a/wl = v2+a/v2 implies vl = wz or wl = a/vz,hence there are a t least two different elements w, = wl+a/v,, i = I , 2, in K such that gk(a, wl)= gk(a, wz). In the second case there are p elements v E H such that w p = 1 and therefore also = 1. If t is a solution of u4+l = a, then {twl vq+l = 1) is the set of all solutions of u4+l = a, hence there are p elements u such that u4+l = a which have equal p-th powers and therefore also equal k-th powers. Again by (9.44), we conclude that &(a, w ) = gk(a, wl) for two different elements w , w1 E K . So in either case gk is not a permutation polynomial. Let z be an indeterminate and K(z) the field of rational functions in z over K . Since K(z) is an extension field of K, (9.44) also holds for z E K(z). We differentiate (9.44) and obtain
+
d ( a , z+(u/z)) (1 -(u/z'))
=
kzk-'-k(ak/zkfl),
09
21 1
SEMIGROUPS OF PERMUTATION POLYNOMIALS AND GROUPS
hence
(9.47) where h(z) E K [ z ] .Hence zk-lgL(a, z+(a/z)) = kh(z), thus (9.47) holds for every z # 0 of an arbitrary extension field of K. Let gk be a regular permutation polynomial, then (k, q2- 1) = 1. If s E K , then let u E H such that u+ (a/u) = s. Then (9.47) implies gL(a, s) = (k/uk-’)h(u) whence (k,p(q2-1)) = 1. Conversely if (k,p(q2-1)) = 1, then gk is a permutation polynomial of K . Suppose there is some s E K such that gL(a, s) = 0, then, for u+(a/u) = s, we have h(u) = 0. Hence (z8-a) h(u) = ( u ~ ) ~ -= - u0,~ therefore
u2 = a.
kak-’ = 0, a contradiction since p polynomial.
But then
h(u) =
k-1
C ak-’
=
j=O
.Y k. Hence gk is a regular permutation
9.5. For an arbitrary b E K, (9.42) implies gk(ab2,x) =
IkI21
C ( k l k -t)
f=O
(kJ
bkb-(K-zt)Xk-zf = bkgk(a, (I/b)x). Hence, if char K = 2, then every polynomial gk(a, x ) E K [ x ] ,a J: 0, can be obtained from composition of gk(1, x ) by linear polynomials ux E K [ x ] ,and if char K # 2, composition of either gk(1, x) or gk(u,x), u being a fixed non-square of K , by linear polynomials ux E K [ x ] . In the second case, we can also obtain every &(a, x)E K [ x ] ,a f 0, from composition of g k (1, x) by linear polynomials ux E H [ x ] where H is an extension field of K of order q2. Thus, in a certain sense, all the Dickson polynomials gk(a, x), a f 0, can be “reduced” to the Dickson polynomial g k ( l , x ) = hk(x).We will therefore restrict ourselves to this particular type for the remainder of this section. If K is the field of rationals, then (9.43) shows that gk(1,x) = = 2tk(x/2)= gk where gk is the polynomial which has been introduced in 9 3.3. Hence, for an arbitrary field K, the set v = { g k ( 1 , x) I k==(l) is just the P-chain over K of Cebyshev polynomials of Q 3.32, which proves .(-a)‘
9.51. Proposition. The set V = {h, = g n ( l , x )l n z= 1) is a commutative subsemigroup of S which is isomorphic to the multiplicative semigroup I of positive integers. 9.52. Proposition. Let K be any finite field of order q and characteristic p , W the semigroup consisting of the permutation polynomials of V , W1 the
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COMPOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
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4
semigroup of all regular permutation polynomials of V , and v = q2- 1, for p = 2, v = (q2-1)/2, for p # 2. If C(1) denotes the subgroup of d(Z 1 (v)) generated by { 1 mod v, - 1 mod v, q mod v, -q mod v } which is of order 2, for q =s3, and of order 4 otherwise, then a V is isomorphic to the factor semigroup of the multiplicative semigroup of Z 1 (v) mod C(1), and OW = aW1 is isomorphic to the,factor group of &(ZJ(V))mod C(1). Proof. 8k = ohk defines an epimorphism 6 :I -, oV.The kernel Ker 6 of 8 is a congruence on I . Let C(1) be the congruence class of I E Z under Ker 6. Then kEC(1) if and only if 8k = 61 which is equivalent to hk(s)= h,(s), for every sE K. If M(1) is defined as in the proof of Th. 9.43, then (9.44) shows that hk(s)= h,(s), for every s E K , is equivalent to uk+(l/uk) = u'+(l/u'), for every uEM(1). This is true if and only if uk = u' or uk = 1/u', for every u E M(1). Let w be a generator of the multiplicative group of H , then (9.46) implies
or = wn(q-1) 1. (9.5) Hence uk = u' or uk = 1/u' for every u E M(1) if and only if wk(q+l) = w l ( q t l ) or W k ( q + l ) = w-'(q+l) and Wk(q-l) = w,(q-l) or Wk(q--l) = w-&2-1). Therefore k E C(1) if and only if k is a solution of one of the following four systems of congruences : ~ ( 1 =) { u l u
= ~m(q+1)
krlmodq-l k=lmodq+l
k = 1 mod q-1 kz-Imodqfl
k E - I mod q-1 k r Imodq+l
k G -1 mod q-1 k=-Imodq+l
If v is defined as in the proposition, a straightforward computation shows that the solutions k are the positive ones among the integers 1+ tv, lq + tv, -Zq+ tv, -I+ tv where t runs through the integers. Hence if k = 1 mod v, then k E C(1). If J is the multiplicative semigroup of Z I (v), then ~ ( mod 1 v) = ah, defines an epimorphism 7 :J oV. In order to simplify our notation, we will subsequently write a for a mod v and C(a)for the congruence class of a under Ker q. Then C ( l )= {I, -1, Iq, -14) = lC(1). ButC(1) = {I, -1, q, -q} is a subsemigroup of the units of Z 1 (v), hence C(1) is a group. By Th. 9.43, h, E W if and only if (k, q2- 1) = 1 which is equivalent to (k, v) = 1. Hence 7 maps the group E of units of Z I (w) onto aW, and the ideal kernel of 7 :E -r OWis E n C(1) = C(1). Furthermore, by Th. 9.43, hk E W , if and only if (k,p(q2- 1)) = 1. Since ( p , q2- 1) = 1, every element of E contains some positive integer
-
0 10
PERMUTATION SPECTRA OF POLYNOMIALS
213
k with ( k ,p(q2- 1)) = 1 , thus a w l = OW.If IC(1) I < 4, then a = 1 mod ZI where a = - 1, or q, or -q. An easy calculation shows that this is the case if and only if q 6 3, and that q 4 3 implies lC( 1) I = 2. 10. Permutation spectra of polynomials 10.1. Let A be any algebra of a variety 23, X = {XI, . . ., x,} a set of indeterminates, and 0.a subset of the congruence lattice 2(A) of A . For any 0 E 2 ( A ) , q ( 0 ) : A A j 0 shall denote the canonical epimorphism, q (0)( X ) : A(X, 23) ( A 10)(I,@) shall denote the extension of q (0)to a composition epimorphism and C,(0) :C,(A) C,(A 10)the corresponding epimorphism. For any f€C,(A), we define the &-permutation spectrum of f by -+
+
-+
Spec(@,f) = (0E G 1 Ck(0)fis a permutation polynomial vector of A 10). Similarly we define the &-permutation spectrum and strict &-permutation spectrum of f C A(X, 23) by
f)= (0 E 6I (Strict) Spec (6,
~(0 (X)f ) is a (strict) permutation polynomial of A 101.
If A is an Q-multioperator group, then by ch. 6, Q 3, there is a bijection from 2(A) to the ideal lattice 9 ( A ) of A , thus we can interpret & also as a subset of 9 ( A ) and 0 as an ideal of A in our definitions. Permutation spectra have so far been studied just in the case n = 1 for certain Dedekind domains. This section is to work out the most important of these results. 10.2. Let 23 be the variety of commutative rings with identity, then every algebra R of 23 is a multioperator group, thus we will use the interpretation of spectra in terms of ideals of R. As in ch. 3, the composition of polynomial vectors will be denoted by 0 ,and i f f € R[x,, . . .,x,], and g = (gl,. . .,gk) is a polynomial vector, we will write f ( g , , . . .,g k ) = f o g as in ch. 3, $ 2 .
10.21. Proposition. Let R be a commutative ring with identity and & the . . .,xk] and f, g E C,(R), and if we set Spec (&, ) = Spec, then: a) Strict Spec f C Specf, b) Spec (f o g) = Spec f nSpec g, Spec (fo g) 2 Spec Spec g, Strict Spec ( f o g ) 2 Strict spec Spec g.
set of all ideals D of R such that R I D isJinite. If .f€R[x,,
J’n
fn
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COMPOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
CH.4
c) r f D E Spec f (Spec f, Strict Spec f ) and C 2 D, then C E Spec f (Spec f, Strict Spec f ) .
Proof. a) follows from ch. 3, Prop. 12.23. b) D E Spec f fl Spec g if and only if Rl D is finite and C,(q(D))f, C,(q(D))g are permutation polynomial vectors ofR 1 D which is true if and only if R I Dis finite and C,(q(D)) (f o g) is a permutalion polynomial vector of R 1 D whichis equivalent to D E Spec ( f o 9). If D E Specf fl Spec g, then R I D is finite, q(D)( X ) f is a permutation polynomial, and C,(q(D))g is a permutation polynomial vector of R ID. Therefore there are functions f,,. . .,f,E F,(R I D ) such that p(aq(D)( X ) f ,f,,. . .,f,)= zis a permutation of (RI D), and p@(a)Ck(q(D))g= p is also a permutation of ( RI D)k. Hence np = p(aq(D)(X)(fog), hz, . . .,h,), for some hi E Fk ( R 1 D), is a permutation of ( RI D),, thus q(D)( X )( fg)~is a permutation polynomial of R I D. Similarly wecan prove the third assertion of b). c) If D E Spec f , then R I D is finite, and f is a permutation polynomial vector mod D.Then R I C is finite, thus by Prop. 4.2, f is a permutation polynomial vector mod C whence C E Spec f. The proof of the other two assertions runs along the same lines. 10.22. Lemma. Let R be a Dedekind domain and Spec be dejined as in Prop. 10.21. Then: a) I f E = C D where C , D are comaximal ideals in R, then E E Spec f (Specf, Strict Spec f)if and only if C,D E Spec f (Specf, Strict Spec f). b) I f P is a prime ideal and Q = P", e w 1, then Q E Spec f (StrictSpec f ) if and only if P2E Spec f (Strict Spec f ) .
Proof. a) E E Spec f if and only if R jE is finite and f is a permutation polynomial vector mod E. By the Chinese remainder theorem and Prop. 4.21, this is true if and only if R I C and R I D are finite and f is a permutation polynomial vector mod C and mod D, i.e. C , D E Spec f. Similar arguments work for Spec f and Strict Specf. b) Q C Spec f if and only if R I Q is finite and f is a permutation polynomial vector mod Q. By ch. 6, Lemma 4.52, and Cor. 4.35, this is true if and only if R 1 P2 is finite and f is a permutation polynomial vector mod P2,i.e. P2E Spec f . The argument for Strict Spec is similar. 10.23. Theorem. Let R be any Dedekind domain, @ ' = p(f)the set of all prime ideuls of Spec f and $7. = D(f) the set of all prime ideals Q E p(f)
0 10
PERMUTATION SPECTRA OF POLYNOMIALS
215
with Q2ESpec f. Then Spec f is the set of all ideals D of the form
D
.
= PIP2.. . P r e y . . Q?
+
where r s =- 0, PI, , . .,P,,Q,, . . . , Q, are pairwise distinct prime ideals, such that PiE P(f)-C(f), i = 1, . . ., Y, and Qj E D(f),j = 1, . . ., s, ej == 1. This assertion remains valid if f is replaced by f and Spec by Strict Spec.
Proof. This is an immediate consequence of Prop. 10.21 c) and Lemma 10.22. 10.24. Lemma. Let P i (0), R be any idealof p(f),p(f)resp. Then PED(f) ifand only if af(ul, . . .,uk) 0 modP, for every (ul, . . .,u k ) E Rk,P E D(f) if and only i f there is no (ul, . . .,uk)E Rk such that aif(ul, . . .,uk) = 0 mod P , i = 1, . . ., k , resp.
+
Proof. This is nothing other than Prop. 4.31 and Prop. 4.34. 10.25. Th. 10.23 shows that Spec f and Strict Spec f are completely determined by the pair (9, Q) of sets of prime ideals. We will call (9,El) the basis pair of Spec f, Strict Specf, resp. The following two questions arise : a) How can one find the basis pair for a given polynomial vector f or a polynomial f ? b) What pairs (P, of prime ideal sets can be basis pairs for suitable polynomial vectors or polynomials? We do not intend to attack this problems in general, but will give some results for the case where k = 1 and R is the Dedekind domain of rational integers. The reader may work out how far these results carry over to the domains of integers in algebraic number fields of finite degree.
a)
10.3. Let R be the domain of rational integers and R [ x ] the polynomial ring over R in one indeterminate x.The three notions of &-permutation f ), for f E R[x], has to be spectra clearly coincide. Thus only Spec (6, considered. Since R is a principal ideal domain, every ideal D in R can be written as D = ( d )where d a 0 is a unique integer. Thus we can describe Spec (&, f)as a set of non-negative integers. 0. will again denote the set of all ideals D of R such that RID is finite, and Spec f = Spec (Q, f).
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COMPOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
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Rewriting the results of 10.2 we obtain: Spec ( f o g ) = Spec f nSpec g.
(10.3)
If d E Spec f and cld, then c E Spec f , and Spec f is uniquely determined by its basis pair which consists of the set '$ = '$(f)of all primes of Spec f and the set El = Qf) of all primes q E ' $ ( f ) with q2E Spec f. Moreover a prime p E '$(f)is in D(f) if and only iff '(u) $ 0 mod p , for every u E R, i.e. if ~ ( p(x) ) f is regular. 10.31. Remark. If '23 is the set of all ideals of R, then '23 = &U{(O)}. By 3 5.33, the permutation polynomials of RI(0) = R are just e x f c , e = f 1, thus we know Spec ('23, f)as soon as we know f and S p e c 5 10.4. Theorem. Let S = ( R [ x ] ; 0 ) be the semigroup consisting of the polynomials over R with the polynomial composition o as operation, H the subsemigroup of S which is generated by the linear polynomials ax b, the powers x" with odd n =- 1, and the Dickson polynomials g,,(a, x), a # 0, (n, 6) = 1, n =- 1, and L the subsemigroup of H which is generated by the polynomials ax+ b and g,(a, x) of H . Then '$(f) is infinite for every f E H . Z f f E H , t h e n E l ( f ) i s i n J n i t e i f a n d o n l ~ i i f f E L .
+
Proof. Let f E H , then f is of the form
.f = f 1 o f z o . * .
os,
(10.41)
where each.fi equals either some ax+b, or some xn, or some g,(a, x) which satisfies the conditions of the theorem. Hence (10.3) implies
w-1= '$(A)n P(f>n * n '$3(f,), am = a(f,)nn ( jn~. . . nCCA). * *
(10.42) (10.43)
Since R 1 ( p ) is a finite field for every prime p , we can apply our results of $5 9.3 and 5 9.4. Prop. 9.3 implies '$(ax+b) = { p 1 p E 9, a $ 0 mod p } where '$ denotes the set of all primes, hence $(ax+ b) consists of almost all primes. Prop. 9.32 implies that '$(x") = { p I p E '$ and ( p - 1, n ) = I}. If n = r y . . . r: is the prime factor decomposition of n, then '$(x") is the set of all primes p such that p $ 1 mod r,, i = 1, . . ., s. Th. 9.43 and Prop. 9.32 imply that '$(g,(a, x)) = UU V where U = { p 1 p E '$, a g O m o d p , ( p 2 - 1 , n ) = 1 ) and Y = { p I PEP, a = O m o d p , ( p - 1, n) = l}. If n = rf' . . . r: is the prime factor decomposition of n, then
§ 10
21 7
PERMUTATION SPECTRA OF POLYNOMIALS
U is the set of all primes p such that p $ f 1 mod Y;, i = 1, . . .,s, and a g 0 mod p while V is the set of all primes p such that p g 1 mod ri, i = I , ..., s , a n d a = Omodp. Suppose now that f € H , and [f]= m = u: . . . u2v;" . . . zf is the prime factor decomposition of [f]where ul, . . .,us are those primes which occur in the degrees of the factors x" in (10.41) but not in the degrees of the factors g,(a, x) while vl, . . .,v1 are the remaining primes. Let Q ( f ) be the set of all primes which belong to those residue classes C(b)modu, . . . u,vl . . . v, for which b g4 1 modu;, i = 1, . . .,s, b g 1 mod vi,i = 1, . . . , t. Then p(f)is obtained from p ( f )by adding and removing a finite number of primes. Indeed, any prime of these residue classes belongs to p(x'), for every x" in (10.41), and to '$(g,(a, x)), for every g,(a, x) in (10.41) whence, by (10.42), all of these primes which are not divisors of a, for some a x + b of (10.41), belong to p(f).Conversely by (10.42), all primes of P(f) except finitely many are contained in !$(f). Hence p ( f )and p ( f )differ by only finitely many primes. By hypothesis, ui # 2, i = 1, . . ., s, vj f 2, 3, j = 1, . . ., t. Hence by the Chinese remainder theorem, there are exactly (ul - I ) . . . (us- 1) (ol-2) . . . -2) residue classes C(b) mod u1 . . . u&vl . . . v, for which b g 1 mod ui, i = 1, . . ., s, b $ + I mod vi, i = 1, . . ., t, and (24-2) . . . (u,-2) -(v1-3) . . . (c,-3) residue classes are prime residue classes. By Dirichlet's theorem, each of these classes contains infinitely many primes (but the other residue classes contain only primes of the set {ul, . . ., us, vl, . . ., v,}). Hence p ( f )is infinite. We have O(ax+ b) = !@(ax+ b), D(Y) = Cp and O(g,,(a,x)) = {PI pE p, a 8 0 mod p , ( p ( p 2 - l ) , n) = 1} by Prop. 9.3, Prop. 9.32, and Th. 9.43. Thus if n = I . : . . . :Y is the prime factor decomposition of n, then D(g,@, x)) = {PE%(g,(a, 4)I a f 0 mod p , P F ri, i = 1, . . ., s}. By (10.43), i f f € H - L , then D(f) = Cp. If, however, f € L then by (10.43) we see that p ( f )and difl'er by only finitely many primes, hence Qf) is infinite.
+
(zit
a(f)
10.5. The question now arises whether there are any other polynomials f € R [ x ]apart from thosein H for which p ( f )is infinite. It was SCHUR who stated the conjecture that this was not the case, and he himself could prove, that this was true for polynomials f of certain degrees, e.g. polygave a general proof of nomials of prime degree. Recently M. FRIED SCHUR'Sconjecture which leans heavily on rather intricate tools and deep
218
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4
results in the theory of complex functions and is therefore beyond the scope of this book. We will state FRIED'S result and two consequences of it which will follow immediately from Th. 10.4 and its proof. 10.51. Theorem. '$(j')is infinite if and only iff E H , and Q(f)is infinite if and only i f f E L. 10.52. Corollary. Every infinite '$(f)can be obtained by adding to and removing from some set T of primes ,finitely many primes. Such a set T of primes consists of the primes of all prime residue classes C(b) mod 1 where I = u1 . . . u p 1 . . . v,, with ui, vj pairwise distinct primes, ui f 2, vj # 2, 3, b g 1 mod ui, i = 1, . . ., s, and b $ & 1 mod vi, i = 1, . . ., t. 10.53. Corollary. If '$(f) is infinite, then from $( f) by only finitely many primes.
a(f)is either empty or dgers
10.6. Theorem. Let Q = {ql, . . ., qs}, 93 = {pl, . . ., p,} beJinite disjoint sets of primes (possibly empty). Then there exist monic polynomials f E R[x]of arbitrarily large degrees such that '$(f)= QU 3 and Q ( f ) = Q.
Proof. If QU% is empty, then f = x ~ ~ - - - x ~ *n- '==, 2, has the required property. Suppose QU % is not empty. Set q = q1 . . . q, if D is not empty, q = 1 otherwise, and p = p 1 . . . p , if % is not empty, p = 1 otherwise. Throughout the proof the sum over an empty set will be 0 while the product over an empty set will be 1 . We choose positive integers u, v such that u =- v and xu-xu E { p q } which is possible because of the finiteness of R[x][{pq}.Furthermore we choose k z= 2 integral such that w = k u p q f l is a prime-by Dirichlet's theorem, there are infinitely many such k . Let = Xkupq+l-&m+l (10.6) Then g E {pq} and g' E {pq}. Furthermore let r
h
=
g+
C m=l
S
(Pq/pm)xPm+
n=l
(Pq/qn)x,
which is a monic polynomial of degree w. It follows easily that h G ( p q / q i )x mod {qi} and h' = (pq/qi)mod {qi}.Thus his a: permutation polynomial mod qi and moreover qi E Q(h), i = 1, . . .,s. Similarly,
REMARKS AND COMMENTS
219
h = ( p q / p Jx mod ( p L }and h' = 0 mod {p,},hencep, E p(h),butp, 4 Q(h), i = 1, . . .)Y. Suppose P(h) is infinite. Then by Th. 10.51 (we need just the weaker result of SCHUR)h E H . Since [h] = w is a prime, h is of the form a(a,x+bb,)"+ b or of the form ag,,,(c, a,xfb,)+b. If b, # 0, then such a polynomial contains a non-zero term dxW-', and if b, = 0, then such a polynomial contains, apart from the leading term, only a constant term or it contains a non-zero term dxWp2.But (kupqf 1)-(kwpqf 1) = kpq(u-u) ==4, and kwpqf 1 =- max(p,, p2, . . . , p , , 1), hence h is certainly not a polynomial of this form, contradiction. Therefore Q(h) is finite, thus Q(h) = QU%US where S = {b,, . . .,b,} is a finite set of primes disjoint from QU W such that Q C Q(h) C QU S.Set b = b,b, . . . b, and choose a positive integer z such that z = 1 mod p q , z = 0 mod b, and positive integers I, m such that I z m and x l - P E { p q b } . If e = x b q -xmP4 f z x , then QU% C Q(e), but SnQ(e) = 4. By (10.42) and (10.43), if d = h o e , then $ ( d ) = QU%, Q(d) = Q. Let
n r
k=
2=1
(p;-l)
n S
qi(q;-l)
and h, = gn(l, x ) . If (n, k ) = 1, then
J=1
p(h,) 2 QU% and Q(hn) 2 Q by the proof of Th. 10.4, whence by (10.42) and (10.43), if , f = doh,,, then Q(f)= QU% and Q ( f ) = Q. Clearly f is monk and there are infinitely many n with (n, k ) = 1. This proves Th. 10.6. Remarks and comments
$5 1 , 2 . If R is a commutative ring with identity and o means the composition of polynomials, then the set R[x] of all elements of the polynomial can also be viewed as an algebra ( R [ x ] ; o), i.e. as ring ( R [ x ] ;+, a semigroup, as an algebra ( R [ x ] ; , o >, i.e. as a near-ring, as an algebra ( R [ x ] ;-,0 ) . and as an algebra ( R [ x ] ;f , o ) , i.e. as a composition ring. In this book we treat just the first and the last case. To the best of our knowledge the third case has never been looked at, but there are a few papers whichconsider the near-ring (R[x]; +, 0 )-mainly for R being a field-or certain subnear-rings of it (ORE[l], [2], CARCANAGUE [l], [ 2 ] , RIHA[l], CLAYand DOI[l]). The semigroup ( R [ x ] ;0 ) has been studied mainly for R being a field, and apart from the topics of our # 1-3, also the automorphism group of this semigroup has been examined (FADELL and MAGILL [l], SAIN[2]).
->
+
.,
220
COMPOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
CH.
4
Most of the results and problems referring to (R[x] ;o ) also make sense for the superassociative system (R[x,, . . .,+I; x ) , x being the composition of polynomials, or for the semigroup q ( ( R [ x , , . . .,xk]; x ) ) , but for k =r 1 , almost nothing has been done in this direction save GOODSTEIN [l], [2] on generating systems of (R[x,, . . ., xk]; x ) . Th. 1.34 and Th. 2.46 have been essentially proved by RITT[I], for the case that K is the field of complex numbers, by means of rather deep methods of the theory of complex functions. The purely algebraic proofs of these theorems for arbitrary fields of characteristic zero which are [ l ] (Th. 1.34) and H. LEVI contained in our book are due to ENGSTROM [l] (Th. 2.46). FRIEDand MACRAE [ l ] have given a different algebraic proof of Th. 1.34. This proof admits also a generalization of the main statements of Th. 1.34 to fields of characteristicp > 0 as long as the degree [f] of the polynomial f is less than p. Apart from this result nothing is known so far about the decomposition into indecomposable polynomials over fields of characteristic p =- 0, nor over integral domains other than fields. Q 3. A systematic study of permutable elements in the semigroup (K[x] ; o ) where K is a field of characteristic 0 is due to JACOBSTHAL[ l ] who gave a proof of our Th. 3.33 for this case. For arbitrary fields, KAUTSCHITSCH [l] could prove this theorem. He also found a similar result for the semigroup of all formal power series in one indeterminate without a constant term over a field with respect to the composition. Th. 3.53 in its general form is due to NIEDERREITER. AFHALLSTROM [ 11 introduced the semipermutability of elements in (K[x]; 0 ) which is an interesting generalization of permutability.
Q 4. For the case of polynomials in one indeterminate over the ring of
rational integers, this section is due to NOBAUER [ l ] (see also NOBAUER [2], CAVIOR [2], ZANE[l], KELLER and OLSON[l], for general Dedekind domains we refer to SCHONIGER[l]). Polynomial vectors in several indeterminates were investigated by NOBAUER [4] for the ring of rational integers, for arbitrary Dedekind domains by AIGNER [l]. NOBAUER [22] also introduced and investigated permutation polynomials and strict permutation polynomials in more than one indeterminate over rings, this paper contains the conditions of 5 4. Moreover, there is a proof in this paper that, in the ring of rational integers and for arbitrary
REMARKS AND COMMENTS
221
a # 0, every permutation polynomial mod (a) is a strict permutation polynomial mod (a), this proof is, however, wrong. So it remains an open problem, even for most of the factor rings of the rational integers, whether or not every permutation polynomial is a strict permutation polynomial. A very interesting characterization of permutation polynomial vectors over the field K of real numbers is due to BIAZYNICKI-BIRULA and ROSENLICHT [ 1] who used topological methods : A k-dimensional polynomial vect or f over the real numbers is a permutation polynomial vector if and only ifthe mapping p'f : K k K k induced by f is injective. P. CHOWLA [l] studied an interesting class of polynomials related to permutation polynomials. 8 5. Comparing Th. 5.21 with Ch. 5 , Th. 3.3, one may expect that Th. 5.21 is open to a considerable generalization. This, however, has not been done yet. The semigroups V k ( R ) M )and even more U , ( R ) M ) have been examined by NOBAUERfor the case where R is the ring of rational integers, first for k = 1 (see NOBAUER [l], [2], [5]),then also for k =- 1 (see NOBAUER [4], [9]). If R is an arbitrary Dedekind domain, then results were obtained by LAUSCH[I] for k = 1, and AIGNER[l] for arbitrary k. Our presentation follows AIGNER'Spaper. Recently the unit group &(Z I (n) [x])of the semigroup Z 1 (n) 1x1 with respect to the composition was discussed by SUVAK[l] (Z denotes the ring of rational integers).
-
$6 6,
7. For R being the ring Z of rational integers and k = 1, ideal power semigroups were originally investigated by NOBAUER [7], [9]. The Pf)was discussed by FERSCHL structure of special cases of the groups Jk (P", and NOBAUER [l], FEICHTINGER [I] for R = Z and k = 1, and DIRNBERGER [l] for R = Z and k =- 1. For more general classes of rings, we refer [l], and AIGNER[l]. Our presentation to LAUSCH[2], SCHITTENHELM follows partly AIGNER'Spaper and sometimes makes use of the papers of LAUSCHand DIRNBERGER.
6 8. As announced in the remarks and comments to Ch. 3 , § 12, we now give some references for the extensive literature about permutation polynomials in one indeterminate over finite fields. For a survey of the work [4]. During this period on this subject prior to 1920 we refer to DICKSON it was DICKSONhimself who contributed substantially to this subject (see DICKSON [l], [2], [3]).Recently, mainly CARLITZ and his school, but also several other authors, have taken up this subject again and widened
222
COMPOSITION AND POLYNOMIAL FUNCTIONS OVER RINGS AND FIELDS
CH.
4
our knowledge through some interesting contributions (CARLITZ[I], [3], [4], [5], [6], [7], CAVIOR[l], S. CHOWLA 113, S. CHOWLAand H. ZASSENHAUS [l], LAWKINS111, LONDONand ZIEGLER[l], RBDEI[l]). There is also some recent work on permutation polynomials over finite and STARKOV [l], LIDL[l], [2], fields in several indeterminates (KURBATOV LIDLand NIEDERREITER [l], NIEDERREITER [Ij, [2], [3]). Our proof of Th. 8.21 is due to GWEHENBERGER [l], Th. 8.31 is a result of MACCLUER[l] (for the case that K is a finite field there is a much El]). Th. 8.81 was proved by simpler proof of this theorem by WILLIAMS HAYES111 (for a similar result see DAVENPORT and LEWIS[l]), and the results of 9 8.8 are also due to HAYES[l]. CARLITZ and WELLShave proved Th. 8.92.
Q 9. Papers on groups of polynomial permutations over finite fields,
mainly for one indeterminate, are numerous and deal with generating sets of such groups (CARLITZ [I], [6], WELLS[2]), with the distribution of the minimal possible degrees of the polynomials representing the permutations of these groups (WELLS[3]) and various other questions (AHMAD [l], CARLITZ and HAYES[l], FRYER111, [2]). Th. 9.23 is due to WELLS[l], but our proof stems from GWEHENBERGER [l]. Related results can be found in AHMAD [2]and FILLMORE [l]. The generalization of the groups d o f Prop. 9.32 from finite fields to finite residue [3], DICKSON [l], [2] class rings of the integers can be found in NOBAUER introduced the permutation polynomials that now bear his name, in these papers our Th. 9.43 has essentially been proved. Prop. 9.52 is due to NOBAUER [27]. The generalization of the group OW of this proposition from finite fields to finite residue class rings of the integers was recently [ 11. Dickson polynomials discussed by LAUSCH,MULLERand NOBAUER were generalized to the case of several indeterminates recently by LIDL and WELLS[l].
8 10. The theory of spectra in this section was started by NOBAUER [24], [25] for the case of one indeterminate and the rational integers. We follow closely these papers. For contributions to SCHUR’Sconjecture prior to FRIED’Ssolution of this problem, see SCHUR[l], WEGNER [l], KURBATOV [I], [2]. Spectra in more than one indeterminate and spectra over varieties other than the variety of commutative rings with identity have so far not been investigated.
CHAPTER 5
COMPOSITION OF POLYNOMIALS AND POLYNOMIAL FUNCTIONS OVER GROUPS
1. The concept of length 1.1. Let % be the variety of groups as defined in ch. 1 , § 2.4, Q . the set of its operations, G any group, and X = {xl,. . ., x,} a set of indeterminates. As inch. 1,$9.2, we set G(X,$' 3) = G [ X ]and also F ( X , 23) = F ( X ) which denotes the free group with free generating set X . From specializing the general definition of o in ch. 1, Prop. 6.41, we obtain the canonical epimorphism ok(G): G [ X ] -+ P, (G) which describes the connection between polynomials and polynomial functions over G . Let F : G F ( X ) be the homomorphism that maps every g E G onto the identity e of F ( X ) and L : F ( X ) + F ( X )the identity automorphism of F ( X ) . By ch. 1, Th. 4.31, there is a unique homomorphism A : G [ X ] F ( X ) such that E = 291,1 = Ap2 which means - if-we observe the definitions of 91, p 2 - that Ag = e, for every g E G, and Axi = xi, i = 1, . . ., k . Clearly A is an epimorphism. We set 2 = &(G) which we call the length epimorphism (of G [ X ] ) The . normal subgroup A,(G) ker a,(G) of F ( X ) is called the length of G [ X ]and will be denoted by L,(G).
-
-+
1.11. Lemma. H
2
G impZies Lk(H) = Lk(G).
-
Proof. Let 6 : H G be any isomorphism and 8 [ X ]: H [ X ] G [ X ] its extension to an isomorphism according to ch. 1, Prop. 4.5. Then &(H) = &(G) 8 [ X ]whence L,(H) = A,(G) #[XI ker ok(H).But by diagram fig. 3.1 of ch. 3, 8[X]kero,(H) = keru,(G) which proves the lemma. -+
1.12. Proposition. Lk(G) is a verbal subgroup of F ( X ) . Proof. By ch. 6 , § 6.72, it is sufficient to show that w(x,, . . ., xk)E L,(G) implies w(wl(xl, . . ., xk), . ., wk(xl, . . ., x,)) E L,(G), for any k elements wi(xl, . . ., xk)E F ( X ) , i = 1, . . ., k. Let f(x,, . . ., x,) E ker ok(G) such that Ak(G)f(xl, . . ., x k ) = w(xl, . . ., xk). By definition of kero,(G), we 223
224
COMPOSITION AND POLYNOMIAL FUNCTIONS OVER GROUPS
CH.
5
have f ( w , ( g , , . . ., gk), . . ., w&, . . ., 8,)) = 1 , for all g,, . . ., gk E G whence .f(w,(x,, . . ., xk), . . ., wk(xl, . . ., x,)) E ker ak(G). Therefore J%(G)f(W1(X1,* . ., x,), . . ., w,(x,, . . ., x,)) = w(wl(xl, . . ., x,), . . ., w,(x,, . * .,I) EL,(G). .Y
1.13. Proposition. G [ X ]i ker ck(G)ker &(G) s F ( X ) Proof.F(X)IL,(G)-~,(G)G[X] I &(G) kera,(G)rG[X 1 kera,(G)kerA,(G) by the second isomorphism theorem of group theory.
1.14. Proposition. I f N isa normalsubgroup of G, then Lk(G)C L,(GIN).
- -
Proof. Let 8 : G G I N be the canonical epimorphism and @[XI: G [ X ] (G 1 N ) [XI its extension to an epimorphism which fixes X elementwise (see ch. 1, Prop. 4.5). Then the diagram fig. 5.1 is commuBut tative. Hence L,(G) = &(G) ker a,(G) = &(GI N ) 8 [ X ]ker a,(G). 6 [ X ]ker bk(G) C ker a,(G 1 N ) , by ch. 3, diagram fig. 3.1 whence M G ) c Lk(GIN). G[XJ
14(G)
FIG.5.1 1.2. Proposition. Let G,, G, be any two groups. Then L,(G,XG,) = Lk(Gl)
nLk(G2).
Proof. By Prop. 1.14 and Lemma 1 . 1 1, Lk(GIXG,) C Lk(Gl)nL,(G,). Conversely let MI L,(G,)nL,(G,). Then there exist f, E ker o,(G,), i = 1, 2, such that I,,(G,)h = w , i = 1, 2. Let p2 : F ( X ) ( G , X G , ) [ X ] be the homomorphism which fixes X elementwise and ii : Gi G , X G,, i = 1, 2, the inclusion monomorphisms (see ch. 6, 5 6.4). Since ch. 1, Prop. 4.5 remains valid if we replace “epimorphism” by “homomorphism” and “onto” by “to” as one can easily see, ii can be extended to a homo-
- -
41
THE CONCEPT OF LENGTH
225
morphism ii[X] : Gi[X] -,(G, x G,) [XI which fixes X elementwise. Let = (il[Xlfl) (c,[Xlf,) (v2w)F1 E (G, X G,) [XI. Then ak(G, X G2)f maps every element of G, x G, onto 1, thus fc ker ak(Gl x G2). Moreover A,(G, xG,)f = w . Hence w E &(G1 X G,) ker a,(G1 X G,) = Lk(G1X G2).
f
1.21. Proposition. Let G = G, X G,. Then ker ;Ik(G)ker a,(G) j ker a,(G) s ker Ak(G1) ker a,(G,) I ker ak(G1)X ker Ak(G2)ker a,(G,) I ker a,(G,).
226
COMPOSITION AND POLYNOMIAL FUNCTIONS OVER GROUPS
CH.
5
1.22. Corollary. If GI, GZ are finite groups and G = G1XG2, then the decomposition homomorphism z of P,(G) is an isomorphism if and only i f Lk(G1) Lk(G2) = Proof. We know from ch. 3, Prop. 3.53, that z:P,(G) Pk(Gl)XPk(G2) is a monomorphism. Thus z is an isomorphism if and only if IP,(G)I = I Pk(G1) I I Pk(G.2)I, i.e. +
1 I
11
GD-1I ker %(GI = G,[X 1 I ker Ok(G1) G,KI I ker u,(G2> 1. This is equivalent to
1
G[X] 1 ker a,(G) ker &(G) ker o,(G) ker &(G) I ker a,(G) = ~
=
2 ~
i=l
~
11
G,[X] I ker uk(GI)ker &(GJ ker g,(GJ ker 2, (Gi) I ker ok(Gi)~.
By Prop. 1.21 and Prop. 1.13, this is true if and only if \F(X)ILk(G)I = F(X)IL,(G,), iF(X)lLk(G2)i, and by Prop. 1.2, if and only if F(X)I Lk(Gl)nL,(G,)i = /F(X)IL,(G,)I l F ( X )lLk(GZ)l. By the first isomorphism theorem, this is equivalent to saying ILk(G1)L,(G,) I Lk(G2) = ‘ F ( X )IL,(G,) which holds if and only if Lk(G1)Lk(G2)= F ( X ) .
1
1
1.3. Proposition. Let G be any Jinite group and p a prime. Suppose Ap(X) is the verbal subgroup of F ( X ) generated by xf and X ~ ~ X ; ~ X , Xfor ~, I X I >- 2, and generated by xp,for X = {x}, then L,(G) c A J X ) if and only if G possesses some central p-chief ,factor. Proof. Let H I K be a central p-chief factor of G. Since by Prop. 1.14, L,(G) E L,(G I K ) , the ‘‘if” part of the proposition will be proved if we show that L,(G) G A p ( X )whenever G has some central minimal normal p-subgroup N which, by ch. 6 , § 6.51, is cyclic of order p. Suppose w E L,(G). Then there exists f €kero,(G) such that A,(G)f= w. Let m,, . . ., mk E N , then 1 = ,f(m,, . . . ,m,) = f ( l , . . ., 1) w(ml, . . ., m,) = w(ml, . . ., mk) since N is central in G. Hence w = 1 is a law for N . But every k-variable law for N is of the form Y = 1 where 21 E A,(X) (see ch. 6, Lemma 6.73). Hence w E A,(X) and L,(G) 5 ,4,(X). Conversely let L,(G) 5 A p ( X ) and x E X . Then, since the subgroup of F ( X ) generated by x is a free group, we have Lk(G)nF(x) c A , ( X ) n F ( x ) and therefore L1(G) c A,(x). Let N be a minimal normal sub-
01
THE CONCEPT OF LENGTH
227
group of G and L ( G , N ) = { w E F(x)1 there exists f~G[x] such that I1(G)f = w andf(n) = 1, for all n E N } .Then Ll(G, N ) is a subgroup of F(x) and L I ( N ) C L1(G, N ) . We distinguish two cases :Case a): N is nonabelian. Then N = Nl x N , x . . . XN,, Ni z Nj, i, j = 1, . . ., k , and Ni is simple non-abelian (cf. ch. 6,§ 6.51). By a result which will be proved in 8 2.43, Ll(Ni) = F(x) whence Prop. 1.2 implies that L,(N) = F(x). Let Ll(N)o Ll(G 1 N ) be the set of all elements of F(x) which we obtain from substituting the elements of L1(GIN)into the elements of Ll(N). Clearly L I ( N )oL1(G1 N ) c L1(G).Since x E Ll(N),we obtain L1(G IN) E L1(G) 5 AJx). Case b): N is abelian. Let L,(G, N)oL1(GjN) be the set of all elements of F(x) which we obtain from substituting the elements of Ll(G 1 N ) into the elements of L ( G , N). Then Ll(G, N ) oLl(G I N ) s L,(G) E A,(x). Since p is a prime, we have either L,(G, N ) E AJx) or L,(G I N ) E AJx). If L,(G, N ) E A,(x), then N is a p-group, for otherwise there exists some prime q # p such that A,(x) c L,(N) E L,(G, N ) s A,(x), contradiction. Let K be the prime field of characteristic p: and regard N as a KG-module where G acts on N by conjugation (cf. ch. 6, $ 7.5). The annihilator An N of N in KG is a maximal two-sided ideal in KG since KG 1 An N is a simple ring (see ch. 6, $7.4). Let (k,gl g E G ) E An N , k, E K, and ki integers representing k, E K 2 Z I ( p ) . Then (g9hg-l I g E G ) E G[x], where the factors are arranged arbitrarily, maps every element of N onto 1 . Thus xXkbE L,(G, N ) c A&x) = [x”].H e nc e plz ki, i.e. k, = 0. But the set I = {C (k,gIg E G ) I k, E K, C k, = 0} is a two-sided ideal in KG such that An N c I. Therefore An N = = I , by the maximality of An N. Hence g - 1 E An N , for every g E G, thus gng-l = n, for all g E G, n E N, i.e. N is central. The proof concludes with an induction argument on the length I of a chief series of G. If I = 1, then G is simple. If G were non-abelian, then by a), F(x) = L,(G) c A,(x), contradiction. Hence G is a simple abelian group and thus obviously possesses a central p-chief factor. Suppose now that the proposition is true for I- I instead of I, and let N be a minimal normal subgroup of G. Then by a), b), N is a central p-chief factor of G, or L,(GIN) E AJx). In the second case, GIN has some central p-chief factor by induction which is isomorphic to some central g-chief factor of G .
n
2
1.4. Proposition. If a k + I-ary operation x is dejined in F ( X ) by 1cw0w1 . . .wk = wo(wl, . . .,wk), then the algebra ( F ( X ) ;Q, x) is a k-dimensional
228
COMPOSITION AND POLYNOMIAL FUNCTIONS OVER GROUPS
CH.
5
composition group with selector system and the epimorphism Ak(G): G [ X ]-, F ( X ) is a composition epimorphism. In particular, ker Ak(G) is a full ideal of G [XI.
Proof. By ch. 1, Cor. 9.22, we have F ( X ) = {l}[X]. Thus by ch. 3, Prop. 2.24, ( F ( X ) ; SZ, x ) is a k-dimensional composition group with selector system. Clearly A,(G) is the unique extension of the epimorphism G (1) to an epimorphism from C[X] to { l}[X] fixing X elementwise whence it is a composition epimorphism, by ch. 3, 9 3.22. The last assertion follows from ch. 3 , s 4.1. -+
2. Distributively generated composition groups and polynomial functions over groups 2.1. In ch. 3 , § 1 1 . 1 , the functor @ from thecategory of k-dimensional %-composition algebras to the category of 1-dimensional %-composition algebras has been introduced, 23 being any variety. Taking the variety of groups for %, the functor @ becomes a functor from the category of k-dimensional composition groups to the category of near-rings. Some special classes of composition groups and the effect of on these classes will be investigated now. First we define: Let G = ( G ; f , -, 0, x ) be a k-dimensional composition group where + is the (not necessarily commutative) group operation, - the inverse operation, 0 the identity, and x the composition. An element d € G is called a distributive element if xd(a, b,) . . . (ak bk) = xda, . . . a,+ xdb, . . . b,, for all a,, . . . , ak, b,, . . ., bk E G . A kdimensional composition group G = ( G ; -, 0, x) is called a distributively generated (d.g.) k-dimensional composition group if there exist -, 0 ) = [{diliEI}] distributive elements diE G , i € Z , such that ( G ; - i.e., G regarded as an ordinary group has a generating set consisting of distributive elements. For k = 1, we obtain the well-known concept of a distributively generated near-ring. Clearly every homomorphic image of a d.g. k-dimensional composition group is again a d.g. k-dimensional composition group.
+
+
+,
+,
2.11. Proposition. If G is a d.g. k-dimensional composition group, then (T(G) is a d.g. near-ring.
$2
DISTRIBUTIVELY GENERATED COMPOSITION GROUPS
229
Proof. Let (G; f , -, 0) = [{di1 i E Z}], di E G distributive. Then ( @ ( G ) ; -, 0) = [U({(di,O,. . .,O), (O,d,, ...,O), ..., (O,O, ..., d i ) } l i € Z ) ] . Hence it suffices to show that any element (0, . . ., di, . . ., 0) € @(G) is distributive w.r.t. the composition o in P ( G ) . Let (al, . . ., ak), (bl, . .,bk) E @(G), then
+,
~
(0,. . ., di, . . ., 0) 0 [(a,, . . ., ak)+ (b1, . . .,bk)] =
. . ., di, . . ., O)o(a,+b,, . .., ak+bk) = (0, . . ., xdi(a, + b,) . . . (ak +bk), . . ., 0 ) = (0, . . ., %dial . . . ak+xdibl . . . bk, . . ., 0) = (0, . . ., xdp, . . . ak, . . ., O)+(O, . . ., xdib, . . . bk, . . ., 0) = (0, . . ., di, . . ., 0) o (al, . . ., ak)+(0, . . ., di,. . ., 0) o (bl, . . ., bk).
= (0,
2.12. Lemma. Let G be any d.g. k-dimensional composition group and A a normal subgroup of (G; +, -, 0). Then A is a full ideal in G i f and only i f x a c ,... c,€AandxcO . . . a . . . OEA, f o r a n y a E A , c , , ..., ck, cEG. Proof. Let A be a full ideal in G, qE A, c,, . . ., ck€ G, then by ch. 3, Q 5.2, xac, . . .c k -xOc, . . . ck E A . But xOc, . . .ck = 0 since x is superdistributive, thus xac, . . . ck € A . Moreover if c € G, then again by ch. 3, Q 5.2, xc0 . . . a . . . 0-xc0 . . . 0 E A . But since c is a sum of distributive elements and additive inverses of distributive elements and xd0 . . . 0 = 0, for any distributive element d E G , superdistributivity again yields xc0 . . 0 = 0. Conversely, let A satisfy the hypothesis of the lemma, C,, . . ., c k E G, a € A . Then X(Co+a)C, . . . ck-%oCl . . . Ck = XCoCl . . . ck +xac, . .. ck-xcocl . . . ck E A since x is superdistributive and A is nor-, 0). Moreover if 1 4 v =sk then xco.. . (c,+a) . . . ck mal in (G; - x c o . . .c, . ..ckE A whenever c, is distributive, for x c o . . .(c,+a). . .ck -xco. . .c,. . .ck = xc,. ..c,. . .ckf %coo...a. . .O --xco . .. c, . . . c k , and if c, is not distributive, then c, is a sum of distributive elements and additive inverses of distributive elements, and again the superdistributivity of x yields the same result as is easily checked. Hence by ch. 3, Q 5.2, A is a full ideal.
.
+,
2.13. Proposition. Let G be a d.g. k-dimensional composition group with selector system sl, . . .,sk.Then @(G) is a simple near-ring if and only ifG is a simple k-dimensional composition group.
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COMPOSITION AND POLYNOMIAL FUNCTIONS OVER GROUPS
CH.
5
Proof. Assume that G is not simple, and let A be a non-trivial full ideal in G. Then Ak is a normal subgroup of V ( G ) . Moreover if ( U 1 , . . . , a , ) E A k 7 ( g l , ...,gk),(hl, ...7hk)EGk,then(gl+a,, .--7gkfak)O (hi, ...) hk)-(gl, gk)o(hl, - . * ) hk)EAk and (gl, - - -gk)O , (hl+ul, . . ., h,+a,)-(gl, . . ., gk)o(hl, . . ., h,)€ Ak whence Ak is a non-trivial full ideal of @(G). Therefore @(G) is non-simple. Conversely suppose that G is simple and A is a non-zero full ideal in e ( G ) . Then there is a non-zero element (a,, . . ., ak) E A . Assume that a, # 0, then the set A iof all i-th components of elements of A contains a non-zero element. We will show that Ai is a full ideal in G . Clearly A, is a subgroup of (G; , -, 0) and is moreover normal in G. Let cl, . . ., c,EG, aEA,, and (al, . . ., a,-l, a, . .., a,)€ A . Then (al, . . ., U j - 1 , a, U i + l , .. ., Uk)O (Cl, . . ., C k ) = (xalcl . . . C k , . . ., xacl.. .ck, . . ., xakcl. . .ck) € A , hence xacl. . .ckEA,, forallc,, . . ., c k E G. Let c E G, then (0, . . ., lcc0 . . . Os,O . . . 0, . . .,0) o (al, . . .,a, . . .,a,) = . . a k , . . I ,0) = (0, . . xco. . .a. . -0, . . 0) E A (0, . . .,xxco. . .osjo. . by the superassociativity of x, hence xc . . . OaO . . . 0 E A,, for all c € G . By Lemma 2.12, A iis a full ideal of G whence A i= G by simplicity of G . Hence, for any a E G, there exist elements a,, . . ., . . ., a, E G such that (al, . . ., aiPl,a, a,+l, . . ., a k ) E A . If (0, . . .,Si, . . ., 0) E V ( G ) has s, as its j-th component, j = 1, . . ., k, then (0, . . ., si, . .., 0 ) o (al, . . ., ai-l, a, u ~ +.~. .,, a,) = (0, . . ., a, . . ., 0) E A , by Lemma 2.12 where a occurs as the j-th component. But ((0, . . ., a, . . ., 0)luE G, j = 1, . . ., k} additively generates G whence A = G(G). Hence Q(G) is simple.
+
- 9
2.2. Let N be a normal subgroup of a group G and k a 1 an integer. By P, (G, N) we will denote the set of all mappings obtained by restricting the action of the elements of Pk (G) to N k while P,(G, N ) = {yEP,(G, N ) ] p ( l , 1 , . . ., 1) = 1). Clearly Fk(G, N ) E F,(N). The set F, (G, G) = { y E P, (G)I p(1, 1, . . ., 1) = l} is a composition subgroup of the k-dimensional composition group Pk(G) and the mapping from p k (G, G) onto F, (G, iV) which maps every element of p k (G, G) onto its restriction to Nk, is a composition epimorphism whence F,(G, N ) is a composition subgroup of the k-dimensional composition group Fk ( N ) . HenceV(P,(G,N))isasubnear-ring of Q(Fk(N)).ForNbeinga minimal normal subgroup of a finite group G, some structural results on f ( P k(G, N ) ) will be obtained. We have to distinguish two cases : a) N is
--
(
12
DISTRIBUTIVELY GENERATED COMPOSITION GROUPS
23 1
abelian, then N is an elementary abelian p-group, for some prime p. b) N is non-abelian. 2.3. First we deal with case a) and view N as a KG-module where G acts on N by conjugation and K is the prime field of characteristic p . In particular, we will write C[(Cczijgj)nj 1 nj E N, gi E GI instead of n(nginj*"gZ:lI nj E.N, gi E G) where ccij are integers. The proof of the following proposition requires some ring-theoretical results (see ch. 6,f 7). 2.31. Proposition. Let G be a jnite group and N an elementary abelian minimal normal p-subgroup of G, for some prime p . Then @(pk(G, N ) ) is a simple ring.
Proof. Let cp : @(Fk(N)) Fl(Nk) be the composition isomorphism of ch. 3, Lemma 11.21.All we have to show is that y@(Fk (G, N)) is a simple ring. Let y E p @ ( P k ( G , N)), then, by ch. 1, Th. 9.21, --L
v(%,
' .7
n ~ =)
(
k
k
j=1
(C("ij1gi
I gi E G))nj,. . ., C (z(%jkgiI gi j=1
(2.3) and any such mapping belongs to p@(Pk (G, N ) ) ,as one can easily check. Let An N be the annihilator ideal of N in KG, then we may regard N also as a faithful irreducible KG I An N-module. Hence by a well-known theorem of representation theory, KG 1 An N is a simple ring, thus a full matrix ring over some finite field. On the other hand, (2.3) shows that p@(Pk ( G , N)) is isomorphic to the full k xk-matrix ring over KG I An N, hence also isomorphic to some full matrix ring over a finite field and thus is a simple ring. 2.4. Lemma. Let G be any$nite group and N a non-abelianminimal normal subgroupofG.ThenPk(G,N)= {pEFk(N)IV(1, ..., 1) = I}.
Proof. We apply ch. 1, Prop. 12.5, putting A = N k and taking for H the set of those mappings of Pk(G, N) which map Nk to N. Then H is a subgroup of F such that conditions a) and c) are satisfied. Let nl, n2 be two distinct elements of Nk, then Ep, # Ei&, for at least one projection Ei € H whence also condition b) holds. Hence H = F k ( N ) and thus pk(G,N) = {(pEHlp(1, . .., 1) = 1 } = { y € F k ( N ) \ y ( l ,. .., 1) = 1).
232
COMPOSITION AND POLYNOMIAL FUNCnONS OVER GROUPS
CH.
5
2.41. Remark. If G is any group and N is a normal subgroup of G, then pk(G, N) is a d.g. k-dimensional composition group. Indeed, [{gtig-'l i = 1, . . ., k, gE G}] = (pk(G, N); ., -l, 1) and gtig-l is distributive, for all g E G and all projections ti. 2.42. Proposition. If G is any finite group and N a non-abelian minimal normal subgroup of G, then Q(Pk (G,N))is a simple d.g. near-ring.
Proof. Let U be a non-zero full ideal of Pk (G, N). If we identify N with then UN is a subgroup of the group of constant functions of Fk(N), (Fk(N); -l, 1) since, for p,, pz E U, n,, nz EN,we have
.,
.
(nlnz> = p1(x(nlhn:l)p2 1 . . 1) (nlnz)E UN,
(PI%) ( V Z ~ Z= )
by Lemma 2.12 where 5, is the first projection of Fk( N ) and thus n1t1n;' E pk(G, N).We now apply ch. 1, Prop. 12.5 setting A = Nk and H = UN. Clearly condition a) is satisfied. Let r E G, y E U , then r-lt,r E pk(G, N) whence, by Lemma 2.12, x(r1t1r)y 1 . . . 1 = r-Llyr E U , thus p E H and r E G implies r-lyr E UN. Hence also condition c) holds. Suppose there exist elements nl, nzENk such that nl f nz while ynl = ynz, for all y E UN. WLOG, we can assume that nl z (1, . . ., 1). For any n EN, we choose q(nl, n) E Fk(N) such that q(nl, n)nl = n and q(nl, n)n = 1, for n # n,. By Lemma 2.4, q(nl, n) E pk(G,N). Hence, for any n E Nk, we have xyq(n,, 5,n) . . . q(nl, tkn)E UN since U is a full ideal of pk(G, N). Therefore = "V'T(n1, [In)
* * .
= xy?;i(n1761n) .
q(nI3 Ekn)nl 5kn)%
= V(I>
I,
a
-7
'1-
Therefore UN C N, i.e. U L Nand U would be the zero-ideal of pk (G, N). Hence also condition b) is satisfied. Therefore UN = Fk(N), and this implies that pk (G,N ) = pk (G, N) n UN = u(Pk (G, N)nN ) = U. Hence p k ( G ,N )is simple. Prop. 2.13 yields the result. 2.43. Corollary. If G is a finite simple non-abelian group, then LJG) = F ( X ) , for any k.
Proof. As in 0 1, let ak(G) : G[X] -,Pk (G) be the canonical epimorphism and &(G) : G[X] F(X)the length-epimorphism. Then, by Prop. 1.4,
-.
43
O N POLYNOMIAL PERMUTATIONS OVER GROUPS
233
ker &(G) is a full ideal of G [ X ] ,hence ak(G) ker &(G) is a full ideal of
P, (G). Therefore aJG) ker L,(G)n P, (G, G) is a full ideal of P, (G, G).
By the proof of Prop. 2.42, F,(G, G) is simple, thus a,(G) ker A,(G)n P,(G, G) equals either {I} or F,(G, G). In the first case, we arrive at a contradiction when taking any g E G, g i1 and h E G such that h does not centralize g, for then, with the notation of Prop. 2.42, for nl E Gk, n, # (1, 1, . . ., l), weobtain 1 #rl(n,,h)-lg~(n,,h)g-lEo,(G)ker;l,(G)n Pk (G, G). Hence P, (G, G) C ak(G) ker &(G) C Pk (G). But the group Pk (G) I P, (G, G) is isomorphic to G, hence is simple, and 5, E P, (G, C), (x(aS,)b, . . . b,) (xab, . . . bk)-l = ab,a-l$ P, (G, G), for b, # 1, shows that P,(G, G) is not a full ideal of Pk(G) by ch. 3, 0 5.2. Hence a,(G) ker R,(G) = Pk(G), therefore ker a,(G) ker &(G) = G [ X ] .Thus, by Prop. 1.13, L,(G) = F ( X ) .
3. On polynomial permutations over groups 3.1. Let G be a group and X = {xl,. . ., x,} a set of indeterminates. In ch. 3, 5 11.45, the semigroup U,(G) of all polynomial permutations of Gk has been introduced. If we recall ch. 3, $9 11.42, 11.43, we note that U,(G) consists of all polynomial function vectors f of q ( P k ( G ) )such that g;f is a permutation of Gk, and that, for finite G, U,(G) coincides with the group of units of the near-ring @(P,(G)). Let G,(X) = {f€G [ X ] I f (1, 1, . . ., 1) = l}. Then G o ( X )is a composition subgroup of the k-dimensional composition group G [ X ] . By Prop. 1.4, G o ( X ) n ker &(G) is a full ideal of G,(X). Moreover ker a,(G) E Go(X) and G,(X)( ker a,(G) Pk(G,G) which, by Remark 2.41, is a distributively generated k-dimensional composition group with a selector system. Let x : G,(X) G,(X)I ker a,(G) be the canonical composition group epimorphism. Then x ( C , ( X ) n ker &(G)) is a full ideal of G,(X)/kera,(G). Hence ker a,(G) ( G , ( X ) n ker Lk(G))is a full ideal of G,(X). Let
-.
be the canonical epimorphism. Then pG is a composition group epimorphism whence q ( p G ) is a near-ring epimorphism from a distributively generated near-ring with identity, by Prop. 2.11. As a consequence of ch. 6. Prop. 8.51, we obtain:
234
COMPOSITION AND POLYNOMIAL FUNCTIONS OVER GROUPS
CH.
5
3.12. The restriction of V ( p G )to &(Q(G,(X)l ker ok(G)))will be denoted by &(@(pG)).With this notation, we now prove
3.2. Proposition. Let G
=
ker &( q ( p G ) )
G I X G Z be a finite group. Then ker
e(@(pGl)) x kr&(q ( p G , ) ) .
Proof. Let 6 : G,(X) G,(X) I ker ok(G), 8;: Gio(X) G,(X) I ker o,(G,) be the canonical epimorphisms, i = 1, 2, and c J X ]:G J X ] (GIXG2) [XI the homomorphisms as in the proof of Prop. 1.2. Then L,[X]G,(X)E G,(X), i = 1, 2. If n i [ X ]: G [ X ] G,[X] denotes the exGi, i = 1, 2, to composition epitensions of the projections x i : G morphisms, then x i [ X ]G,(X) c G,(X). Let y : V ( G ,X G,) @(Gl) X q ( G J be the isomorphism as in ch. 3, Lemma 11.13, and define a mapping q : ker & ( ( G ( p G J )x ker &(CF(pG,)) ker & ( v ( p G ) )as follows : If (u,, &) E ker &(@(pG,)) x ker & ( e ( p G 2 ) )then , choose counterimages Ui of u; under @(Si) such that U;E p?(G;,(X)n ker lk(Gi)).This choice can be made, indeed, since if bi is any counterimage of ui under V(6J, then bi E z@(ker ok(Gi)(G,(X)n ker 3Lk(Gi))). We now set p(ul, u2) = Q(8) ( ~ ~ ( c J ii,s-lQ(~~[X])i&). X]) Then if g E Gk = @(G) and yg = (a, b), we have +
-
--f
-
+
-t
-
Y [ ( V ( ~ I [ XU~X-~J;F(LZ[XI)&) I) 0 91 = =
~ [ ( ( i a ( ~ i [ ~ I ) U ~ t 0, -4)- ~S) ( ( ( ~ ( ~ Z [ X I ) Fog)] -~UZ)
=
((GI o a)a-la, bb-l(iiz o b))
‘(a(
=
(GI o a, UZ0 6).
Hence @(o,(G)) ( ( ~ ( L , [ XGlx]) I ~ [ X ] ) U ,is) independent of the choice of the Gi,hence q(ul, &) is well defined. Since ui E &(@( Gio(X)1 ker ok(Gi))),Ui is a permutation polynomial vector of G:, hence the last equation shows that ~ ( L J XLtlF-1(?(~2[XI)& ]) induces a surjective mapping from Gk into itself and thus is a permutation polynomial vector of Gk. Hence ~(u,,u,) E & ( v ( G , ( X )I ker ok(G))).
03
ON POLYNOMIAL PERMUTATIONS OVER GROUPS
Hence q-l is an isomorphism and so is p.
235
236
COMPOSITION AND POLYNOMIAL FUNCTIONS OVER GROUPS
CH.
5
3.3. Theorem. Let G be a finite group, q :G H an epimorphism, and P,(q):P,(G) P,(H) the extension to a composition epimorphism. Then V(Pk(7))U,(G) = U,(H). --f
-+
Proof. By ch. 3, Prop. 11.51, @(P,(q)) U,(G) E U,(H). Let n = (1, 1, . . ., 1) E Gk and 1, = (El, . . ., Ek) E P,(G). By definition, P,(C, G) = {p' t P,(G)fp'n = l} whence Pk(q)Pk(G,G) E P,(H, H ) . Conversely, if y E P,(H, H ) and x E P,(G) such that Pk(q)x = y , then q(xn) = y n = 1. Thus (xn)-l x E P,(G, G) andP,(q) ((xn)-Ix) = q(xn)-lP,(q) x = y . Hence Pk(q)Pk(G,G) = P,(H, H ) , therefore @(P,(q)) @(Pk(G,G)) = @(P,(H, H ) ) . By 9 2.2 and 9 3.1, P,(G, G) is a d.g. composition subgroup of P,(G) containing the selector system of P,(G), hence by Prop. 2.11, @(Pk(G, G)) is a d.g. subnear-ring of @(P,(C)) containing the identity of @(P,(G)) which shows that &(@(Pk(G,G))) E d(@(P,(G))) = Uk(G) since G is finite. Clearly @(P,(G, G)) is also finite, therefore a result of LAUSCH (ch. 6,Prop.8.5l)implies that @(P,(q)) d(@(Pk(G,G ) ) ) = d ( V ( P k ( H , H ) ) ) . Hence @(P,(q)) U,(G) 2 &(@(P,(H, H ) ) ) = {f E U,(H))f o n = n}. Moreover @(P,(q)) U,(G) 2 {@)/ $ € H k } . But for any f € U,(H), we have f = (f 0 n) 1,0 (f 0 n)-lf, thus @ ( ~ , ( q ) U,(G) ) L? U,(H). 3.31. Theorem. Let G I , G2 be two Jinite groups and G = G1XG2. Then the decomposition homomorphism y2@(z2)for P,(G) maps U,(G) isomorphically onto U,(G,) X Uk(G2)if and only if either L,(G,) Lk(G2)= F(X), or k = 1 and L1(G1)L1(G2)= [ x 2 ] . Proof. By ch. 3, Prop. 11.61, yz@(zz) maps U,(G) monomorphically to U,(Gl)x U,(G2), hence also surjectively if and only if I Uk(G,)1 1 Uk(G2)1 = I U,(G) I. But, for any group G, f € U J G ) has a unique representation f = $9 0 g where 9 € Gk and g E @(Pk(G,G)) U,(G) = &(G(P,(G, G))), and any such element belongs to U,(G). Hence
n
I u,(G) I
=
I ckI I e(@(P,(G GI)) i
= I G I,
I e(ia(G,(x) I ker a , @ ) ) )
by 9 3.1. Thus y2@(z2)acts surjectively on U,(G) if and only if ~
' I-I 1 d ( @ ( ~ i o ( I~kero,(Gi))) ) i
e ( l ~ ( ~ oI ker ( ~~ )J G ) ) )=
2
1=1
and this is, by Prop. 3.2, equivalent to
1 &(
(?(PG))
8(@(G&X) I ker G 3 ) )= =
fi
1=l
~
~
&(v(~GJ)
&(@(G~,(x) I ker Ok(Gi>>)j.
83
237
ON POLYNOMIAL PERMUTATIONS OVER GROUPS
For an arbitrary group G , we have ker uk(G) c Go(X), therefore ker ok(G)( G , ( X ) n ker Ak(G)) = G o ( X ) nker uk(G)ker &(G). By the first isomorphism theorem of group theory and Prop. 1.13,
Go(X ) 1 ker Ck(G)(Go(X ) n ker A,(G)) >
z G , ( X ) ker o,(G) ker A,(G) 1 ker uk(G)ker A,(G)
2
F ( X ) IL,(G)
since G,,(X) ker &(G) = G [ X ] . Since ker o,(G) and ker A,(G) are full ideals of G [ X ] and L,(G) = Ak(G)ker uk(G) is a full ideal of F ( X ) , and the isomorphism theorems also hold for ideals of multioperator groups, this isomorphism is a composition isomorphism. Hence / our last condition is satisfied if and only if ! d ( G ( F ( X ) l L , ( G ) ) )= 2
fl [ d((?(F(X)IL,(GJ)) ~.If w E F ( X ) , G will denote the coset
wL,(G).
r=l
We define a mapping 8:Aut (F(X)IL,(G)) P ( F ( X ) / L k ( G >by ) 8r = (aZ1, . . ., ~2,). Certainly 8 is injective. Moreover 8(aB) = (a(/%,), . . >a(p2,)) ( P Z l o (a213 . ., E Z k ) , . . ., @2, 0 (ap,, . . ., E Z k ) ) = 8po 8a and if F denotes the identical automorphism, then 8 e = (Zl, . . ., 2,). Hence 8 a E d(@ ( F ( X )1 L,(G))). Conversely if (wl(Zl, . . .,a,), . . .,wk(Zl, . . ., a,)) E e(i?(F(X)IL,(G))), then the mapping a : F ( X ) / L , ( G ) -+ F ( X )I L,(G) defined by a( w(Z1, . . .,Z,)) = w(wl(Zl, . . ., Z,), . . ., wk(Zl, . . .,ZJ)is a well-defined automorphism. Thus 8 maps Aut F ( X )ILk(G) bijectively onto & ( ( T ( F ( XIL,(G))) ) and our last condition is equivalent to +
.
1 Aut F ( X ) ILk(G) I
2
=
IT I Aut F ( X )I
i=l
Lk(G1)
1’
(3.3)
We set wL,(G,) = i = 1 , 2. Let 8, be the mapping we obtain from 8 by replacing G by G I . Since, by Prop. 1.2, L,(G) = L,(G,)nL,(G,), F ( X ) ILk(Gl)X F ( X )IL,(G,) defined by the mapping [ :F ( X )I L,(G) 5S = (lG,2S) is certainly a composition monomorphism. Thus G(5): (?(F(X)IL,(G)) -(~(F(X)IL,(G,)XF(X)IL,(G,)) is a composition mo+
-
exists a composition monomorphism x P ( F ( X ) l L k ( G 2 )which ) maps the identity w.r.t. the composition of the first algebra to the identity w.r.t. the composition of the second one and therefore d(Q(F(X)IL,(G))) into e(G(F(X>iL,(Gl)))Xd(G(F(X)IL,(G,))). If a E AutF(X)ILk(G) and q8a = (ul, u2),then we set I I E = (8;lu1, 8 ~ ~ Thus 4 ) .II : AutF(X)I L,(G) Aut F ( X )1 &(GI) X Aut F ( X )I L,(G2) is a monomorphism. nomorphism.
Hence
there
7 :@(F(X)IL,(G)) i?(F(X)IL,(GJ)
+
238
COMPOSITION AND POLYNOMIAL FUNCTIONS OVER GROUPS
CH.
5
Suppose now that (3.3) is satisfied, then ~tis an isomorphism. The k-generator group G = F(X)IL,(G) has normal subgroups Ni= L,(G,)IL,(G), i = 1, 2, which satisfy N l n N , = 1. Let iGil, . . ., iGik be a generating set for F ( X )lLk(Gi).SinceL,(G,) is a full ideal of F ( X )and F ( X ) 1 L,(Gi) is finite by Prop. I . 13, there exists ai E Aut F ( X ) I L,(G,) such that aiiZj = iGij, j = 1, . . ., k. Set a = ~ - l ( a , a,). , Then q8.a = (6,cr,, 8 , ~ ~ Set ) . 8a = (Cl, . . ., C,) then this is a generating set of F ( X ) I L,(G), and 1.5. J = 1 .G.. 0 ' j = 1, . . . , k. Hence Gj _c lGljn 2G2j, i.e. vjLk(G) E wljLk(G1)nw,~L,(GJ. Therefore the hypothesis of ch. 6, Lemma 6.9 is satisfied. The lemma implies that either F(X)IL,(G) = Lk(G1) I XL,(GJ I L,(G) i.e. F ( X ) = L J G J L,(GJ, or /F(X)IL,(G,)L,(G,)I = 2. Since Lk(G1)Lk(G2)= V is a full ideal, in the second case xi 4 V , thus if k 1 1, we would have x2x;l E V whence x1 E V , contradiction. Therefore k = 1 and L1(G1)L1(G2)= [x2]. Conversely let Lk(G1)L,(G,) = F ( X ) . Then by Cor. 1.22, the decomposition homomorphism z, of P,(G) is an isomorphism, and by ch. 3, 5 11.3, y 2 t 7 ( z 2 )is an isomorphism. If, however, k = 1 and Ll(Gl)Ll(Gz)= [x'], then let &(GI) = [x'l] and L1(GZ)= [x'~].It follows (11,12) = 2. &(G) = L1(Gl)nL1(G2)implies L1(G) = [x'] where I is the least common multiple of 11 and 1,. Since I Aut F(x) 1 [x"]1 = p(n), y being Euler's p-function, (3.3) holds if p(Z) = p(Zl) p(Z2). WLOG, assume that 11 = 2n1, 1 2 = 2'n2, r z= 1, n1, n2 odd positive integers, (nl, n2) = 1. Thus I = 2*nlnz,hence p(Z) = 2'-l p(nl) p(nz) = p(Zl) p(IJ as required. 3.32. Corollary. IfG1, G2 are twofinite groups and (1 G I / ,I Gzl) = 1, then y 2 e ( z 2 )maps U,(Gl x G,) isomorphically onto U,(Gl) X U,(G,). Proof. (IGll, IG,l) = 1 implies L,(C,)L,(G,)
=
F(X).
4. Further results on the group of polynomial permutations over a finite group 4.1. Let G be a finite group and U,(G) the group of its k-dimensional
polynomial permutations which, by 5 3. I , coincides with the group of units of the near-ring (T(P,(G)). This section is devoted to various structuralproperties of U,(G). Since (T(P,(G)) is, in general, not a d.g. near-ring, but @(F,(G, G ) ) is, by Remark 2.41, it is of advantage to investigate first of all the group U,(G) = d(tT(P,(G, G ) ) )= U,(G)n tT(P,(G, G ) ) in order to utilize some results of ch. 6 on d.g. near-rings.
84
239
FURTHER RESULTS ON THE GROUP
4.11. Theorem. Let ]GI # 1. If Ok(G) is soluble, then G is soluble and either (i) k = 1 and, for every chief factor HI K of G, the group Aut, (HI K ) of automorphisms of HI K induced by the inner automorphisms of G is abelian or IHIKi = 4 or 9, or (ii) k = 2 and G is a supersoluble (2, 3)-group.
Proof. Let Uk(G)be soluble. Since the mapping y : G Uk(G)defined by yg = (g[,g-l, . . . ,gE,g-l) is a homomorphism with ker y = Z(G), the centre of G, GI Z(G) is soluble and so is G. Let HI K be a chief factor of G, then H ( K is a p-chief factor, for some prime p. One checks easily that (HI K)k is an V ( P k ( G ,G))-group in the sense of ch. 6 , s 8, where the r61e of S is taken by the set of elements (1, 1 , . . .,g-lE,g, . . ., 1) and the action of V ( P k ( G ,G)) on (HI K)k is defined by +
(PI,
. . ., pk) 0 (h1K . . .,
= (pi(h1, .
. ., hk) K , . . ., pk(h,, . . .,h k ) K ) .
(HI K)k is even a minimal T ( P k ( G ,G))-group since V ( P k ( G ,G ) ) contains all the elements of the form (1, 1, . . ., Ej, . . ., 1). The mapping o : (T(P, (G, G ) ) F1((HIK ) k )defified by (of) o lj = f o 4, f E @(Pk (G, G ) ) , lj E (HI I Y )is~ a near-ring homomorphism whence ker cr is a full ideal of Q(Pk(G, G)). With the notation of 5 2.3 and the result therein, we find that, for any p E P, (G,G),
-
v(h1, . . ., h J K
k
=
C (C.ij(giK) j=1
IgiKE G I K)hjK.
Hence as in 5 2.3, o ( y ( P k(G, G ) ) is isomorphic to the full k X k-matrix ring over K,(G I K ) I An (HI K ) where K, denotes the field of p elements. But by 0 2.3, K,(G j K ) I An ( H j K ) is a simple ring, thus is isomorphic to some full matrix ring of degree m over some finite field L containing K,. Therefore oiY(Fk(G,G)) is isomorphic to the full matrix-ring Lkmof degree km over the finite field L of characteristic p. By ch. 6, Prop. 8.51, o&((Y(Pk(G,G))) = d ( o v ( P k ( G ,G ) ) ) z &(Lk,). Since Uk(G)is soluble, also &(Lkm)is soluble. By a well-known result on linear groups this is the case only if either (i) km = 1, or (ii) km = 2, p = 2 or 3 and L = K,. (i) k m = 1 impfies k = 1 a n d m = 1. Hence K , ( G ( K ) ( A n ( H I K )z L. Since Aut, (HI K ) is isomorphic to a subgroup of the multiplicative semigroup of K,(G I K ) I An ( H 1 K ) , it is cyclic, therefore abelian.
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(ii) km = 2. Case 1: k = 1, m = 2. By our previous results, KJG I K ) I An (HI K ) is isomorphic to the full matrix ring R z of degree 2 over K,. Hence HI K is a faithful irreducible R2-module. By ch. 6 , § 74, this implies that H I K is a vector space of dimension 2 over K,. Hence HlKI = 4 o r 9, since p must be 2 or 3. Case 2: k = 2, m = 1. The same argument as in Case 1 shows that dimKn( H I K ) = 1 whence !HIKi = 2 or 3. In either case k = 1 or 2. If k = 1, then condition (i) of Th. 4.11 is satisfied while if k = 2, then every chief factor of G has order 2 or 3 whence G is a supersoluble (2,3)-group. 4.12. Theorem. If G is soluble and either (i) or (ii) of Th. 4.11 is satisjied,
then U,(G) is soluble.
Proof. By induction on I G 1. If I G 1 = 1, then 1 Uk(G)1 = 1, hence U,(G) is soluble. Let N be a minimal normal subgroup of G , then the canonical epimorphism 7 : G G I N induces an epimorphism @ ( P k ( ( q ) )= Uk(7) : U,(G) U k ( G I N ) .Since GIN also satisfies (i) or (ii), U,(G IN) is soluble by induction, and it remains to show that ker U,(q) is soluble. Now ker Uk(q)= {u E Uk(G)1 g-l(uo g) E N k , for all 9 E Gk}, thus u o g = gn(u, g), n(u, g) E N k . Furthermore, for any n E N k , 8 E Gk, we have uo (gn) = (uo g) z(u, g)n where z(u, g) is some endomorphism of N k . Moreover z(u, g) is bijective, and p = a15ila25i2 . . . U , ~ , ~ UE~Pk + ~( G ) implies F(gln1, . . .> g,n,) = a1g,,a2git . - - asg,,as+l(a& . . . asgps+l)-l nr,(a2gr2. . .a,g,,a,+,) (a3gr8.. .a,grsas+l)-l. . . hence (p 0 g1-l (p 0 gn,qJ = (p o g)-' (9:o gnl) (p o (9o gn2) since N is abelian. Therefore u o (gn) = gn(u, g) z(u, g)n, for all u E ker U,(q). For each g E Gk, we obtain some mapping 6(g) : ker U,(q) -+ Sym N k , the symmetric group of N k , defined by (6(g)u)n = n(u, g)z(u, g)n since uE Uk(G). Moreover g( 6(g) (ulo u2))n = (1110 uz) 0 (gn) = u10 (g( 6(g)u2)n) = g( 6(g)ul) * (Kg)uz)n = g( a(g)u10 6(g)uZ)n hence 6(g) (ul 0 UZ) = 8(g)ulo 6(g)u2. Therefore 6(g) is a homomorphism. Furthermore n(ker 6(g) I g E G")I = 1 whence ker U,(q) can be embedded into the direct product of all S(g) ker U,(q). All that remains is to show that 6(g) ker U,(r) is soluble, for all g E Gk. We choose g E Gk arbitrary and define : 6(g) ker Uk(q) Aut N k by q(6(g)u) = Z(U, 9). This is a well-defined mapping, for if 6(g)u = 6(g)b, then (6(g)u)e = (d(g)b)e, e = (1, 1, . . ., l)ENk. Hence n(U, g) = n(b, g), thus z(u, g) = z(b, 9). Moreover n(u o b, g) z(uo b, g)n = (a(g)u 0 6(g)b)n = (6(g)u) n(b, g) z(b, g)n = n(u, 9) z(u, 9) n(b 9) (z(u, 4 ) 0
-
-t
1
-.
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z(b, g))n. If we set n = e, then the first factors cancel and z(u o b, g) = z(u, g)oz(b, 9). Hence rp is a homomorphism. If 6(g)uE ker rp, then (6(g)u)n = n(u, g)n, hence ker rp is isomorphic to some subgroup of
N k and therefore soluble. But @(g) ker Uk(q)is a subgroup of Aut Nk. If (ii) holds or (i) holds and IN1 = 4 or 9, then Aut N k is isomorphic to GL(2, 2) or GL(2, 3) which is soluble. If, however, k = 1, and Aut, N is abelian, then any two elements z(u, g) commute since every z(u, g) in the endomorphism ring of N can be written as a sum of elements of Aut, N , and again p6(g) ker Uk(q)is soluble. 4.13. Corollary. If G is afinitegroup, then Uk(G) is soluble ifand only if G is soluble and either (i) k = 1 and, .for every chief factor H I K of G, Aut, (HI K ) is ubelian, or IHIKl = 4 or 9; or (ii) k = 2 and G is a supersoluble (2,3)-group; or (iii) I GI = 1 . 4.2. Proposition. If G is afinite group, then the radicalJ of p ( P k ( G ,G)) is nilpotent. Proof. Let HI K be a chief factor of G, then as in the proof of Th. 4.11, (H 1 K ) k is a minimal p(Pk (G, G))-group. If f € J , then by definition of J , f annihilates (HIK)k, i.e. fo(h,, . . ., h,)E K k for all (hl, . . ., hk)E H k . Hence if 1 is the chief series length of G, then J' consists of the zero mapping only (ie. the mapping where every element has the image (1,l . . ., 1). Thus J is nilpotent.
4.21. Proposition. Every minimal q ( p k(G, G))-group is q ( P k(G, G))isomorphic to some H k I K k where H 1 K is some chief factor of G. Proof. By ch. 6, Prop. 8.43, every minimal (J(Pk(G, G))-group is isomorphic to some @(Pk (G, G))i Ji, for some idempotent i E @(Pk (G, G)) where i is non-zero, since J is nilpotent by Prop. 4.2. But i cannot annihilate H k I K k , for every chief factor HI K of G, otherwise i is the zero mapping So we can choose some chief factor H IK such that i(HkIKk) # K k . Hence there exists m € H k ( K kwith ( t y ( F k(G, G))i) m # K k . Since (T(Pk (G,G))i)mis also an P ( F k ( G , G))-group, the minimality of H k I K k implies (@(Pk(G, G))i)m = Hkl K k . The mapping
I
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f i -,(fi)m, f E ( T ( p k ( G ,G)) is an (?(P,(G, G))-homomorphism from ( ? ( p k (G, G))i to H k 1 K k and is surjective, and its kernel contains Ji, by ch. 6, Prop. 8.43. Hence, by the minimality of G ( P , ( G , G))ijJi, the kernel equals Ji. Therefore p ( p k(G, G))i Ji z H k I K k . ~
n
I c q P k ( G , c,)(H, I K k ) , the intersection being taken over a fullset of pairwise non-G(pk(G, G))isomorphic factors H 1 K k where HI K are chief factors of G.
4.22. Corollary. With the notation of ch. 6 , § 8, J =
4.3. Proposition. Let G be anyjinite abelian group. Then U,(G) is isomorphic to a semidirectproduct of Gk by a group isomorphic to &(M(k,exp G)) where M(k, exp G ) denotes the ring of k x k-matrices over the integers modulo exp G.
Proof. Every p E [F(F,(G, (3)) can be written uniquely as p = . . . i"?, . . ., l=ff"l. . . tp) where aG is an integer, 0 =saij < exp G, and it is evident that 9 : V ( P k ( G ,G)) M(k, exp G) defined by 8cp = (aGmod exp G) is an isomorphism from the multiplicative semigroup of ?(pk(G, G)) to the multiplicative semigroup of M(k, exp G). ( ~ , 2 &(M(k,exp GI). If we set z = Hence Uk(G) = 8 ( V ( P k GI)) (El, . . ., t k ) then for any p E Vk(G),c E Gk, we have p G cx = (rp G c ) o~ p Since every element of U , (G) is of the form gx G cp where cp E V, (G) and g E Gk, it follows that C = (cz 1 c E Gk} is a normal subgroup of U , (G), and clearly C 2 Gk, moreover c a k ( G ) = u k ( G ) and Cn Vk(G) = (1). --t
4.4. Let F ( X ) be the free group on X = {xl, . . ., xk} and Z+ the additive group of integers. Then q i : F ( X ) Z+ defined by qixj= a, 1 -c i, j ==k , where 13, is the Kronecker symbol describes a group homomorphism. Let M(k, 0) be the ring of kxk-matrices over Z, and G a group. Then [,(G) : @(G[X]) -.- M(k, 0) defined by rk(G)f = Ck(G)(A, . . .,f k ) = (a,) where a,. = qjlbk(G)i.is a near-ring epimorphism as one can easily check. -f
4.41. Theorem. I f G f (1) is afinitep-group, then f E @ ( G [ X ] ) is ff permutation polynomial vector if and only if p does not divide det Ck(G)f.
Proof. By induction on I GI. If I G I = p , then G is abelian, hence for any f E @(G[x]) with 5'k(G)f = (aij),we have G(u,(G))f = (g15p1. . . 5Zk, . . .,
04
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FURTHER RESULTS ON THE GROUP
gkt(lL1. . . Ep).By Prop. 4.3, f is a permutation polynomial vector if and only if pldet 5,(G)f. Suppose now that I G I > p, and Nis a minimal normal subgroup of G . Then IN1 = p and N C Z(G), the centre of G . Let q : G GIN be the canonical epimorphism and q [ X ]: G[X] (GIN) [XI its extension to a composition epimorphism. If f is a permutation polynomial vector of G, then by ch. 3, § 1I S,@(q[X])f is also a permutation polynomial vector of G lN. By induction, p does not divide det Ck(G IN) f$(q[X])f = det Sk(G)f. Conversely let pfdet Ck(G)f, then pldet Ck(GIN) f$(q[X])f, hence by induction, @(q[X])f is a permutation polynomial vector of GIN. Thus if g, g1 E Gk and gg’; 6N k , then f o g # f o 9., If, however, 99’; E N k , then g1 = ga with 8 E N k G Z(G)k. Hence fog, = f o g 3 = ( f o g ) ( b o a ) where b = (tF1. . . t z k , . . ., 5 F . . . tp). Since p7det (aik), b is a polynomial permutation of Z(G)k,by Prop. 4.3, whence f o 9 1 = f o g implies a = e, i.e. g1 = g. Hence f is a permutation polynomial vector of G . -+
-+
4.42. Corollary. If G # (1) is a finite nilpotent group, then f E f$(G[X]) is a permutation polynomial vector if and only i f (det {,(G)f, I GI) = I. Proof. G is a direct product of its Sylow p-subgroups Gpi,pi being pairwise distinct primes. Repeated application of Cor. 3.32 together with Th. 4.41 yields the result. 4.43. Theorem. Let G # (1) be a jinite p-group. Then I Uk(G)I = I GL(k, p) 1 p*,for some integer t 0, and Uk(G) is a group extension of
some p-group by the group GL(k, p).
-
Proof. Let N be a maximal normal subgroup of G, q : G G IN the canonical epimorphism and @(Pk(q)):Q(Pk (G)) Q(Pk(G IN)) the corresponding near-ring epimorphism. If Q ( P k(q,q ) )denotes the restriction of @(Pk(q))to @(Pk(G, GI), then certainly @(Pk(q, q ) )@(Fk(G, G)) = p(pk(GIN,GIN)) and ker @(Fk(q, 7)) = IQ(P,(c, c , ) ( G ~ ( N ~Since ). G is a p-group, every chief factor H 1 K of G is a group of order p, being contained in Z(G I K ) . Hence an element of @(Pk (G, G)) is contained in I q p k ~ cq)(Hk , I K k ) if and only if, for every polynomial vector f representing this element, we have C,(G)f = p A where A is some integral kXk-matrix. Thus, by Cor. 4.22, ker@(Pk(q, q ) ) = J , the radical of @(Pk(G, G)). Since (t:, . . ., 5,”)EJ and, by Prop. 4.2, J is nilpotent, -t
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the (additive) order of every element of J is a power of p whence I J I = ps for some integer s. Now V(F,(7, T))f E &(@(P, (G 1 N , G I N ) ) ) if and only if fE&(@(F,(G, G ) ) ) , by Th. 4.41. Hence l&(Q(P,(G, G ) ) ) (= &(Q(pk(GIN, G I N ) ) ) I J 1. Since G 1 N is abelian, Prop. 4.3. implies I U,(G)I = IGIk l d ( V ( F , ( G , G ) ) ) I= l G l k IGL(k,p)l IJi=lGL(k,~)lp'.By Th. 3.3, @(P,(T))U,(G)= U,(GIN). Since / G I N / = p, the group U, (G IN) has a normal subgroup C such that U, (G IN) I C = GL(k,p ) , by Prop. 4.3, hence the remaining assertion of the theorem follows from the second isomorphism theorem.
i
I
4.44. Corollary. If G is afinite group, then U,(G) is nilpotent if and only if either 1 G I = 1, or k = 1 and G is a 2-group.
Proof. If 1 G I = 1, then I U,(G) I = 1, and if k = 1 and G is a 2-group, then by Th. 4.43, I U 1 ( G ) /= (GL(1, 2)12' = 2', hence U1(G)is nilpotent. Conversely let I G I # 1 and U,(G) be nilpotent. Since the set of all (at1, . . ., at,) constitutes a subgroup of U,(G) isomorphic to G , G itself is nilpotent, i.e. G is a direct product of its Sylow pi-subgroups Gpi.By Cor. 3.32, every U,(G,J is nilpotent whence, by Th. 4.43, every GL(k, pi) is nilpotent thus k = 1. Let G be any finite group and p E Z(Ul(G)). Then since 5 - I and g5, g E G , belong to Ul(G), we have p-l = [-lop, = q05-l and gp = g t o p , = p o g t . Hence p(1) = p,(gg-l) = gp,(g-l) = gp-l(g) thus p(g) = p,(l)-Ig, for any g E G . Therefore p = at, a E G whence (-la-] = a5-l. We conclude that a2 = 1 and see that, for odd I G 1, the group U1(G) cannot be nilpotent. Hence G has G$ as its only Sylow p-subgroup, i.e. G is a 2-group.
5. Characterization of classes of groups by properties of their permutation polynomials 5.1. Theorem. A finite group G is abelian if and only can be written as p, = aE', a E G , 1 an integer.
if every p, E U l ( G )
Proof. The "only if" part of the theorem is evident. Suppose that every polynomial permutation cp of G can be written as p = atk. Then, for each b c G , there exists some integer l(b) such that t b = bt'(b), i.e. b-ltb = E'('). Hence the group of inner automorphisms of G is abelian, i.e., GIZ(G) is abelian whence G is nilpotent (of class G 2). Moreover
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b'(b)-l= 1 whence the order o(b) of b divides I(b)- 1. Since G is nilpotent, there exists h E G such that o(h) = exp G whence exp G/(l(h)- 1). Hence E@) = 5 and h-lth = E. Therefore h E Z(G) and exp Z(G) = exp G . For z E Z(G), b E G, we have z ' ( ~ )= b-?zb = z, hence exp G 1 (l(b)- 1) and E'(b) = 5 . Therefore Eb = b5, for all b E G , i.e. G is abelian. 5.2. Theorem. Let k z 0 be an integer, X = {xl,. . .,xk}, and G a finite group. r f t for every f € G ( G [ X ] ) ,(det Ck(G)f, [ G I )= 1 implies that f is a permutation polynomial vector, then G is nilpotent. Proof. a) Assume k = 1. We use induction on IG/.Certainly the theorem is true for 1 G 1 = 1. Suppose the hypothesis of our theorem holds. Now let I G 1 =- 1, and M be a maximal subgroup of G such that 1 M 1 and 1 G I have the same prime divisors. By ch. 1, Th. 9.11, M [ x ] is a subgroup of G [ x ] ,and for any f €M [ x ] ,certainly (C1(M)f, 1 M I ) = 1 implies (Cl(G)f, IG 1) = 1. Thus i f f € M [ x ] and (Cl(M)f,I M i ) = 1, f is a permutation polynomial of M and by induction, M is nilpotent. Now suppose that M is a maximal subgroup of G such that I MI and I G I have not the same prime divisors, and let pi, i = 1 , . . ., r, be those primes with pi/lGI, p i { ] M ] .Let f c M [ x ] and (Cl(M)f,IMI) = 1. Then we can choose an integer I with I = 0 mod IMI, I E 1 -Cl(G)f modp,, i = 1 , 2, . . ., r, and set g = x y Then Cl(G)g = l+Cl(G)f whence Cl(G)g = Cl(G)f mod /MI and C1(G)g = 1 mod pi. Therefore (Cl(G)g, IGl) = 1 and by assumption, g is a permutation polynomial of G, furthermore g(m) = m'f(m) = f(m),for all m M . Hence f i s a permutation polynomial of M . By induction, M is also nilpotent in this case. Therefore G is a group all of whose maximal subgroups are nilpotent. By ch. 6 , § 6.53, G is nilpotent or 1 G j = p"#, p # q primes, G has a normal Sylow p-subgroup P,and all Sylow q-subgroups of G are cyclic. We want to show that the second alternative is impossible: Assume, that G is not nilpotent. Let f € (GIZ(G))1x1, (Ci(GIZ(G))f, IGIZ(G)I) = 1, 8 : G GIZ(G) be the canonical epimorphism and @[XI :G [ x ] -. (GI Z(G))[XI its extension to a composition epimorphism. We choose g E G [ x ] such that (Cl(G)g, IG I ) = 1 and al(G 1 Z(G))8[xIg = al(G 1 Z ( G ) ) f .Such a g can be found by a procedure similar to that used in the first part of the proof. Then g is a permutation polynomial of G whence f is a permutation polynomial of G IZ(G). Unless Z(G) = 1, G 1 Z(G) is nilpotent by induction and so would 4
24 6
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G be, contradiction. Therefore Z ( G ) = 1. Let Q = [h] be a fixed Sylow q-subgroup of G, then G = PQ. Since Z ( G ) = 1 , ch. 6, Q 6.53 implies I Q j = q and P i s elementary abelian. Moreover if PI is a normal subgroup of G and PI is properly contained in P,then PIQ is nilpotent, thus abelian whence PI G Z(G). Hence P is a minimal normal subgroup of G which we will regard as an irreducible KG-module where K is the field of order p and G acts by conjugation as usual. Let T be the representation of G over K afforded by P . If T(h)had 1 as an eigenvalue, then there would exist some 1 # r E Psuch that hrh-I r whence Y E Z(G),contradiction. Hence T is a non-trivial representation. Let f ( z ) = u,+u,z+ . . . + u p be the characteristic polynomial for T(h), then f(1) # 0. We choose integers vo, wl, . . ., v, representing uo, . . ., us E K, respectively, and an integer m such that m f 0 modp, m = l-vo- . . . -vs modq. Let .f= X r n ~ " ~ h x.V. ~. hx"sh-s E G [ x ] . Then Cl(G)f = m f wo+ . . . +us whence [ ~ ( G ) f 0g modp and Cl(G)f= 1 modq. Therefore (Cl(G)f,IGl) = 1, hence by hypothesis, f is a permutation polynomial of G. On the other hand, for any t g P, we have f(t) = tmtaohtvlh. .. ht".h-" = tvo(htV1h-l) -(h2tvW2). . . (h'Ph-"). Or when switching to additive notation, f ( t ) = f(T(h))t = 0, by the CAYLEY-HAMILTON equation (see ch. 6, Q 7.2) which contradicts the just obtained result that f is a permutation polynomial. Hence G is nilpotent. b) k =- 1. If G is not nilpotent, then by a) there exists a polynomial f ( x l ) E G [ x l ]such that (Cl(G)f(xl), 1 GI)= 1 andf(xl) is not a permutation polynomial. Let f = (f(x,), xz,. . ., x k ) E p(G[X]), then f is not a permutation polynomial vector, but (det tk(G) f, I G 1) = (det C,(G)f(xl), IGl) = 1. 5.3. Theorem. If G is a finite nilpotent group of class c(G) =s2, then Ul(G) is the set of all polynomial functions p= a"b, a, b E G, (1, 1 G 1) = I.
Proof. Certainly every polynomial function of this form belongs to U,(G). Conversely let p E U,(G), then p = a15'1a,5'2. . . ar5'rar+l,ai E G, li integers. Since c(G) s 2, the commutator group of G is contained in Z(G), and by ch. 6, Q 6.53, we have [ab, c] = [a, c] [b, c] and [a, bc] = [a, b] [a,c], a, b, c E G. Hence p = a, =
. . . ar+,5'1+ ... +'r[E'l, a,] . . . a,+,5'l+ ... +"[E, a i a $ + 1 2
as] . . . [5'1+ ... +'r ar+,l . . . all+',+ ... + I , I. r+l
[5'1+'2,
2
ByCor.4.42,(CI=,li, IGI) = 1,hencethereexistsh E Gsuchthath':=l'i
=
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5.31. Corollary. I f G is a finite nilpotent group of class 2, g, Euler's vfunction, and H(G) = {g E G 1 g - Y g = E"@, for some integer e(g)}, then
I Ul(G)I
=
I G l 2 d e x p G) I H(G) I
Proof.ByTh.5.3, U 1 ( G ) = { u 5 ' v ) u , v E G , 0 s Z ~ e x p G , ( Z , e x p G ) = 1 } . Amongst these I G l2 p(exp G) (not necessarily pairwise distinct) functions, we have to determine under what conditions at'b = c5"d. Suppose this holds for a , b , c , d € G , O s Z , m i e x p G , ( I ,e x p G)= ( m,e x p G )= 1, then a = cdb-' whence dbF'5'bd-l = Em, and if 11 E 1 mod exp G, then dbP15bd-l = Elm. Hence bd-l = h E H(G), thus d = h-lb, c = ah, and m = le(h)modexp G. If conversely, d = h-lb, c = ah, h € G(H), and m = le(h) mod exp G, then cE"d = ahE"(')h-lb = a(hh-lEhh-l)' b = a5'b.
5.32. The converse of Th. 5.3 is not true as the following example shows: Let G = S3, the symmetric group on three letters. The polynomial functions a<*lb, a, b E S3, belong to Ul(S3) and are pairwise distinct since Z ( S 3 )= { l}. Hence I U1(S3)I z=72. However, the cosets mod As, the alternating group on three letters, are blocks for Ui(S3) as a permutation group on S 3 . Since there are exactly two blocks of three elements each, weconclude that 1 U1(S3)j s 2! (3!)*=72. Hence 1 Ul(S3)(= 72 whence every mapping of U1(S3) has the form a5'b. S3 is, however, not nilpotent. Yet there is a partial converse of Th. 5.3 if we restrict the prime divisors of I G I : 5.33. Theorem. If G is a finite group of odd order and every p E U,(G) is of the form p = ayb, a, b € G, then G is nilpotent of class c(G) == 3. lffurthermore (/GI, 6) = 1, then c(G) ==2. Proof. By ch. 6, 0 6.53, it suffices to show that a and b-lab commute, for any two elements a, b E G. Since (1 G I,2) = 1, xg-lxg = xg2o x2o xg-l is a permutation polynomial, for any g E G. Hence there exists an integer I(g) and an element a(g) E G such that Eg-lEg = a(g)-lE'(g)a(g), (Z(g), I G 1) = 1. Let l(g) be a solution of I(g)Z(g)= 1 mod 1 G 1, then @)g-lEj(g)g =
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Remarks and comments
$6 1-5. Polynomial permutations on groups were originally discussed NOBAUER and SCHWEIGER [l], [2], for k = 1, and by NOby LAUSCH, BAUER [26], for k > 1. The concept of length in the case of one indeterminate (in a form which is somewhat different from that in our book) has been introduced by S. D. SCOTT[l] who also proved Prop. 1.3 for this case. Some of the methods of Q 2 originate from FROHLICH [l]. The case k = 1 of Th. 4.12 has been proved implicitly in LAUSCH, NOBAUER and SCHWEIGER [l], the proof for the general case in our book is due to LAUSCH. Th. 4.11 has also been proved by LAUSCH, after some preliminary [5]. The characterizations of certain classes work had been done in LAUSCH of groups by properties of their permutation polynomials in Q 5 are due to LAUSCH, NOBAUER and SCHWEIGER [2], LAUSCH and SCHWEIGER [l], and LAUSCH[3]. An explicit computation of the group U1(G), for various finite non-abelian groups G, has been performed by SCHUMACHER [l]. For a connection between U1(G) and the automorphism group of G, we can only refer to TROTTER [l].
CHAPTER 6 APPENDIX
1. Sets 1.1. In this section we will give a brief survey of those basic concepts and theorems on sets that are used in the book. For more detailed information and proofs of the theorems, we refer to standard texts, e.g. KAMKE[l]. A set is a collection of objects, these being called elements or members of the set. As usual we write a E A to indicate that a is an element of the set A , and a 6A in the opposite case. A set B is called a subset of the set A if every element of B is an element of A , we write in this case B E A . The set P ( A ) of all subsets of A is called the power set of A. If B is a proper subset of A, i.e. there exists a E A, a $ B, then we write B c A . The union A U B of the sets. A , B is the set of all elements which belong to at least one of the sets A; B. The intersection A n B of A , B is the set of allelements which belong to both A and B and A. B = { a a E A, a 6 B } is the difference of A, B. The empty set 4 is the set consisting of no element. 4 is a subset of every set. Two sets A, B are called disjoint if A n B = 4. Let Z be an arbitrary set. If for every i E I, there is some set Ai defined, then we speak of the family ( A , i i E I ) of sets. U (Ail i f I ) = { a ] a f A i for at least one i E I } is called the union of this family while n ( A ,I i C I ) = { a I a E A , for all i E I } is called the intersection of this family. It is well-known that intuitive set theory sometimes leads to contradictions, thus one has to introduce the concept of classes for “very large” sets. In this book, however, the distinction between classes and sets is, in fact, not essential.
1.2. Let A, B be two sets. A mapping (or function) p; from A to B (9: A B ) is a rule which assigns some element pa E B to each a E A . B is called the range of p. Sometimes the mapping p will also be written as a -qa. The element ga is called the image of a under Q., and if ya = b, a E A, then a is called an inverse image or counterimage of b under q. -t
249
250
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The set consisting of the images of all a E A will be denoted by a,A. The mapping p : A B is called injective or an injection if every b E B has at most one counterimage under while 9; is called surjective or a surjection if a,A = B and bijective or a bijection if p is injective and surjective. A surjective mapping is also called a mapping ,,onto”, whereas a mapping ,,into” is not necessarily surjective. Sometimes we use the following +
Lemma. If A is ajinite set and p : A bijective.
-.A is injective or surjective, then a, is
Let 9- : A B be any mapping and A , c A . Then the mapping pl: A1 B defined by pla = pa, for every a E A,,is called the restriction of 4p to A1 while a, is called an extension of p l to A. Sometimes we write 91 = elA,. Let a,: A B be a mapping and C E A n B . We say that p fixes C (elementwise) if 9;c = c, for all c E C. If a, : A B,y :B C , then ypa = y(ya) defines a mapping ya, : A C being called the product or the composition of the mappings y , p. For any ) both sides of the three mappings y , p, we have (xy)p = ~ ( y pwhenever equation are defined. The mapping E : A A with &a= a, for all a E A, is called the identical or identity mapping of A . E satisfies e p = p, a,& = y, for every mapping p to or from A, respectively. If qj: A B is bijective, then p-] :B A defined by p-l(pz) = a, for all a,a E B, a E A, is again bijective. p - l is called the inverse of p. A (finite) diagram consists of a finite family of sets and a finite set of mappings between the sets of this family which can be drawn in such a manner that the sets are represented by points and the mappings a,: A B by arrows linking the points which represent A, B. A diagram is called commutative if any two sequences of arrows between two points A and B represent one and the same mapping, e.g. diagram fig. 6.1 is commutative if xy = a,. A family of elements of the set A with index set Zis a mapping p : Z A . We set pi = a, and write (ajliE I ) for this family. Any restriction cpl of to a subset Z, of Z is called a subfamily of (a,1 i E I ) . Sometimes we also speak of systems and subsystems instead of families or subfamilies. If I is the set of positive integers, then (a,i i E Z) is a sequence, and if I is finite, then (a, i i E I ) is called a finite sequence. -+
+
+
-+
+
+
x,
+
+
--f
+
+
25 1
SETS
C
FIG.6.1
1.3. Let (A[IiEZ) be a family of sets. The Cartesian product D = n ( A ,1 i E I ) of this family is the set of all families (aiI i E I) of elements of U ( A , l i E Z ) such that aiE Ai, for all i E I . If d = (ailiEI)E D and j E Z, then aj E Aj is called the j-th component of d. If I = {I, 2, . . . , k}, then we write A , X A 2 X . . . X A , for JJ(A,liE I ) and if Ai = A , i = 1 , . . .,k, then we write Ak for A , x . .. x A,. We call Ak a Cartesian power of A . Let k =- 0 be an integer and A a set. By a k-place function or k-ary operation on A , we understand a mapping p :Ak A . A subset B G A is said to be closed under yif y(b,, ;. .,b,) E B whenever (bl, . . .,bk)E Bk. -t
1.4. Let A be a set. A binary relation O on A is a subset of A2. For (a, b) E O we will write aOb. A binary relation 0 on A is called an equivalence relation if 0 is reflexive, transitive, and symmetric, i.e. the following three conditions hold, for any a. b, c E A : (i) aOa; (ii) uOb, bOc imply aOc; (iii) aOb implies boa. Equivalence relations on A are strongly tied up with partitions of A. A partition of the set A is a set of non-empty subsets of A such that every a E A is an element of exactly one of these subsets. The subsets are called classes or blocks of the partition, and the subset containing a E A is called the class of a, denoted by C(a). Every element of a class C of a partition is called a representative of C . By a (full) system of representatives of a partition of A , we understand a subset R E A such that R n C consists of exactly one element, for every class C . The connection between partitions and equivalence relations of A can now be expressed by a
252
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Theorem. Let 8 be any partition of A. r f we dejine the binary relation @(a) on A by: aO(8)b if and oniy if a, b are in the-same class of 8, then O(8) is an equivalence relation. The mapping p from the set of all partitions of A to the set of all equivalence relations on A dejined by pa = O(8),for every partition 9, is bijective. The classes of the partition, corresponding to the equivalence relation 0, are called equivalence classes under 0. 1.5. Let A be any set. A binary relation =son A is called a partial order (relation) if =s is reflexive, transitive and antisymmetric, i.e., for all a, b, c E A , the following conditions hold: (ij a == a ; (iij a =s 6, b =s c imply a =s c ; (iiij a =S b and b =s a imply a = b. The pair ( A ; 4) is called a partially ordered set. We write a -== b if a =sb but not a = b. A partial order relation =son A is called a total order (relation) if also (ivj a =s b or b =sa, for all a, b € A , is satisfied. (A; =s) where =s is a total order relation, is called a totally ordered set or a chaiii. Let ( A ; =s) be a partially ordered set. An element aE A is called a maximal element of A if a =sb implies a = b and a is called a minimal element of A if b == a implies b = a. a E A is the greatest element of A if b 4 a, for all b E A, and a is the least element of A if a =s 6, for all b E A . Clearly A has at most one greatest (least) element being denoted by max A (min A). Let B be a subset of the partially ordered set (A; ==). Then B 2 n ( = s ) is a partial order relation on B which will again be denoted by =s. a € A is an upper (lower) bound of B if b =sa (a =s b), for all b E B. A very important theorem gives a condition for the existence of maximal elements : Zorn’s lemma. Let ( A ; =s) be a partially ordered set such that every totally ordered subset ( B ; G ) of A has an upper bound, Then A has a maximal element.
In most cases this theorem is applied when A is a subset of the power set P ( M ) of some set M and a e b means that a is a subset of b.
01
SETS
253
Let ( A ; s)and ( B ; e)be partially ordered sets. A mapping p : A B is an order homomorphism if al s a2 always implies pal =s puz. A surjective order homomorphism is called an order epimorphism. If A = B, then p; is an order endomorphism. If rp is bijective and p - l is also an order homomorphism, then p is called an order isomorphism. Let (A,; s),i = 1, . . ., k , be partially ordered sets and D the Cartesian product of these sets. We define a relation < on D by (al, . . ., ak)s (b,, . .., bk) if and only if there is an index I < i =sk + I such that a, = b,, . . ., = bi-,, a, ibi. Then =s is a partial order relation on D,the so-called lexicographic partial order relation of D.If all ( A i ; s) are totally ordered sets, so is (D;s). --t
1.6. The sets A , B are called equipotent if there exists a bijective mapping
p : A -, B. Equipotence of,sets is reflexible, transitive, and symmetric.
The class of all sets which are equipotent to some set A is called the cardinality or cardinal j A I of the set A . If moreover rp :Z -+ A is a family of elements of A with index set I , then the cardinality of this family will mean I yZl. If A is a finite set, then the cardinality I A 1 is called finite and if A is infinite, then 1 A I is called infinite. The finite cardinalities can be identified with the non-negative integers. The cardinality IN I of the set N of positive integers is denoted by 80 and a set A with lAI = K~ is called a countable set. Let (mi 1 i E I ) be a family of cardinals. The sum and product of this family are well-defined cardinals if we define them as follows: Take any family (Ai1 i E I ) of sets such that 1 Ai 1 = mi, for all i E I, and A i n Ai = Cp, for i ij, and set C(m,liEZ) = I U ( A i l i E I ) ~ .Take any family ( A i l i C Z ) of sets such that I Ai1 = mi, for all i E Z, and set n ( m iI i E I ) = j n ( A iI i E I ) ~. Important special cases are ml+mz and mlmz, the sum and the product, resp., of two cardinals m, and m,. If mi = m, for all i E Z , then C ( m , l i E I ) = /I1 m. Let m, n be two cardinals, and I a set with 111 = n. If ( m ii E/I ) is a family of cardinals with mi = m, for all i E I , then we define mn= n ( m i I i E I ) , which is well-defined. For any set A , IP(A)j = 2!4. Let m,n be two cardinals. We write m s n if and only if there exist sets A , B with I A 1 = m,I B I = n such that there is an injective mapping y :A B. If C is an arbitrary set of cardinals, then =s is a total order relation on C . If n is a finite cardinal, then n < 8 0 ,and if n is infinite then 8 0 == n. If B is a subset of A , then IBI 6 ] A / .If ( A i l i E I ) is a family of
-.
254
APPENDIX
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6
sets, then IU(AiliEZ)I = s ~ ( l A i l l i € ZIf) . (miliEZ), ( n i l i € I )are families of cardinals such that mi =z ni, for all i E I, then C(miI i E I ) =z C(ni 1 i E Z) and n ( m iI i E I) =S n ( n iI i E I ) . If ml== m2 and n is an arbitrary cardinal, then my =S mt. If m or n is an infinite cardinal, then m f n = max (m,n) and if moreover m # 0, n # 0, then mn = max (m, n). If m is an infinite and n is a finite cardinal, then m” = m.For any set A we have IA 1 < IP(A)[. 1.7. A partial order relation -c on the set A is called a well ordering if every non-empty subset of A has a least element w.r.1. e.A partially ordered set ( A ; e)is called well ordered if =sis a well ordering. Theorem. (Well ordering principle). On every non-empty set A, one can define a well ordering =s.
Two well ordered sets A, B are called isomorphic if there exists an order isomorphism p : A B. “Isomorphic” for well ordered sets is reflexive, transitive, and symmetric. The class of all well ordered sets isomorphic to some given well ordered set A is called the ordinal of A . The ordinal of the empty set is denoted by 0, the ordinal of the set {1,2, . . ., n} ordered by the usual order relation of the integers will be denoted by n. Let a, B be two ordinals. We write a =s /3 if and only if there exist well ordered sets A, B with ordinal a, B, respectively, such that there is an injective order homomorphism p : A -,B. If 0 is an arbitrary set of ordinals, then =S is a well order relation on 0. If a is an arbitrary ordinal, then the ordinal of the set ( G ; e)of all ordinals y a is just a. -f
1.8. Induction is frequently used throughout the book. The principle of this powerful method will now be described briefly : Let a be a non-negative integer and the statement p(m) be defined for all integers m a. Suppose that (i) p(a) holds and (ii) if y(n) holds, for a == n -=c m, then ~ ( mholds. ) ) for all integers m 3 a. Then ~ ( mholds An important generalization of induction is the transfinite induction :
02
LATTICES
255
Let a, 6 be ordinals with a < S and the statement p(p) be defined for all ordinals p with 6 == p 2 a. Suppose that (i) q(a) holds and (ii) if p(y) holds for cc =sy ip, then q(p) holds. Then p(p) holds for all ordinals p with 6 a p a a. 1.9. A binary (i.e. 2-ary) operation k on a set M (written with infix notation) is called a ssociative if x % (y +% z) = ( x +% y ) +c z, for all x, y, z E M ; commutative if x + % y= y*x, for all x, yE M ; and idempotent if x % x = x for all x E M . Let % and o be binary operations on M . The if (x++ey)oz= operation o is called right (left) distributive w.r.t. (xoz)%(yoz)(if zo (x % y ) = (zox)%(zoy)), for all x, y , zEM. Let ++c be a binary operation on M . An element i E M is called a right (left) identity for % if a+ci = a (if i%a = a), for all a E M . If i is a right a > well as a left identity for %, then i is called an identity for % . There exists at most one such identity. 2. Lattices 2.1. All those results from lattice theory which are used in this book will now be reviewed in brief. For more detailed information, we refer to standard texts (e.g. SzAsz [l]). An algebra ( L ; U , n) where U , n are binary operations is called a lattice if both U and n are associative and commutative and if rnoreover the “absorption laws” aU ( a n b) = a, a 1 7(aU b) = a are satisfied for all a, b E L. There is some close relation between lattices and certain partially ordered sets. Let ( A ; s)be any partially ordered set and B a subset of A. A least (greatest) element of the set of upper (lower) bounds of B - which, of course, need not always exist - is called a least upper (greatest lower) bound of B. We will write U (b 1 b E B ) for the least upper bound of B and n( b I b E B ) for the greatest lower bound of B. Theorem. Let ( L ; U, n>be a lattice. W e dejne a binary relation == on L by: a s b if and only if aU b = b. Then ( L ;=s)= 8L is a partially ordered set where U(a, b) and n(a, b) exist, for any subset {a, b} C L , and U (a, b) = aU b, n (a, b) = an b. K conversely, ( L ;e)is a partially ordered set where U (a, b) and n (a, b) always exist, then we obtain a
256
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APPENDIX
6
lattice ( L ; U , i I ) = qL when defining aU b = U (a, b), a n b = n (a, b). Moreover q(8L) = L, for any lattice L, and z9(qL) = L, for any partially ordered set L where U (a, b) and n (a, b) always exist.
2.2. A lattice L is called a complete lattice if U(b1 b E B ) and n(b1 b E B ) exist in the corresponding partially ordered set 8L, for any B G L. For proving completeness of a given lattice one sometimes uses the Lemma. A partially orderedset ( A ; =s)corresponds to some complete lattice i f and only i f A has a greatest element and, for any non-empty subset B C_ A , there exists n( b I b E B). Let L be a complete lattice. A subset M of L is called a complete sublattice of L if, for every subset B G M , we have U (bI b E B ) E M and n(bi b E B ) EM. 2.3. A lattice L is called a distributive lattice if U is right distributive w.r.t. n and n is right distributive w.r.t. U . Let ( L ; U , n ) be a lattice. A right identity for U - there exists at most one such identity - is called a zero of L and denoted by 0, and similarly a right identity for n is called an identity of L, denoted by 1. An element a E L is a zero (identity) of L if and only if a is the least (greatest) element of the corresponding partially ordered set 8L. Let L be a distributive lattice with zero and identity. L is a Boolean algebra if, for any a E L, there exists a unique element a' EL such that aUa' = 1 and a n a ' = 0. 2.4. Let L , M be two lattices. A homomorphism (epimorphism, isomorphism) p :L M from the algebra L to the algebra M is called a lattice homomorphism (epimorphism, isomorphism). If p; : L M is a lattice homomorphism (epimorphism), then p is also an order homomorphism (epimorphism) from 8L to 8M, but the converse is not always true. is, however, a lattice isomorphism if and only if 9: is an order isomorphism. Let L, M be complete lattices. A lattice homomorM is called complete if p- U (b I b E B ) = phism (epimorphism) p : L U ( ~ b I b E B ) a n d ~ n ( b I b E B=) n(gbIbEB),forallsubsetsB C L. --f
+
+
2.5. An algebra ( S ; il) is called a semilattice if tative, and idempotent.
n is associative, comrnu-
03
257
MULTIOPERATOR GROUPS
3. Multioperator groups
3.1. This section is to exhibit what is needed in the book from the theory of multioperator groups. Since just very few books contain some material on this subject, we give the proofs in full. Assume + is a 2-ary operation, written with infix notation, - is a 1-ary operation, and 0 is a 0-ary operation. Let 8 = {mi I i E Z} be a family of operations, indexed by the set Z of all ordinals L < o where o is an arbitrary ordinal. Let T = {niI i E Z} be the type of 8.We define : An algebra G = (G; + , -, 0, Q) is called an 8-multioperator group if 1) (G; + , -, 0) is a group with + as group operation, - the operation of forming the inverse, and 0 the identity; 2) c o i O . . . 0 = 0, for any m i c 8 with ni =- 0. Thus e.g. every group (G; f , -,O) and every ring (R; -,O, .) is a multioperator group.
+,
3.11. Proposition. For any 8, the class \%J1 of 8-multioperator groups is a variety.
This follows from the fact that “(G; stated by means of laws.
+, -, 0) being a group” can be
3.2. Assume G = (G; +, -, 0, Q) is an 8-multioperator group. Let 0 be any congruence on G and C(a) the congruence class of the element a E G under 0. We define the kernel ker 0 of 0 by ker 0 = C(0). A subset A of G is called an ideal of G if there is some congruence 0 on G such that ker 0 = A. If yis a homomorphism from the Q-multioperator group G to some algebra H , then the ideal kernel ker p; of p will be the ideal ker Ker p. 3.21. Proposition. Assume that A = ker 0. Then A is a normal subgroup of ( G ; +, -, 0 ) and 0 is induced by the partition of this group into the cosets of the normal subgroup A. Proof. By hypothesis A = C(0)whence A is a subgroup. Moreover cOb i f a n d o n l y i f c E b + A b u t a l s o i f a n d onlyif cEA+b. Thus b + A = A f b = C(b) whence A is a normal subgroup of ( G ; +, -, 0) and also the second statement follows.
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6
3.22. Corollary. The mapping which maps every congruence 0 of the O-multioperator group G onto its kernel, is a bijection from the congruence lattice 2(G) to the set of all ideals of G.
Proof. Evident. 3.23. From ch. 1, Q 1.5 and the definition of an ideal, we see that the subsets (0) and G of G are ideals, the so-called zero ideal and unit ideal, respectively. Together they are called the trivial ideals of G. Cor. 3.22 shows that the multioperator group G is simple if and only if G has no non-trivial ideals. If A is an ideal of G and 0 the congruence such that ker 0 = A , then we will write GI A for the factor algebra G 10. Let Q(G) be the set of all ideals of G. The set-theoretical inclusion G is a partial order relation on Q(G). If 01,O2E i?(G), then O1 =sO 2 if and only if ker 01C ker 0 2 . Hence the mapping ker : 2(G) -+ (R(G); G ) is an order isomorphism and ker-l is also an order isomorphism. Therefore (Q(G); & ) is a complete lattice. Let M = {Ai 1 i € I } be a non-empty subset of Q(G) and set ker-lAj = Oj. Then by ch. 1,§ 1.6, the greatest lower bound of M is ker n(0,I i E I ) where n means the set-theoretical intersection. Since a E ker Oi if and only if a E ker O,, we conclude that the greatest lower bound of M is the set-theoretical intersection f' (Aj1 i E I ) which therefore is again an ideal. The least upper bound S of M is ker @ where @ is the least upper bound of the set ker-I M . Thus a E S if and only if a@O. By ch. 1 , $ 1.6, this is true if and only if there are congruences Oil, . . .,0, in ker-l M and elements cl, . . ., c , - ~E G such that aOjlcl, clOi2cZ,. . .,cr-,Oi7O. This is equivalent to a-c, = ailE Ail, c1-c2 = aizE Aj2, . . ., C , - ~ - - C , - ~ = a,,-, E -4jr-l,crP1= ai, E A , , i.e.
n
n
a = ai,+aj2+ . . . +ai, where r
==
1,
aj,,EAjY,
z1
=
1, . . .,r. (3.2)
Thus S is the set of all elements of the form (3.2). This set is called the sum of the set M = {Ail i E I } of ideals and is, of course, itself an ideal. Hence 3.24. Theorem. The set 9(G) of all ideals of the S-multioperator group G being partially ordered by set-theoretical inclusion, is a complete lattice, the so-called ideal lattice of G. If M is a non-empty subset of @(GI, then the
03
259
MULITOPERATOR GROUPS
greatest lower bound of M is the set-theoretical intersection of the ideals of M and the least upper bound of M is the sum of the ideals of M . The 9(G) is a lattice isomorphism. mapping ker : S(G)
-
3.25. Let P be a subset of G. The ideal of G generated by P is the inter-
section of all ideals of G containing P.
3.3. Proposition. Let G be an Q-multioperator group, D an ideal of G, and 0.the sublattice oj' 9(G) consisting of all ideals B 2 D of G. Then the canonical epimorphism 8 G G I D induces a lattice isomorphism from G to the lattice B(G 0). Moreover 8 maps the set of all subalgebras of G containing D bijectively to the set of all subalgebras of G D. -t
~
Proof. We set @ = ker-l D. Let a be the lattice isomorphism of ch. 1, Th. 1.71. Then (ker)a(ker-l) :G R(G1D) is a lattice isomorphism. But a € (ker) a(ker-l)B means a(a ker-1 B)O which is equivalent to a=8u,u(ker-lB)O,forsomeu€G,i.e. aE8B.Hence(ker)a(ker-l)B=83, for every B E G. Let U be any subalgebra of G 1 D and EU the set of the inverse images of its elements under 8. Then EU is a subalgebra of G containing D. On the other hand, 8 maps every subalgebra of G containing D to a subalgebra of GID. Moreover 88 and && are the identical mapping whence 8 is bijective. -f
3.4. Lemma. A subset A of the Q-multioperator group G is an ideal of G if and only if a) A is a normal subgroup of (G; t,-, 0), b) if wi is an arbitrary operation of Q with ni =- 0, then, for any a E A , (gl, . . ., g,J E G"i, and 1 e v == nj, we have
wig1 . . . gu-l(gv+a)gu+, . * . gni-wjgl . . . gv-lgvgv+l . . . gni E A* Proof. Let A be an ideal of G. Then A = ker 0 whence by Prop. 3.21, Ais a normal subgroup of (G; -, 0) and 0 is induced by the partition of G into the cosets of A . Hence (g,,+a)@g,,, therefore wigl.. . (g,+a) . . .gni@ojg,.. . g,, . . . gni i.e. b) holds. Suppose now that a), b) are satisfied. Then the equivalence relation being induced by the partition of G into cosets of A is some congruence 0 and ker 0 = A . Hence A is an ideal.
+,
260
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3.41. Remark. Lemma 3.4 shows that for a group, regarded as a multioperator group, the ideals are just the normal subgroups while for rings, regarded as multioperator groups, the ideals are the two-sided ideals in the sense of ring theory. 3.5. Proposition. (First isomorphism theorem for multioperator groups). Let G be any Q-multioperator group, U a subalgebra of G , and A an ideal of G. Then U f A is a subalgebra of G, A is an ideal of U + A , U n A is an ideal of U, and U+ AIA 2 U j U n A . Proof. By the first isomorphism theorem for groups, U + A is a subgroup of G . Since 0 E A , we have o,E U + A for any w, with nz = 0, and if u,+ a,, . . ., uni+ani E U+ A , and w, is an n,-ary operation of Q, n, =- 0, then by Lemma 3.4, wI(ul+a,) . . . ( ~ ~ , + u , J - w ~ .u.,. uniE A whence U+ A is a subalgebra of G . A is an ideal of Uf A and U n A is an ideal of U, by the first isomorphism theorem of group theory and Lemma 3.4. Also by the first isomorphism theorem of group theory, 0: : U + A [ A + UI U n A , defined by cx(u+A) = u + ( U n A), is a group isomorphism. One checks easily that cc is also a homomorphism w.r.t. the operations 0 ,E 0.
-.
3.51. Proposition. (Second isomorphism theorem for multioperator groups). Let y : G H be any epimorphism of Q-multioperator groups, B an ideal of H , and A = {a€G / q a EB } . Then A is an ideal of G and H J Bz G / A . Proof. By the second isomorphism theorem of group theory and Lemma 3.4, A is an ideal of G . Also, by the second isomorphism theorem of group theory, the mapping a : GI A HIB, defined by a(g+ A ) = yg+B, is a group isomorphism. One checks easily that a is a homomorphism also w.r.t. the operations m iE Q.
-
4. Rings 4.1. Ring-theoretical definitions and results that are used in this book will now be compiled, but proofs will be given only for less widelyknown lemmas. Moreover some important facts about roots and partial derivatives of polynomials over commutative rings with identity - e v e n though they are well-known-will be proved for the sake of
04
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completeness since they are frequently used in this book. We refer the reader to standard books like REDEI [2], VAN DER WAERDEN [I], [2] and ZARISKI-SAMUEL [I] for some further information. A ring is an algebra (R; , .) with two binary operations , such that (R;+) is a commutative group whose identity is 0; is associative and right and left distributive w.r.t. . If is also commutative, then R is called a commutative ring, and if has a (left) identity, then R is called a ring with (left) identity. Let R be any ring. An element a E R is called nilpotent if a" = 0, for some integer n =- 0. a # 0 is a zero divisor if there exists 0 # b E R such that ab = 0. An idempotent a E R is an element with a2 = a. An (integral) domain is a commutative ring R with identity and without zero divisors, i.e. ab = 0 implies a = 0 or b = 0, for all a, b E R. A (not necessarily commutative) ring K with identity 1 such that {a E K I a # 0} forms a group w.r.t. the multiplication is called a skew-field, and a commutative skew-field is a field. Every field is an integral domain and every finite integral domain is a field. Moreover every finite skew-field is a field, A commutative ring with identity is a field if and only if it is a simple algebra with at least two elements. Let D be any integral domain, then there is (up to isomorphisms that fix D) exactly one field K that contains D as a subring such that every d E K can be written as d = ab-l, a, b E D. K is then called the quotient field of D. An integral domain D is called integrally closed if every x E K that satisfies some equation x"+dn-lx"-l+ . . . +d,x+d, = 0, d, E D belongs to D where K is the quotient field of D. Let D be any integral domain. If 1 ED has no finite order in the additive group of D,then D is said to be of characteristic 0, and if 1 is of additive order p , then D is of characteristic p . Such a number p is always a prime. We write char D for the characteristic of D. If char D = p > 0, then (a+ b)P = up+ bP, for all a, b E D. If R is a ring with identity, then a unit of R is an element e E R such that de = ed = 1, for some dE R. The set of all units of R constitutes a group w.r.t. the multiplication. In a commutative ring R with identity, b E R is called an associate of a E R if b = ae, for some unit e E R. The relation "being associated with" is an equivalence relation on R.
+
+
-
-
-
+
-
262
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6
4.2. Let
R be a commutative ring with identity. An element b E R is a divisor of a E R if a = bc, for some c E R. If b is a divisor of a, we say b divides a or a is divisible by b and write bla. If bla, then b is called a proper divisor of a whenever b is not associated with a, and is called a non-trivial divisor if b is neither associated with a nor a unit of R.A non-zero element p E R is called an irreducible element of R if it is not a unit of R and has no non trivial divisor. (Sometimes we shall also say prime element instead of irreducible element). An integral domain R is a unique factorization domain (UFD) if every non-zero non-unit a E R is a product of a finite number of irreducible elements and, for any two factorizations of such an element a E R into irreducible elements, there is a bijection between the two sets of factors such that each factor of the one factorization is mapped to an associate. If R is a UFD, then the polynomial ring R [ x ] is also a UFD. Let R be a UFD. If p C R is an irreducible element and a E R,then the number of associates of p occurring in a decomposition of a into irreducible factors is called the multiplicity of p in a. The element a is square free if every irreducible element p E R has at most multiplicity 1 in a. Let a,, . . ., a, be elements of the UFD R. An element d E R is a greatestcommon divisor (g.c.d) of a,, . . .,a,, denoted by (a,, . . .,a,) if d/a,, i = 1, . . ., r and if c/ai, i = I , . . ., r, implies cld, for any c E R. An element v E R is a least common multiple (1.c.m.) of a,, . . .,a,, denoted by [a,, . . ., a,] if a,/v, i = 1 , . . ., r, and if ai/w, i = 1, . . ., r, implies z'lw, for all w E R. Greatest common divisors and least common multiples exist for any elements a,, . . ., a, E R, and are uniquely determined up to associates. If the irreducible pE R has multiplicity m i in a , i = 1, . . ., r, then p has multiplicity min (mi]1 ==i =s r ) in (al, . . ., a,) and multiplicity max (mil 1 -c i -c r) in [a,, . . .,a,]. If a, b E R, then (a, b) [a, b] = ab. The elements a, b E R are called relatively prime if (a, b) = 1. The elements al, . . ., a, E R are called pairwise relatively prime or coprimal if (ai, aj) = 1 , for i # j . If (a,b) = d, then aid, bld are relatively prime. 4.3. Let R be any ring. A non-empty subset I E R is a left (right) ideal of R if a, b E I implies a-b E I and a E I , r R implies ra E I (ar E I ) . An ideal of R is a left and right ideal of R (this definition coincides with
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that in 0 3 if we regard R as a multioperator group ( R ; +, -, 0, .)). If Z is an ideal of R, then a@(Z)bif and only if a-b E I defines some congruence @(I) on R, and 0 is a bijection from the set of all ideals of R to the set of all congruences on R . For a@(Z)b, usually a E b mod I is written and RIZ means RI@(I). The ideals {0}, the zero ideal, and R, the unit ideal, are called the trivial ideals of R . For the remainder of this subsection, let R be a commutative ring with identity. If a,, . . .,a, E R,then the set (a,, . . .,a,) = { t p , .. . t,a, I ti E R} is an ideal of R, the ideal generated by a,, . . ., a,. a,, . . ., a, is called a finite ideal basis of the ideal Z if Z = (al, .. ., a,). I is a principal ideal if I = (a), for some a E R ; we then write u = v mod a instead of u = v mod (a). Clearly (a) = Ra. If A, B are ideals of R, then their intersection A n B and their sum A B = {a+ b [ a E A, b E B } which is sometimes denoted by (A, B ) are also ideals. Moreover AB = ,qb, I a, E A , b, E B, I finite} is again an ideal, the product of A , B. Intersection, sum, and product of ideals satisfy the commutative and the associative law, and the product is distributive w.r.t. the sum. If A = (a,, . . ., a,), B = (bl, . . ., b,), then AB = (a,b,, alb2, . . .,albs, a,$,, . ...,arb,, . . ., arb,). A" shall mean the product of n equal ideals A. Two ideals A, B are called comaximal if A + B = R. The ideals A,, . . ., A , are called pairwise comaximal if A,, Ai are comaximal whenever i # j . If A,, . . ., A, are pairwise comaximal, then A, . . . A, = A,n . . . nA,, and any system of congruences x = a, mod A,, i = 1, . . . ,Y, has a solution in R which is unique mod A,. . .A, (this is the well-known Chinese remainder theorem). Moreover if C(a) denotes the congruence class of a under the congruence @(A, . .. A,) and C,(a),i = 1, . . ., Y, denotes the congruence class of a under @(Ai),then @(a) = (C,(a), . . .,C,(a)) defines a ring isomorphism p :R [A, . . . A,
+ +
+
(1,
-f
(RIA,)X
. . . X(RIA,).
An ideal P of R is called a prime ideal of R if RIP is an integral domain while an ideal Q is a primary ideal of R if every zero divisor in RlQ is nilpotent. If Q is a primary ideal of R, then {b E R lb" E Q, for some integer m > 0} is a prime ideal of R, the so-called radical of Q or the prime ideal associated with Q.If Q,P are ideals of R , then Q is primary with associated prime ideal P if and only if: (i) Q c P,(ii) if b E P,then b" E Q,for some integer rn =- 0, and (iii) if ab E Q, a 6 Q, then b E P.
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cn. 6
If there is a positive integer n such that P" E Q, then the least such integer is called the exponent of Q. An ideal N of R is called nilpotent if N" = (0) for some integer n =- 0. 4.4. A principal ideal domain (PID) is an integral domain R such that every ideal of R is principal. Every PID is a UFD. If a,, . . ., a, E R and d = (al, . . ., a,), then there exist elements t,, . . ., t, E R such that d = tlal+ . . . trur.An ideal (a) of R is a non-trivial prime ideal if and only if a is irreducible. For a f 0, the factor ring R I (a) is a field if and only if a is irreducible.
+
A Euclidean domain is an integral domain R such that there exists a function p from R . (0) to the set of non-negative integers with (i) b fa implies p(b) =sp(a), for any non-zero elements a, b t R, and (ii) if 0 # b, a E R, then there exist q, r E R with a = bq+r and either r = 0 or ~ ( r<) p(b). Every Euclidean domain is a PID. The common examples of Euclidean domains are the ring of rational integers and the polynomial ring K [ x ] over any field K. 4.5. A commutative ring R with identity is called noetherian if every ideal
of R has a finite ideal basis. If R is noetherian, then the polynomial ring R[x,, . . .,xk] over R is also noetherian. A Dedekind domain is a integral domain R where every ideal of R is a product of a finite number of prime ideals of R.Every Dedekind domain is noetherian and integrally closed. Every PID is Dedekind. The best known examples of Dedekind domains are the integral domains consisting of all algebraic integers in a finite extension field of the field of rational numbers. In particular, the ring of rational integers is Dedekind. If A E B for two ideals of a Dedekind domain R, then there exists some ideal C in R such that A = BC. Let R be a Dedekind domain. Then every non-trivial ideal A of R can be represented as a product of non-trivial prime ideals, and this representation, the so-called primary decomposition of A , is unique up to the order of the factors. If P is a non-trivial prime ideal of R, and A is a non-zero ideal, then the number of factors P in the decomposition of A into prime ideals is called the multiplicity of P in A. If a prime ideal P has multiplicity m,in the ideals Ai, i = 1, . . ., r, then P has the multiplicity min (m, I 1 =z i =z r) in the sum and the multiplicity max (mil 1 =si =sr) in the intersection of these ideals. Hence any two
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powers PT, P" of different non-trivial prime ideals P I ,Pz are cornaximal An ideal Q of R is primary if and only if Q is a power of some prime ideal P, and this prime ideal is just the radical of Q. Let R be a Dedekind domain, and A a non-trivial ideal of R. Then RIA has only principal ideals, and if 0 # a E A , then there exists b E A such that A = (a, b). Subsequently let R be a Dedekind domain. 4.51. Lemma. Let P be a prime ideal of R. If a E R and the class C(a) under O(P) is a unit of R I P, then the class Cl(a) under O ( P ) is a unit of RI P".
Proof. By hypothesis, there is b E R,p E P such that ab = 1 - p whence ab(1 + p + . . . +pep') = 1-p". 4.52. Lemma. Let P be a non-trivialprime ideal of R and e =- 0 an integer. Then there exists a bijective mapping from Pe/Pe+lto RIP.
Proof. Since every ideal of R I P e f 1 is principal, so is PejPecl. Say Pel Pe+l = (C(r))where r E P" and C(r) is the class of r under O(Pe+l). Hence the equation C(u) = C(r)C(x) has a solution x E R , for any C(a)E P'JPe+l. One verifies easily that the set 8C(a) of all solutions x of this equation is a class under O(P), that 8C(a) = .BC(b) implies C(a) = C(b),and that every element of R I P is an image under 8.
1
4.53. Lemma. Let P be a non-trivial prime ideal of R such that R I Pi is jinite, e a 2 an integer, a EP and a 4 P 2 . If W1 is a vector system mod P, We-l a vector svstem mod PeP1 (cf. ch. 4 ) then the sets W = {u+ab[uEW1.bEWe-,} and (b+ue-lUIuEWl,bEWe~l} are vector systems mod P".
w=
Proof. Let : R I P l = r , then byLemma4.52, l W I = l r l = l W I I I W e - l l = fl(re-l)k = If in W , ufab = u l f a b l mod P", then u = u1 mod P whence u = u, and b = b l , hence W is a vector system modP'. A similar argument shows that V is a vector system mod P". 4.6. Let R1,R2 be commutative rings with identity and R = R I x R ~ . Then for every ideal A E R, there exist ideals A, of Ri,i = 1,2, such that A = A , X A 2 . Moreover A" = AYXA;.
266
CH. 6
APPENDIX
A commutative ring R with identity is called the inner direct product or the direct sum of its subrings R1, R2 if the mapping q :R1x R2 R, defined by p(rl, r2)= r1f r2 is an isomorphism. If d E R is an idempotent, then R is the inner direct product of its subrings ( d ) and ( 1 -d) and if, conversely, R is the inner direct product of its subrings R1, R2, then there is an idempotent d € R such that R 1 = ( d ) and Rt = (1 A commutative ring R with identity is called directly indecomposable if, in any representation of R as an inner direct product of subrings R,, R,, we have I Ri 1 = 1, for at least one i. +
-a).
4.7. Let R be a commutative ring with identity, R [ x ] the polynomial ring in x over R , and f E R [ x ] .An element a E R is a root off if f ( a ) = 0. 4.71. Lemma. If a is a root off, then the polynomial x--a is a divisor off in R [ x ] .
Proof. We can write f = (x-a)g+r where g E R [ x ] , r E R. I f we substitute u for x,then r = 0 by the principle of substitution. 4.72. The greatest integer k such that ( x - u ) ~is a divisor o ff is called the multiplicity of a as a root off. 4.13. Proposition. Let R be any integral domain, 0 # f E R [ x ] apolynomial of degree [f],a,, i = 1, . . ., r, distinct roots of f , and ki, i = 1, . . .,r, the multiplicities of these roots. Then k,+ . . . k, == [f ] and f is divisible in R[x] by the polynomial (x-al)kl . . . (x-a,)kr.
+
Proof. The first statement follows from the second one and ch. 1, Prop. 8.31. We have f = (x-aJk1g1. Suppose f = (x-al)kl . . . (x-ai-Jkt-T (x -ai)'gjlwhere 0 =s1 -= k,. Then, since R [ x ] is an integral domain, we have (x-ui)kg-'g = ( X - U ~ ) .~.~. (x-ui-Jk(-lgi,, for some g E R [ x ] .If we substitute ai for x, and observe that R is an integral domain, we obtain gi,(ai)= 0 whence f = (x-a$l . . . ( ~ - a , ) ' + ~ &for , some gil E R [ x ] .
-
4.74. Proposition. LetR be any integraldomain of characteristic zero, anda a root of the non-zero polynomial f E R[x] of multiplicity k . Then a is a root o f f ' of multiplicity k - 1.
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Proof. By hypothesis, f = ( ~ - a ) ~where g g E R[x]and g(a) # 0. Hence f ’ = k(x-a)k-lg+(x-a)kg’ = (x-a)k-l(kg+(x-a)g’). If ( ~ - - a were )~ a divisor off’, then a would be a root of the second factor whence kg(a)=O. But then k . 1 = 0, a contradiction. 4.75. Corollary. If R is an arbitrary commutative ring with identity and a
is a root o f f of multiplicity k =- 1 , then a is also a root o f f ‘ .
Proof. As in Prop. 4.74. 4.8. Let R be a commutative ring with identity and R[x,, . . ., xk] the polynomial ring over R in xl, . . .,xk. For 1 =si == k, we define mappings a/ax, = ai:R[xl,. . ., xk] R[x,, . . ., x,] by: If f E R[x,, . . .,x,] and f = C a(ll, ,,,k)x?.. . xk is the normal form off according to ch. 1, Th. 8.21, then aif = C lia(ll,,.,,wx? . . . $-’ . . . The mapping a, is called the i-th partial derivation of R[x,, . . ., xk] and aif is called the i-th partial derivative off. If k = 1, then we write x instead of X I , d/dx for a/ax, andf’ for (d/dx)f. +
e.
,,
4.81. Theorem. Let R be a cornmutative ring with identity, R[x,,
. . ., xk]
ai the i-th partial derivation of R[x,, . ..,xk], the composition of polynomials, and f, g, g,, . . .,gk E R[x,, . . ., xk]. Then (i) aia = 0, for a E R, (ii) a,(f+g> = a,f+a,g, (iii) aj(fg) = (a j f k +f(aig) (iv) ailcfgl . . . gk = CY!,(.(a, f ) g , . . . gk) aig, (“chain rule”). a polynomial ring over R, 1c
Proof. (i) and (ii) are obvious. Since (ii) and the distributive law for rings hold, we have to prove (iii) only for f = ax: . . . xk, g = b q l . . . x;r“, a, b E R. Then a,(fg) = (li+m,)abx~+ml . . . xfi +mi- 1 . . . ~ kl k + ~=k (aif)g+f(aig). Similarly, since (ii) holds and x is right superdistributive w.r.t. +, we have to prove (iv) only for f = ax? . . . xk, a c R. Then xfg, . . . g, = ag? . . . g i . Repeated application of (iii) leads to aixfgl . . . gk
=
k
k
v=1
v=l
C iYag;. I . . g)-l . . . g t a i g v = C [ x ( a f ) g , . . .gk]aig,.
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CH.
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6
-
fc R[x,, . . ., x k ] , r z= 1 an integer, 1 =si, s k, v = 1, . . ., r. Then the mapping ail a, . . . a,: R[x,, . . ., xk] R[x,, . . ., xk] is called a partial derivation of order r of R[x,, . . .,xk]and ailEli, . . . a, f a partial derivative of order r off. 4.82. Let
4.83. Lemma. Zf [ f ] = n, then all partial derivatives of order t
=- n o f f are
zero. Proof. The application of a partial derivation decreases [f ] by at least 1, or aif = 0.
aito a polynomial f
either
4.84. Theorem (Taylor's formula). Assume that f E R[x,, . . ., xk] with [ f ] = n and.al, . . .,ak,g,, . . .,g, are eZements of a commutative ring S
with identity containing R as a subring. Then
k
f(alfgl,
akfgk) =f(al,
+(1/2!)
il=l
k
C il,
(ail aj,J')
* * -9
i,=l
k
+(lIn!)
* * * )
il, i.,
C
(ail
. . ., in=l
aj,
(ai,f>(al,
...,'k)gii
a,Jgilgi2+ . * .
.. . a i n f )
. . ak)gil . . . gin. - 3
Proof. By Th. 4.81, (ii), Lemma 4.83, and the right superdistributivity of x w.r.t. , Th. 4.84 is true for all f €R[x,, . . ., xk] if it is true for f = ax? . . . xk, a € R . The left-hand side of Taylor's formula is then a(a1+gl)" . . . (ak+gk)C. Using the distributive laws, we expand this expression. Then, for any k-tuple (rl, . . .,rk) of non-negative integers with r,+ . . . +rk =s[ f ] , the term with second factor g? . . . g;c" has as its first factor (Il!/rl!( z , - - r J ! ) . . . (lk!/rk!(Ik-rk)!)aa~-rl . . . a i - ~if ri G li, i = 1 , . . ., k, and zero otherwise. On the right-hand side of Taylor's formula, the term with second factor g;' . .. g;;" has as its first factor (l/(rl+ . .. + r k ) ! )((r,+ .. . +rk)!/rl! . . . r k ! )(il!/(ll-rl)!) . . . (Zk!/(Zk-rk)!)aat-r~ . . . atMrkif ri ==Zi, i = 1 , . . ., k , and zero otherwise.
+
4.85. Let R[x,, . . ., xk] be the polynomial ring in x,, . . ., x k over a commutative ring R with identity and f = (fl, . . . , J f k ) an element of R[xl, . . ., xkIk.The Jacobian determinant af of f is the determinant of the kxk-matrix Of = @,A) where s means the row and t the column index.
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5. Fields
5.1. First we are going to review standard concepts and results on fields that are used in this book (for details cf. RBDEI[2], VAN DER WAERDEN [ 11, ZARISKI-SAMUEL [l]). Then we treat, in more detail, a few results on the rational function field K(x) which are crucial for some sections of ch. 4, and are not so well-known, as we believe. Let K be any field. A subset S of K is called a subfield if S is a subring of K which is a field. S is a proper subfield of K if S c K. A proper subfield S of K is called a maximal subfield of K if there is no proper subfield S1 with S c S,. If (S,1 i E I ) is a family of subfields of K, then n (S,I i E I ) is also a subfield of K. Let S be a subfield of K and U a subset of K, then the intersection of all subfields of K containing S U U will be denoted by S(U), and we say that S ( U ) is obtained from S by adjoining U . A field P is called a prime field if P has no proper subfield. Up to ring isomorphism, the prime fields are just the field of rational numbers and the factor rings Z I ( p ) where Z is the ring of rational integers and p is a prime. Any field K has exactly one prime field P amongst its subfields, being called the prime field of K. Let K be a field. A field L containing K as a (proper) subfield is called a (proper) extension field of K. The extension field L of K is called a simple extension of K if there is u EL such that L = K(u). Any extension field L ofK can be viewed as a K-vector space w.r.t. addition and multiplication in L, the dimension of which is called the degree [L :K] of L over K. If [L : K] is finite, then L is called a finite extension field of K. If a E L , then [K(a): K] is called the degree of a over K. If L is a finite extension field of K, and Mis a finite extension field ofL, then M is a finite extension field of K and [M : K] = [M :L] [L : K]. Let L and L1 be extension fields of K. An isomorphism p :L -+ L1 is called a K-isomorphism if p fixes K; if L = L1, then 9 is called a K-automorphism. Two extension fields L, L1 of K are called K-equivalent if there is a K-isomorphism from L to L1. 5.2, Let K be any field and K[x] the polynomial ring in x over K. A trans-
cendental element over K is an element a of an extension field of K such that f € K[x], f ( a ) = 0 implies f = 0, otherwise a is called algebraic over K. Let a be algebraic over K. Then there is exactly one monic irreducible polynomial p E K[x] such that p(a) = 0, and f ( a ) = 0 implies p i f , for
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any f E K[x]. The polynomial p is called the minimal polynomial of a over K. [ p ] equals the degree of a over K. An extension field L of K is called an algebraic extension of K or algebraic over K if every element of L is algebraic over K. Otherwise we speak of a transcendental extension of K. If L is a finite extension field of K, then L is algebraic over K, and there exists a finite subset {aL,. . ., a,} of L such that L = K(a,, . . ., ar). Conversely, if a,, . . ., ar C L are algebraic over K, then K(a,, . . .,a,) is a finite extension field of K. A field L is called algebraically closed if every element which is algebraic overLbelongs to L. If K is an arbitrary field, then there exists an extension field C of K which is an algebraic extension of K and algebraically closed, and any two such fields are K-equivalent. C is called an algebraic closure of K. Let L be an extension field of K. Two elements a, bEL are called conjugate over K if they are algebraic over K and have the same minimal polynomial over K. Let L be a finite extension field of K, a EL, ul,. . ., u, any K-basis for L. Then au, = '&bp,, b, € K, i = 1, . . ., n. The determinant of the nXn-matrix (bij) is independent of the choice of the basis u,, . . ., u, and is called the norm C;n, K(u)of a E L over K. We have laz,,,(a,u,) = OZL,,(aJ @ZL,,(a2),if a,, u2E L. If a E L, [K(a) : K] = rn, [L:K(a)] = n, and c is the constant term of the minimal polynomial of a over K, then faz,,,(a) = ( - l ) m n C n . Let K be any field and f € K[x] a non-constant polynomial. Then there exists an extension field L of K such that f is the product of linear factors in Llx] and such that L is obtained from K by adjoining the roots of f in L. Such a field L is called a splitting field for f over K. Any two splitting fields for f over K are Kequivalent. An extension field L of K is called a normal extension of K if L is algebraic over K and every irreducible polynomial f € K[x], which has one root in L, is the product of linear factors in L[x]. An extension field L of K is a finite normal extension of K if and only if L is a splitting field for .f over K, for some f € K[x]. If L is a finite normal extension of K and M is a subfield of L containing K, then every K-isomorphism from M to any subfield of L can be extended to an automorphism of L. For any finite extension field L of K, there exists a finite normal extension N of K which contains L such that there
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is no proper subfield M of N which contains L and is a normal extension of K. N is called a least normal extension field of L I K . 5.3. Let K be any field. A separable polynomial of K [ x ] is an irreducible polynomialf€ K[x] such that f ' # 0, otherwise f is called inseparable. An element separable over K is an element a algebraic over K such that the minimal polynomial of a over K is separable; otherwise a is called inseparable. An algebraic extension L of K is called separable over K if every a € L is separable over K, otherwise L is called inseparable. If char K = 0, then every algebraic extension of K is separable. If L is an algebraic extension of K, then the subset S of L consisting of all elements of L which are separable over K is a subfield of L containing K. If L is a normal extension of K, then S is also a normal extension of K. Let L be a finite extension of K, char L = p , and S the subfield of L consisting of the elements of L which are separable over K. Then there is an integer e == 0 such that apeE S , for every a EL. The least such e is called the exponent of the extension L I K.
5.4. Let K be any field and n =- 0 an integer. An n-th root of unity over K is an element z of an extension field of K which is a root of the polynomial x"- 1. A primitive n-th root of unity is an n-th root of unity which is not an m-th root of unity, for all m -= n. If char K does not divide n, then there are always primitive n-th roots of unity over K, and for any primitive n-th root of unity z, the degree of z over K divides p(n), p being the Euler 9-function. 5.5. Let L be any finite normal separable extension of the field K. Then the set G of all K-automorphisms of L is a finite group w.r.t. the composition of mappings, the so-called Galois group of L over K . We have I G I = [ L : K ] . The fundamental theorem of Galois theory tells us that there is a bijective mapping from the set of all subgroups of G to the set of all subfields of L containing K . An abelian (cyclic) extension of K is a finite normal separable extension L of K such that the Galois group of L over K is abelian (cyclic). If L is an abelian extension of K, then every subfield of L containing K is a normal extension of K . If K is a field and I? a positive integer such that char K does not divide n and K contains n n-th roots of unity, then the splitting field of any polynomial Y - a E K[x] over K is a cyclic extension of K . If, conversely, L = K(a) is a cyclic extension of K,
272
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6
[ L : K] is a prime q # char K , K contains q q-th roots of unity, and (T is a K-automorphism of L generating the Galois group of L over K, then there exists a q-th root of unity z K such that the “Lagrange resolvent’’ 1 = a+z(oa)+z2(02a)+ . . . +z4-l(oq-’a) satisfies lq E K and K(1) = L. 5.6. Let K be a finite field. Then char K = p, a prime, and [ K 1 is a power of p. Conversely if p‘ =- 1 is a power of the prime p , there is, up to isomorphism, exactly one field K of order p‘. If K is a finite field, then {a E K I a f 0} is a cyclic group w.r.t. the multiplication in K. If Kis a finite field and 1 K 1 = q, then every element of K is a root of the polynomial xq-x E K [ x ] . If K is a finite field of order p“, then every subfield S of K has order pf where f / e and conversely for every pf with f i e , there exists exactly one subfield of K of order pf. If [ K I = p“ and S is a subfield of K of orderpf, then K is a finite extension field of San d [K: S ] = e& Every automorphism 9 of K is of the form 6a = a*’, a € K, r = 0, 1, . . ., e-1, and the S-automorphisms of K are those automorphisms of K for which r is divisible by .f. 5.7. Let K be any field. The quotient field Q of K [ x ]is called the field of rational functions in x over K and is denoted by K ( x ) . K[xl is a subring of K ( x ) whence K ( x ) is an extension field of K . An “intermediate field of K ( x ) ” is a subfield of K ( x ) which contains K . 5.71. Theorem. If Y E K ( x ) , r Q K , then r is transcendental over K and K ( x ) is algebraic over K ( r ) . I f r E K ( x ) , then K ( r ) = K ( x ) if and only if r is of the form (ax+b)/(cx+d) and r ( K . If rE K l x ] , then the degree [ K ( x ): K(r)] equals the degree [r] of r. 5.72. Theorem (Luroth). Every intermediatefield S # K of K ( x ) is a simple transcendental extension field of K, i.e., there is an element r E K ( x ) which is transcendental over K such that S = K(r). 5.73. In the subsequent subsections, we will derive a few results on intermediate fields of K ( x ) which are not too well known. Throughout these subsections we assume that char K = 0 (though some results of these subsections will also hold for arbitrary fields K ) .
§5
FIELDS
27 3
5.8. Lemma. Let U be an intermediatejeldof K(x) which contains apolynomial f E K [x] of positive degree. Then there is a polynomial r E K [XI of positive degree such that U = K(r).
Proof. By Th. 5.72, U = K(r) for some r E K(x), whence r = g/h, g, h E K[x]. If [g] < [h], we put s = l/r, then U = K(s), and if [g] = [h], then we put s = l/(r-a) = h/(g-ah) where a E K is chosen in such a way that [g-ah] -= [h]. Again U = K(s). Therefore wecan assume that [g] =- [h]. Moreover we may assume that (8, h) = 1, otherwise we cancel by the g.c.d. of g and h. Iff€ U, thenf= u(r)/v(r), u, E K[x], whencefv(r) = u(r). aixi, w = Cy='=, bixi, a, # 0, b, # 0. Then Let u =
z?o
(5.81) Since the degrees of the polynomials on either side of (5.81) are equal and [g] =. [hl,wehave[f]+m[h]+n[g] = n[h]+m[g] whencem =. n. By(5.81), h divides a,f. Since (8, h) = 1 and K[x] is a UFD, we conclude [h] = 0 whence r E K[x]. Clearly [r] > 0. 5.81.Lemma.Let g € K [ x ] a n d k E K ( g ) n K [ x ] . T h e n there existspE K[x] such that k = p(g). 5.82. Corollary. Let g E K[x], [g] > 0, and h t K(g)n K[x] be of minimal positive degree. Then K(h) = K(g). Proof. If [g] = 0, then the lemma is obviously true. If [g] > 0, then k = u(g)/v(g), u, v E K [ x ] , whence u(g) = kv(g). On the other hand, u = vpft, p, tfK[x] such that [t] -= [v] or t = 0. Hence u(g) = v(g) p(g) t(g). This implies (5.82) 4 g ) (k-P(d) = t ( d .
+
By Th. 5.71, t f 0 implies t(g) # 0, thus v(g) z 0 and k-p(g) z 0. By (5.82), this contradicts [t] < [v]. Hence t = 0, u(g) = v(g)p(g) which implies k = p(g). Let h be as in the corollary. Then [h] =s [g], and by Lemma 5.81, also [h] a [g], thus [h] = [g].Moreover, by Lemma 5.81, h = p(g), for some p E K[x], whence [p] = 1. Therefore p = ax+& 0 # a, b E K and K(h) = K(d.
214
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5.83. Lemma. Let f ( x ) = aoxrR+alxrn-l+ . . . be a polynomial of K [ x ] of degree rn # 0. Then there exists exactly one monic polynomial g of deg-
ree n in K [ x ]such that [f-aog']
(r-1)n orf-aog'
i
= 0.
+
Proof. If g = x"+ blxn-l+ . . . + b,-lx b,, then g' = xrn+ (rb,)x'"-l+ (rb,+w,(b,))xr"-2+ . . . +(rb,+w,-,(b,, . . . , b,-l))x'"-"+. . . where w,(x,, . . .,xi),i= 1, . . .,n - 1 , is a polynomial over the ring of rational integers. If we additionally define wo = 0, then f-a&' = C ~ X ' "+- ~ c ~ x ~ " -.~. .+ cnxrn-,+ . . . where ci = a, -a,[rb, wj-,(bl, . . .,b i J ] , 1 G i s n. Clearly the system ai-ao[rbi+wi-l(bl, . . ., b,-,)] = 0, 1 =si G n, of equations in b,, . . ., b,, has exactly one solution in K.
+
+
+
5.84. Theorem. Let f, g C K [ x ] be polynomials of positive degree such that
f)n
the intermediate field D = K( K(g) of K ( x ) contains a polynomial of positive degree, and H = K ( f , 8). Then: a) I f k is a polynomial of minimal positive degree in D, then D = K ( k ) and [ k ]equals the least common multiple v of [f 1, [g]. b) If h is a polynomial of minimal positive degree in H , then H = K(h), and [ h ]is the greatest common divisor x of [f], [g]. c) [H : K( f ) ] = [K(g): D] and [H : K(g)] = [K(f): D]. d) IfDisamaximalsubfieldof K ( f ) a n d o f K(g), then K ( f ) a n d K ( g ) a r e maximal subjields of H .
Proof. Since both D and H contain a polynomial of positive degree, the first saternents of a), b ) follow from Lemma 5.8 and Cor. 5.82. I f p E K [ x ] ,0 Z a E K , then K(ap) = K(p). Hence we can assume that J g, h, k are monic.
K(k) FIG.6.2
§ 5
275
FIELDS
By Th. 5.71, [ k ] is divisible by [ f ] ,[ g ] whence v / [ k ] .Let [k] = rv. By Lemma 5.81, there exist polynomialsp, q € K [ x ]such that k = p ( f ) = q(g). Since k,f, ga r e monic,p and q are monic. Moreover IpI [ f l = [ql[ g ]= [kl. Since vd = [ f ][g], we have [ p ] = r v / [ f ]= r [ f ][ g ] / l f ] d= rm where m = [ g ] / d , and similarly [q] = rn wheren = [ f ] / d . By Lemma 5.83, there are monic polynomials s, t E K [ x ] , [s] = m, [ t ] = n such that [ p - f ] < ( r - l ) m , [ q - t ' ] - = ( r - l ) n . H e n c e [ p ( f ) - s ( f ) ' ] - = ( r - l ) m [ f ]= (r - I)v, and similarly [q(g)- t(g)'] -= (r - 1)v. Hence [ ~ ( f )-' t(g)*]-= (r - I)V, i.e. [s(f) - t(g)l [s(f>'-l+s(f)'-2t(g j . . . t(g)'-'I -= ( r - 1)v. Every term of the second factor on the left-hand side is monic + i[t][ g ] = (r - 1 - i)v + iv = (r - 1)v. and of degree (r - 1 - i) [s][f] Since char K = 0, the second factor as a whole is thus of degree (r - 1)v. Hence s ( f ) - t ( g ) = 0, therefore s ( f ) = t(g) = w E K [ x ] where [w] = m [ f ] = v and w E K ( f ) n K ( g )= K ( k ) . By Lemma 5.81, [w]2 [ k ] whence v 3 rz', thus r = 1 and [ k ] = v. Furthermore But [K(h):K ( k ) ] = [K(f)( g ) :K ( f ) ][ K ( f ):K(k)]. K ( g ) = K ( k ) ( g ) and [ K ( g ):K ( k ) ] = [ k ] / [ g ]= v / [ g ]= n implies [ K ( f )( g ) : K (f ) ] == n. On the other hand, [ K ( f ) :K(k)] = m, (m, n) = 1, a n d m and n divide [ K ( h ): K ( k j ] . Hence [ K ( h ):K ( k ) ] = mn. Therefore [h] = [ K ( x j :K(h)] = [k]/mn= v/mn = d. Thus a), b) hold. Since [f][ g ] = vd, c j follows. Finally, suppose that the hypothesis of d) is satisfied and that U is a field between K(f)and K(hj. By Lemma 5.8, U = K(I), I E K [ x ] .We have K ( k ) C K ( l ) n K ( g ) c K(g).If K ( I ) nK(g) = K(g), then K(1) = K ( f , g ) = K(h). Suppose now that K ( l ) nK ( g ) = K(k). Since K(1, g ) = K(h),a ) and b)imPlY that ([4[gl)= ([fl, [gl)and "11, [gl] = " f l , [gl] whence [I1 [gl = f ][g]. Therefore [1] = [ f ] ,hence K(Ij = K ( f ) .
+
+
r
5.85. A chain of fields KOc K , c . . . c Kr is called a maximal chain if every member is a maximal subfield of the subsequent one. 5.86. Theorem. Let f E K [ x ] be of positive degree and
. . . c s, c,,s, K(f) = To c T, c . . . c T,, c T,,,,
K ( f ) = soc s, c
= K(x),
= K(x)
be two maximal chains of subfields from K ( f ) to K(x). Then a) m = n.
276
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b) The degrees [Si : Si-Jcan be paired 08with the degrees [Tj : 7;-,]. c) Thefirst chain can be transformed into the second one by applying finitely many times the following procedure: Take any member of the first chain diferent j r o m Soand S,, and replace this member by a diflerent field such that the new chain becomes again a maximal chain. Proof. WLOG, we may assume m G n. Suppose that i is the greatest index such that Si= Ti. We proceed by induction on m+ 1 -i. If m+l -i = 0, nothing has to be proved. Now let m+l -i = e =, 0. Then the two chains are
Since Sm-e+l whereS,-,+, f- Tm-e+2.Let H be the field Sm-e+2(Tm--e+2). is a maximal subfield of Sm-e+2 and Tm-e+2,we have Srnpefl S,-,+,n Tm-e+2.By Lemma 5.8, Sm-e+2,TrnPe+$ satisfy the hypothesis are maximal subfields of Th. 5.84 whence d) implies that SmPe+,,Tn,-e+2 of H . Now we take any maximal chain H c H I c . . . c K ( x )from H to K ( x )which can always be done since [ K ( x ):HI is finite, and consider the two maximal chains -
Rl:Soc S , c . . . c Sm--e+lc Sm-e+2c H c Hl c . . . c K(x),
-
,Q2
: Soc S , c
. . . c Sm-e+l c Tm-e+2c H c.H , c . . . c K ( x ) .
By induction, a), b), c) hold for the chains $, and 91.Hence the number of members in R2 equals the number of members in 91.Again by inducand $2. By Th. 5.84 c), also the chains tion, a), b), c) hold for the chains 9, @I, satisfy a), b), c) whence the theorem is proved.
s2
5.87. Remark. The number of different maximal chains of subfields from K(f)to K(x) is finite.
Proof. By Th. 5.71, K(x) is algebraic of degree [f ] over K(f ). Since char K = 0, K( x )is separable over K ( f ) . Let N be a least normal extension field of K(x)1 K ( f ) (see 3 5.2). Then N is of finite degree over K ( f ) and is separable over K(f ). By the fundamental theorem of Galois theory, the number of fields between K ( f ) and N is finite, therefore - a fortiori -
§6
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SEMIGROWS AND GROUPS
277
the number of fields between K ( f ) and K ( x ) is finite. Hence there is only a finite number of maximal chains from K ( f ) to K(x). 5.88. Let
f, g be two non-constant polynomials of K [ x ] such that K ( g ) C K ( f ) . OZf,g(h)will denote the norm OZbz,f,,Kk,(h) of an element h in the finite algebraic extension K ( f ) of K ( g ) . Then Mfig(h> equals - up to the sign- the [K(f): K ( g ) @)I-th power of the constant term of the minimal polynomial of h over K(g) and @Zflg(h1h,)= @Zfig(hl)@Zfig(hz), for all hl, h2 E K ( f ) . 5.9. Let K be a field and K ( x ) the field of rational functions in x over K. Let a : K ( x ) K ( x ) be defined by: If r = u/v,u, v E K[x], then ar = (vu‘-uv‘)/v2. a is well-defined since, if r = ul/vl is the unique representation of r as a quotient of polynomials u1, 211 E K[x] such that (ul, vl) = 1 and el is monic, then u = tul, v = tvl, t E K [ x ] .Hence (vu’-uv’)/v2 = (v,u~-u,v;)/v:. a is an extension of the derivation d/dx of K [ x ] , we will write therefore a = d/dx, set (d/dx)r = r’, and will call r’ the derivative of r. -t
5.91. Theorem. Let r, s E K ( x ) and;€ K [ x ] .Then: (i) ( r f s ) ’ = r ’ f s ’ ,
(ii) (rs)’ = r’sfrs’, (iii) f ( r ) ’ = f ’ ( r ) r ’ .
Proof. We set r = u/v,s = ul/vl and compute either side of (i) and (ii). (iii) follows easily from (i) and (ii). 6. Semigroups and groups
6.1. Very little is used in this book from the theory of semigroups, but quite a lot from group theory. Both will be reviewed now, details can be found in books like HUPPERT[l], KUROS[l], SPECHT[l]. At the end of this section we will also prove a few lemmas which are not so well-known. A semigroup is an algebra ( S ; -)with an associative binary operation If is also commutative, S is called a commutative semigroup. If S is a semigroup and U, V are subsets of S , then the “complex product” UV is defined to be the set {uv I u E U, v E V}. U“ will denote the complex a.
278
APPENDIX
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6
product of n factors U. If U is an arbitrary subset of S , then the subsemigroup [ U ] generated by U is just [ U ] = U (U"I n = 1,2,3, . . .). Let S be a semigroup with identity 1. If u E S , then w E S is called a left (right) inverse of u if wu = 1 (uv = 1). If v is a right as well as a left inverse of u, then v is called an inverse of u. Every element of S has at most one inverse. The elements of S that possess inverses are called the units of S . The set of all units of S is a subsemigroup E = d(S) of S , and E is a group. If S is a commutative semigroup with identity 1 and H is a subsemigroup of S that is a group such that 1 EH, then the set of all subsets aH of S is a semigroup w.r.t. complex multiplication, the so-called factor semigroup of S modulo H . If IS I is finite, then the factor semigroup is of order / S l / J H l . Let S be any semigroup. A left (right) regular element of S is an element a E S such that au = av (ua = wa) implies u = v, for all u, w E S. a is called regular if a is right as well as left regular. An idempotent is an element a E S such that a2 = a. A partially ordered semigroup is a semigroup S together with a partial order relation =s such that a =sb implies ax G bx and xu 4 xb, for all x E S . If =sis a total order relation, then S is called a totally ordered semigroup.
6.2. A group is a semigroup G with identity such that every element of G possesses an inverse. A semigroup with identity such that every element has a left inverse is a group. An abelian group is a commutative group. Let U be a subgroup of G. A subset aU, a E G, of G isca lled a left coset of G modulo U while the subsets Ua are called right cosets of G modulo U . The set of all left (right) cosets of G modulo U is a partition of G. The cardinality of the set of all blocks of this partition is equal for left and right cosets and is denoted by [G : U ] . If G is a finite group and U is a subsemigroup of G, then U is a subgroup of G. The order of any subgroup of a finite group G divides the order of G . A proper subgroup U of G is called maximal if U c V implies V = G , for any subgroup V of G. Let G be a group and g E G . The order of g is the order of the subgroup [ g ] generated by g . A torsionfree group is a group with no element g f 1 of finite order. Let p, q be two distinct primes, then a group G is called a p-group if the order of every element of G is a power of p, and G is called a ( p , q)-group if the order of every element of G is a prod-
06
SEMIGROUPS AND GROUPS
219
uct of powers of p and q. A group G is called cyclic if G = [g], for some g t G. Any group of prime order is cyclic. If G is a finite cyclic group, then G has exactly one subgroup of order d, for any divisor d of I G 1. If the order of the elements in a group G is bounded, there exists a least positive integer n such that g” = 1, for all g E G, and n is called the exponent exp G of G. An elementary abelian group is an abelian group G with exp G = p where p is a prime. The set End G of all endomorphisms of a group G is a semigroup w.r.t. the composition of mappings. The set Aut G of all automorphisms of a group G is a group w.r.t. the composition of mappings. 8 E Aut G is called an inner automorphism of G if 8g = ugu-l, for all g E G and some u E G. The set In G of all inner automorphisms of G is a subgroup of Aut G. Two subgroups H I , Hz (elements hl, h2) of G are called conjugate if 8H1 = Hz(8hhl= h2), for some 8 E In G. If G is a finite cyclic group, then 1 Aut G I = q ( \G I) where p? is the Euler v-function. Let G be any group and Q E End G. If we regard mi E 9 as a 1-ary operation on G, then, since mi 1 = 1, for all miEQ, G together with Q is an Q-multioperator group (G; Q), in short an Q-group. The subalgebras of (G; Q) are called Q-admissible subgroups of G. If R is the set of all endomorphisms, automorphism;, inner automorphisms of G , resp., then the 0-admissible subgroups of G are called fully invariant, characteristic, normal, resp. If T is a subset of G and [Q] is the subsemigroup of End G generated by 0, then the subalgebra [TI of (G; Q) generated by T is the set of all elements g E G of the form g = (milt:;) (mi,t:;) . . . (m,,tz) where n is some integer, mlkE[Q], ti,ET, ck = 1, k = 1, . . ., n. If [TI = G, then T is called a n Q-generating set for G. An n-generator 0-group G is an Q-group with a generating set of cardinality n. A 1-generator S-group is called monogenic. 6.3. N Q G will mean that N is a normal subgroup of G. The normal subgroups G and (1) of G are called the trivial normal subgroups of G. A non-trivial normal subgroup N of G is called a minimal normal subgroup of G if M c N implies M = {l}, for any normal subgroup M of G and is called a maximal normal subgroup of G if N c M implies M = G, for any normal subgroup M of G. If N Q G, then the partitions of G into left cosets and right cosets modulo N coincide and the equivalence relation corresponding to this partition is a congruence 8N on G. The mapping 8 is a bijection from
280
APPENDIX
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6
the set of all normal subgroups to the set of all congruences on G, and we will write G IN for G I8N. A group G is simple if and only if G has just the trivial normal subgroups. A finite abelian group G is simple if and onlyifIGI = 1 o r j G I = p , f o r s o m e p r i m e p . First isomorphism theorem. If U is a subgroup and N is a normal subgroup of G, then UN is a subgroup of G, U n N is a normal subgroup of U, and U N ( N = UI U n N . An isomorphism a : U N l N -.- U ( U n N is given by a ( u N) = u ( U n N ) , uE U. Second isomorphism theorem. Let p : G -.- H be an epimorphism from the group G to thegroup H , N a H , and A4 = {g E GjpjgE N } . Then M
If N a G, then every a E In G induces an automorphism of N , and the set of all automorphisms of N induced by automorphisms of In G is a subgroup of Aut N . Let G be an Q-group. Then the Q-admissible normal subgroups of G are just the ideals of the multioperator group (G;Q), by Lemma 3.4. If V is an arbitrary and N a normal Q-admissible subgroup of G , then VN is the Q-admissible subgroup generated by VU N . 6.4. Assume that (Gi I i E I ) is a family of groups and let G = n ( G i1 i E I ) be the direct product of this family. For any k E I , the mapping i , :G, G defined by ikg = (a(i)iiE I ) where a(k) = g and a(i) = 1, for i # k , --t
is a monomorphism from G, to G, the so-called inclusion monomorphism from G, to G. A group G is called the inner direct product of its subgroups U,, . . ., U, if p : U,X . . . x U, G, p(ul, . . .,u,) = uluz . . . u, is an isomorphism. Subsequently “direct product” will always mean “inner direct product”. A finite group G is elementary abelian if and only if G is the direct product of a finite number of groups of order p where p is some fixed prime. Let G be a group, N a normal subgroup, and H a subgroup of G. Then G is called the semidirect product of N by H if NH = G and N,Q H = {l}, and H in this case is called a semidirect factor or a retract of G. A group extension of a group N by a group H is a group G such that N 4 G and GIN 5 H . --t
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6.5. Let G be a group. The set Z ( G ) = ( z E GI zg = g z , for all g E G )
is called the centre of G. The centre is a characteristic, thus a normal subgroup of G and In G G iZ(G). If G is a finite p-group, then I Z ( G ) I z 1. A subgroup U of G is called central if U G Z(G).An element z G centralizes g G if zg = gz. Let G be a finite group. The intersection of all maximal subgroups of G is denoted by @(G)and is called the Frattini subgroup of G. @(G) is a normal subgroup of G . If G is a p-group, then G I@(G) is elementary abelian. 6.51. Let G be any group. If K and H are normal subgroups of G such that K c H and HI K is a minimal normal subgroup of G I K, then HI K is called a chief factor of G. H ( K is called a central chief factor if HI K C Z(G 1 K ) and a p-chief factor if H 1 K is a p-group. A finite series G = H , 2 H I 2 . . . 2 H,, = (1) of subgroups of G is called a composition series of G if Hi is a maximal normal subgroup of i = 1, . . ., M, and is called a chief series of G of length n if each H i is normal in G and Hi-l I Hiis a chief factor of G, i = 1, . . ., n. The factor groups H i - , / H i are called the factors of the series.
Theorem. Let G possess a composition series. Then G possesses a chief series, any two chief series of G have the same length and, up to the order and isomorphism, the same factors. Every chief factor of G is the direct product of a Jinite number of isomorphic simple groups.
A finite group G is called soluble if all the chief factors of G are abelian (and hence elementary abelian). G is called su~ersolubleif all the chief factors of G are cyclic (hence of prime order). Therefore every supersoluble group is soluble. Every subgroup and every factor group of a soluble (supersoluble) group is soluble (supersoluble). If N 4 G and N and GIN are soluble, then G is soluble. 6.52. Let G be a finitegroup a n d p a prime dividing I G 1. Ifp" is the greatest power of p dividing 1 G I, then any subgroup of G of order pa is called a Sylow p-subgroup of G. A Sylow p-subgroup always exists and any two Sylow p-subgroups of G are conjugate. 6.53. Let G be a group and g , h E G. Then we define the commutator [g, h] of g and h by [g,h] = g-lh-lgh. If S , T a r e two subsets of G, then
282
CH.
APPENDIX
6
[S, T ] = [[s, t ]1s E S , t E T ] i.e. the subgroup generated by all [s, t ] . The subgroup [G, GI = G’ of G is called the commutator subgroup of G . The lower central series of G is the series Go = G, Gi= [GtP1,GI, i = 1,2, . . ., of subgroups of G. A group G is called nilpotent if G, = {l}, for some n. The least such integer n is called the class of G. Every subgroup and every factor group of a nilpotent group is nilpotent. Every minimal normal subgroup of a nilpotent group G is contained in Z(G). A finite group G is nilpotent if and only if G is the direct product of its Sylow subgroups. Every finite p-group is nilpotent. Every finite nilpotent group is supersoluble and a fortiori soluble.
Theorem (Schmidt-Rtdei-Iwasawa). Let G be a finite non-nilpotent group such that every proper subgroup of G is nilpotent. Then: a) IGI =paqb where p # q are primes, a, b =- 0, G has a normal Sylow p-subgroup, and the Sylow q-subgroups are cyclic. b) I f S is a Sylow p-subgroup or a Sylow q-subgroup of G, then @(S) G Z(G). Let G be a nilpotent group of class 2. Then G’ E Z(G) and [ab, c] = [a, cl [b,c], [a,bc] = [a,b][a, c],for all a, b, c E G. Theorem (Levi). in any group G, ab-lab = b-laba, for all a, b E G, then G is nilpotent and of class =s3. If moreover G has no elements of order 3, then G is of class == 2. 6.6. Let M be any set. Then the set of all mappings from A4 to M is a semigroup S ( M ) with identity w.r.t. the composition of mappings. This semigroup is called the symmetric semigroup of M . An element of S ( M ) is called a permutation of M if it is a bijective mapping from M to M . The permutations of M are just the units of the semigroup S ( M ) whence the set of all permutations of M is a group w.r.t. the composition of mappings. This group is called the symmetric group Sym M of M . If M is finite, then I Sym MI = 1 MI !. Any subsemigroup of S ( M ) is called a mapping semigroup on M , and any subgroup of Sym M is called a permutation group on M . Let G be a permutation group on M . If a, b E M , we will write a b if and only if there is some z E G such that na = b. is an equivalence
-
-
86
SEMIGROUPS AND GROUPS
283
relation on M yand the blocks of the corresponding partition of M are called orbits of G on M . Let G be an arbitrary group and a € G. Then (p)g = ug, gE G , defines a permutation pa of G, and p :G Sym G is a monomorphism. The permutation group eG on G is called the regular representation of G. Suppose that M = { 1,2, . . .,n}. Then -t
is a normal subgroup of Sym M of order n!/2 and is called the alternating group on M. 6.61. Let G be any mapping semigroup on A4 and H a mapping semigroup on N . Suppose that a f G, ,LI(m)E H, for all m E M y then w(m, n) = (am,B(m)a) defines a mapping w :M x N M x N . The set W of all mappings of S ( M x N ) of this kind is a mapping semigroup on M X N which is called the wreath product of H by G and is denoted by HwrG. If G and H are permutation groups, so is HwrG. Let G, H be permutation groups on M , IV, resp. and W = HwrG. If w E W and w(m,n) = (am, P(m)n),;then we set 80 = a. Then 8 :W G is an epimorphism and ker 8 is isomorphic to the direct product (H(m)1 m f M ) where H(nz) = H, for all m E M . If M and H are finite, then HwrG is soluble if and only if G and H are soluble. Let G be a finite and H an arbitrary group. Let R = { ( g , y)l g E G, p :G H} and define a binary operation on R by (gl, 91)(gz, y~2)= (glgz, y ) where yg = (ylg)yz(ggl).Then (R;-) is a group which is called the regular wreath product of H by G and also denoted by HwrG. -t
-
n
-
6.7. Let F be a free group, then any two free generating sets for F a r e of equal cardinality which is called the rank r(F) of F. Every subgroup of a free group is again a free group. Let @ be the variety of groups. Then a free product in 8 exists for any family of groups. If (GiI i E I ) is a family of groups, then their free product in @ will be denoted by % (GJi€ f), or if i = 1, . ..,n, by GI* . . . %G,. If F is the free group with free generating set {xili E I } and F(xJ is the free group with free generating set {xi}, then F = ++c(F(xi)I i E I). If G is a free product of the family (GJiE I ) and every Gi is a free product of the , G is a free product of the family (Hg/iEI, family ( H G / j C J ( i ) ) then
284
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6
j E J ( i ) ) . The free product of free groups Fi is itself a free group F, and
the rank of this free group equals the sum of the ranks r(Fi). Let G be a group and (Ail i E I ) be a family of subgroups of G. Then G is the free product of this family if [ U (Ail i E I ) ] = G and if every g E G, g # 1, can be uniquely represented in the form g = a, . . . a,,, n 2 1 , ak # 1 , k = 1 , . . ., n, a , € A,, ik z ik+l, k = 1, . . ., n-1. The KUROSHsubgroup theorem gives information on the subgroups of a free product. We need the following special case of this theorem: Let G = x (G,I i E I ) be a free product of a family of groups and H a normal subgroup of G. Then H can be represented as
where Uik z H f l G , , for all k E K,, IK,] = [G:G,H],and F is a free group such that r ( F ) + [ G : H ] + C ( I K , /IiEI) = IZI[G:H]+l. Let (GjI i E I ) be a family of groups and (piI i € Z) a family of monomorphisms from a group B to G,. Then there exists a group A such that (i) B is a subgroup of A , (ii) for every i € I , there is a monomorphism of which is y i l , (iii) [ U(yiGi1 i E Z ) ] = yi:Gi A the restriction onto G, and (iv) (yiG)fi[(U ykGkI k E Z, k # i)]= B, and such that every group with these four properties is a homomorphic image of A . The group A which we obtain if we perform the embedding of B into Gi by q,,for all i E I, and then the embedding of the groups so obtained into G by 'yi is called a free product of the family (Gjj i E I ) with amalgamated subgroup B. +
6.71. Let F ( X ) be the free group with free generating set X = { x i / iE I}, W = {wjlj E J } a set of words in Y = {yil i E I } , yi being indeterminates,
and R the normal subgroup of F ( X ) which is generated by all words wj(xi), j € J . If, for all if I , we denote the class C(xJ of the factor group F ( X ) I R also by y,, we obtain a group G which is called the group generated by the family (yiI i € I ) and defined by the relations wj = 1, j € J . If H is an arbitrary group with a generating set {ai1 i E I } which satisfies the relations wj(ai)= 1, j E J , then the mapping 6 : Y H , 8yi = a,, i E I, can be extended to an epimorphism from G to H . Let us consider a special case of this construction: Let m i 1, 2 be a non-negative integer, Y = {a, b}, W = {a2,b", (ab)'}. Then the group D , generated by {a, b} and defined by the relations a2 = b" = (ab)' = 1
-
06
285
SEMIGROUPS AND GROUPS
is called a dihedral group. If m = 0, then D, is an infinite group, and if m w 0, then ID, I = 2m, a is o f order 2 and for m # 0, b is of order m. 6.72. Let Y = {yjli E Z} be a set of indeterminates, W = {wj(yj,,. . . yjn,)I j E J } a set of words in Y over the group operations, G an arbitrary group, and W ( G )= { wj(g,, . . .,g,J I j E J , gk E G}. The subgroup I W/(G)1 is called the verbal subgroup of G generated by W . If F is a free group, then the set of all verbal subgroups of Fcoincides with the set of all fully invariant subgroups of F. 6.73. Lemma. Let N be ajinite elementary abelian p-group, Y = {yl, . . ., y k } a finite set of indeterminates, A, the verbal subgroup of the free group F(x,, . . ., xk) generated by yf and y;ly;'ylyZ, for k a 2, and by y:, for k = 1 , and w(yl, . . ., yk) = 1 a Zaw for N . Then w(x,, . . ., xk)E A,. Proof. Since every finite elementary abelian p-group is the direct product of cyclic groups of orderp, w(yl, . . .,Y k ) = 1 is also a law for the elementary abelian group N1 of order pk. The group G, generated by yl, . . .,yk and defined by the relations yp = 1, i = 1 , . . .,k , y;lyY1yiyj = 1, i , j = 1, . . ., k, is elementary abelian of 'order pk, hence isomorphic to N1. Thus w(yl, . . ., y k ) = 1 is also a law for G whence w(xl, . ., xk) is contained in the normal subgroup R of F(x,, . . ., x k ) which is generated by the elements xf and x~:'x,~'xixj. But since these elements are also contained in A, and A, is fully invariant, hence normal in F(x,, . . ., xk), we have R E A,.
.
6.8. Proposition. Let G be a group, A, B subgroups of G and p : A an isomorphism. Then there exists an extension group H of G and t such that pa = t-lat, for all a E A.
-.
B H
6.81. Corollary. Two elements a, b of a group G are conjugate in a suitable extension group of G if and only if they have the same order. Proof. The corollary is an immediate consequence of the proposition. Let K = G[u] and L = G[w] where u, w are indeterminates, and set U = [GU u-lAu] E K . Then U = G % u-IAu since every wE U can be represented in the form w = g,,u-lalug,u-la,u . . . gr-,u-larugr where r a 0 , a j E A , aj # 1 , g j E G , gj # 1 , j = 1 , ..., r-1, and this repre-
286
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6
sentation is unique, by ch. 1, Th. 9.1 1. Similarly for V = [GU vBw-l]E L, we have V = G f vBv-l.Hence there is an isomorphism a : U - V which fixes G such that a(u-lau) = v(pa) v-l, for every a E A . Let H be the free product of K and L with amalgamated subgroup U according to this isomorphism, then G is a subgroup of H , and if we put t = uv,then t-,at = v-lu-lauv = v-]w(pa)v-lv = pa, for all a E A .
6.9. Lemma. Let G be an n-generator group andN1, N2 normal subgroups for each pair of generating sets {elN1, of G such that N l n N 2 = (1). . . .,e,N,} and {flN2,. . .,f,N2} for G IN,, G IN2 resp., there exists a generating set {gl, . . .,g,) for G such that gi E eiN1nf;.N2,i = I , . . .,n, then ] 2. either G = N1 X N 2 or [G :N I N ~= Proof. Suppose that {g, ,. . .,g,} is a generating set for G. Then {g,Nl, . . .,g,N,) generates GI Nl and {giN2,g g 2 , . . ., g,N2) generates GIN2. Hence there are g: E G, i = 1, . . ., n, such that [gl, . . ., gi] = G and g; E glNlngiN2, g] E giN,ilg,N2. Then g , K 1 = (glgi-l>(gk;') E N1N2 whence GIN N is cyclic. Moreover [g;'N,, . . ., gT1N1]= GIN,, hence $, we can find g j E G, i = 1, . . ., n, such that [g;', . . .,g,] = G and g,!' E g;lN1ngiN2. Thereforeg: = (gig;'-') (g]'gi)E N2N, = N,N, whence IGINlN2( =S 2. I ,
6.91. Lemma. Let n be a positive integer, G an n-generator SZ-group, and N a finite SZ-admissible normal subgroup of G. Then, for each SZgenerating set { L N , . .. , f , N } of the SZ2-factorgroup GIN, there exists an !2-generating set {el, . . .,en}of G such that ei CAN, i = 1, . . .,n.
Proof. Let S be the set of all 0-subgroups 'V of G such that VN = G. If V E S, then A N = viN, for some viE V, i = 1, . . ., n. By the first isomorphism theorem for multioperator groups, there is an isomorphism a : G I N .+ V l V n N such that a(vN)= v ( V n N ) , v E V , whence the set of all vi(VnN)= vn viN= V n j p is an Q-generating set for VI V n N . Therefore V = [ ( v n f , N ) U ( v n f , N ) U . . . U ( V n f , N ) U ( V n N ) ] . Since N is finite, we conclude that S is also finite. Hence S contains only finitely many maximal Q-subgroups and every proper Q-subgroup in S is contained in at least one maximal Q-subgroup in S. If every maximal Q-subgroup of G contains N, then S = {G} whence [f 1 , . ..,jJ= G. Otherwise let M,, . . .,M, be just those maximal Q-subgroups of G which
07
LINEAR ALGEBRA AND REPRESENTATION THEORY
287
do not contain N , i.e. all maximal Q-subgroups in S. For an arbitrary Q-subgroup U of G, we set 0, for UN # G, .(U) = 1, for UN = G.
{
Then the number of all systems {el, . . ., en>,
e,EJ;N,
i = 1 , . . ., n,
(6.9)
which are contained in U, is given by I NT'I U I"E(U).For if UN = G , then U n X N = v i ( U n N ) , for some viE U and if UN # G, then at least one of the U n L N is empty. Hence the number of all systems (6.9) that are not contained in any proper Q-subgroup of G is qiv) = IN]"+
c.. .
k = l il-=irc
( - i ) k I N n M i l n ...nM;,,~n&(M,~n...nM~k). -=ik
@(N) does not depend on the particular choice of the generating set {fJV, . . . , f n N } . Since G is an n-generator Q-group, there exists such a generating set which yields a system (6.9) that is not contained in any proper Q-subgroup of G . Hence @(N) # 0.
7. Linear algebra and representation theory 7.1. This section is devoted to linear algebra and representations, but will be reviewed just briefly. For further information, we refer to standard books, such as CURTIS-REINER [l], HUPPERT[l], R ~ D E[2]. I Let R be an arbitrary ring and m, n positive integers. An mXn-array
of elements uik E R, i = 1, . . ., m, k = 1, . . .,n, is called an m x n matrix over R. A brief notation for such a matrix A is A = (aik). The sum A + B of any two mxn-matrices A = (aik), B = (bik) is defined to be the mxn-matrix A + B = (a,+b,). If A = (aik)is an m x n-matrix and B = (bik)is an n Xp-matrix, then we define the product AB to be the mXp-matrix AB = (cik)where c;k =
n
r=i
uitb,. If A = (aik)
288
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6
is an m X n-matrix and c E R,then cA and Ac are m x n-matrices defined by cA = (ca,), Ac = (aikc). The m x n-zero matrix Om, is the matrix (aik) with aik= 0, for all pairs (i, k), and if R has an identity 1, then the m X midentity matrix is the matrix Em = (aik) where a,, = 0, for i # k, and dik = 1, for i = k, i. e. 6, is the so-called Kronecker symbol. If, in any of the following equations, one side is defined, so is the other side and the equation holds : A+B
B+A, (A+B)+C = A+(B+C), (AB)C = A(BC), A(B+C) = AB+AC, (B+C)A = BA+CA, c(A+B) = cA+cB, (c+d)A = cA+dA, (cd)A = c(dA), =
where A, B, C are matrices over R and c, d € R.If A is an mXn-matrix, then A+O,, = O,,+A = A, E,A = AE, = A. These equations show that, for any m, the set of allmXm-matrices over R is a ring w.r.t. + and as just defined which is called the full m xm-matrix ring R, over R. If R has an identity, then R, also has an identity. Then the group of all units of R, is called the general linear group GL(m, R) of dimension m over R. Theorem. if S is a skewfield, then every full matrix ring S , over S is simple. Conversely, if R is a simple ring such that I R I $ 1, ab # 0, for at least one pair (a, b) of elements a, b E R, and every descending chain L, 2 L, 3 L, 2 . . . of left ideals Li in R is finite, then R is isomorphic to a full matrix ring over some skewJield S . 7.2. Let R be a commutative ring with identity, A = (aik) an m x m matrix over R , and M = {1,2, . . ., m}. We define the determinant IAl of A by
IA I
=
C(~
.
( ; ~ ) . a. a,,,,, ~ ~In~E Sym a ~ M~) , ~
E(Z)
= -1
E(Z) =
1
if if
z $ A,;
nEA,.
Sometimes we write I A1 = det A. If A , B are any two mXm-matrices over R, then j AB I = I A I 1 B 1. Moreover the mapping det : R, -+ R is an epimorphism of (multiplicative) semigroups. A matrix A C R, is a unit of R, if and only if 1 A1 is a unit of R. A matrix A C R, is called unimodular if 1.41 = 1. The set SL(rn, R) of all unimodular matrices of R, is a subgroup of GL(m, R).Let A E R, and b be an m x 1-matrix over R, then q = Ax+ b, x E R", defines a
07
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mapping n E S(Rn'). The set of all these mappings is a subsemigroup of S(Rm), the so-called inhomogeneous linear semigroup of dimension m over R . In this semigroup, the subset of all mappings with 1 A I being a unit is a group, the inhomogeneous linear group of dimension m over R . If R is a finite field of order p', then we write GL(m, R ) = GL(m, p t ) . We have 1 GL(m, p') I = (p"'- 1) (p"'-p') . . . (p"' -p("-l)'). The group GL(m,p') is soluble if and only if m = 1 or m = 2 and pf = 2, 3. GL(m, p') is nilpotent if and only if m = 1. Let A be an mxm-matrix over R and set Em = E. Then x E - A is an mxm-matrix over R[x] whence IxE-A( is a polynomial of R [ x ] , the so-called characteristic polynomialof A . If IxE-AI = ~ ~ + a ~ x " .-. .~ + fa, then A" f alA"-' . . . a,-,A+ a,E = Om, (CAYLEY-HAMILTON equation). The roots of the characteristic polynomial of A are called the eigenvalues of A . Let A be an m Xm-matrix over the field K and o the m X 1-matrix where all elements are 0. Then the "system of linear equations" AT = o has a solution g # o in K" if and only if A is singular, i.e. I A1 = 0.
+
+
+,
7.3. An abelian group ( M ; -,.O) is also called a module. Let R be a ring with identity. An R-module (or left R-module) is an abelian group ( M ; f, -, 0) together with a mapping from R X M to M , (r, m) rm, such that (r+s)m = rm+sm, r(m+n) = rmfrn, (rs)m = r(sm), l m = m, for all Y , s E R,m, n EM. If K is a field, then a K-module V is called a vector space over K or K-vector space. The elements of V are called vectors, and the identity 0 of V the zero vector. The family v,, . . ., v, E V is called linearly independent if r,v,+ ~.. f r n v , = 0 implies ri = 0, i = 1 , . . ., n ; otherwise linearly dependent. If, for any integer d a 0, there exists a family of d linearly independent elements in V , but every family of n =- d elements is linearly dependent, then we set d = dim V = dim, V and call d the dimension of V . Then every family of d linearly independent elements is called a basis of V. If m < dim V and vl,. . ., v,, are linearly independent, then there exist elements v,,,+~,. . ., vnE V such that v,, . . ., vn is a basis of V. If vl. . . ., v, is a basis of V , then any w E V can be uniquely represented as +
n
rkvk, r,
w = k=l
E K . If wl, . . ., w, E V and wi =
n
k=l
r i p k ,then
are linearly independent if and only if det (rik)# 0.
MY*, . . . , w,
290
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6
Let V be any vector space over K . An endomorphism (automorphism) of V is an endomorphism (automorphism) p of ( V ; + , -, 0) such that p(kv) = kp(v), for all k E K, v E V. If p1+ p2and p1p2 are defined, for endomorphisms PI,9 ; of ~ V , by (p1+y2)v = p1v+p)2v, (plp2)v = p 1 ( ~ 2 ~ ) , then the set Hom, (V, V ) of all endomorphisms of V is a ring with identity. The units of Hom,(V, V ) are just the automorphisms of V , they form a group Aut V. If dim V = n, then Hom,(V, V ) Y K, as rings, and Aut V 2 GL(n, K ) . 7.4. Let M be an R-module. A submodule of M is a subgroup U of ( M ; , -, 0) such that ru E U, for all r E R, u E U. The submodules (0) and M of M are called the trivial submodules. M is irreducible if M has no non-trivial submodules. An isomorphism (homomorphism) p :M .+ N of R-modules is an isomorphism (homomorphism) from ( M ; , -, 0 ) to ,N; , -, 0) such that p(rm) = rp(m), for all Y E R, m E M .
+
+
+
Let M be an R-module. Then the set An M = {YE Rlrm = 0, for all m E M } is an ideal of R, the so-called annihilator of M . M is called faithful if An M = (0). If M is an arbitrary R-module, then M is a faithful R I An M-module under the action ( r + An M)m = rm, r E R, in E M . If R is a ring where every descending chain L1 I)L2 3 . . . of left ideals is finite, and if R possesses a faithful, irreducible R-module M , then R is a simple ring. Let K, be the full n X n-matrixring over a field K . Then every irreducible K,-module can be regarded as a vector space over K and its dimension is n.
7.5. Let G be a group and V a vector space over the field K . A homomorphism 6 : G Aut V is called a representation of G on V . If Sg = 1 for all g E G , then 6 is called a trivial representation. If dim V = n and a : Hom,(V, V ) K, is an isomorphism, then a6 is called a matrix representation of G over K afforded by V. Let G be a finite group, K any field, and KG the set of all formal sums c ( k , g I g E G ) , k, E K . In KG we define x k , g + X1,g = x ( k g + Q g , ( x k , g ) (Cl,g) = C(Ckhlh-,,IhE G)g. Then (KG; is a ring with identity which is called the group ring of G over K . Let V be a vector space over K and 6 a representation of G on V . If we define (Ckgg)v= Ck,((Sg)v), for any Ck,g E KG, v E V , then V becomes a KG-module. The representation 6 of G is called irreducible if V is an -+
+
+,
a)
88
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NEAR RINGS
irreducible KG-module. All these concepts are used in the special situation that K = K,, the field of prime order p , Y = N, a minimal abelian normal p-subgroup of G regarded as a K,-vector space by means of n l f n 2 = n1n2, (k1)n = nk, for any integer k == 0, and (6g)n = gng-l. 8. Near-rings
8.1. Near-rings and distributively generated (d.g.) near-rings, in particular, have been defined in ch. 3,§ 1.3 and ch. 5 , s 2. This section shall present some basic facts on d.g. near-rings as they are applied in ch. 5. Since there does not exist a book on near-rings so far, every result will be proved here. Let (A; +, .) = A be a d.g. near-ring with an identity 1 for and S a generating set for (A; +, -, 0) consisting of distributive elements. An (additively written) group M together with a mapping @ : A X M -+ M, p(a, m) = am, is called an (A, S)-group or-if S is kept fixed-an A-group if (al+a2)m = alm+u2m, al, uz E A, m E M , s(ml+mz) = sm1+sm2,: sES, ml, m2EM, (s1s2)m= sl(szm), s1, s2 E S , m E M , lm = m, mEM. a,
Let M be an A-group. A subgroup N of M is called an A-subgroup of M if unEN, for all a € A, n E N . A non-zero A-group M is called a minimal A-group if M contains no non-trivial A-subgroups. If M is an A-group and N is a normal A-subgroup of M , then the factor group MIN becomes an A-group by virtue of a(m+N) = am+N, and MI N is called an A-factor group of M . Let M1, M z be two A-groups. An A-homomorphism (isomorphism, epimorphism) is a group homomorphism (isomorphism, epimorphism) q ? : M1 M 2 such that q(arn) = a(y-m), for all a € A, m E M . If rp : M1 M z is an A-epimorphism, then ker rp is a normal A-subgroup of M 1 and M 2 is A-isomorphic to the A-factor group M1j ker q?. Note that A itself is an A-group through the left multiplication by elements of A. If U is an A-subgroup of A and b E A, then Ub is also an A-subgroup of A. We say that A satisfies the minimum condition for A-subgroups if every set of A-subgroups of A has a minimal element w.r.t. to the inclu+ -f
292
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6
sion of sets, or equivalently, if every chain U1 3 Uz 3 . . . of A-subgroups is finite. Throughout this section, we will assume that A is a d.g. near-ring with identity 1 and minimum condition for A-subgroups. 8.2. An element a € A is called nilpotent if a" = 0, for some positive integer 11. An A-subgroup U of A is a nil A-subgroup if every uE U is nilpotent. An A-subgroup U of A is nilpotent if there exists a positive integer n such that U" = {0}, where U" means the complex product. Let U,,. . ., U,, be A-subgroups of A. Then UloU z o . . . o U,, is defined to be the A-subgroup of A generated by UIU,. . . U,,. This does not necessarily imply (Ul o U 2 )o U 3 = U1o (UZo U3), but U1o ( U zb) = ( U l o Uz)b, b E A, is always true since Ulo Uz is the set of all finite i = 1,2. sums ~ a j u j u v uaj2 jE, A, uijE Ui, 8.21. Theorem. Every nil A-subgroup of A is nilpotent.
Proof. Let U be any nil A-subgroup of A and set U(') = U, U(")= U("-')o U("-l), n = 1,2, . . . . The minimum condition implies U ( k )= U(k+l),for some k . Suppose U ( k )f {0}, then there exists an A-subgroup I of A minimal w.r.t. U ( k ) ~#I {0}, again by the minimum condition. Choose b E Z such that U(k)b # (0).Then U(k)bis an A-subgroup of A which is contained in I and U ( k o) (Uck)b) = (U(k)o U"))b = U(k+l)b = Uck)b # {O}, whence Uck)b= I . Hence there exists an element uE U ( k ) such that b = ub. Therefore b = ub = u2b = . . . = u'b = 0, for some r, since u E U and U is a nil A-subgroup, contradiction. Hence U ( k )= {0}, therefore U Z k= (0). 8.3. Let M be an A-group and X any set of elements in M . The set I,(X) = {a E A1 ax = 0, for all x E X} is called the left annihilator of X i n A. If X = {x}, we will write l,(X) = l,(x). 8.31. Proposition. Let M be an A-group and X a set of elements in M . Then IA(X)is an A-subgroup of A.
x E X , al, a2E A, alx = azx = 0, then ( a l f a z ) x = alxfazx = 0, and (-al)x = -alx = 0. Moreover if a E A, then (aa~)x= a(a~x>= 0 since (A; +) is generated by S . Proof. Let
88
NEAR RINGS
293
8.32. Theorem. Every non-nilpotent A-subgroup U of A contains an idempotent element e # 0. Proof. Since the minimum condition holds for A, it suffices to prove the theorem for the case that every proper A-subgroup of U is nilpotent. Since U is non-nilpotent, there exists a non-nilpotent element uo E U, by Th. 8.21, hence U = UUO,for if UUOc U, then UUOwould be nilpotent whence ui would be nilpotent, contradiction. Hence there exists u, E U such that uo = q u o , and U I is non-nilpotent, otherwise we would have zio = uluo = u$o = . . . = 4uo = 0, for some s, contradiction. Hence U = Uu,, and again there is a non-nilpotent element u2E U such that u1 = u2u1. If we continue this procedure, we obtain a sequence U O , u1, u g , . . . of non-nilpotent elements in U such that U = Uu, and ui= ui+,ui, i = 0, 1,2, . . . , This yields a chain lA(uo)2 IA(ul) 2 . . . of A-subgroups of A, and by the minimum condition, we have I,(u,+,) = lA(uk). Since u, = uk+lu,, we conclude (u,+,-l)u, = 0, hence u,+,- 1 E lA(u,) = ~ J U , + ~ ) . Therefore (u,+,-1)uk+, = 0 i.e. u,+, 2 = uk+,. Hence ukilE U is an idempotent.
8.33. Corollary. Every minimal non-nilpotent A-subgroup U of A is of the form U = A e where e is an idempotent of A. 8.34. Proposition. Every set {el, e2, . . .} of non-zero idempotents of A such that ei+, E IA({el, . . ., e,}), i = 1, 2, . . ., is finite. Proof. Suppose the opposite is true, then by the minimum condition, iA({el, . . .,e,}) = lA({el, . . ., e,,,}), for some k. But then ek+rE iA({el, . . .,e,,,}) whence eE+l = 0, contradiction.
8.35. Corollary. There exists afinite set {Ae,, . . ., Ae,,} of minimal nonnilpotent A-subgroups of A, ei idempotent, i = 1, . . ., n, such that e,+, E lA({el, . . .,ei}), i = 1, . ..,n - 1, and such that there is no nonzero idempotent e,,, E IA({el, . . ., e,}). Proof. Since A is non-nilpotent as 1 E A, there is a minimal non-nilpotent A-subgroup Ael of A. Take any minimal non-nilpotent A-subgroup Aez of IA({el}), then e, E IA({el}). Then choose a minimal non-nilpotent A-subgroup Ae, of lA({el, e,}), etc. By Prop. 8.34, this procedure must
294
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6
terminate after finitely many steps, i.e. there is no non-zero idempotent en+, E l,({e,, ..., en})for some n. 8.36. Proposition. Let {Ae,, . . ., Ae,} be a set of minimai non-nilpotent A-subgroups of A such that all e, are idempotents, ei+l E I,({e,, . . .,e;}), i = 1,2, . . ., n-1, and there is no non-zero idempotent en+,€ lA({el,. . ., en}). Then L = IA({el, . . ., en}) is a nilpotent A-subgroup of A, and every element a E A can be written uniquely as U = Ul+U,f . . . +U,+I, UkEAek,k = 1, . . ., n, IEL. Proof. Suppose L is not nilpotent. Then by Th. 8.32 and Cor. 8.33, L contains a minimal non-nilpotent A-subgroup Ae,,, where en+, is an idempotent, and en+,E lA({e,, . ..,en}),contradiction. The second statement will follow from: Every element a € A can be written uniquely as a = a,+ . . . fai+ll, UkEAek, k = 1, 2, . . ., i, and ~ L ~ ~ A (.{.e.,,e,}). , We will prove this by induction on i: for i = 1, we have A = Ae,+l,(e,) and -ae,+u~~,(el). since, for any a € A, u = ae,+(-ae,+a) Moreover Ae,n I,(e,) = 0, hence this decomposition is unique. Let j Z= 1. By induction, a = a,+ . . . +aj+$, a,€ Ae,, k = 1, . . ., j , $ € l,({e,, . . ., ej}>.If we put I,+, = -bej+,+ 5, then aj+l = bej+,E Aej+l and Ij+,€lA({e,, . . ., ej+l}). Suppose that a = a,+ . . . +aj+,+$+: = a;+ . . . +aJ!+,+$+,, a,, ahEAek, k = 1, . . ., j + l , &+,, $+lC x,({e,, . . ., ej+,}). Then (a,+,+$+,)ek = 0 = (u;+,+<+,)ek, k = 1, . . ., j . By induction, a k = a;, for k = 1, . . ., j , and u ~ + ~ + $ +=, a;+,+(+,. Hence -a:J f l +aj+,= $+,-$+, E Aej+,nlA(ej+J Therefore aj+, = aj+,, ! I + ,
=
$+l.
8.4. The intersection of all A-subgroups U of A such that there is a
minimal A-group M with U = I,(M) is called the radical J(A) of A .
8.41. Proposition. J(A) contains every nilpotent A-subgroup of A. Proof. Suppose not, then there exists some nilpotent A-subgroup U of A and minimal A-group M such that UM # (0). Then Urn # {0}, for some m E M whence Urn = M . Hence, for some uE U, m = urn = u2m = . . . = u'm = 0, for some s, since U is nilpotent: contradiction. 8.42. Proposition. Let M be an A-group. Then I,(M) is a normal A-subgroup of A and I,(M)a L I,(M), for all a E A.
58
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295
Proof. By Prop. 8.31, l,(M) is an A-subgroup of A. Let x E fA(M), a € A, m € M . Then (-a+x+a)m = -am+xm+am = 0. Moreover, (xa)m = x(am) = 0. 8.43. Proposition. Suppose the radical J of A is nilpotent, and Ae is a minimal non-nilpotent A-subgroup of A, e idempotent. Then Je = A e n J , and Je is a proper A-subgroup of Ae which contains every proper A-subgroup of Ae. Furthermore Je is a normal subgroup of Ae, and for every minimal A-group M , there exists some minimal non-nilpotent A-subgroup Ae of A, e idempotent, such that M is A-isomorphic to the A-factor group Ae 1 Je. Proof. By Prop. 8.42, J is a normal A-subgroup of A, and Ja C_ J , for all a € A, thus Je C J , whence Je C A e n J ; but also A e n J C Je since e is an idempotent. Therefore Je = A e n J , hence by the first isomorphism theorem of group theory, Je a Ae. Since J is nilpotent, also A e n J is nilpotent, hence A e n J is properly contained in Ae. Let U be an arbitrary proper A-subgroup of Ae, then U is nilpotent. By Prop. 8.41, U C J whence U G A e n J = Je. Now let M be a minimal A-group, {Ae,, . ., Ae,} a set of minimal nm-nilpotent A-subgroups of A as in Cor. 8.35, and L = IA({e,, . . ., e,}). Then M = AM = Ae,M+ . . . Ae,M+LM, by Prop. 8.36. Since L is nilpotent, L G J whence LM = (0). Therefore there exists an idempotent ej such that AeiM # (0). Thus we can find an element m E M such that Ae,m + {0}, hence Ae,m = M by minimality of M . If we define an A-epimorphism p : Ae, M by p(ae,) = ae,m, then M ”= Ae, I ker p. ker cp is a proper normal A-subgroup of Ae, whence ker p = Je,, i.e. M z Aei/Je,.
.
+
+
8.5. We recall ch. 3, Prop. 5.11, which tells us that every near-ring ( A ; +, -, 0, -) may be regarded as a {-}-multioperator group. Let A = ( A ; +, -, 0, ., 1) be a near-ring with identity and &(A)the group o f units of the semigroup ( A ; 1 ) .
.,
8.51. Proposition. Let A be a d.g. near-ring with identity 1 and minimum B a near-ring epimorphism. If condition for A-subgroups, and :A ker p isfinite, then p&(A) = d(B). +
Proof. By the homomorphism theorem, it suffices to show: If Z is a finite
296
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6
ideal of A and uf I a unit of A I Z, then there exists an element i E I such that u f i is a unit of A. By the minimum condition, there exists a positive integer k = k(v) such that Avk = Awk+', for all v E A . Let u, v E A such that IIU = 1, then A = Avkuk = Awk+luk = Av whence 1 = av, for some a € A . But u = l u = avu = a whence 1 = uv. Since S additively generates A , we can regard A as a monogenic S-group with S-generator 1. By Lemma 3.4, I is a finite S-admissible normal subgroup of the S-group A , and 1 f l is an S-generator of the S-factor group A I I. Since u+Z is a unit of A1 I and S additively generates A , we conclude that u s Z is an S-generator of A l l . Hence by Lemma 6.91, there exists an S-generator u f i of A where iEZ.Thusbyg6.2,l =Cf=,t,(ufi)where t i € A , i = 1, ..., k , whence ~ ( u f i )= 1, for some II E A , i.e. u+i is a unit of A .
9. Miscellaneous
9.1. In this section, we collect various definitions and results from quite divergent branches of mathematics which are used at some place in this book. Not all of these results are well-known, but since some of the proofs are far beyond the scope of this book, we will restrict ourselves to giving references. First we deal with the concepts of category and functor: Let 6 be a class of objects A , B, C , . . ., and suppose that, for any ordered pair (A, B ) of objects in 6, there is a set Mor (A, B ) such that Mor ( A , B ) f l Mor (C, 0)= 4 if ( A , B ) f ( C , 0). For any ordered triple (A, B, C ) in 6, let w(A, B, C) : Mor ( A , B ) x M o r (B, C ) Mor (A, C ) be defined to be a mapping (we will write w(A, B, C )( 5 g) = gf) such that: For any four objects A , B , C , D c G and all fEMor (A,B),gE Mor (B, C ) , h E Mor (C, D),we have h(gf) = ( h g ) f ;for any A E 6, there exists a morphism I, E Mor ( A , A ) such that f l A = f, for any BE 6 and f €Mor ( A , B ) ; and 1,g = g, for any C € 6 and g E Mor (C, A ) . Then the triple consisting of the class 6, the class of all Mor ( A , B), and the class of all w(A, B, C )is called a category which will be denoted briefly by 6. As an example, we may take any variety 8 for 6, and the set of all homomorphisms p : A B for Mor ( A , B ) while w ( A , B, C ) will be the composition of mappings.
-.
-.
19
MISCELLANEOUS
297
Let (B, Mor,, 0,) = B and (9Mor,, , w,) = 9 be two categories, 9a mapping and @(A, B ) :MorE(A, B ) + Mor, ( @ ( A ) , G(B)) a mapping. For any pair ( A , B ) of objects in B we will write @(A, B ) f = If W g f ) = V ( g )G ( f ) for all A , B, C E 6, f E Morg ( A , B), g E MorE (B, C ) , and V ( l A ) = for all A E B, then the pair consisting of the mapping @ and the family @(A, B ) is called a covariant functor from the category 0. into the category 9.
V :0
-t
Wf>.
9.2. Waring's formula. Let Z be the ring of rational integers, y,, . . ., Yk indeterminates and n a positive integer. For every positive integer i, let W?) E Z[y,, . . .,y k ] be the sum of those terms in the expansion of (yl+ .. . + y k ) i into monomials cyy . . . y? for which e,+2e2+ . . . +kek = n. Then the polynomial W , = C;=,[(-l)'n/i]W!") belongs to Z [ y , , . . ., y k ] . If x,, . . ., xk are indeterminates and at = (- 1)' c ( X i , . . . Xi,1 i, < . . . < it), t = 1, . . .,k , then Wn(Ul, . . .,ak) =
+
x;+ . . . +x;.
We refer the reader to R ~ D E[2] I for a detailed proof.
9.3. We continue with a few number-theoretical results that can be found in almost every text on number theory. Let Z be the ring of rational integers, n a positive integer, (n) the princi,pal ideal with basis n, and Z I (n)the corresponding factor ring of order n. The group &(Z I (n))of units of Z 1 (n)is called the group of prime residue classes mod n. We call ~ ( n=) &(Z1 (n)) the Euler pfunction. If n = ab and (a, b) = 1, then q(ab) = p(a) p(b), and if p is a prime and e Z= 1, then ~ ( p "=) p"(1 - 1/p). Let q Z- 1 be a fixed integer. Then any integer k == 0 can be written uniquely as k = a,,qm+a,,~lqm-l+. . . +a, where m == 0 and 0 s u, q, i = 0, 1, . . ., m. This representation is called the q-adic expansion of k and s,(k) = a,,+ . . . f a , is called the sum of digits of this expansion. If k and 1 are arbitrary positive integers, then s,(k+ I ) =ss,(k)+s,(Z), and if we put A(i) = max (A\qa divides i), then EL1 = (k-s,(k))/(q- 1). In contrast to these very elementary results, the famous DIRICHLET theorem is very deep. It is used at the end of ch. 4 and can be stated as follows:
/
1
298
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If a,b are positive rational integers and (a, b) = 1, then the arithmetic progression a n f b , n = 1,2, 3, . . . contains infinitely many primes. 9.4. WEIL[I] has proved the RIEMANN hypothesis for algebraic function fields over finite fields. Two of its consequences are the theorem of LANGand WEILon the number of rational points of varieties in finite fields and a theorem of CARLITZ and WELLSwhich we need here. Let K be a finite field of order q, and the equivalence relation on K X K X K defined by: (bl, b2,b3) (al, a2,a3) if and only if there is 0 # I € K such that bi = la,, i = 1,2,3. Then a special case of the LANG-WEIL theorem is the following
-
9.41. Theorem. Let u E K [ x ,y , z ] be an absolutely irreducible form of degree d > 0 and n the number of non-equivalent solutions in K of the equation u(x, y , z) = 0. Then
where k(d) is a constant which depends only on d. For a proof, we refer to LANGand WEIL[l]. Now we state the CARLITZ-WELLS result:
9.42. Theorem. Let K be a finite jield of order q ; a,, . , ., a,, b,, . . ., b, ajbi, for i # j ; k , k,, . . ., k , non-zero elements of K such that aibj i positive integers and n the number of solutions in K of the system
i = 1, . . ., r,
y? = ai+biXk, of equations in x, yl, . . .,y,. In-ql =s Cql”, for some C z 0.
Then
n = q+ 0(q1”) (q --c -),
i.e.
For a proof, we refer to CARLITZ-WELLS [l]. 9.5. Finally we state some elementary results from analysis : The complex exponential function ez, for z = ip, q being real, satisfies eiP = cos p+ i sin p which is a special case of EULER’S formula and implies DE MOIVRE’S equation (cos p + i sin rp)” = cos np i sin np, for any integer n. We have cos ( n - y ) = -cos p. If, for any differentiable real function f, afdenotes the first derivative, then a cos p = -sin p, and if g is also a differentiable
+
REMARKS A N D COMMENTS
299
function, then a(fog) = @ f o g ) ag. If f is a polynomial in x over the field of real numbers and 0 is the canonical mapping from the ring of polynomials to the ring of polynomial functions, then a(af) = qf’. Remarks and comments
0 1-9. The concept of a multioperator group is due to HIGGINS[l]. K U R O[2] ~ contains the elements of the theory of multioperator groups. 9 5.8 follows ENGSTROM [l], Th. 5.86 was also proved by a different method in the paper by FRIEDand MACRAE[I]. Prop. 6.8 is due to HIGMAN, B. H. NEUMANN and H. NEUMANN [I], Lemma 6.91 is a result of GASCHUTZ [I]. The theory of d.g. near-rings with minimum condition as contained in § 8 has been developed by LAUSCH 171, using some ideas of BEIDLEMAN [I], [2]. Prop. 8.51 was proved by LAUSCH [6].
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AUTHOR INDEX
Abian, A. 72 Aczd, J. 43 Adler, I. 130 Ahmad, S. 222 Aigner, M. 132,220, 221 Allenby, R. B. 72 Andreoli, G. 44, 72
Fadell, A. G. 219 Feichtinger, G. 221 Ferschl, F. 221 Fillmore, J. 222 Foster, A. L. 45, 131 Fried, M. D. 135,217, 218,220,222,299 Frohkh, A. 46, 248 Fryer, K. D. 222 Fujiwara, T. 71
Beidleman, J. C. 299 Berman, G. 130, 131 Bialynicki-Birula, A. 221 Birkhoff, G. 41 Bokut’, L. A. 71 Burke, J. C. 131
Gaschiitz, W. 299 Gerstenhaber, M. 72 Giber, R. W. 42 Goodstein, R. L. 12,220 Gratzer, G. X, 11, 41, 42, 46, 71, 72 Gruber, P. 132 Gwehenberger, G. 222
Cahen, P. M. 43 Carcanague, J. 219 Carlitz, L. 43, 132, 202, 221, 222, 298 Cavior, S. R. 220, 222 Chabert, J. L. 43 Chowla, P. 221 Chowla, S. 222 Clay, J. R. 131, 219 Cohen, S. D. 192 Cohn, P. M. 11, 41, 71 Curtis, C. W. 287
af Hallstrom, G. 220 Hayes, D. 202, 222 Heisler, J. 45 Hermite, C. 190, 191 Higgins, P. J. 299 Higman, G. 299 Hion, Ja. V. 130 Hoang Ki 72 Hosszu, M. 43 Hule, H. 42, 43, 70, 131 Huppert, B. 277, 287
Davenport, H. 222 Dicker, R. M. 130 Dickson, L. E. 132,191,202,221,222 Dirnberger, J. 221 Doi, D. K. 131, 219 Dorge, K. 70 Dueball, F. 43
Isaacs, I. M. 72 Jacobsthal, E. 220
Engstrom, H. T. 220, 299 Erdelyi, M. 71, 72
Kaiser, H. K. 44, 45 313
314
AUTHOR INDEX
Kaloujnine, L. 133 Kamke, E. 249 Kautschitsch, H. 220 Keller, G. 220 Kempner, A. J. 132 Kertbz, A. 44 Knoebel, A. R. 45 Kovacs, L. 42 Kowol, G. 190 Kurbatov, V. A. 222 Kurob, A. G. 41,277, 299 Lagrange, J. L. 43 Lang, S. 298 Lausch, H. 46, 132, 221, 222, 236, 248, 299 Lawkins, W. F. 222 Levi, H. 220 Levin, F. 72 Lewis, D. J. 132, 222 Lidl, R. 44, 222 Litzinger, M. 132 London, D. 222 Lyndon, R. C. 71 MacCluer, C. R. 222 MacRae, R. E. 135, 220, 299 Magill, K. D. 219 Mannos, M. 131 Marczewski, E. 71, 72 Maurer, W. D. 46 Menger, K. 130, 131 Milgram, A. N. 131 Mitsch, H. 44, 130, 132 Mlitz, R. 131 Miiller, W. 112, 222 Mycielski, J. 71 Neumann, B. H. 71, 72, 299 Neumann, H. 42,299 Newton, I. 43 Niederreiter, H. 220, 222 Niven. I. 132 Nobauer, W. 44, 45, 130, 131, 132, 133, 220, 221, 222, 248 Olson, F. R. 220 Ore, 0.219
Philipp, W. 131 Pixley, A. F. 45 Quackenbush, R. W. 45 Redei, L. 43, 45, 222, 261, 269, 287, 297 Reiner, I. 287 Rhodes, J. L. 46 Riha, W. 219 Ritt, J. F. 220 Rosenlicht, M. 221 Rothaus, 0. S. 72 Rousseau, G. 46 Rudeanu, S. 44,72 Ryll-Nardzewski, C. 71 Sain, B. M. 132, 219 Samuel, P. 44, 261, 269 Schiek, H. 72 Schinzel, A. 138 S-chittenhelm,R. 221 Schoniger, W. 220 Schweiger, F. 248 Schweigert, D. 44, 45, 132 Schweizer, B. 130 SchuE, H. K. 41,44,70,72 Schumacher, F. 43, 248 Schur, I. 217, 219, 222 Scognamiglio, G. 43 Scott, S. D. 248 Scott, W. R. 70, 71 Shafaat, A. 41, 44 Shoda, K. 70, 71 Sierpinski, W. 45 Silverman, R. J. 131 Skala, H. 130 Sklar, A. 130 Slominski, J. 41, 70 Solomon, L. 72 Specht, W. 217 Spira, R. 43 Starkov, N. G. 222 Straus, E. 132 Stueben, E. F. 131 Suvak, J. A. 221 Szasz, G. 255 Szele, T. 43, 45
AUTHOR INDEX
Trotter, H. F. 248 van der Waerden, B. L. 43, 70, 261, 269 Warren, L. J. 132 Weglorz, B. 71 Wegner, U. 222 Weil, A. 298 Wells, C. 205, 222, 298 Wenzel, G. H. 71
Werner, H. 45 Whitlock, H. J. 130 Williams, K. S. 222 Zane, B. 220 Zariski, 0. 44, 261, 269 Zassenhaus, H. 130,222 Ziegler, Z. 222
315
SUBJECT INDEX
A-epimorphism 221 A-factor group 291 A-generating set 60 -, minimal 60 A-group 291 -, minimal 291 A-homomorphism 291 A-isomorphism 291 algebra 1 -, absolutely algebraically closed 69 -, algebraically closed 56 -, equationally compact 71 -, finite 1 -, free 9 -, hemiprimal 45 -, mixed algebraically closed 56 -, (m, n)-algebraically closed 71 -, m-polynomially complete 34 -,iocally (S, T)-complete 45 -,non trivial 38 -, of polynomial functions 20 -, polynomially complete 38 -, polynomially hemicomplete 45 -,polynomially incomplete 38 -,polynomially semicomplete 38 -, primal 45 -, semiprimal 45 -, simple 4 -, ( S , T)-complete 45 -,weakly algebraically closed 56 -, weakly mixed algebraically closed 56 algebras, similar 1 algebraic closure 270 algebraic element 269 algebraic equation 47 algebraic inequality 51 algebraic system 47 -, finite 56 -, maximal 52 -, solvable 47 316
-,unrestricted 69
algebraic systems, equivalent 50 alternating group 283 annihilator 290 ascending family 6 associate 261 associated elements 261 A-subgroup 291 -, nil 292 -, nilpotent 292 automorphism 2 -, inner 279 basis of a vector space 289 basis pair of a spectrum 215 bijection 250 block 251 Boolean algebra 256 cardinal 253 cardinality 253 -, finite 253 -, infinite 253 Cartesian power 251 Cartesian product 251 category 296 Cayley-Hamilton equation 289 &basis 61 &dependent set 59 C-derivation family 108 cebyshev polynomial 140 Cebyshev solution I41 centralizing element 281 centre of a group 281 chain 252 - from a to b 5 -from Y to w 10 chain rule 267
SUBJECT iNDEX
characteristic of an integral domain 261 chief factor 281 -,central 281 chief series 281 Chinese remainder theorem 263 &-independent set 59 class 249 - of a nilpotent group 282 - of a partition 251 comaximal ideals 263 commutator 281 commutator subgroup 282 complex product 277 component 25 1 Composition - of functions 75 - of mappings 250 - of polynomials 77 composition epimorphism 78 composition extension 80 composition group 74 -, distributively generated 228 composition homomorphism 78 composition isomorphism 78 composition lattice 75 composition monomorphism 78 composition ring 74 composition series 281 congruence 2 -, separating 48 -, trivial 4 congruence lattice 4 conjugate elements 270 constant 78 coprimal elements 262 corresponding linear factor 143 counterimage 249 covariant functor 297 D-core of a full ideal 96 decomposition homomorphism 82 Dedekind domain 264 degree - of an extension field 269 - of a field element 269 - of a polynomial 27 derivation 108 derivative of a full ideal 94 determinant 288 D-full ideal 95
diagram 250
-,commutative 250
Dickson polynomial 209 difference of sets 249 dihedral group 285 dimension of a vector space 289 direct limit 6 direct product 6 direct sum 266 directly indecomposable 266 Dirichlet’s Theorem 297 distributive element 228 divisor 262 -, non trivial 262 -, proper 262 domain 261 eigenvalue 289 embedding 2 embedding isomorphism 2 enclosing congruence 87 enclosing ideal 92 endomorphism 2 epimorphism 2 -, canonical 3, 6, 21 equipotent sets 253 equivalence class 252 equivalence relation 251 Euclidean domain 264 Euler’s formula 298 Euler‘s p-function 297 exponent - of an extension field 271 - of a group 279 - of a primary ideal 264 extension -, abelian 271 -, algebraic 270 -,cyclic 271 -, inseparable 271 -, least normal 271 -, normal 270 - of an algebra 1 - of a mapping 250 -, separable 271 -,simple 269 -, transcendental 270 extension field 269 -, finite 269
317
318
SUBJECT INDEX
factor algebra 3 factor semigroup 278 family 249 field 261 -,algebraically closed 270 -,finite 272 -, intermediate 272 - of rational functions 272 form 26 -, cubic 26 -, linear 26 -, quadratic 26 formal power series 44 Frattini subgroup 281 free generating set 9 free product 11 - of SOUPS 283 - with amalgamated subgroups 284 free union 11 full congruence 84 full function algebra 20 full ideal 90 full matrix ring 288 full system of representatives 251 function 249 -, compatible 45 -, conservative 45 -, constant 20 -, k-place 251 fundamental theorem of Galois theory 271 Galois group 271 general linear group 288 general Toot 50 generating set 2 Gratzer’s polynomial functions 42 greatest common divisor 262 greatest element 252 group 278 -,abelian 278 -, absolutely algebraically closed 69 -, algebraically closed 66 -, cyclic 279 -,defined by relations 284 -, elementary abelian 279 -,free 283 -,nilpotent 282 - of prime residue classes 297 -,simple 280
-, soluble 281 -,supersoluble 281 -, torsionfree 278
group extension 280 group ring 290
homomorphic image 2 homomorphism 2 homomorphism theorem 3 ideal
-, nilpotent 264 -, primary 263 -, prime 263
- of a multioperator group 257 - of a ring 262 -, trivial 258 ideal basis 263 ideal kernel 257 ideal power semigroup 180 idempotent - of a ring 261 - of a semigroup 278 identity 250 - of a binary operation 255 - of a lattice 256 identity matrix 288 image 249 inclusion monomorphism 280 independent set 72 indeterminate 6 induction 254 -,transfinite 254 inhomogeneous linear group 289 inhomogeneous linear semigroup 289 injection 250 inner direct product - of ~ ~ O U P280 S - of rings 266 integral domain 261 -,integrally closed 261 integral valued polynomial 132 interpolation 43 -, local 43 intersection of sets 249 inverse image 249 inverse - of an element 278 - of a mapping 250
SUBJECT INDEX
irreducible element 262 isomorphism theorem - for algebras, second 5 - for groups, first 280 - for groups, second 280 - for multioperator groups, first 260 - for multioperator groups, second 260 Jacobian determinant 268 Jacobson radical 131 K-automorphism 269 K-equivalent 269 kernel 4 K-isomorphism 269 Kronecker symbol 288 Kurosh subgroup theorem 284 Lagrange resolvent 272 lattice 255 -,complete 256 -. distributive 256 - of A-varieties 17 -. order polynomially complete 45 lattice epimorphism 256 lattice homomorphism 256 -, complete 256 lattice isomorphism 256 law 7 least common multiple 262 least element 252 left annihilator 292 left coset 278 left ideal 262 left identity 255 left inverse 278 left regular element 278 length of a chief series 281 - of an ideal power semigroup 186 - of a poIynomia1 group 223 - of a word 37 length epimorphism 223 Levi‘s theorem 282 linearly dependent 289 linearly independent 289 lower bound 252 -, greatest 255
-
319
lower central series 282 Luroth’s theorem 272 mapping 249
-, bijective 250 -,identical 250 -, injective 250
-, surjective 250
mapping semigroup 282 matrix 287 -,singular 289 -, unimodular 288 matrix representation 290 maximal chain of fields 275 maximal element 252 minimal element 252 minimal polynomial 270 minimum condition 291 mixed algebraic system 51 -,finite 56 -, solvable 51 mixed algebraic systems, equivalent 52 module 289 de Moivre’s equation 298 monomorphism 2 multioperator group 257 multiplicity - of a factor 262 - of an ideal 264 - of a root 266 near-ring 74
-, distributively generated 228
n-generator a-group 279 nilpotent element - of a near-ring 292 - of a ring 261 norm 270 normal form 23 normal form system 23 normal subgroup 279 -,maximal 279 -, minimal 279 -, trivial 279 a-generating set 279 &group 279 -,monogenic 279
320
SUBJECT INDEX
operation 1
-, associative 255
-, binary 255
-,commutative 255 -,distributive 255 -, idempotent 255 -, k-ary 251 -,0-ary 1 -, right superdistributive 73 -,superassociative 73
orbit 283 order - ofanalgebra 1 - of a group element 278 order endomorphism 253 order epimorphism 253 order homomorphism 253 order isomorphism 253 ordinal 254
@-algebraicallydependent 58 parametric word vector 125 -, semigroup generating 125 part of S 119 partial derivation 267 partial derivative 267 partial order 252 -, lexicographic 252 partial product 136 partition 251 P-chain 154 - of Cebyshev polynomials 155 - of powers 155 P-chains -, conjugate over a domain 160 -,conjugate over a field 155 p-chief factor 281 Permutable chain I54 permutation 282 permutation function 120 Permutation group 282 Permutation polynomial 120 - mod D 161 - mod D,strict 161 -,strict 121 Permutation polynoniial function 120 -, strict 121 Permutation Polynomial vector 116 - mod D 161 Permutation spectrum 213
-, strict 213 p-group 278 polynomial 12 -, absolutely irreducible 192 -, characteristic 289 -, exceptional 192 -, indecomposable 135 -,inseparable 271 -, linear 27 -,monic 26 -,monic in x 193 -,normed 136 -, regular 204 -,semimonic in x 193 -, separable 271 polynomials -, conjugate 155 - in two variables, conjugate 193 polynomial algebra 12 polynomial function 20 polynomial function vector 114 polynomial permutation 116 polynomial symbol 41 polynomial vector 1I3 power permutation polynomial 208 power set 249 power solution 140 (P,4)-grOuP 278 primary decomposition 264 prime element 262 prime factor decomposition 136 prime field 269 principal ideal domain 264 product - of cardinals 253 - of ideals 263 - of mappings 250 projection - from a direct product 6 -, i-th 20 q-adic expansion 297 quotient field 261
radical - of an A-group 294 - of a primary ideal 263 range 249
SUBJECT INDEX
rank - of a free group 283 - of a word 6 - of a word, minimal 6 rational function 44 reducible pair 30 reduction of a polynomial 191 regular element 278 regular representation 283 relation -, antisymmetric 252 -, binary 251 -,reflexive 251 -,symmetric 251 -, transitive 251 relatively prime elements 262 representation 290 -,irreducible 290 -, trivial 290 representative 251 residue polynomial ideal 81 - m o d D 101 restriction of a mapping 250 retract 280 s e m a n n hypothesis 298 right coset 278 right ideal 262 right identity 255 right inverse 278 right regular element 278 ring 261 -, commutative 261 -, noetherian 264 - of polynomials 101 with identity 261 R-module 289 -, faithful 290 -,irreducible 290 root 266 root of unity 271 -,primitive 271
-
Schmidt-Rkdei-Iwasawa theorem 282 selector system 73 semidirect factor 280 semidirect product 280 semigroup 277 -,commutative 277 -, partially ordered 278 -,totally ordered 278
321
semilattice 256 semipermutability 220 separable element 271 sequence 250 set 249 -, countable 253 -, empty 249 -, partially ordered 252 -, totally ordered 252 -, well ordered 254 sets, disjoint 249 skew field 261 solution - of an algebraic system 47 - of a mixed algebraic system 51 specialization 125 specialization vector 125 splitting field - of a polynomial in one variable 270 - of a polynomial in two variables 193 - of a polynomial in two variables, minimal 193 square free element 262 (S)-root extension 49 -, greatest 49 . (S)-root extensions, equivalent 49 standard solution 138 -,trivial 140 standard solutions, conjugate 139 (S, T)-completeness defect 45 subalgebra 1 subdirect product 6 subfamily 250 subfield 269 -, maximal 269 -,proper 269 subgroup -,central 281 -, characteristic 279 -, fully invariant 279 -, maximal 278 -, normal 279 -, verbal 285 -, a-admissible 279 subgroups, conjugate 279 sublattice, complete 256 submodule 290 -, trivial 290 bubset 249 -, closed 251 proper 249 -I
322 substitution principle 21 subsystem 250 subword 7 sum - of cardinals 253 - of ideals 258 superassociative system 74 surjection 250 S Y ~ Op-subgroup W 281 symmetric group 282 symmetric semigroup 282 system 250 system of representatives 251 Taylor’s formula 268 total order 252 transcendental element 269 tri-operational algebra 74 type 1 union of sets 249 unique factorization domain 262 unit - ofaring261 - of a semigroup 278 unit ideal 258 upper bound 252 -, least 255
SUBJECT INDEX
%-algebraically dependent set 58 value of a polynomial 21 variety 7 -, degenerate 7 -,semidegenerate 13 %-composition algebra 74 vector 289 vector-space 289 vector system mod P“ I71 %-extension 15
Waring’s formula 297 well ordering 254 well ordering principle 254 word 6 word algebra 7 word problem 23 wreath product 283 - of parametric word vectors 127 - regular 283
zero divisor 261
zero ideal 258 zero matrix 288 zero of a lattice 256 zero vector 173, 289 Zorn’s lemma 257
