Higher Algebra (2 Volumes)

which thel"8 exists at least aile representation of two elements of K [Xl'

... , xn]

L of
9 ~cith g(x l' ... , XII)

=F O. there

c01Tesponds as function value the quotient f(x l ··· .. XII)) of the g(x l • .•• ,x" function values of f and g. Analogous to Theorem 12 the prinriple of substitution is valid here:

Theorem 13. For the field K (J'l'

...•

K(.l·l' ... ,x;) of the rational tUIle/iollS O{

XII) and the

ring

over K ill the sense of analysis H'e have ill virtue of the correspondence defilled in Def. 12 that the analogue to Theorem 12 is 'I:alid except that here any possible nOlZvalidily of Section 2. (8') alLcays implies the nOllvalidity of Section 2, (&). Proo)'.· a) In order to prove that Section :? (c:) is satisfied. it must be shown that the function in the "ense of analysis corresponding by Def. 12 to an element

a quotient L. Now, g.

ifL9

and

tg' are

Xl'"

• ,XII

two representations (satis-

fying these conditions) of

that

r g,

existing by Section 3, (1), it follows by Theorem 12

52

1. I. RLngs, Ftelds, Integral Domains

is also valid. Hence by the assumption of DeI. 12 regarding g I(x l , ••• , xn) r(x), ... , xn) and g it follows further that ( ) = '( . ). g ~, ••. , Xn g Xl"'" Xn b) By going back to the formulae, Section 3, (2) and (3) and applying Theorem 12, it likewise follows that Section 2, (3), (4), (5) are satisfied.

un

follows iust as in the c) The validity of Section 2, proof of Theorem 12 under c); for the precise statement and derivation, see Vol. 3, Section 5, Exer. 1. d) That Section 2, ef') need not be valid is shown by the same example as above. It is obvious that this situation arises if and only if there exists at least one element g in K [Xl' ••• , xn] such that g

=!= 0

but g(x l , ••• , XII)

=

0 for all

Xl"'" XII

in K.

On the one hand, if this is the ease, the element ~ in K(x l , . . . , xu) g has the property that there is no system of elements xl" .. , Xn in K for which it tan be represented as a quotient whose denominator has a function value different from zero; for, by Section 3, (1)

:f

is its most general quotient representation,

where f is an arbitrary element of K [Xl' ... , x,,]. In such a case there ,vould be no function in the sense of analysis corresponding to~, since by Def. 12 the function values are undefined g

for every xl"'" x" in K. On the other hand, if there exists no such g in K [Xl' ... , XI']' then a function value can be associated according to Dei. 12 to the quotient

.Lg for

at least one system

of elements Xl" •• , XII in K. ()n the basis of Theorem 13 the remarks about I [Xl' ... , XII J made in connection with Theorem 12 may accordingly be

5. Formulation 01 the Basic Problem of Algebra

53

applied to K(x 1, · · . , X n), too. Hence our notational lonYemilln shall also be yalid for the elements of K(x 1••••••1',,),

5. Detailed Formulation of the Basic Problem of Algebra By llleans of the concepts explamed in the preceding. we will no\y give an exact fmInulation of the basic problem of algebra specified in the introduction. According to the formulation of the introduction. we obtain an "equation" formed by means of the four elementary operations between known and unknown elements of a field K. if we submit two calculating processe;:; applied to the unkno\vns Xl' ••. , Xn and given (known) elements of K and ask for which systems of elements Xl' ••• , XII in K do both processes yield the same result. This implies that the unknowns Xl' ••.• x" have at first the character of indeterminates, and that the submitted calculating processes are two elements cp and cp' in K(xl' ... , .Tn). The question implied in the "equation" refers then, in a certain analogy to the last developments of Section 4, to the replacement of the indeterminates Xl"" , Xn by systems of elements Xl' ••• , XI! in K and amounis to asking for which of these systems of elements is the equation
is designated by
"'"'

x,,)~


XJI)'

1. I. Rings, Fields, Integral Domains

54

First by applying the principle of substitution to the relation cp - cp' =1jJ the equation cp(Xl' ••. , xn)~

-=-

Before reducing this equation further we must. deal with the following situation. On the one hand, the calculating process leading to 'i' in the sense of the stated problem (which process, according to 'I' = cp - cp', is composed of the two processes given originally and leading to cp and cp/) consists, on closer inspection, of a chain of individual operations. Each operation is an addition, subtraction, multiplication or division of two elements, each of which is either an element of K or one of the XU"'' x ll or is obtained from one of the preceding operations. On the other hand, as 'i' is an element of K(x u " " XII)' its normal representation can be written in the simple form of a quotient

L of g

two elements of K[X j " , " xn]' By the prin-

ciple of substitution the result of substituting a system of elements xl>' . " XII of K is indifff'rent to whether this substitution is made before the process is carried out (in other words, as implied in the sense of the problem, whether we calculate freely from the beginning with the Xl" •• ' X" as elements of K), or whether the substitution in a quotient representation

Lg is

made only after the process has

been carried out. It should be understood, of course, that this statement is valid only if we j'est,'ict olL1"selves to those substitutions fOl' which neithel' the denominat01' g nOl' anyone of the denominators appearing in the successive application of the calculating processes is zel'O, Now, at the outset we are not at all sure that the denominator g is actually different from zero for those systems of elements of K for which the successive denominators of the process were not zero and which would the1'ef01'e be allowable in the sense of the stated pj·oblem. However, it can be shown that among all quotient representations of 'i' there is one (at least) with this property (see Vol. 3, Section 5, Exer. 1). Such a quotient representation 'l'

= 1., g

adapted in a natural way to the problem, is taken as a basis in the following.

55

5. Formulation of the Basic Problem of Algebra

In \'irtue of a quotient l<:pl esemation

1[' 'r

={ g

lof the hind 1ust ,

characterized in detail) the solutlOn of the equation 1,'(.r! ..... .r n ) ~ 0 can now be reduced further. by Section '3 and the

principle of substitution, to the determination of all those solutions of f(:r 1 •••.• xn) = 0 which are also solutions of g(x 1, ••• , xlI)=/=O. Now, since the solutions of the latter inequality are known if those of the equation g(x 1 . . . . . xu) -'- 0 are knowll, the problem reduces itself to considering equations of the form [(4' ... , x... ) --.:. . 0,

r

\,,-here is an element of K (;'t\, .... x,,]. Although, in principle, the common solutions of a number of equations are under control if the solutions of each indiYiduaJ e4.uation are known, it is still useful from a theoretical as well as a practical viewpoint to consider sllch systems of equations as a "'hole. Consequently, we formulate the problem of algebra to be used as our guidfl in the following ,yay: Let K be a field and 1'1· .. ·' t'm elements of K [Xl' .... xn]· There shall be developed l~ methods for obtaining all solutions of the system of equations ff,(x 1, ••• , xn) = 0 (i = 1, ... , m). A systematic complete theory for the solution of this problem in its full generality would require more space than is available in this presentation. Therefore, \ve shall consider here only the following two specializations of the general case: 13

This problem suggests the following two generalizations:

1. The number of equations and unknowns may also be allowed to be

countably infinite; 2. Instead of the field K an integral domain (or even only a ring) may be used as a basis. In recent times both cases have been studied.

56

1. I. Rings, Fields, Integral Domains

1) The elements

11' ... , 1m

are lineal', that is, all coefficients

in the normal rcpresentation (Def. 9 [ 45 ]) of these elemcnts are zero cxccpt at most the II 1 coefficients

+

ao,. .. ,o, al,O, ...,o,··., aO, ... ,O,l ..

In this case we are concerned \..-ith a syslem of equations which can be written in thl') form n

E ail.xk ~ ai

(1)

(i =

1, ... , m),

1:=1

where aik and a, are elements of K.•1.. system of eqnations of the form (1) is called. a system of lineal' equations in K. 2) Let rn = equation of the

11

=

1. \Ve are then concerned with a single

fOI'Ill

E"" a"XC == 0,

*

1:=0

where the ali are elements of K with only finitely many O. \Ve (;an disregard the trivial case where all ai, = 0, since in this case any x in K is a solution of the equation. Consequently,

*

there exists a last a r O. The index r so determined is called the degree of the leftmost element of K [x]. The case r = 0 is also trivial, since the assumption ao =F 0 implies thai there is no x ill K whidl satisfies the equation. Consequently, an equation of the form r

(2)

Ea"xk..:-O

(a,=!=0,

1'>1)

lI:=O

is to be considered. .-\.n equation of the form (2) is called an

algebraic equation Of1'-th degree in K. Tn Vol. 1. Chapters III and IV we will consider the problem 1); in Vol. 2, the subproblem 2).

8U1>-

II. Groups 6. Definition of Groups One speaks of a group if the following postulates are realized:

(a) A set eM of distillct elements is submitted ll"hich contaias either an arbiil'ary finite or all infinite number of elements. See the remarks to Section 1, (a). Here, however, we will not require that 0} have at least two different elements. We designate groups by capital German letters, elements of groups by capital Latin letters.

(b) For any tu'o elemellts A, B of 6), git'ell ill a definite order, one 1"1,le of combinatioll is defined. that is. to et'ery

O1'dered pail' of elements .-:I. B of an element 0 of @S.

6)

there corresponds somehou'

See the remarks to Section 1, (b). We call the rule of combination multiplication, though on occal'ion addition in a domain will also be regarded as a group operation. We write C = AB and call C the product of A and B.

(e) For arbitral'y elemellts of 6) the rLile of combination specified in (b) satisfies the laws: (1) (AB) 0 = A (BU) (associatizoe law); (2) To every ordel'ed pair of elements A, 0 of @S. there exist uniquely determined elemellts Bi and B2 of ® such that ABi = 0 and B2A = 0 (Law of' tlte um'esf1"ict.e.cl and unique rigId; and left diVision).

We note the omission of the commutative law in contrast to its inclusion in both the addition and multiplication postulates of Section 1 for fields. Hence in (2) we must distinguish between right 1 division 1 These designations refer to the position of the quotients Bl,B1•

57

58

1. II. Groups

(determination of B, from AB, tion of B, from B2A

-

=

= C) and left' dIvision (determina-

C). As a consequence the notation.Q cannot

A

be used. On occasion authors write B, = A '"C, B2 = CIA; however, the notation of Theorem 15 is more widely adopted. Naturally, the J'estriction a oF 0 in postulate Section 1, (7) corresponding to (2) is not imposed here; for, since a second rule of combination is not submitted, there is no distributive law [ef. the remark after Section 1, (7)] .

Definition 13. If the postulates specified under (a), (b), (c) are realized tn a set 01, then 01 is called a fJ~'mtl) with r'espect to the rule of combination (b). The number of elements of 01 (whether it be finite or infinite) is called the or(lm' of 01. In pm'ticular, if the commutative law (3) AB=BA is also salisi ied, ® is called an A.belian f/1'01l1). The analogue to Theorem 3 [17] is also valid for groups:

Theorem 14. In every group 01 there exists a uniquely determined element E, called the mtity elmnent or identity of @, 'with the property: AE=EA=A for all A in 0). Pr'oof: For all A, B, ... in @, by (2) there exist in (5) alp-ments E A, E 8 •.. , and FA, F B, .... , such that

AEA=A, BEB=B, .. . FAA =A, FBB=B, .. .

Furthermore, to every pair of elements A, B of ments 0 and D can be chosen such that AC=B, DA r B. By (1) this implies that

(2),

by (2) ele-

1 These designations refer to the position of the quotients B 1,B2•

6. Definition of Groups

59

BEA = (DA)EA = D(AEA ) = DA = B = BEB , F AB = F A(AC) = (F A-4.)C = AC = B = FEB. therefore E.J = E B, FA = F B OIl aCt:Ount of the uniquene.s-" in (:2). Hence EA. E B, ••. are all the same element £: FA. F B • art:' all the same element F; and AE = A. FA = A is yahd for

every A m ~. In particular, nus implIes that for A = F. and £, FE = F, and FE = E. lespectiYely; thelefo1'e. r = 1-'. That E iR determined umquely by the stipulation of the theorem naturally follows from the uniqueness in (2). In regard to division in a group we further prove: Theorem 15. To every element .:1 ulliquely determined element A-I of A, u'lth the property

a

of a group 0:\ there exists of 0).

called the

iUI"f'J".'ile

= A-I A = E. (.AB)-1 = B-1 A-I.

AA-l It is valid that

The elements BI and B2 (right and left quotient of 0 and .:1) specified in (2) are given by Bl = A-IC, B2 =CA-l. Proof: a) By (2) there exist uniquely determined elements ..11 and A2 in ® such that AA.l = ~A =E. By (1) and Theorem 14 it then follows that Al =EA1 = (A2A) Al =-~(A.Al) =~E =~. '1'herefore the elemelH A-I =~ ..11 =.:12 has the property specified in the theorem and is uniquely determined by A. b) From (AB)(B-l A-1) = A (BB-I) A-l = AEA-I

=A.A-l=E and the uniqueness of (AB)-l it follows that (AB)-l= B-1 A-I. c) By (1), Theorem 14, and the proofs under a) the elements Bl = A-l 0, B2 = CA-l satisfy the equations AB1 = C.

60

1. 1I. Groups

B2A =-= 0; therefore, by (~) they are the uniquely determined

solutions of those equations. Analogously to the conventions agreed upon at the end of Section 1, we write ... , A-l, A-I, AO, AI, A2, .,. for ... , A-lA-I, A-I, E, A, AA, ... (integral powers of A). By taking into account the formula (A-I)-I = A., which is valid by Theorem 15, it then follows by the definition of the arithmetical operations in the domain of the integers that AmAn = Am+n, (Am)n = Amn for arbitrary integers -In, n. In particular, Em = E is valid for every integer'l?t.

We formulate the following two theorems especially on account uf later application. 'rhe first is merely a rephrasing of postulate (2). Theorem 16. If A is a fixed element of a group 0), then each of the products AB and BA rtms through all elements of 61, each once, if B does. Theorem 17. The invel'se B-1 runs through all elements of @, each once, if B does. Proof: a) If A is any element of ®, then by Theorem 15 A= (A-1)-I; therefore A is the inverse B-1 of B=A-l. b) From B;-l = B;-l it follows that (B;-l)-l = (B;-I)-l; thorefore Bl = B 2 , again by Theorem 15. III order to prove that a set is a group, eondition (2), namely. Lhat all left and right quotients exist and are uniquely determined, can be replaced by two simpler ('onditions as n consequence of Theorem 18. Under the assumptioll that (a), (b) and (1) are satisfied, postulate (2) is equivalent to the two postulates:

6. Definition

of Groups

61

(2 a) There exists an element E in G) such that AE= A for all A in G) (existence of the 'l'i!Jht-luHul unity element).

(2 b) To every A in 0) there exists an element A-I of 6). such that AA-1 = E (existence of the 1'igllt-lwna iJH'el'se). Proof: a) If (a), lb), (1). C::) are satisfied. then by the preceding (2 a) and (2 h) are also satisfied. b) Let (a), (b), (1), (2a), (2b) be satisfied. Then by l2b) the right-hand inverse of A-I also exists. If it is l_1.-1)-1, then on multiplying the relationA- 1 (A-1)-1.= E on the left hy ,.J it follows from (1), (2 a), P b) that E(A-1)-1 = A. On the one hand, this means that EA = E (A-l)-1 = A. that is. E is als~ I a left-hand unity element; on the other hand. (A-1)-1 = A. therefore A-I A = E. Consequently A-I is also the left-haml inverse of A. By (2 b) as well as (1) this implies that the equations AB1 = 0 alld B2 A = 0 are equh-alent to the relations B 1 = A-I 0 and B2 = OA-1, respectively; namely. the latter are obtained from the former by left and right multiplication. respectively, by A-1 and conversely by A. These equations will therefore be uniquely solved by these expressions Bl' B 2 • Hence (2) is satisfied.

Examples 1. Obviously every ring is an Abelian group with respect to the

ring addition as the group operation. The unity element of this group is the null element of the ring. Furthermore, the elements of a field different from zero also form an Abelian group with respect to the field mUltiplication as the group operation. 2. If the set G; consists only of a single element E and if we set EE = E, then G; is an Abelian group of order 1 with respect to this operation, the so-called identity group or unity g1·OUp. E is its unity element.

1. II. Groups

62

3. If ® contains only two elements E, A and we set EE=E, EA :=AE=A, AA =E, then it is easy to see that ® is an Abelian group of order 2 with respect to this rule of combination. This group arises from the field specified in Section 1, Example 4, if the operation of addition in this field is taken as the group operation and we identify 0 with E and e with A. 4. Let an equilateral triangle be given in space whose three vertices and two surfaces are taken to be distinct. We consider all rotations which take this triangle a
2n;, the other 4n 3 3 about the axis which is perpendicular to the plane of the triangle and passes through the center of the triangle; c) Three rotations B o' B I , B 2, each through the angle :t, about one of the three medians of the triangle. In setting up this set the rotation sense submitted in b) as well as the rotation axes specified in c) may be regarded as fixed in space, that is, not subjected to the rotations of the triangle. Proceeding from a fixed initial position these rotations can be illustrated by their end positions as follows: ~

,

b~~~~b 3'~ '~3~t~,

3

Now, if multiplication in eM is defined as the successive application of the rotations in question, then eM il:. a finite group of order 6 with respect to this multiplication. For, after what has been said, (a), (b) are realized in eM; furthermore, (1) is obviously satisfied; finally, (2 a) and (2 b) are valid, since eM contains the identity

63

7. Subgroups, Congruence Relatwns, IsomorphLSm

rotation E as unity element and to any rotation C the :rotation D generated by reversing the sense of the motion, fOl which CD E is obviously valid. As immedIately implied by the above lllustration, the elements of GJ different from E can be expressed in terms of "4 = A1 and B = Bo as follows: AI = A, A, = A', Bo = B, Bl BA, B, BA.'. Furthermore, the following relations arise from the rule of combination: A3=E, B'=E, AB =B.42; all remaining relations can be deduced from these. The last of these relations shows that GJ is not an Abelian group. The inverses are E-I E, A-I A2, (A')-l A-' = A, B-1 = B, (BA)-I = BA, (BA2)-1 = BA.'. In Vol. 2, Section 4 the application of the group concept pointed out in Example 1 will yield an important msight mto the structure of integral domains and fields. In addltion, at two crucial places (the definition of determinant in Vol. 1, Section 17, and the definition of the Galois group in Vol. 2, Section 15), we will be concerned with finite groups, not necE'ssarily Abelian, whose elements are not at the same time elements of those domains on which we base the solution of the problems of algebra.

=

=

=

=

=

=

7. Subgroups. Congruence Relations. Isomorphism In this section the de\-elopments of Sedion :2 will be applied to groups. As the analogue to Def. 4 [23] we haye:

Definition 14. If the elements of a subset

0

of a group ~

form a gr'oup with respect to the multiplicatIOn defined ill 6). then ~) is called a snbgl'Qnp of ®.

Just as in the case of Theorem 6

[25]

in Section ~. we

prove here

Theorem 19. A subset

if

and only

if the

~ of the group

® is a subgroup of

6)

products as well as the left and right quotients

1. II. Groups

64

of elements of belong to S).

.£5,

as they are defined withm ®, always again

In order to show that the quotients belong to $) it is obviously sufficient by Section 6 to require that 5;:1 contains the unity element E of @ as well as the inverse B-1 of every element B of ;9. For thp case of a finite 0) we even have

Theorem 20. If ill lS a flillte gI'OUP, then the statement ot J.'heorem 19 tS also valid if only the products of elements or ~) (not also the quotients) are taken iI/to account. Pl'OO!: Let A be a fixed element of ~1. Then by Section 6, (2) the elements AB, as well as BA, are all distinct if B runs through the elements of 5;:1, each once. Therefore, the two sets that are generated must each consist of exactly the same elements as are in 5;:1. This implies that all left and right quotients of elements of $) exist in $).

We will carryover Theorems 7 to 9 [26 to 29] and Definitions 5 to 7 [27 to 30] of Section 2 by merely formulating the corresponding theorems and definitions; for thE'- rest. we refe!' to the corresponding prools and expositions of Section 2. Theorem 21. If ,\11' ~)2"" are any [finite 01' in finite 2 11 um-

ber of) subgroups lchatsoever of a group (2), then the intel'section of the sets !Ql' ~)2' ... is also a SUbgl'OUp of ®; this is called the intm'section (j1'QUP or briefly intel'section of the groups S)1' S)2' ... Defillition 15. If ~)1' .);;>2"" are any (finite or infinite number of) subgroups 01 a group ®, then the intersection of all subgroups of ® containing .);;>1' ~2' .•• as subgl'otlps is called the

% Cf. the statement about numbering in footnote 5 to Theorem 7 [26].

7. Subgroups, Congruence Relations, Isomorphism composite

of 5)1,5)2""

65

or also the (fJ'OU]J cOlll]Jo~ed

5)1,5)2' ...

from

=

Definition 16. If an equwalence 1'elatlOll iii a group G) satisftes not only the conditions Sectwl! 2. (u.). W). ('f) but also:

=

=

(1) ..11 A 2, Bl == B2 implies At Bl ..12 B2 • (hen u'e call it a cony,"uellce 1'elatiolt in G) anrl the class!?s ther'eby determined the t'esidue cla.~ses of 0J relatice to It. Theorem 22. Lf'f a congruence relation = be {Jll'tn ill a

group @. In the set G) of residue classes relatil'f! to flus congruence relation let a rule of combi/la/IOII be defliled bll elementwise multiplicatwn. Theil

@

is a group Ifitl< respHt to

this operation; @ is called the residue class (f?,OIlP of GJ relatit'e to the congruence relation =. Theorem 23. The following convention yields an equh'alence relation in the set of all groups: Wrde @ r:s, G)' if and only if 1) @ and @' are equipotent, and 2) the one-to-one correspondence between the elements A., B, ... of G) and A', B', ... of @' can be chosen such that the condition: (2) if A +--+ A', B +--+ B', then AB ~ A' B' is valid. Definition 17. A one-la-one correspondeNce betu'een tleo groups @ and @' with the property (2) lS called all isom01phism between ~ and ~'; G) and ~' themselves are said to be isomorphic. The equivalence r'elatioll Q) r:s, G)' specified in Theorem 23 for gr'oups is called an ismnOTpltis1n, the classes thereby determined the types of the groups.

1. II. Groups

66 Examples

1. Every group contains the following subgroups: a) itself; b) the identity subgroup (Section 6, Example 2) consisting only of its unity element. All other subgroups of @ are called true or proper. 2. All groups of order 1 are isomorphic, that is, there is only one group type of order 1. Within a group Qj there is only one subgroup of order 1; for by Section 6, (2) and Theorem 14 [58] AA = A implies that A = E. Hence it is appropriate to speak of tke identical group (l: and the identity subgroup (l: of Qj. 3. The group specified in Section 6, Example 3, has no proper subgroups. 4. It is easy to confirm that the group Qj of Section 6, Example 4, has the following proper subgroups and no others: a) E,A,A2; b o) E,B; b1 ) E,BA; b 2 ) E,BA2. Let us designate them by ~,5)0' 5)11 5)2' Evidently, the intersection of any two is ~; the composite of any two is Qj. Furthermore, '~o, '~1> '~2 are isomorphic to one another (cf. the remark after Def. 7 I [32] I).

8. Partition of a Group Relative to a Subgroup Equiyalence reiations,and their partitions, which are more general than congruence relations (Def. 16 [65] ) are introduced in group theory, due to the omission of the commutative law. In studying these we will obtain a deeper insight into the nature of congruence relations in groups. They arise as follows: Theorem 24. Let ~ be a subgroup of the group ®. Then each of the following two conventions yields an equivalence relation in the set ®: If Sand S' are elements of ®, then (1 a) S ~S' (Sj) if and only if S

= S' A with A in Sj,

S'(Sj) if and only if S = AS' with A in ~, that is, if the right and left, respectively, quotient of Sand S' belongs to Sj. (1 b) S

(I)

8. Partition of a Group Relative to a Subgroup

67

Proof: Section 2. (a) is valid becam,e E belongs to ~; Section 2. (~), because A and ..1-1 belong to Sj at th~ same time:

Section 2, (y) because A1A~ and ..12 A1 belong to ~"\ with ..1 1' ..1 2' as immediately follows flom Def. 1:1 [6.3] or TheuH'ID 19

[63]. On the basis of Theorem 24 we now define:

Definition 18. The

eqllh'alellce

relatiolts

specified

ill

Theorem 24 are called 1'lgM alld left, respectiLeiy. equlmleJlces relative to s;,; the classes of elements 0/ GJ ihl:reby left and l"iglli. l'espectit'ely. cosets ~or determined the residue clas.'1e.'1p 1'el((til'e to s;,,. the partition of 0) thereby obtailled, the 1';glit and lett, respectil:ely. pm'fition of & 1'elatil'e to s;,; and a complete system of representatil'es for these, a complete riglli and leit, respecth'ely, residue system of & 1'elative t.o s;,. Each of the cosets relative to ~ is generated by an arbitrary element S belonging to it by forming all products SA and AS, respectively, where A runs through the elements of &;>. To indicate their structure we usually denote them by Si;} and ff)S, respectively and (cf. Section 9, Def. 20 [70]). Furthermore, if S1' Sl'" T l • T 2 , • •• 4 is a complete right and left, respectively, residue system of Gj relative to ~, then we write: 0} SiS;> + 8 2 :?) + . " and OJ ~Tl T :?)T2 T .•• for the right and left, respectively, partition of OJ relative to 5;', where the symbol + has the usual significance assigned to it in the theory of sets (i. e., formation of the union of mutually exclusive sets) ,

=

=

3 Translator's note: In the case of groups the designation C08et is used more frequently than residue c1488. The latter nomenclature is sometimes reserved for the special case considered in Vol. 2, Section 2.

4

[26].

Cf, the statement about numbering in footnote 5 to Theorem 7

68

1. 11. Groups

The left as well as the right coset generated by the unity element E, or by any other element of s;;" is obvious~y the group s;;, itself. In the next section, after defining the concept of normal divisor, we will deal in greater detail with the way in which two equivalence relations (1 a) and (1 b) as well as the partitions determined by these equivalence relations are connected.

If ®, and therefore

~,

is a finite group, an especially important conclusion can be drawn from the partition of ® relative to Sj. Thus, from Section 6, (2) we immediately have: Theorem 25. If ® is a finite group of order nand S) is a subgroup of G) of order m, then each of the left as well as the "ight cosets relative to Sj has the same number of elements, . namely, m. If j is the number of cosets, the so-called index of ~ in ®, then n = mj. This means that the order m as well' as the index j of any subgroup of ® is a divisor of the order n of C\). Examples is itself the only coset relative to s;;,; if <M is finite of order n, then m n, j 1. If s;;, ~, then the elements of <M are the cosets relative to Sj; if ® is finite of order n, then m= 1, j=n. 2. In the case of Example 4 considered in Sections 6, 7 we have: mis a subgroup of order 3 and index 2, ~o, s;;,1' ~2 are subgroups of order 2 and index 3. The partitions 1M m+ Bm m+ inB, ® s;;,o + A~o + A2S)o ~o + SjoA + SjOA2 are valid. The right and left equivalences and partitions relative to in must actually coincide, since there are only two cosets, one of which is in; the other must therefore consist of the elements B, BA, BA2 of 1M not belonging to in. However, the right and left equivalences and partitions relative to Sjo are different, their 1. If

s;;, =

<M, then

=

s;;,

=

=

=

=

=

=

9. Normal Divisors, Conjugate Subsets, Factor Groups

69

distinctness being not only the sequence of the classes (which in itself is not to be counted as a distinguishing characteristic). For, A.\)o contains the elements A, BA 2 ; A ~~;-o, the elements A 2, BA; .\)oA, the elements .4., BA; ~OA2, the elements A~, B..P.

9. Normal Divisors. Conjugate Subsets of a Group. Factor Groups As to be seen from the last example of the pre\ious section, . 1 ' (I) (rl th e t wo eqmva ence l'e lahons === and === need not coim:ide in a group ® relative to a subgroup Definition 19. A subgroup

~).

,Ye now define:

~) of the group G) is called a or i1H'at-iaut sllbgJ"()llp of G) if and oilly if the right and left equivalences relative to .'Q are the same, that is, if and only if for every S ill G} til e left cosets 8'0 coincide with the right eosets ~)S.

1tO'J,,})WZ (lil:isO'J'

If .\) is a normal divisor of ~. then the left partition of ~ relative to .\) (except for the undetermined sequence of the classes) is identical with the right partition of ® relative to .\); and, conversely, if the two partitions relative to a subgroup ~ are identical it also follows, according to Section 2, (A), (B), that the left and right congruences relative to S) are the same. Therefore, ~ is a normal divisor of ®. In the case of a normal divisor ~) of ~ we naturally no lon~r need to characterize the left and right equivalences relative to ~, nor the left and right cosets relative to ~, by the designation "left" and "right." If ® is an Abelian group, then by Section 6, (3) and

Theorem 24 [66] the left and right equivalences are surely the same. Consequently, Tbeorem 26. If eM is an Abeliall group, then every subgroup Sj of ® is a normal divisor of G}. In order to obtain a more penetrating insight into the significance of the concept of normal di't'isor, we will introduce

1. ll. Groups

70

another equivalence relation of group theory which refers to the sct of all subsets of a group. For the present, however, we will not connect this equivalence relation with DeI. 19. We will first generalize to arbitrary subsets of @ the designations S.~) and ,'QS, given ahove for the left and right, respectively, congruence classes of S relative to ,'Q, that is, for the set of all elements SA and AS, respectively, where A runs through the elements of \i. This will simplify the notation and the conclusions to be derived. Definition 20. Le-t me and 91 be subsets of the group @. Then by 9R91 we will understand that subset of @ which consists oj' all pZ'oducts AB, where A runs through the elements of we, B those of SJt Since multiplication in @ satisfies the associative law Section 6, (1), we immediately have Theorem 27. The "elementwise multiplication" defined in Def. 20 in the set of all subsets of a group associative law.

m satisfies

the

This implies the validity of Section 6, (a), (b), (1) in the set of all subsets of ~. Yet, in case @ =!= (\; this set of subsets is not a group with respect to the operation of Def. 20. For, if m = E and ~n consists of E and A =!= E, then no subset £ exists such that 9(~ = me. Let T, S be arbitrary elements of (Ii. Then, by Theorem 27 T9JIS has the uniquely determined sense (T9J1)S = T(WlS). Furthermore, we have the relation T'(T'JJIS)S' = (T'T)'JJI(SS'), which will frequently be used in the following.

We now prove: Theorem 28. Let and let us put

me

and

mz '" 9)(' if

ill @. Then, this is subsets of @.

all

my be two subsets of a group

0)

and only if SJ]l' = S-l weB for an 8 equivalence ~'elation in the set of all

9. Normal Divisors, Conjugate Subsets, Factor Groups

71

Proof: Section 2, (0;) is satisfied, since E-l i)RE = iJJC; Section 2, (~), since iJJ1' 8- 1 iJJ1S implies iJJC SiJJC'S-1 (S-I)-1 WI'S-I: Section 2, (y), since ror = S-1 SJRS, m" = ']'-1 9)l'T implies

=

=

=

£m" = ']'-1(S-1rolS)T = (']'-l,s-l)iJJC(ST) = (Srr l m(ST).

The formation of WC' formation of W1 by S.

=- 8-1 WCS

from

\II(

is called the

trans-

On the hasis of Theorem 28 we define: Definition 21. If WI "" iJJY in the sense of Theorem 28. thell iJJc and mr are called cou./ugate sub.~ets of G$. The clas.~es determined by this equivalence relation in the set of all subsets of ® are called the classes o/' conjugate s'Ubsets of m. Regarding these classes the following special cases may be cited: a) The classes of conju,2;ate elements of G$, that is. those dasses which arc generated by a subset containing only one element A of m. Such a class, therefore, consists of the totality of elements 8-1 AS, whore S runs through the elements of m. One such class is generated by the unity element E and contains only the unity element itself. If ill is Abelian, then S-1 AS S-1 SA EA A; therefore all classes of conjugate elements of GI contain only one element. If OJ is not Abelian, there is at least one such class which contains more than one element, since AS == SA implies S-l AS =!= A.

=

=

=

h) The classes of conjugate subgroups of 0..\, that is. those classes which are generated by a subgroup ~j of ®. Such a class, therefore, consists of the totality of subsets 8-1 S')8. where 8 runs through the elements of G$. The designation conjugate subgroups is justified by the follovring theorem:

72

1. ll. Groups

Theorem 29. The conjugate subsets S-l ~)S of a subgroup S) of a g1'OUp ® a/'e again subgroups of ®. Moreover, they are isomorphic to ~) and therefore also to one another. Proof: By Theorem 27 we have (S-l~S) (S-1~)8)= 8- 1 (8)~) 8 = 8-1 8)8, since ~)~) =~) is obvious from the group property, Section 6, (b) of S) and by Theorem 16 [60] . According to Def. 20 this relation says that all products of elements of 8-1 8)8 again belong to 8- 1 ~)S. Now, the element E = 8-1 ES belongs to s- t ~8, since E is contained in ~; also, if A' = 8-1 A8 belongs to 8-1 .'i)8 so does A'-l = 8-1 A-1 8, since if A is contained in &) so is A-I. ConsequentIy 8-1 &)8 is a subgroup of ® (Theorem 19 [63] ). Let a correspondence be set up between the element" of ~) and 8-1 &)8 by the convention A +---+ 8-1 A8. Then Section 2, (0) and Section 2, (8) are satisfied, since to every A in 8) there corresponds a unique element of 8-1 ~)8; Section 2, (0'), since every A' in 8-1 &)8 by definition of this subset can be represented as 8-1 AS with A in ~; and Section 2, (a'), since 8-1 A 18 = 8-1 A 2S implies through left and right multiplication by S and S--t, respectively, that A1 = A 2 • Finally, (S-lA I S) (S-IA 2 S) = S-1 (A 1A 2 )S; therefore, condition (2) of Theorem 23 [65] is also satisfied. Consequently, we actually haye that ~) C'..;J S-1 &)S. In regard to the distinctness of conjugate subgroups, we prove: Theorem 30. Tu:o subgroups 8-1 &)S and T-l ~T which are conjugates of the subgroup 8) of the group G:i are surely identical if S(l) T(&)). Hence, if T 1 , T 2 , ••• is a complete left l'esidue system of ® relative to &), then the conjugate subgroups

9. Normal Dwisors, ConJugate Subsets, Factor Groups

73

TIl ~)T1' T"2l S)T2' ... at most a1'e different from one anothel"' that IS. every '1ubgroup of ill conjugate to ~) is zdentical u'ith Ol1e of these. (I)

Proof: If S = T(Sj), WhICh means that S =AT with A in Sj, then, 8-1 SjS = (AT)-1 Sj (AT) = (T-l A-I) ~) (AT) = T-1 (A-l SjA) T = T-1 f;:>T. because by Theorem 16 [60l we obvi(lusly ha,-e A-1 SjA = A-l(:s)A) = ...1-1 :s) = Sj. ,Ve will now relate the concept of normal di'l.:isor to the special classes of conjugate subsets specified in a) and b). We do so by the following two theorems. either of which could also have been used to define this concept. Theorem 31. A subgroup Sj of the group OJ is a normal subgroup of @ if and only if it is identical ~L'ith all its conjugate subgroups, that is, if the class of Sj in the sellse of Def. 21 consists only of S) itself· Proof: The relations SSj = SjS occurring 1ll Def. 19 for the elements S of (}j are equivalent to the relations 1» = S-1 ~>~. This follows through left multiplication by S-1 and S, respecth"ely. Theorem 32. A subgroup Sj of @ is a normal divisor of if and only if it is a union of classes of conjugate elements of @, that is, if all elements of (}j conjugate to A belong to ~ whenever A itself does. P1'ooi: a) Let Sj be a normal divisor of @. Then by Theorem 31 S-1 SjS =~) for all S in ®. Hence Sj contains all elements S-1 AS, where S belongs to @ and A to :s); that is, all elements of ~ coniugate to A belong to the subgroup if A does. b) Conversely, if the latter is the case, then S-1 ~8 and SSjS-1 are contained in ~) for every S in 6>. Through left and right multiplication, respectively, by S we obtain that :oS is @

1. ll. Groups

74

contained in Sf;) and S~ in ~IS; therefore, that ~1 is a normal divisor of (5$.

S~

=

.\)8. This means

By Theorem 31 a subgroup .'Q is also characterized as a normal divisor of ® if ~ is preserved with respect to transformation by all elements S of @ (cf. the remark to Theorem 28). This is why the designation invariant 8Vbgroup was used in Def. 19.

.A subgroup of ® is not always a normal divisor of ®. However, two normal divisors can be derived from any subgroup by the following theorem.

Theorem 33. If Sj is a subgroup of ®, then the intersection and the composite of all subgroups conjugate to Sj are normal divisors of (5$. Proof: a) If A occurs in the intersection ~ of all subgroups of ® conjugate to Sj, that is, in all S-lf;)S, where S runs through the group ®, then for any fixed T in ® T-IAT occurs in all T-l(S-lSjS)T = (ST)--lf;) (ST). By Theorem 16 [60] we again have, if T is any fixed element of ®, that these are all bubgroups of ® coniugate to Sj. Hence by Theorem 32 ~ is a normal divisor of ®. b) If St is the specified composite, then Si contains all S-lSjS. This means, as above, that T-1StT and TStT-l contain all S-lSjS, too. Hence these subgroups of ® are the kind which should be used according to Def. 15 [64] to determine St by the formation of intersections. Therefore Sl is contained in T-1StT and TStT-l for every T in ®. This implies, as in the proof to Theorem 32 under b), that St is a normal divisor of ®. The most important property of normal divisors, which will be of fundamental Significance in our application of group theory in Vol. 2, Section 17, is to be found in the close connection between the normal divisors of a group (5$ and the con-

75

9. Normal DIVisors, Conjugate Subsets, Factor Groups

gruence relations possible in G). In this regard the following two theorems are valid: Theorem 3-1. If $) is a normal divisor of 6), then the (simultaneoHsly j'ight and left) equivalence relative to S) is a congruence r'elation in ®. Proof: By DeL 19, .'l')8 = 8&j for e\'ery S in G). This implies by DeL 20 and Theorem 27 that

(1) (~S)(~T) =Sj(SSj)T =Sj(SjS)T =(~~)(ST) =~(ST). Consequently, all products of elements from two CDsers ;);is. ;~~T relative to ~) belong to one and the same coset, i. e. ,~~(ST). relative to $). This means that Section 7, (1) is satisfied for the equivalence relative to S). Theorem 3;). Every cOllort/ence ndatiol/ ill G) is ideilticI1.1 the (simultaneollsly right and left) equivalence relatil'e to a defillite normal div180/' ~) of G). J) IS the totality of elell/ellis of G) 1chich m'e cOllgnlellt to the llnity element E. that is, the

~cith

coset determiHed by E Wider the cOllgruence relation. Proof': a) The set ~) of elements of ill congruent to E is, first of all, a subgroup of 63. ,\~e will show that the conditions of Theorem 19 [63] (cf. the remark adjoined to it) are satis--1, E B implies fied. First, by (1) in Def. 16 [65] E E = AB; secondly, E = E; thirdly, by (1) in Def. 16 E = A, ..1.-1 A-I implies A-1 E. h) If A = B, then by (1) in Def.16 AB-l = E and B-1 A = E.

=

=

Hence by Theorem 24

=

=

[66]

A

(r)

B(.,») and A

(Il B($).

Con-

versely, by (1) in Def. 16 it follows from each of these relations that A == lJ. Conseqnently, the right and left equi,alences rela.tive to the subgroup ,.;;, hoth coincide with our congruence, therefore with one another. This implies the statement of the theorem.

1. II. Groups

76

By the last two theorems the only congruence relations that exist in a group @ are the equivalence relations in the sense of Def. 18 [67] relative to normal divisors S) of @. In particular, the equivalence relations last specified are not congruence relations if .'0 is not a normal divisor of 6).

We can also express the conclusion of Theorem 22 [65] as follows:

Theorem 36. If

®, then the (simultaneously left and right) eosets of ® relative to S) form a group S") is a normal divisor of

@ th1"Ough elementwise multiplication, the 1"esidue class group

of ® relative to S). @ is also called the facto'}' g1"Onp of 0) relative to ~) and 'Loe write ~ = f.JJ/~·

'fo operate with the elements \lS, ~T, ... of the factor group proceed in accordance with rule (1). For finite OJ Theorem 25 [68] says that the order of 0)/.,{) is equal to the index of S) in ®. Finally, we obviously have @/~)

Theorem 37. If of @, then

@/~)

@

is an Abelian group and S) a subgroup

is also Abelian. Examples

1. The improper subgroups Cl: and @ of @ are always normal divisors of @. For their factor groups we have @/Cl: QQ @ and @/('iJQQCl:.

2. For the group @ considered in Sections 6, 7, Example 4, as already stressed in the statements about @ in Section 8, Example 2, the subgroups S)o' S)1> S)2 are conjugate to one another and are not normal divisors; however, the subgroup In is a normal divisor. This can also be seen by forming the classes of conjugate elements of @. It follows from the formulae of Section 6, Example 4, that these classes have the following composition: a) E; c) B,

b) A,

BA

A2 = B-1 AB; BA2 A-l BA.

= A-2 BA2,

=

9. Normal Divisors, Conjugate Subsets, Factor Groups

Hence the classes of the conjugates of .\)0'

.\),

= A-2~)oA2,

77

.\10 are: '\)2

=A-1'~)oA,

whereas 91 is the union of the classes a) and b). The factor gl'OUP @;9c is Abelian of order 2 (cf. Seetion 6, Example 3). 3. The Abelian group @ of rational numbers = 0 with respect to ordinary multiplication has, for instance, the following subgroups: the group \1,5 of positive rational numbers and the group U of all those rational numbers which can be represented as quotients of odd integers. The following partitions of ili l·elative to \3 and n, respectively, are obviously valid: QS= $+ (-1) $, @l=U+2U+22 11+ ... + 2-1 U T 2-2U+ "', therefore @/\1,5 is finite of order 2, but ilii 11 is infinite.s 4. The Abelian group ili of integers with respect to ordinary addition has, for example, the subgroup .\) consisting of all even numbers. The partition Q)=~)+1.'Q

is valid so that @f~) is again finite of order 2." In Vol. 2, Section 2 we will discuss in detail these and analogously formed subgroups of @ as well as their factor groups.

5 In the case of U that part of the fundamental theorem of arithmetic dealing with the unique factorization of rational numbers into powers of prime numbers is assumed as known in the case of the prime number 2. In Vol. 2, Section 1 this will be dealt with systematically. 6 Here, too, we assume that Theorem 13 of Vol. 2, Section 1 is valid relative to the prime number 2, namely, that every integer g can be uniquely expressed in the form g = 2q + r, where q and r are integers and 0;:;;; r < 2. .\) then consists of the g with ,. =:= 0; 1.\), of the g with r = 1. - Naturally 1.'Q indicates here that 1 IS to be added to the elements of Sj.

III. Linear Algebra without Determinants 10. Linear forms. Vectors. lUatrices Let K be an arbitrary field. \Ve will u:;;e this field as the ground field of linear algebl"a in the sense of Section 5. t1) throughout the remainder of Yol. 1. To simplify our terminology we make the convention that in Chapters III and IV all elements designated by a, b, c, n, ~, .: 'with or without indices shall be elements of K, even though this ma~ not always be expressly stated. Likewise, x,"," :("" shall be elements of K whenever we pass over to the concept of function in the sense of analysis.

Before taking up the main problem. as formulated in Section 5, (1), we will introduce in this section some Loncepts. which in themsehTes are not essential, whose application. however, will extraordinarily simplIfy the following de\'elopments both from a notational and descriptiye point of ,iew.

a) Linear Forms First, we intruduce a special nomenclature for the integral rational functions of Xl"'" XI! which appear on the left sides of the system of equations Section 5, (1) under consideration. Definition 22. An element of K[x1 , ..• , l'lI] u'hose normal

/'epresentation is or also

n

:£ a"x" is called a liueW'j'01'm of

&=1 line«(1' and homoyeneolls

in

Xl' ....

"1' .•. ,

XII

XII'

The significance of linear has already been explained in Section 5 in the ~ase of (1); form or homogeneous shall mean that in the ncrmal representation the coefficient which is designated by a o" .. , 0 in Theorem 11 is zero. The expression lineal' form by itself will always stand for a linear form of the n indeterminates ~'l"" ,:1'" unless qualified otherwise by the context.

79

80

1. Ill. Linear Algebra without Determinants

The follmving two definitions are very important for all further considerations: Definition 23. A linear form f is called a linea~'combin(-f,tion of or Unem'll/ (Zependent on the linear j'orms f 1 , · · · , fm if there exist c1' · · . ,

Cm

to be linem'ly independent of Hence, the null form 0 "'"

m

=i=1 ~ ed. f1"'" tm'

such that t

n ~ OXk

Otherwise, f is said

is certainly a linear combination

~=l

of every system 11 , ••• , 1m of linear forms. To show this we merely have to choose ct , ••• , em = O. On taking this into account we further define:

Definition 24. The linear forms fl"'" fm are said to be if there exist c1" ' " cm, which are not all

linearly dependent m

zero, such that 1: cil. = O. Otherwise,

11"'" fm are said to be

0=1

linearly independent.

=

In pm·ticular (m 1) every linear form f =F 0 is line1arly independent, whereas the form 0 is linearly dependent.

The two entirely distinct concepts of linear'ly dependent on (linearly independent of) and linearly dependent (linear'ly independent) introduced in Def. 23 and 24 are connected by a relation ,yhose proof is so simple that it is left to the reader: 1 Theorem 38. a) If f is linearly dependent all, fl'" .,fm, then f, fl"'" mare linea1'ly dependent. b) If f is linearly independent of fl,· .. ,fm and f 1,· .. ,fm ar'e linearly iI/dependent, then f, f l' ••• , fm are linearly independent.

r

1 Above all it should be made clear that the field property [Section 1, (7)] plays an essential role in this proof, so that even these facts, upon which the following exposition is based, are not generally valid in integral domains. (Cf. point 2 in footnote 13 [55] to Section 5.)

10. Linear Forms, Vectors, Matrices

81

a') If f, fl' ... , fill are linearly dependent alld there is a

°

1'elatio1l cf + Cdl + ... + cmfm = such that 1 has a coefficient C::f: (in particular. this is the case if f1"" "1' are lhlearlp independe71i), then f is lillearly dependent 011 fl" ... f/ll'

°

b') If

I, fl' ... , fill are linearly independent, thell f is linearly

independent of fl"'" fm' aild fl.· ... fm are also lillearly independent. The successive application of b') yields the following two mutually implicative infer!lnces: Theol'em 39. If fl' , .. , fm, J'm+l' ... , fm-t are linearly independent, then fl"'" f m are also. If fl .. ·· . are linearly dependent, then fl' ... , fm, fmh' ... , f",-l are also. Analogous to this, the following two mutually implicative inferences are also valid: .. A+I Theorem 40. Let ti Ea 1: au: x", g, EaI aU. x"

'm

".1

1=1

(i=l, ... , m). Then, if f l' •.. , fm are linearly independ ellt so also are 91- .... Bm and if 91"'" Bm are linearly dependent so also are fl- .... fm' Proof: Let K[Xl' ... , Xn] = Kn. Then the 9, can be described as those elements (linear but not forms) of Kn [x n -\• •••• .Tn-i) whose function values are the elements fi of Kn if the indeterminates x"+ l "'" X.. +l are replaced by the system (0.... ,0). Consequently, by the principle of substitution it follows that the til

r.lation I

c, g, = 0 fo!: the function values also satisfy the rela-

,=1 tion I c,1i =0. m

,-1

Furthermore, we have: Theorem 41. If f 1, ••• , fm are linearly independent, then each of their linear combinations can be linearly composed from these in only one way. If f1'" .,fm are linearly dependent. then each

82

1. Ill. Linear Algebra without Determinants

of their linea?' combinationB can be linea1"ly composed in at least two different ways. Proof: a) If fl"'" fll! are linearly independent, then m

m

m

2: t; Ii = Z Ct Ii implies E (c. - ci) Ii = 0 . Hence, by Def. 24 \=1

\=1

c, -

\=1

ci =0, that is,

c,=ci

for i = 1, ... , m. m

b) If f1" •• , 1m are linearly dependent, that is, Z

Ci

Ii =

0,

i=l

wherein at least one c,

=t= 0, then

m

t=

E di Ii

also implies that

i=1

m

t = \=1 E (d, + C,) Ii,

wherein at least one d,

=t= d, + cl .

By Theorem 41, for example, the special system of n linear n

forms

Xl>""

x n • from which every linear form

~ akxk

can be

k=1

linearly composed, is linearly independent. For, by Theorem 11 this representation is unique, since it is the normal representation.

Finally we have: Theorem 42. If 91, ... ,91 are linear combinations of fl .... , fm, then every linear combination of U1"'" Ul is also a linear

combination of fl"'"' fm· m

Proof:

l

Yh = 1: ckdi (k = 1, .. .,1) and g = Z Ck Yk imply i=1

g=

k=1

1 [ek (.E CHti)] .E [cl Clr Cki) ti]'

h=1

.=1

<=1

k=l

In the following we will frequently apply the rule used above regarding the interchange of the orde?' of summation, which is based on the addition postulatE'S, Section 1, (1), (3), (5). Due to its validity the parentheses and brackets may he omitted without misunderstanding in the case of such operations.

b) Vectors According to the conception underlying the construction of K[x l , · · " , xn] from K in Section 4, c) and d). the linear forms

10. Linear Forms, Vectors, 111 atTices

83

n

E alexk' in particular, are formally nothing else but systems k=1

(a l , •.. , an) of elements which are subiect to the distinctness criterion and rules of combination given in Section 4, (1 a) to (3 a), where the abbreviating designations Xl' ... , Xn were introduced for the special systems (e, 0, ... , 0), ... , (0 .... , O. e). "Without using these abbreviations the laws. Section 4. (1 a) to (3 a), insofar as they refer to linear forms (which are now to be our only obiects of study) and to the elements of the ground field, amount to the following: (1 ) (aI" • " an) = (al, .. "a;,) if and only if, al: = al, for k = (2) (~"'" an) (bI , ' , " b,,) = (~ bl , ' , ., a" -[- b,,), (3) a(~". " a,,) = (a~, ' , ., aa... ).

+

+

1, , , "

n,

Now, in linear algebra we have to handle not only systems of coefficients of linear forms but also systems of n elements of the ground field which are to be substituted for the indeterminates

xn in the linear forms, In doing so we frequently have to characterize these substitution systems by (1) as well as admit them in the right-hand formations of (2) and (3). This end could actually be attained merely by formally thinking of the substitution systems as systems of coefficients of linear forms. With respect to such an interpretation the substitution systems would be distinct if their associated linear forms were distinct. and the right-hand formations in (2) and (3) would be effected by carrying out the rules of combination on the left with these linear fOTIns. However, such a way of thinking would complicate the terminology unduly, Moreover it could be very confusing, since in the case of the expression linear form we usually think of the possibility of l'eplacing the indeterminates by elements of the ground field, which is naturally not the case for the "auxiliary linear forms" last mentioned, It is therefore advisable to devise a brief terminology which will make it possible to apply the formal rules (1) to (3) in another way, Xl" •• ,

1. Ill. Linear Algebra without Determinants

84

Definition 25. Systems of n elements satisfying the distinctness criterion and the rules of combination (1) to (3) are called n-te'l'1netl vecW'l'S. IV e designate them by the small German letters corresponding to their terms. For example, (a l , ••• , an) is designated by a; (ail'"'" ain), by ai' etc. Unless otherwise specified, a vector is always understood as being n-termed. By (2) the subtraction of vectors must naturally also be defined without restriction and uniquely. This can actually be done by tae formula analogous to (2): (a l , . · · , a n ) - (b1,.·.,b n ) =(al-b!>' .. ,a,,-bn ),

either by assuming that the rule of combination (2) satisfies postu. lates Section 1, (1), (3), (6), or simply in virtue of the formal identity with linear forms. The vector (0, ... , 0) corresponding to the null form and playing the role of the null vector may again be designated by 0.

Since vectors and linear forms are formally the same, we can also think of the concepts introduced in Def. 23, 24 as explained for vectors; consequently, the analogues to Theorems 38 to 42 must also be valid when formulated in terms of vectors. Written out in detail the statements "a is linearly dependent on a I , ••• , am" and .• a1, ••• , am are linearly dependent" mean according to Def. 23, 24 that the relations m

(4) ~ Ciaik = i=1

m

ak

and (5) ~ Cjaik ;=1

== 0, respectively,

for k = 1, ... , n

are valid, where the latter contains at least one Cj =1= 0. The special n linearly independent vectors (e, 0, ... , 0), ... , (0, ... , 0, e), which correspond to the linear forms Xli"" X n' are also called the n unit vectors and designated by el>"" en' There •

eXIsts then for any vector tt the representation

n ~ ak ek k=!

in terms of

these unit vectors. Through these representations we are naturally brought hack (except for the difference of notation between ek and xl) to the standpoint of linear forms.

10. Linear Forms, Vectors, Matrices

85

Hitherto the conyentions set up for vectors have been formally the same as those in the case of linear forms. l\ow. however, a convention is made which has no counterpart from the standpoint of linear forms. Definition 26. The inner p1'oduct aD of

tiCO

vectors a aild 0

n

is defined as the element l: al:"k' .1:=1

In contrast to (3) in the inner Rroduct both factors are vectors, while the result of forming the inner product is not a vector but an element of the ground field. - In particular, we have e for k= k'} aet = at, eke.!:' = { 0 for k =f= k' , aO = O.

Theorem 43. The illner product of vectors is an operation satisfying the rules

ao

= ba, e(ab) = (ea) b =a (co), (a + 0) c =;ac+ bc.

By DeI. 25, 26 this immediately follows from postulates Section 1, (1) to (5). P1'00f:

The successive application of the last of these rules naturally gives rise to the still more general formula

XaiC' Xao)c =.=1 (\=1 fl

m

On writing this out in detail we obtain £ £

tZi.!:Ct .,=1i=1

=

m

"

£ X au<:"1: • i=l.1:=l

This is the rule mentioned in connection with Theorem 42 regarding the interchange of the order of summation. We will make essential use (Theorem 44) of it.

The inner product makes no use of the field property (Section 1, (7)] of the ground domain K. Hence the last developments are also vali d for vectors over the integral domain K[Xl' ..• , XnV 2 To such vectors there would then correspond linear forms 1(;1>".' ;n) of the integral domain Kn[l;w··, ;,J over Kii = K[x 1, • • ,x ll ] ; however, for our purposes we have no need of this interpretation (cf. the exposition before Def. 25).

86

1. Ill. Linear Algebra without Determinants

'l'he Yertor 7: of the indeterminates is the only vector of this kind we need. By using this vector we can also designate 11 linear form !(x j ,. • • , XII) by f(,£). In doing so we agree to mterpret ~ as a vector of the ground domain, and to use the symbols and in accord with the conventions given in the footnote to Theorem 12

=

=

[50).

By Def. 26 any linear fOIm f(x l •

••• ,

x ll )

== E" aJoxJo also

has

Jo~l

=

the representation f(t) ol as an inner product. On the basis of the formulae of Theorem 43 this representation leads to an extraordinarily simple rule for operations with the function values of a linear form. \Ve especially stress the following fact in connection with the remark after Theorem 43: Theorem 44. It fCl) is a linear form. then for a linear comm

.-1

bination ~ = E Ci!. of 61' ... , 6m the formula m

f("b) = E c,/{!.)

'-1

is t'alid, that is, the function l,alue of f for a linear combination of m tlectors is the corresponding linear combination of' the m function values for these vectors. Proof: If fCl)

== 0;. then by Theorem 43

m

m

m

«,1

.... 1

,-1

m

m

feE Ci~i) = a(E ~i"b') = E a(c,"bi) = E Ci(0"b.) = E Cite!,). '~1

(-I

In our remarks before Def. 25 we had in mind particularly the facts and calculations which occur in Theorem 44 and its proof. In view of Theorem 44 the appropriateness of the introduction of vectors becomes obvious. Finally, in line with the remarks of Section 4, we point out that in the case of linear forms the formal function concept of algebra coineides with the function concept in the sense of analysis.

87

10. Lmear Forms, Vectors, Matrices

Namely, on passing over to the linear forms the condition in question, Section 2, (e'), is satisfied in the sense of analysis by Theorem 45. For linear forms f and U over K, the relation

nrJ == 0(6) is equivalent to the ?'elation fW = 0(6) fOl' all 6: in K. Proof: a) It is clear that the former relation implies the latter. b) If f(6) O(l;) for all 6 in K, then in particular f(ef..) g(cf..} (k = 1, ... , n). Now, fCr.) U6 implies f(c,,) = ael. = al.' Hence the corresponding coefficients of f and U must be the same, that is f(tJ U(I;).

=

=

=

=

c) Matrices In the systems of coefficients from the left sides of systems of linear equations we encounter systems of m n-termed vectors. which we can think of as combined into an (m n)-termed vector. These (m 1t)-termed vectors can also be thought of as formed by combining, in some other way, n m-termed vectors, where each vector consists of the coefficients of a fixed indeterminate. It is advisable to introduce a special terminology for these two combining processes as well as, com'ersely, for the decomposition of an (m n)-tel'med vector in one of these two ways. For this purpose we define:

Definition 27. An (m n)-termed vector, insofar as it is thought of as being formed by combining m n-termed or 11 mtermed vertors into a rectangular orray

(~~l. a

a~~ ), briefly Caw)

: .' : .. : .. .

m1 • .•••. amn

U i: ~ ~ :::),

is called an (m, nj-rmved matrix. The horizontal and vertical component vectors al'e called the'l"()l()s and emu/mns, respectively. of the matrix. We also designate matrices by the capital letters c01"responding to their terms.

88

1. Ill. Linear Algebra without Determinants

For example, (a"..> is designated by A;

(a"..>,

by A,... The

(?n, n)-rowed null matrix corresponding to the (mn)-termed null

VEretor can again be designated by O. - We will also omit the addendum (m, n)-rowed when the numbers m and n follow from the context. According to Def. 27 the concept (m, n)-rowed matrix is narrower than the concept (mn)-termed vector, just as the concept "the integer l mn decomposed into factors" is narrower than the conl.'E'pt "the integer l." In the case of matrices the distinctness criterion and the rules of combination, namely, the analogues to

=

(1), (2), (3)

(l')(lZtk) = (air.) if and only if (2 ')(lZik) + (bir.) = (aik+ bik) (3 ')a(aik) = (aa~k)

a.r. = aik}(i -1

m)

k=l""'n ' -

, ... ,

do not actually bring this out. The restriction lies rather in the rectangular array imposed on the (mn)-termed vector, whereby the terms standing in each row or column are conceptually combined.

As a rule the index i is used for numbering the rows; k, for the columns. Hence, if an (m, n)-rowed matrix Calk) is submitted, then (akJ stands for the (n, m)-rowed matrix

(al~~::. :. a~~~) n •. ·• mn

obtained by interchanging the rows and columns; for, here it is the first index which enumerates the columns, the second index the rows.

Definition 28. 'l'he (11, m)-rowed matrix (ak.) obtained from an (m, n)-rowed matrix (a,k) by interchanging its rows and columns is called the tJranspose of (aik)' If (a;.,~) is denoted by A, then (aka is denoted by A'. Besides the rules of combination (2') and (3'), in the l>o-called matrix calculus there is still another extremely important rule of combination which is used to obtain a new matrix from two given

11. Nonhomogeneous and Homogeneous Systems oj Linear Equations

89

matrices. We are referring to the so-called matrix product which can be defined, however, only within the set of all matTices (not only those with fixed 1n and n). The formation of matrix product actually contains the formation of inner vector product as a special case; 3 however, it does not simply amount to the inner product of the vectors corresponding to the matrices. Even though the so-called matrix calculus in qnestion plays a very important role in lineal' algebra and especially contributes far more than the vector notation towards the clear statement of the developments and results of linear algebra, we still must refrain from investigating it further in the limited space at our disposal. For such a treatment we refer to more extensive texts. I

11. Nonhomogeneous and Homogeneous

Systems of Linear Equations We next begin the systematie treatment of the problem formulated in Section 5, (1). Besides investigating the proper system of linear equations

(J)

f,(Xr, ... , xn) =

"

).'ou;{CJ:~ tlt

J:=1

(i = 1, ... , m)

we consider independently the system of linear equations

,.

(II)

f.t:11., •.. , x,,) == Eaa,xJ:"":'" 0

(i = 1, ... , m).

k=l

(H) is said to be the system of homogeneous equations associated to (J), whereas (.1) is said to be nonhomogeneous. The fact that (J) and (H) have been assigned opposite names implies that we do not wish to regard, as it seems natural 3 From the standpoint of the product of matrices the two factors of the inner vector product are a (1, n)-rowed and an (n, i)-rowed matrix and the result a (l,l)-rowed matrix. The latter is formally, but not conceptually, an element of the ground field. 4 See also Vol. 3, Section 10, Exer. 3, as well as many other exercises in the following sections of Vol. 1 and Vol. 2.

90

1. ill. Linear Algebra without Determinants

to do at first, the ::;peGial case of (J), where all at = 0, as formally identical with (H). On the contrary, in order that the results to be deduced may be neatly formulated, we make the following convention, which technically distinguishes (H) from this special case of (J): the null vector 6=0, which is always a solution of (H) (the so-called identical solution), shall not be counted as a solution of (H). In particular, therefore, we say that (H) cannot be solved if the null vector is its only solution. However, we regard the null vector as an admissible solution for the mentioned special case of (J). By the matrix of (J) and (H) we understand the (m, n)rowed matrix: A = (all,).

By means of the concepts developed in Section 10 the existence of (J) and (H) for a system Xl>"" x" can also be expressed as follows: The linear combination of the columns of A with the coefficients Xl>' •• , X 1/ yields the vector a formed by the right-hand sides of (J), or the null vector, respectively. By this convention the solvabiiity of (H) becomes equivalent, in particular, to the linear dependen-ce of the columns of A. (Cf. formulae, Section 10, (4), (5), [84] , which refer, however, in this sense to the system of equations with the matrix A'.) The problem of linear algebra Section 5, (1) can accordingly be formulated in this case as follows: Find all possible linear combinations of a given system of vectors which yield a particular vector; in particular, find all linear dependences of a given system of vectors. It is important to keep this interpretation in mind, since it will be frequently used in the following.

Finally, besides (J) and (H) we will also have to take into consideration the transpose system of homogeneous equations formed with the transpose matrix A' = (a,,,):

91

11. Nonhomogeneous and Homogeneous Systems of Lmear Equations

(H') f,,(x;., ... , x;")

= E'" aikxi

.

i=l

°"

(k = 1, ... , n).

Originally (J) alone was to be investigated. The mdependent consideration of (H) is iustified by the following theorem:

Theorem 46. If (J) can be solved, then all remaining solutions t..r of (J) are obtailled by adding to allY fixed solution 6(~) of (J) all solutions t.H of (H): therefore. they have the form t.J=&~O)

+ tH.

Proof: a) By Theorem 44

[86] it follows from f,(6jO») = a,.

that f.(t.(J) + tH) = f,(t.)))) + /'ct.}]) =a, + therefore, all t.J = &.)0) + t.ll are solutions of (J).

f.(t.H) =0,

°

=a,:

b) If 1.(6J) = at, Ut.5°») = at, then it likewise follows that

6(J») = 0. Therefore, if 6J

*

then 6J - 6}0) =!ll is a solution of (H). This means that any solution 61 of (J) distinct from 6)0) can actually be \yritten as 1) = 65°) + r.H. Theorem 46 reduces the problem of linear algebra to the following two subproblems: J) Determination of olle solution of (J): H) Determination of all solutions of (H). On the one hand, in the case of eH) we have f,(6J -

t.)o),

Theorem 47. If h· .. , 65 are solutions o{(H) . .so also are all their' linear combinations. Proof: By Theorem

implieb

.

,

a

l86]

.

f,(!])=O

Ii (1: cl Il) = E Cdit'6i) = E c]o = 0 i=1

i=1

i=l

Ci=l, .... s)

(i = 1, ... , m).

On writing out the proposed system of equations it actually turns out that it is the coefficient akl which is in the i-th row and k-th column and not all., as one might believe at a first glance. It is worth while in the following to visualize the equations of (H') as written side by sioe with each individual equation running downwq,rds, to show the generation of (H') from the matrix A.

92

1. lIf. Linear Algebra without Determinants

We now state Definition 29. A system 61"'" 61 of linearly independent solutions ot' (H) is called a system of fu,ndamental solutions of (H) if every solution of (H) is a linear combination of

61' .... , 6S' By Theorem -17 and DeL 29 the totality of solutions of (H) is identical with the totality of linear combinations of a system of fundamental solutions of (H); moreover by rl'heorem 41 lSi] the rep.resentations of the solutions in terms of the fundamental solutions are unique. Consequently, problem H) reduces to the problem of determining a system of fundamental solutions of (H). Whether such a system actually exists in all cases remains undecided for the time being; 6 it will be decided affirmatively only later (Theorem 50 [104]). If (H) cannot be solved, that is, if there exist no linearly independent solutions of (H) (cf. the remark to Def. 24 [80] ), then we say, so as to conform with our convention, that (H) has a system of fundamental solutions consisting of 0 solutions. That the latter case actually occurs can be illustrated by the system of equations all x 1 = 0, where m = n = 1 with all =!= o.

On the other hand, in the case of J) the following necessary condition for solvability exists. Later (Theorem 49 [102]) it will be shown that this condition is also sufficient.

'rheorem 48. A necessar'y condition for' the solvability of m

(J) is the following: If any linear dependence :E xiti = 0 exists i=l

between the linear forms on the left, then the corresponding m

relation :E Xiai =() must eLiso be valid for the right sides . • =1

c It could very well be that to any system of linearly independent solutions of (H) there would still exist another solution of (H) linearly independent of these.

11. Nonhomogeneous and Homogeneous Systems oj Linear Equations

93

Proof: Let us assume that (J) can be solved. Then a \'ector I: exists such that the function values are f,(x) = az• InHence by III • the principle of substitution 1::xi h = 0 implies that 1: xi.a. = 0 is also ,'alid. '=1 .=1 ~

Now, by Section 10 a linear dependence

m

:2x:/, = 0 of the linear

l=l

forms

It is equivalent to the linear dependence

t

m

:8

x; a, = 0

between

1=1

the corresponding vectors a" i. e., the rows of.4.. Since this amounts to saying that ;( is a solution of (H'), we have Corollary 1. The condition of Theorem 48 can also be expressed as follows: For every solution ;( of (H'), 6'a 0 must be valid. If we assume the existence of a system of fundamental solutions of (H'), then by Theorem 43 [85] we aIM have in addition:

=

Corollary 2. The condition of Theorem 48 can also be expressed as follows: For the solutions 6[ of a system of fnndamental solutions of (H') 6; a 0 must be valid. These corollaries justify the introduction of (H') within the range of our considerations, since they show that (J) is not only related to (H) as shown in Theorem 46 but also to (H').

=

The problems J) and H) under discussion can next be subdivided into a theoretical and a pl'actical part as formulated below: J th ) to prove that the necessary condition given in Theorem 48 for the existence at a solution of (J) is also sufficient: J pr) to determine a solution of (J) if it can be solved: H th ) to p1'ove the existence of a system of fundamental solutions of (H) i H pr ) to detel'inine a system of fundamental solutions of (H). We have seen that in the case of (J) we are only concerned vl'ith one solution; while in the case of (H), with all solutions. Hence the investigations and results relative to (H) will naturally take up more space in the following than those relative to (J), which was

94

1. III. Linear Algebra without Determinants

originally to be our only object of study. Due to hmitations space we will not always be able to express according to Theorems and 48 the meaning of the results to be found for (H) in terms (J). The reader should not fail to realize this meaning clearly every individual case.

of 46 of in

12. The Toeplitz Process The theoretical parts J th ) and H th ) of the problems specified at the end of the pre\-ious section can be completely solved by a process given by Toeplitz,7 which we will explain in this section. For this purpose \ve set up the following three definitions:

be

Definition 30. Two systems of linear equations are said to if they have the same totality of solutions.

eqnit'alellt

This is naturally an equivalence relation in the sense of Section 2, (I), However, we have no need of the partition thereby determined. This partition is significant only in the calculus of matrices, where the equivalence itself can be described in terms of relations between the matrices of the systems of equations.

Defiuition 31. The length of a linear form f(x l , . . • , xn) is defilled as 1) the numbel' 0 if f -= 0; 2) the index of its last coefficient (taking the natural ordering Xl' ..• ,x" as a basis)

different from zero if f =f:: O. This implies that the length k of a linear form f(xl>"" x,) is always 0 ;;;;; k ;;;;; n, and that k ::::: 0 is equivalent to f ::::: O.

Definition 32. .d linear form of length k

~

1 is said to be

n01wwlized if its "'-th coefficient is e. 1 Cf. O. Toeplitz, Ober die Aufl6sung unendlich vieler linearer Gleichungen mit unendlich vielen Unbekannten (On the Solution of an Infinite Numbe1' oj Linear Equations with an Infinite Number of Unknowns), Ren
95

12. The Toeplitz Process

The objective ot the Toeplitz process is to find a system of equations equivalent to (J) or (H) whose totality of solutions is especially simple to surYeY. It will be assumed that (J) satisfies the necessary condition for solvability of Theorem 48 [92]. Since this condition is always valid if a = 0, there will be no loss of generality, from the point of view of the process mentioned, if (H) is regarded as the special case a = 0 of (J). Consequently, it is sufficient to subject (J) alone to the process. We will again consider (H) separately from (J) only for the theorems dealing with the solvability and the solutions of systems of linear equations to be derived from this process in the next section.

The proof of the following existence statement forms the content of the ToeplHz process.

Theorem of Toeplitz. If (J) satisfies the necessary conditioll [92] and not all linear fOl'ms f. 0, then there is a system of equations equivalent to (J)

fol' solvability of Theorem 48

=

gl (-l;)

(Jo)

{

== bll:l1. + .. + b1,k, -1 Xk,-1 + Xk, 8

..:

0

hI

~~(~~ .• •b~ ~o ~ ~ ~:,~ -:l.~k~-:: :- .X:: .. .b~ gr{~) = b'l Xl + .... + b"kr-l Xkr-1 + X};, ...:.. b, :

:

:

•

0

0

••

0

:

••

with the properties: (1) r is the maximal numbe7' of linearly independent f;,

(2) 01"'" Or are linearly independent, (3) 01"'" Or are normalized, (4) The lenoths k 1, •• 0, kr of 01'"

.,g,.

satisfy the relations

8 In order to bring out the structure of (J o) clearly we avoid, here and at similar places in the following, using the summation symbol even though it may be possible to do so. - The awkward notation for the case k! = 1 cannot be readily avoided. We make the convention that a sum U t + ... + Uk-t for k = 1 is to be considered as nonexistent, and eventually replaced by 0; also, that statements about U t , •• 0, u1.-t for k = 1 are to be considered as void.

96

1. Ill. Lmear Algebra wuhout Determinants

1 :::;; le1 < k2 <

... < k~ <

n.

Proof: The proof will be handled in two .steps. In the first step, which is of a preliminary nature, we will prove the existence of a system of equations (Jo) equivalent to (J) with the properties (1), (2). In the second step we will give the proper existence proof for a system of equations (Jo) equivalent to (Jo), and consequently also to (J), with all properties (1) to (4)

specified in the theorem. Flrst Step

The fust step consists in the application of the following lemma to the linear forms fl on the left in (J): Lemma 1. To any [finite 9 ] system of linear forms f" u'hich are not all 0, there exists a maximal system of linearly independent f" that is, a subsystem containing the greatest possible number of linearly independent f" the so-called maximal number of linearly independent f,.

Proof: Let fl' ... , fin be a system of linear forms in which not all f, = 0. We now apply the fact that every non-empty set of natural numbels ;;;;; m contains a greatest number, to the set of numbers obtained by counting all linearly independent subsystems of the system f l' . . . , fm. Since by assumption this set of natural numbers ;;;;; m is not empty - (according to Def. 24 [SO) the number 1 surely belongs to this set) - there must be a greatest possible number in this set. The corresponding subsystem must contain the greatest possible number of linearly independent forms. This completes the proof of Lemma 1. It should be noted that in order to formulate later results uniformly it is convenient to say in the case excluded in Lemma 1,

• Cf. Section 13, Lemma 2 [104].

12. The Toeplitz Process

97

where all I, = 0, just as with respect to Def. 29 [92] , that the maximal number of linearly independent I, is 0, and that a maximal system of 0 linearly independent I, exists.

Lemma 1 will be used to prove the equivalence of the system (Yo), set up below, to (J). This applicatlOn depends on the following lemma, which we formulate especially because of its frequent use in the following. Lemma 2. The totality of linear combinations of a [finite lO ] system of linear forms fit which are not all 0, is identical with the totality of linear combinations of a maximal system at linearly independent f,. Proof: Let f'l' .... f'r be a maximal system of linearly independent linear forms of the system fl'" ., fm. By Theorem 42 [82] the proof rests on the following facts:

a) Every f" (v = 1, ... , r) is a linear combination of f1' ... , 1m. - This is clear since f" occurs among fl' .. " fm. b)

Every

f'l' ... , f'r' - I f

f, (i = 1, ... , m) is a linear combination of i is an ii' then, as above, this is clear. If i is

not an i" then f" f, 1, ... , f, r is a subsystem of f1' ... , fm with a greater number ~. + 1 of linearly independent f, than the maximal number r. Therefore, f" " 1, ... , f, r are linearly dependent. By a') in Theorem 38 [801 the linear independence of i'l' ... , fir then implies that f, is linearly dependent Oll

f'l" .. , f'r' This completes the proof of Lemma 2. The hypothesis of the Theorem of Toeplitz enables us to apply Lemma 1 to the linear forms f 1 , ••• , fm on the left in (J)_ Now, the totality of solutions of (J) is independent of the ordering of the equations; in other words, any rearrangement 10

Cf. Section 13, Lemma 2 [104].

98

1. Ill. Linear Algebra without Determinants

of the equations of (J) takes (J) into an equivalent system of equations. Hence we are permitted to assume without loss of generalityll that fl' ... , i, is a maximal system of linearly independent f,. From the linear forms of this maximal system we form the system of equations (i = 1, ... , r).

ft.(x) . a.

(Jo) has then the properties (1) and (2) of the Theorem of Toeplitz. Moreover, (Jo) is equivalent to (J). This is clear if l' = m, since in this case (Jo) is the same as (J). Let l' < m. Then, on the one hand, any solution of (J) is a fortiori also a solution of (Jo)' On the other hand, by Lemma 2 there exist relations of the form r

(5)

fj=Eciit.

(j=r+l, ... ,m),

>=1 and by Theorem 48 [92] , which by the assumption of the

Theorem of Toeplitz is valid for (J), there exist at the same time the relations r

(6)

ai = E ci,ai i=l

(i

=

r + 1, ... , m).

By the principle of substitution (5) and (6) yield that any solution of (Jo) is also conversely a solution of (J).

Second Step The basic idea of the second step consists in the repeated application of the following lemma to subsystems of linear forms on the left in (Jo): 11

This is only a matter of notation.

12. The Toeplitz Process

99

Lemma 3. Every system l2 of linearly independellt lil/ear to'l'lUS contains one and only one normalized lineal' combination of smallest positive length. Proof: Let f1' ... , tv be linearly independent linear forms. Ko\\', in any set of non-negative integers ",hidl does not consist of the number 0 alone there is a uniquely determined smallest positive number. \Ve apply this fact to the set of lengths of all linear combinations of fl'" ., f." Since this set of non-negative integers does not consist of the number 0 alone. due to the linear independence of the fl' .... f." there is a linear combination

g = cd! + ... + cpt~ =bl~ + ... + b.l;x.l; (b.l;-;- 0) of fl"'" f., with smallest positive length !.:. and k is hereby uniquely determined. If 'oYe set e_ c, bE =-g=g, =-=Ci, =-=bz,then blc b" bk 9 =Cdl + c.t. =bl~ + ... + b"-lX"_l + X1& is a normalized linear combination of f l' .'.• , f., of smallest positive length. Kow, if g' = CUI + ... + c;tv=b~~ + ... + bLIXlc_l + xk is another one of this kind (by the remarks, its length must again be f,;), then by subtraction it follows that

+ ...

g-g'

= (cl-cDtl + ... + (cv-c;) tv = (b1 - b~) ~ + ... + (blc- 1 - bk- 1) Xk_l'

Therefore g - g' is a linear combination of fl.· .. ' f., with a length < k. Due to the choice of f,; this length must be O. Therefore g - g' = 0, that is. g = g'. This completes the proof of Lemma 3. 12 Naturally "finite". Infinite systems of linearly independent linear forms are not defined. Moreover, due to Lemma 1 [103] to be given in Section 13 they cannot on principle be meaningfully defined so as to make sense.

100

1. Ill. Linear Algebra without Determinants

By means of Lemma 3 we next prove the following lemma. The crucial point of the second step consists in its application to the linear forms on the left in (Jo)' Lemma 4. Let fl" .. ,11' be a system of linearly independellt linear forms of Xl'" . , Xn · It i~ possible to order f1"'" fl' so that there exists a system of linear forms gl"'" Ur with the properties (2), (3), (4) of the Theorem of Toeplitz which are

the following linear combinations of the fl' ... , f,· Y1 = cult + ................ + C1~t, { (7) Y2 = CUll + ... + C2• r- l Ir-1

g" = Crltl~ .......····"·····"··"""··" (8 )

Clr , cZ.f_l, ••• , C,1

=F O.

Proof: We apply Lemma 3 in r steps, 1) to r), to certain r subsystems of the system f l' •.• , fl' This is possible, since by Theorem 39 [81] any such subsystem is linearly independent. 1) a) Let

+

91 = Cu fl + ... + Clr I, === Qn:li. + ... + bt.1:,-1 X1;1-1 X1;, be the normalized linear combination of f l' . . . ,1r of smallest positive length k 1• Naturally 1.:1 ;;;;:; 1. b) Since gl =F 0, it is naturally linearly independent. r) Since Ul =1= 0, not all Cli= O. \Ve can assume, therefore, that the ordering of the f 1, ••• , 1r is chosen so that cl1' =F O.

g2 = en!l + ... + c2.r-d,_1 == bn:li. + ... + bZ,k._1 xk,_l + :1'1;, be the normalized linear combination of f l' • . . , 1"-1 of smallest positive length k 2 • Then k2> k 1• First of all, since the linear combinatrons of fl"'" fr-1 are contained among those of fl' ... , [,-1' Tn the definition of k1 precludes that le2 < le 1• But. if k2 = kl were valid, then g2 could be regarded as a lineal' combination of f l , ... , 11' identical with Ul on account of the 2) a) Let

12. The Toeplitz Process

101

unique determination of 91' By Theorem 41 [81) and the fact thai cIr =!= 0, this result would contradict the linear indepE'ndence of f t , ••. , fJ' b) {J1' 92 are linearly independent. For otherwise, on account of 1) b) and a') in Theorem 38 (80), g: would be a linear combination of gt' which obviously contradicts k2 > k 1 . c) Since g2 =!= 0, not all C2• = O. Since the assumption made under 1) c) regarding the ordering refers only to the position of flO we can assume the ordering of fl' .•• , ' r _ 1 as chosen so that c2 , 1-1 =!= O. 3) to (r -1)) As above. r) a) As above. Naturally, leT;;;;; n. b) ,\s above. c) Since gT =!= 0, Crt =!= O. The system g1' ... , gr resulting from 1) to r) has the properties (2) to (4), (7), (8). This completes the proof of Lemma 4. We next apply Lemma 4 to the linearly independent system of linear forms f l' .•. , fT on the left in (Io). As above there is no loss in generality if we assume13 that the equations of (Io) are arranged in accord with Lemma 4. 'We now adopt the notation introduced in Lemma 4; in addition we set hI =eua,. C17t!r { (9 ) hz = en a,. + C2, '-::~ a,=-l h, = Crla!. ....... ..' ................

+ ............... + + ...

and form the system of equations with these bi , •.• , b, as the right sides and g1' ... , gr as the left sides, namely, (Jo) gi(6)=bi (i=l, ...,r). By the preceding this system has then the properties (1) to (4) of tho Theorem of Toeplitz. 13

This is only a matter of notation.

102

1. III. Linear Algebra without Determinants

In order to complete the proof we only have to show in addition that (Jo) is equivalent to ( 0 ) and consequently also to (J). ?\ow, on the one hand, any solution ;\; of (Jo) is also a solution of (Jo)' since by (7). (9) and the l)rinciple of substitution f;(?;) = ai implies 9;(;\;) = bi (i = 1, ... ,1'), On the other hand, by (7). (9) and the principle of substitution 9i(;\;) = bi (i = 1.... , 1') implies, above all, that

" .. + Clrir("{.) =Cll~ + ....... + Clr ar + ., + c2,r-11,-1(;\;) =C21~ + ... + ......... C2,r_1 ar_l .....

CuM;\;) + .. C21 ft(;\;)

Crl 11 (6) ..·........ ·· ............ ·

=

C'l ~ ........ .

By (8), however, it can be deduced by proceeding successh'ely upwards through this system that fie;\;) = ai (i = 1, .. , r). Hence every solution of (Jo) is conversely also a solntion of (Jo)' This completes the proof of the Theorem of Toeplitz. For the application to (H) in the next section we further add:

Corollary. The (H) associated to (J) is equivalent to the (Ho) associated to (Jo).

Proof: If a = 0, then by (9) we also have 1) = 0, that is. (Jo) goes over into its associated homogeneous system (Ho)'

13. Solvability and Solutions of Systems of Linear Equations vVe will now use the Theorem of Toeplitz to solve the two problems Jth ) and HtlJ formulated at the end of Section 11, as well as to deduce some results concerning the solutions of (H), going beyond Hth)' J th ) is solved through the proof of the following theorem: Theorem 49. For the solvability of (J) the necessary condition of Theorem 48 [92) is also sufficient.

13. Solvability

0/ Systems 0/ Linear Equations

Proof: a) The statement is trihal if all

103

Ii =

U. i\amely, since in this case the special linear dependences fi = 0 exist between the 1;, the condition of Theorem 48 says that it must also be true Lhat all ai = O. This means, however, that every ! is a solution of (J). b) If not all fi = 0, then by the Theorem of Toeplitz [95] it is sufficient to show that the (Jo), appearing there, always has a solution /;. This, however, is easily established by means of condition (4) of that theorem. First of all, Xl" ••• , X",_l 14 are arbitrarily chosen; then xk, can be (uniquely) determined such that the first equation exists no matter how the remaining XI.may be chosen. Next, X"',+l,"" XI.-,-l 14 are arbitrarily chosen; then x", can he (uniquely) determined such that not only the first but also the second equation exists no matter how the remaining x" may be chosen, etc. After the determination of

x"r the

XIc r+ 1, ••• , Xn

are still to be chosen arbitrarily. Then the /; so determined is a solution of (J 0)' therefore also of (.J). For the solution of lith) we first prove the following two lemmas: Lemma 1. There are at most n lineal'ly independent lin ear 101'ms in n indeterminates. Proof: If fl"'" fT is a system of )' linearly independent

linear forms, then by Section 12, Lemma 4 [100] there exist, in particular, r integers k 1, ••• , kr satisfying condition (4) of the rrheorem of Toeplitz. Obviously the number of these can be at most n. Therefore r ;;;; n. That the maximal number n is actually attained is shown by the n linearly independent linear forms Xl" •• ,x".

=

= +

14 In regard to the cases kl 1, k2 kl 1, . .. cf. footnote 8 [95] in Section 12 to the Theorem of Toeplitz.

104

1. Ill. Linear Algebra without Determinants

Lemma 1 immediately yields the following facts regarding the number r occurring in the Theorem of Toeplitz. This number is important for the later development of the theory of (J), (H), (H').

Corollal'Y. The maximal number r of linearly independent forms am01!g 1m linear forms of n indeterminat~s \ satisfies )the rows of an (m, n)-rowed matrzx J not only the self-evident inequality 0 ~ r ~ m but also the in-

equality 0 ~ r ~ n. Lemma 2. Lemmas 1

(96)

and 2 I [97] of Section 12 are also valid for an infinite system 0/ linear f070ms which do not

all vanish, that is, for any such system there exists a subsystern 15 consisting of the greatest possible numbe7' of linearly independent linear forms, and the totality of all linear combinations of the forms of sitch a subsystem is identical with the totality of all linear combinations (each combination containing only a finite number15 ) of linear forms of the entire system. Proof: By Lemma 1 proved above the number of linearly

independent linear forms in a subsystem picked at random from the infinite system of linear forms is ~ n. The proof of Section 12, Lemma 1, can then be completely transferred if the number 1'1 is used instead of the number m appearing there. Furthermore, the proof of Section 12, Lemma 2 immediately carries over. By the methods then at our command we would not have been able to prove Lemma 2, since it would not have been established whether the number of linearly independent linear forms in every subsystem is bounded (cf. footnote 6 [92] after Def. 29).

H th ) itself can now be solved by the proof of the following theorem: Theorem 50. There exists a system of fundamental solutions of (H). 15

Cf. footnote 12 [99] to Section 12, Lemma 3.

13. Solvability of Systems of Linear Equations

105

Proof: a) If (ll) cannot be soh-ed, the theorem is trhial (cf. the remark regarding this after Def. 29 19:2J).

b) If (ll) can be solved, then the totality of its solutions forms a finite or infinite system of vectors which do not all vanish. By Lemma 2, expressed in terms of vectors, there exists a linearly independent subsystem of this system such that all vectors of the entire system are linear combinations of the vectors of this subsystem. By DeL 29 [92] this means that there exists a system of fundamental solutions of (ll). By Lemma 1 we can, for the present, only say that the number s of fundamental solutions of (ll) satisfies the inequalit;\r o ~ s ;;:;; n. But, in the meantime. we are unable to Ray which of these possible values s has and, abo\-e all, whether this number s is an illVal'iant for all systems of fundamental solutions of (ll). That the latter is actually the case CUll now be inferred from Lemma 1:

Theorem 51. All systems of fundamental solutions of (H) have exactly the same number of solutions. Proof: a) If (H) cannot be solved, the theorem is trivial.

b) If ell) can be solved, let 61"'" 65 and 6;"'" ~;. be two systems of fundamental solutions of (H). Then there exist representations of 6, in terms of the 6' of the form Ie

¥

r. =E Ci.l:!~

(i = 1, ... , s).

1>=1

Now, if s' < s, then by Lemma 1 the s rows of the (s, s')-rowed matrix (e,k) would be linearly dependent as s'-termed vectors; consequently, the corresponding linear dependence would also be valid for the 6,' By Def. 29 [92] , however, the 6i are linearly independent. Hence s' < s cannot be valid. Similarly, the representations of the 6' in terms of the 6, imply that s < s' cannot be valid. Consequently, s' = s.

"

106

1. Ill. Linear Algebra without Determinants

'rhe conclusion of Theorem 51 already goes beyond problem B th ) even though its proof did not take advantage of the 'rheorem of Toeplitz to the same extent as the solution of J tIJ ) in the proof of Theorem 49. Through the full use of the Theorem of 'roeplitz we now can go beyond Theorem 51 and actually determine the exact number of fundamental solutions of (R). We prove Theorem 52. The number of fundamental solutions of (B) is n - r, where r is the maximal number of linearly independent fl' i. e., the maximal number of linearly independent rows of the (m, n)-rowed matrix of (B). Proof: The number n - r will turn out to he the number of indeterminates xl' ... , Xn different from the xk" ••• , x k ,. specified in the Theorem of Toeplitz. a) If all f, = 0, therefore r = 0 (cf. the remark to Section 12, Lemma 1 [96) ). the theorem is trivial. For. in this case all ~ are solutions of (H), therefore by DeL 29 [92] the n - 0 = n linearly independent unit vectors e1••••• en form a system of fundamental solutions of (li). b)lf not all fl 0, then by the Theorem of Toeplitz (and its corollary [102]) it is sufficient to show that the number of fundamental solutions of the system (Ro) of homogeneous equations associated to the (Jo), appearing there, is n - r. We can now form the totality of solutions of (Ro) on the basis of condition (4) of the Theorem of Toeplitz [95] in exactly the same way as we have indicated a solution of (Jo) in the proof 1.0 Theorem 49. 1) First of all, in order to satisfy the first equation of (Ro) with respect to any given x t = ;1' ••.• Xk,-l ;k,_116 we must

=

=

=

=

16 Regarding the eases kl 1, k2 k1 + 1, . .. cf. footnote 8 [9S] in Section 12 to the Theorem of Toeplitz.

13. Solvability of Systems oj Linear EquatIOns

107

by the linear homogeneous expression in ~l!"" ~k,-l [C1:, = (- bll ) ~l + (- b1,1:1-1) ~1:,-1 i the first equation will be satisfied by such Xl' ••• , XI" no matter how the remaining Xl may be chosen. 2) Next, in order to satisfy the second equation of (Ho) with replace

Xk,

+ ...

respect to any given xk,+l = ~kl"'" Xk._l = ;k,_2 16 we musl replace xk, by the linear homogeneous expression in ~l""­

gk,-2 X.I:.

= (-b21 ) ~l + ... + (- b2,k,_1) ~kl-l + (-b 2k,) [ ( - bll ) ~1 + ... + (- b1,l:,_1) ~.I:,-l] + (-b2,Ml) ~kl. + .. , + (- b2,k.-I) ~k.-2 = (-b21 + b2k, bn );l + .. ,+ (-b 2,.1:,_1 + b2k, b1,k,_1) ~.I:l-1 + (-bZ,k,+I) ~k. + .. , + (- bU.-I) ~k.-2.

The second equation as well as the first will be satisfied by such no matter how the remaining 3) to 1') As aboye.

Xl' • , " X ""

Xr..

may be chosen.

After the )'-th step XI' +1 =~/_ -r+1o .• .• ; x" = :;,,_,17 can still - tr be arbitrarily chosen. The xl' ... , Xn so determined will satisfy ~r

all equations of (Ho)' To recapitulate: Let those of the indeterminates different from XI.;.•• .. , X/_ be designated bv XI ' 1 ' " ' ' 1 'r ... (Ho) is satisfied with respect to any given ~T'7'

Xl'··.' XI. ' tl

XII

Then

~kr+l =~1 Xhn

by substituting for pressions in

;1"'"

:1'k,,' •. ,

-

rfn-r

xJ<,. certain linear homogeneous ex-

~1I-r

17 In case these do not appear the last; which does occur has at any rate the index n - r. The limiting case r := n, namely, where no ~ appears, will be considered by itself under c).

108

1. Ill. Lmear Algebra without Determmants

Xkr

= OCIr;l + ......... + Oin-r,r;n-r

(whose coefficients Uk. are uniquely determined by the coefficients b,,, of (Ho)); conversely, the system (Ho) will be satisfied in this way in the case of any given ;1' ... , ;n-J' This means that the totality of solutions of (Ho) is generated by the n - l' special solutions h =(e,O, ... ,O;

!"n-r = (0, ... , 0, e;

Ctn-r,l, . . -, I'X n_ r ,.,.)

through arbitrary linear combinations (with the ;1' ... , ~n-I as coefficients), whereby the Xl. are to be thought of as arranged :lS above. Now, by Theorem 40 [81] the linear independence of the Cn -t·)-termed unit vectors implies that t1"'" tn-r are also linearly independent. Hence the latter form a system of fundamental solutions of (Ho) of n - r solutions, as stated. c) The proof carried through under b) must be supplemented for the limiting case r = rI. In this case ;1"'" Sn-, do not appear at all, and by condition (4) of the Theorem of Toeplitz the set of numbers lel , ••• , le, is identical with the sequence 1, ... , n. Consequently, the r steps under b) yield that X"""" x"r' i. e., Xl' ••. , Xn are all to be set = 0 in order to satisfy (flo)' rfhis means that (lIo) cannot be solved; in other words, it possesses a system of fundamental solutions of 0= n - n = n - r solutions, as stated in the theorem. Theorem 52 says that the number of fundamental solutions of the system of equations (H) increases as the number of linearly independent rows of its matrix A decreases. Hence the more linear dependences there are between these rows, or, since a linear dependence between the rows of A is equivalent to a

13. Solvability

0/ Systems 0/ Linear Equations

109

solution of (H'), the greater the totality of solutions of (H'). it is, therefore, to be expected that a relation exists betwcen the number of fundamental solutions of (H) and (H'). By Theorem 52 this relation must also be expressible as one between the maximal numbers of linearly independent rows and columns of A. 'We now prove in this regard the following two facts: Theorem 53. The maximal number of linearly independent rows of a matrlX Ais equal to the maximal ilumber of lilu:arly independent columns of A.

Theorem 54. Let s be the number of fundamental solutIOns of a system of homogeneolls equations (B) consisting of m equations, alld s' the number of fundamental solutions of its transpose (B') consisting of m' eqtlatiolls. Then the follmcing j'elation

lS

valid:

m+ s.=m' + t. Here, for once, we have written m' for the number designated otherwise by n, for the sake of symmetry.

Proofs: 1) (Theorem 53.) Let rand r' be the maximal num-

bers specified in the theorem for the (m, m')-rowed matrix A. a) For A = the theorem is triviaL since in this case r = 0, r' = 0 (cf. the remark to Section 12, Lemma 1 [96]). b) For A =1= we can apply Section 12, Lemmas 1, 2 [96, 97] to the rows a1 , ••• , am of A. Since the rearrangement of the rows will in no way alter any linear dependences which may exist either between the rows or between the columns. we are permitted to assume18 without loss of generality that the rows are arranged so that a1 , . . . , aT is a maximal system of linearly independent rows. Now, to start with, let r < m. Then by Section 12, Lemma 2, there exist corresponding to Section 12,

° °

18

This is only a matter of notation.

110 (5) rn -

1. Ill. Lmear Algebra without Determinants r relations between the rows of the form r a, =}; c,.a. (j =r 1, ... , m) .

+

• =1

These say that the m - r YectOls (- C,+l, 1, ... , - C,+l,,.; e, 0, ... ,0)

(- em!

, ... , -

e",,.

;

0, ... ,0, e),

which are linearly independent by Theorem 40 [81], are solutions of the (H') belonging to A. On applying Theorems 50 to 52 to (H') the number of fundamental solutions rn - r' of (H') is at least rn - r, that is, rn - r' ;:;; rn - r, r' ;;;;; 1'. In case r = rn this is also valid by Lemma 1. On applying analogous considerations to the columns of A it likewise follows that r ;;;;; T". Consequently, r = r', 2) (Theorem 54.) By Theorems 52, 53 ,ye have s = rn' - r, s'=m-r, that is, m+s=m'+s' (=m+m'-r). Theorem 53 clarifies the fact established above in the Corollary to Lemma 1. For, by Theorem 53 the "non-self-evident" inequality o :;;; r :;;; m' between a row-number and a column-number goes over into the" self-evident" inequality 0;;;;; r' :;;; m' between two columnnumbers. By Theorem 54 the "circle" of our considerations regarding (J), (H), (H') is closed: By Theorem 46 [91] (J) is linked with (H), by Theorem 54 (H) with (H'), and by Theorems 48 [92], 49 (H') with (J).

14. The Case m=n It is of interest for the investigations in Chapter IV and also for the applications to specialize the results obtained in Theorems 46 to 54 [91 to 109] , regarding the solvability and solutions of systems of linear equations, to the case where the number m of equations is the same as the number n of unknowns, namely, to the case where A is an (n, n)-rowed matrix. Now, in the previous

III

14. The Case m=n

section we have seen that the numbers m and n alone say practically nothing about the totality of solutions of (H) (and also of (J), as seen by Theorem 46 [91]); on the contrary, we had to include the above number r to determine thlS totality. Hence special results are not be expected here unless a definite assumption is made regarding r. WE' will therefore introducE' another specialization besides 'm n, which will distinguish m'Zy between the limiting case r m n and the case 0;;:;; 1" < m n (without further distinction in the latter case). 19 Accordingly we make the following convention whose full significance will become clear only through the developments in Chapter IV; for the time being it is made only to make our terminology briefer. Definition 33. An (n, n)-rowed matrix A is said to be regular or singular according as r = n 01' 0 ;;:;; r < n, where l' has the significance of Sections 12, 13. The facts contained in Theorems 52 to 54 [106 to 109] about (H) can now be summarized as follows: Theorem (52, 53, 54) a. If A ~s an (n, n)-1'owed 'matrix, then either its rows as well as 'bts columns a're linearly independent or its rows as well as its coht'mns are lineai'Zy dependent; that is, either both systems of equations (H) and (H') belonging to A cannot be solved or both can be solved. The former or the latter is valid according as A is regtGlar or singular. The facts contained in Theorems 46 [91] ,48 [92], 49 [102] about (J) yield the further result: Theorem (46, 48, 49) a. The system of equations (J) with (n, n)-1'owed matrix A can be uniq~tely solv6d for any arbitrary vector a on the right if and only if A is regular. Proof: a) Let A be regular. 1. Then by Theorem 49 [102] (J) can be solved for arbitrary a, sinee by Theorem (52,53,54) a (H') cannot be solved; therefore, the restrictive condition of Theorem 48 (Corollary) [93] with regard to (1 does not exist.

=

= =

=

19 The other limiting caS€ r special consideration.

=0

is too trivial to warrant a

1. Ill. Linear Algebra without Determinants

112

2. Furthermore, by Theorem 46 [91] (J) can be uniquely solved in this case, since by Theorem (52, 53, 54) a (H) cannot be solved. b) Let A be singular. 1. Then by Theorem (52, 53, 54) a there exists a solution t (=1= 0) of (H'). If xi =1= 0 is in this solution, therefore:!;' ei = xi =1= 0, then by Theorem 48 [92] (J) cannot be solved for the vector a == e,; therefore, it cannot be solved for any arbitrary VE'ctor a. 2. Furthermore, by Theorem 46 [91] (J) can then, if at all, not be uniquely solved, since by Theorem (52, 53, 54)a (H) can be solved. These two theorems further yield through the elimination of the alternative (that is, the conb'adictory opposed statements) "A is regular" or "A is singular": Corollary, There exists the alternative: Eithet· (J) can be

solved without 1'estriction and uniquely, or (H) and (H') can be solved. Finally, for the first case of this alternative, that is, for regular A, we can make an elegant statement concerning the dependence of the solution 6 of (J) (which always exists in this case and is uniquely determined) on the vf'ctor a on the right. This statement will go beyond the results of Section 13. In this connection we designate a by 6* and prove: Theorem 55. Let A = (aik) be an (n, n)-?'owed j'egular matrtx.

Then a uniquely determined (n, n)-rowed matrix A * exists such that if 6 ~s a solution (which always exists and is uniquely detennined) of the system of equations (J )

*

ft

1: ai&x" = x. "=1

(i = 1, ... , n)

•

with the mat1'ix A the dependence of the solution on the x7 on the rig ht is given by the formulae (~*)

'" x"* = x, 1:n ail;

1:=1

(i=I ....,n)

'

with the 1natrix A ". A~' too is reg1tlar and (A *)* = A, that is, the system of eq1£atiC'1!s corresponding to the formulae (:;S*)

"

....

1: a,,,x~ = x· (i = I, ..., n)l .c=l .. • as its unknowns and xi on the right which has A* as its matrix, '(J *)

xl;

113

14. The Case m:;:::;n

side, is solved by the formulae corresponding to the system of equations (J) 11

.1:a;;,xJ;=.r~

(3)

.1:=1

(i=l, ... ,n)

•

with the mat)'ix A. Proof: 20 a) Let the n vectors .1, (at", .. . , a;;k) be the solutions of (J) in the case where the right-hand sides are e,,(k = 1, " ., n). Then Theorem 44 [86] immediately implies that the linear

a: =

..

Of<

Of<

combination ! = J: xJ; a.• .!: is a solution of (J), and therefore the k=1

solution, corresponding to the linear combination

~

=

11 * J::r}: e.j;. But,

.1:=1

on writing it out, the representation of 6 in terms of the a ~. k goes over into (~"). Therefore, there exists an (n, n)-rowed A'" with the property specified in the first part of the theorem. b) Let X:, (u,i) be another matrix with this property, so that (:;:s.,) and the C§*) formed with X:, always yield the same ); on the right for all ('. Then for the special case );* ek the k-th columns a~_k and a:.]. of X" and k-' coincide (k 1, .. . ,n), and therefore Ai, = A *. This means that A * is uniquely determined by the property specified in the first part of the theorem. c) Conversely, if I: is arbitrarily chosen and, in addition, 6* determined so that (:J) exists, then by a) (:;:s ") also must exist (since in this very case 6 is the solution of (J) for the 6':' thereby determined). In other words, (~) yields a solution of (J':') for any arbitrary 6' Hence by Theorem (46,48, 49)a its matrix A~' is regular, and further (A *)" A. In view of the property given in Theorem 55 which characterizes the matrix A --' we define in addition: Definition 34. The matrix A * uniquely detej"mined by TheM'em 54 through an (n,n)-1"owed j"egular matrix A is called the 1'(!sQll"(mt 1fwtJ·i;x' of A.

=

=

=

=

20 For a better understanding of the connection in this proof the reader should set, as is done in the theorem, the phrase system of equations before each (J), (J*) and the word formulae before each (~), (~*).

15. Importance of Linear Algebra without Determinants Through the results of Sections 11, 13 the problem of linear algebra Section 5, (1) is completely solved from a theoretical point of view. This follows from the fact that we have obtained for the system of equations (J) a necessary and sufficient condition for '>olvability ( Theorems 48 [92], 49 [102]) as well as precise information regarding the structure of the totality of solutions (Theorem 46 [91] I together with Theorems 50 to 52 [104 to 106]). From a p"actical point of view, however, we have made very little progress. That is to say, the specified results make it possible neither to decide whether an explicitly given system of equations (J) can be solved nor to determine in the solvability case the totality of solutions, because the application of these results requires in general, that is, for infinite ground fields, infinitely many trials. By Theorem 48 (Corollary 2) [93], Theorem 49 [102] and Theorem 46 [91] these two points could be effected by a finite -process if only the two problems J pr) and Hpr) cited at the end of Section 11 could be solved by a finite process. Now, at a first glance this seems actually possible by means of the Toeplitz process; for in the proof to Theorem 49 [102, 103] and Theorem 52 [106, 107] this process yielded obviously finite constructions of a solution of (J) and of a system of fundamental solutions of (H). For this, however, it is assumed that the coefficients b". of (J o) are known, and the process a!. Section 12, the Toeplitz process itself, leading to their determination, is in genera} infinite in the first as well as in the second step. Thus, in the first step (Lemma 1 [96]) the decision whether a subsystem selected from the i, is linearly independent (a decision that must actually be made finitely often for the determination of a maximal system of linearly independent f, and the associated maximal Ili1lmber r) requires the testing of all possible, in general infinitely many. existing linear combinations of the I,; this is also true in regard to making a decision in the second step (Lemma 3 [99]) as to which linear combination of the fi has the minimal length.

114

15. Importance of Algebra without Determmants

115

In particular, this means that even the number /' and consequently the number of fundamental solutions of (H) cannot in general be determined by finitely many trials. From a practical standpoint the results of Theorem 52 (l06] are not to be regarded as a determinatwn of the number of fundamental solutions of (H), since to know l' is equivalent to knowing all linear dependences between the rows of A, that is, all solutions of (H'). Hence from a practical point of VIew only a Ci1'C'ular connection is given, such as is expressed in Theorem 54 [109] which ties (H) to (H') (cf. also the remark at the end of Section 13). Naturally, the preceding remarks also apply to the results obtained in the special case of Section 14. In particular, Theorem 55 [H2] is on this account not applicable form a practical vie"wpoint; for, though the existence of the resolvent matrix .4 ~ IS established, no finite process is given for its construction. In spite of the extremely important theoretical insights arising from Theorems 46 to 54, for the practical applications direct methods must still be developed for deciding in finitely many steps whether (J) can be solved, and in the solvability case for constructing the totality of solutions. Only in this way would the complex of facts obtained so far receive its desired conclusion from a practical (and naturally also theoretical) standpoint. Such methods are developed by the theory of determinants, starting from the special case dealt with in Section 14, and then passing on to the decision of solvability and the construction of solutions in the most general case. As a rule, the foregoing results are derived by means of the theory of determinants. We have not used this approach here for two reasons. On the one hand, if the concept of determinant were put at the fore of the above investigations, it would look rather extraneous, having nothing to do with the problem to be solved. Thus the results obtained through this approach would appear surprising and loosened from their context. Instead, the methods we have used are adapted throughout to the problem and the connective thread between Theorems 46 to 54 stands out very clearly. On the other hand, however, the complex of theore'lnS of linear algebra developed without dete'rminants has received special attention in

116

1. Ill. Linear Algebra without Determinants

modern times, since it is only these theorems, with all their proofs, which can be transferred nearly word for word to the corresponding problems in the case of infinitely many equations with infinitely many unknowns and to the theory of linear integral equations closely connected with this. For such problems the concept of determinant, except for special cases, proves to be too narrow. Besides, the beauty and completeness of the theory without determinants, as developed in the preceding, is enough to justify its separate treatment.

IV. Linear Algebra with Determinants 16. Permutation Groups In the proofs of the preceding chapter we have frequently made rearrangements of the rows or columns of a matrix merely for reasons of notation. The concept of determinant to be introduced in this chapter IS now based m a factual manner on such rearrangements or, to be more exact, on certain relations connected with them. Before developing the theory of determinants we must, therefore, first of all become familiar with these relations. '1'he concept of rearrangement or permutatwn is a pure settheoretic concept. It arises from the fact that every set is equipotent to itself [Section 2, (II) l, so that every set corresponds biuniquely to itself in at least one way (namely. the correspondence which maps eyery element onto itself). A permutation is formed by considering any arbitrary correspondence of this kind. Definition 35. A IH31-m/utation of a set M is allY one-io-one correspondence of M onto itself with a definite mapping rule; or apply the permutation means to replace each element of M by the element corresponding to it. to cat'rlJ

(J'Ut

Def. 35 implies that to distinguish permutations according to the correspondences based on them we should consider the mapping rule. Therefore, we call two permutations equal if and only if to each element there corresponds the same element under both permutations. In order to describe a permutation uniquely we could naturally give all the mappings as well as all the substitutions (transitions) to be made in carrying them out; these are merely two different ways of looking at one and the same formal fact. Obviously, the permutation is independent of the order in which the individual correspondences are given.

117

118

1. IV. Linear Algebra with Determinants

Regarding the permutations of a set we now prove: Them'em 56. The totality of permutations of a set form a group, if the product of two permutations is defined as the permutation generated by applying one after the other. The unity element of this group is the permutation which takes every element into itsplf; the reciprocal of a permutation is obtained by reversing the mapping rule. Pl'oof: Section 6, (a) is satisfied in the sense of the previous remarks. We will now prove that Section 6, (b) is satisfied. Let a be mapped on a' by the first permutation and a' on a" by the second. Then the successive application of these two permutations, that is, the actual replacement of a by a", is again a permutation. This is valid for every pair of permutations that may be chosen. Section 6, (1) is valid, since (logical) substitutions satisfy the associative law; Section 6, (2 a) und (2 b) are obviously satisfied in the way stated in the theorem. Theorem 57. If M and M are equipotent sets, then the groups of permutations of M and M are isomorphic. Proof: If every permutation of M is associated to that permutation of M which is generated by carrying out a biunique transition from M to M, then this correspondence satisfies condition (2) of Theorem 23 [65]. Since it is easy to work out the details, it is left to the reader. On the basis of Theorem 57 the type of the permutation group of M is determined only by the cardinal number of M (see Section 2, (II) and Def. 17 [65 J ). In particular, for finite M the type is determined only by the number of elements of M, If isomorphic groups are not to be distinguished, we can then define:

16. Permukltion Groups

119

Definition 36. The group of all permutations of a finite set of n distinct elements is called the symmetl'ic g1'OUp 1 of n elements. It is denoted by G n • Here, we will be Since every set of n the particular set of sufficient to base the designate by

occupIed exclusively with this group Sli' elements can be associated biuniquely to n numerals 1, ... , n, by Theorem 5j it is study of @in on this set of numerals. \\e

(;1: ::;J '

briefly

(:J

(i

=1, ..., n),

that permutation of the numerals 1, ... , n which takes the numeral i into PI (i=l, ... ,n). If a1, . . . ,a" is any set of 11 elements which through the numbering of its elements corresponds biuniquely to the set of numerals 1, .... 11. then the above permutation can also be regarded as a permutation of the 11 elements al' ... , an, nfl.mely, as that permutation by which a. goes into ap , (i=l, .. . ,n). The biuniqueness imposed on permutations in Def. 35 [conditions Section 2, (8), (8'), (e), (e')], applied to the above notation

(p:)

(i = 1, ... , n), is the precise formulation of the

following statement: Pi"'" Pn are the numerals 1, ... , II apart from their arrangement or in any arrangement. This will frequently be used. The totality of arrangements of 1, ... , n

1 The designation symmetric group is to be understood as meaning that "something" is symmetric, in the usual sense of the word, relative to n elements if it remains unchanged by the application of all permutations of these elements. For example, in Section 4 it was in this sense that we called [:1;1> ••• , :l;n] symmetric in :1;1' ••• , :1;". Cf. also Theorem 131 [329] (Theorem of symmetric functions) •

120

1. IV. Linear Algebra with Determinants

corresponds biuniquely to the totality of permutations of 1, ... ,11. 2

By the remark to Def. 35 a permutation is completely indifferent to the order in which its individual transitions are given. Hence in giving the above permutation we could Just as well use

qn), (qplq,'...•• Pqn

(qi)

(i=l, ... ,n), P',i where q1"'" qn is any arrangement of 1, ... , n. By means of this rpmark the multiplication rule of Theorem 56 for prrmutaHons in @in can be expressed by the formula

briefly

(i = 1, ... , n), and similarly the reciprocal of 8

1

(P:) can be given as (P;}

is naturally the identity group Cf. 8 2 is the Abelian group

consisting of the two elements E

=G;} p =(; ~) (with p~ =E)

(cf. Section 6, Example 3). For n;:;; 3, on the contrary, G n is surely not Abelian; for example, we have (123 .•.)(12 3 ...) (12 3 ...) 213 ... 321. .. =\231. .. ' 2 3 ... ) (1 2 3 ... ) (1 2 3 ... ) _ 321. .. 213 ... - 312 ... · Incidentally, 8 3 is isomorphic to the group of 6 elements considered in Sections 6, 7, Example 4, as can be seen by mapping the rotations, appearing there, onto the permutations of the vertices of the triangle generated by the rotations.

(1

2 In elementary mathematics it is usually the arrangements themselves not the process of their derivation which are called permutations of 1, ... , n. Even though, after the above remarks, this amounts to the same thing, it would, on the one hand, certainly be inconvenient for the formulation of the rule of combination of Theorem 56, and, on the other hand, it would not agree with the literal meaning of pe'rmutation (interchange) as an operation.

121

16. Permutation Groups

We can assume as known from the elements of arithmetic: Theorem 58. S" is finite and has the ordel' n! == 1· 2··· n. In the following we actually need only the finiteness. not the order of @In'

We next separate the permutations of :0 n into two categories. a distinction which is basic for the definition of determinants. For this purpose, and also in other connections to arise later, we must consider subsets of the set of numerals 1, ... , n on which the permutations of en are based. In accordance vd.th the terminology used in the elements of arithmetic, such subsets are called combinations of the numerals 1, ... , n; if they contain v numerals, they are said to be of v-th ol·del·. We designate the combination consisting of the numerals i" ... , i, by {il , . . . , i, J. This notation implies that 1) i p " " i, are different numerals of the sequence 1, ... ,n, 2) {il, ... ,i v } == {i;, ... ,i~} if and only if the numerals i;, ... , are the numerals i1 , ••• , i., except for the orc/e)', therefore can be derived by a permutation of these. This means that a combination does not depend on the arrangement of the numerals. Two mutually exclusive combinations of 1, ... , n whose union is the entire set 1, ... , n are called complementary. The combination complementary to {it> ..• , iv} (1 ;;;; v ;;;; n - 1) is usually designated by {iv+ l " ' " in}' The number of different combinations of v-th

i:

order of 1, ... , n is designated, as usual, by (:) ; its value, which, incidentally, will follow from the proof of not important for us.

Theorem 66

[183], is

First of all we have: Theorem 59. Let 1 ;;;;;

v ;;;;;

n. If a permutation P = (:,) of the

numerals 1, ... , n is applied to the totality of oj' v-th order of these numerals, that bination {il ,

•.. ,

lS,

C)

combinations

if each such com-

iv} is replaced by {Pit"'" P",}, then this

totality of (:) combinations is again generated. In other uords,

122

1. IV. Lmear Algebra with Determinants

P will effect a permutation oj' the set of these combinations. Proof": Obviously, through the application of P we can

generate all the (:) combinations of v-th order of the set Pl' ... ,Pn, which is identical with the set 1, .... n. We now consider, in particular, the combinations of 2nd order of 1, ... ,n. If in any such combination {i, k} we think of the two numerals i and k as written in their natural order lnarnely, i < k is assumed), then this ordering relation will not necessarily be preserved under the application of a permutation, since there could exist very many pairs of numerals i, k with i < k but p, > Pk. This situation is the occasion for separating the permutations of 8 n into two categories; this distinction, as already mentioned, is important for the definition of determinants.

Definition 37. Let n > 1 and P =

(;J

be a permutation of

1, ... , n. The appear'ance of a pair of numer'als i, k with i


but Pi> PTc is called an i1wersio'n of P. P is said to be even or odd according as the number v of its tnversions is even or odd. We set sgn P = (- 1)', theTe!01'e = 1 or as P is even 0'1' odd. 3 For n

= 1, where

only the permutation E

=-

1, according

= (~)

exists, we

will set sgn E = 1. sgn is an abbreviation for the Latin word signum (sign). For real numbers p =!= 0 we set, as is well known, sgn p = 1 or - 1 according as p > 0 or < O.

For

n> 1 it is

easy to see that there actually are even and

8 This is just as in the case of Section 9, Example 4 (footnote 6) [77].

123

16. Permutation Groups

odd permutations. For example. E =

n).

... ( 123 213 ... n

IS 0

dd

(1.1. .... nn)

is e,-en and

.

'Ve now prove a property basic for our application: 'rheorem 60. For two permutations P and Q of 1. ... ,11 !ee have sgn (PQ) = sgn P sgn Q.

Proof: For n = 1, the statement is tri ....ial. Let tl> 1. Then DeL 37 says that in counting the inversions of a permutation

P=~J

(i=l, ... ,n) we must consider all pairs of numerals

i, It: of the sequence 1, ... , 11 with i < k, that is, all combinations

<

of 2nd order {i, k} with the ordering lUle i k for its numerals. If this ordering rule is left out, then a combination {i, k} yields an inversion of P if and only if the integers (different from zero) i - k and P, - PI, have different signs. that is, if i-Ie < O. Accordingly sgn P can also be defined by the P,-PI,

formula

i-k

sgnP=llsgn-p, -

{i,i}

~

Pi

where the product on the right is to extend over all different combinations of 2-nd order of 1, ... , n (without regard to the order in which the two numerals are taken). For, by the remarks, the number of times the factor -1 appears in this product is exactly the number

'V

of inversions of P. Now, if

Q= (~) (i = 1, .• "n), then we haye i-k Pi-Pi sgnQ=llsgn--=llsgn ; {u}

f/i- qi

{i,lI}

Q'1l' -

q1Ji

1. IV. Linear Algebra with Determinants

124

the latter is valid, since by Theorem 59 {Pi' Pk} runs through the totality of different combinations of 2nd order of 1, ... , n if {i, k} does, and because the product is indifferent to the order of the fadors. Now, as is well known, the rule sgn (p q) sgn p sgn q is valid for real numbers p, q =F O. Hence, by termwise multiplication of both products it is also true for sgn P and sgn Q that i-k Pi-Pk i-k sgn P sgn Q = II [sgn - - sgn - - = II sgn - - - . {i.k) P. - Pk qPi - qPk {i.k} qPi - qPk The product on the right-hand side, however, is sgn (PQ), since

=

I

PQ=(:J (i=l, ...

,n).

As an immediate consequence of Theorem 60, which is to be used later, we have: Theorem 61. sgn P .= sgn P-l. Proof: By Theorem 60 sgn P sgn p-l = sgn (PP-l) = sgn E = 1, since E =

(i ::::)

obviously has no inversions.

The concept of inversion defined in Def. 37 and the explanation of sgn P based on this is not determined by the permutation P of the set 1, ... , n alone; in addition, it refers to a definite basic order of this set, namely, the natural order 1, ... , n. This reference to the natural order 1, ... , n as the basic order becomes especially clear if the rule of Def. 37 for counting the inversions of P =

(1.Pl'..·Pn n)

is formulated as follows: Let the upper

row of P be written in the natural order. Then the number of inversions is the number of those pairs of numerals of the lower row whose order is opposite to that in the upper row. If this rule were to be applied in the case of any other ordering of the upper row of P, then, in general, another number of such pairs of numerals of the lower row would be obtained. For example, if we write (1234) 4132

16. Permutation Groups

125

there are in the lower row 4 pairs of numerals {41}, {43), {42}, {32} whose orders are opposite to that in the upper row; whIle in . of numerals {41}, {21} whose orders are ( 314 342 2) 1 t h ere are 2 paIrs opposite to that in the upper row. However, for the determination of sgn P these are not to be distinguished, since both numbers are even. The following theorem shows that this is generally valid.

Theorem 62. sgn P defmed in Def. 37 is independent, w the following sense, of the natural order of the numerals 1, ... , n on which its defimtion is based: If ql"'" qT' is any ordering of 1, .. " n, and P=

(~: : : :;J =(~:: :::~) = (::1:: :~:J, Pqi = q~i

therefore

so that the permutation R =

(i = 1, ... , n),

(1. ... n) thereby introduced speClT1 ••• Tn

fies how P changes the order of ql"'" qn' then sgnP=sgnR. Proof: Let Q=

(1.ql ... ... qn'It) . Then if the equation of permu-

tations given in the theorem is multiplied on the left by Q, we obtain

QP _

(1Pq,.... n ) _ (1. ... ••. P1ft

'It ) _

RQ

Q"1 ••• q~"

Consequently, by Theorem 60 sgnQ sgnP= sgnR sgnQ (= sgn Q sgn R); this implies the statement of the theorem. since sgn Q =1= O. The inversions of the permutation R introduced in the theorem are obviously to be understood as those pairs of numerals in the lower row of P whose order is opposite to that in the upper row if the numerals of the upper row are written in the sequence ql' ••• , qn; therefore, Theorem 62 proves the statement made beforehand in the remark. - We point to the fact that the permutation R

126

1. IV. Linear Algebra with Determinants

of Theorem 62 is generated from P by transformation with Q-l (d. the remark to Theorem 28 [70]). - By Theorem 62 it now makes sense to speak of even and odd pe1'mutations of n elements without specifying a basic order as a reference system for the "evenness" or "oddness."

In conclusion we call attention to the following additional facts, even though we have no need of them here, because they give us a deeper insight into our classification of the permutations of n elements into even and odd. Theorem 63. The totality of even permutations of n elements (n> 1) form a normal divis01' )l(n of @In of index 2, the socalled altel"1Utting g1'OUp of n elements. One of the two cosets into which @In decomposes relative to 5l(n consists of all the even and the other of all the odd permutations. Consequently, there are just as many permutations of one kind as the other (namelY, exactly

~!).

Proof: This immediately follows from Theorem 35 [75], since the relation P Q, if sgn P = sgn Q, obviously represents a congruence relation in the group @In' The partition of @In, thereby determined, corresponds to the classification into even and odd permutations.

=

That mn is a normal divisor also follows from Theorem 62. In the proof of this theorem we used QP = RQ. On writing this relation in the form QPQ-l R, it says in conjunction with the result sgn P sgn R that if P is even so also are all its conjugates R (Theorem 32 [73]). Conversely, Theorem 62 can also be inferred from the result of Theorem 63 that mn is a normal divisor of index 2 (Theorem 32).

=

=

17. Determinants In the space of this presentation we must forego a genetic introduction of determinants4 based on the methods of Chapter III. Instead we immediately make the following definition:

Definition 38. The determinant of the (n, n)-rowed matrix A = (aik) is defined as the expression I

I A ! = I aile I = where the summation

au ... Uln I • ......... I = E sgn P alp: ... a"p,,",

a,.

a I Pme,. ':ver all permutations P =

eX~;·i;ds

(;J

of the column indices 1, ... , II.

Written out in detail, this means that the determinant A of A is formed as follows: While holding the first (row) index fixed, apply all permutations P = (:;) to the second (column) indices in the product all ... ann of the 1/ terms standing in the so-called principal diagonal of A. Thereby products of the form /ZuJ••.• a,.Pn (there are n! of these) are generated. The determinant is obtained from these by forming the difference

E alp, ..• a"p" - E alp1 •.• a"Pn = E sgn P alp, ... a"Pn

Pinllt"

pnot inllt"

Pin $"

of the sum of all such products corresponding to the even permutations and the sum of all those corresponding to the odd permutations. 4 Through such an introduction starting from the Toeplitz process th~ concept of determinant would no longer appear out of phase with the concept formations and methods of Chapter III, and thereby we would obtain a deeper understanding as to why our present course leads to the same results.

5 See the end of Section 1 for the significance of sgn P := ± 1 as a "factor" before a field element.

127

128

1. IV. Linear Algebra with Determmants

In particular we have for n

=

1: I all II = all' for n

for n

= 3: Ia21a22a23 ~1 ~12:131 t

=

= {

-

31 82 33

=

2:

I aa21l1 aa 12 I = 22

+

a ll a 22 -

a 12 a 211

+

all 0,220,33 a12a28a31 a13a21a32 a ll a n a 32 -0,130,220,31 -Q,U a21a 33'

For n 3 the determinant can also be formed by the following rule: Let the first two columns be adjoined to A once more on the right, then form products according to the 6 lines drawn parallel to the two diagonals of A and subtract from the sum of the products in the direction", (principal diagonal) the sum of the products in the direction / (secondary diagonal). For n 2 a corresponding rule is obviously valid; on the contrary, such a rule is not valid for "l. > R. In addition, we cite the following formulae for arbitrary n since they will be frequently used:

=

0,10 ..... 0 o as .... 0

-= 0102 •.• an;

.

In

o 0 .... . an

eO ..... 0 pal" t'lCU Iar, 0 e ..... 0 00 ..... e

= e.

These follow directly from Def. 38. Historically, (Leibniz, Cramer, etc.) the concept of determinant arose somewhat as follows: In the case of 2 linear equations in 2 unknowns all XI

a 21 x 1

+a

12x

2 == 0,1

+ a22x == a 2

2

the so-called method of multiplication leads to the conditions (a 11 a22 - a 12 ( 21 )'l:1 == a22 a 1 - 0,120,2 (a n a22 -

a I2 ( 21 )'l:2

== aH a2 -

0,210,11

from which it is easy to show the unique solvability for arbitrary 0,11 0,2 if the expression a11 a 22 -

a 12 a 21 ::::

la11al21 a a 21

22

=1= O. Similarly, for

n = 3,4, . " we obtain by the method of multiplication the determinants : a,k as coefficients if we linearly combine the linear forms on the left in a single step with only one indeterminate 'l:k' The expressions, which are easily formed for n = 2, 3, 4, enable us to read off the general rule of formation (if we wish, even to derive

I

17. Determinants

129

them by induction from 11 to n T 1) and lead to the defmlhon gIVen above. Here we must forego such an inductive approach to determinants. Instead, in a strictly deductit'e manner we wiE derh"e the indicated connection with the problem of solving equations in linear algebra for the case m n (Sections 20, 21) after developing in Sections 17 to 19 the most important properties of determinants as defined above.

=

In Def. 38 the rows and columns of A playa dIfferent loie. This is however only an apparent difference, since we ha\"e Theorem 64. An (n, n)-rowed matrix A and
= i A'

. The determhwllt

of

A

depends, therefore, on the rows of A in the same tray as all the columns of A, alld it can be defined not only by the formula of Def. 38 but also by the formula

I A I =Pin@;n 1: sgn P ap,l ..• ap ' " .. where the summatioH extends oeer all permutatlOlls

of the row indices. Proal: By Dei. 38 we have

I A'l =1:sgnPtlpl1 ... ap ... Pin@;,.

"

Since the ordering of the factors of a product is arbitrary, we may always apply the permutation p-l in each summand to the n factors. If p-l =

(P;) =

(;J =

Q, then we obtain

I A' I = 1: sgn P-al g"

•• anq •

Pin ISn

"

On the one hand, sgn P = sgn Q (Theorem 61 [124]); on the other hand, Q runs through the whole group Sn if P does, every element once (Theorem 17 [60). Hence we also have

I A'l =Esgn Qalg•••• 0:n2: QinlSn

=

I A I.

11

From the earlier standpoint the symmetry relative to the rows and columns in Theorems 53, 54 [109J and Theorem (52, 53, 54)a

1. IV. Linear Algebra with Determinants

130

[111] was not very plausible. From the of determinants it is finally brought of the determinant i A relative to the exhibited in Theorem 64. This will be clearly in the following.

I

standpoint of the theory back to the symmetry rows and columns of A brought out even more

While Theorem 64 compares the dependence of the determinant I A I on the rows of A with that on the columns, the following theorem states something about the dependence of the determinant 1 A 1 on the ordering of the rows or columns of A. Theorem 65. If A1 is generated from the (n, n)-rowed matrix A by a permutation R of the rows (columns), then

i All = sgn RIA I ' therefore, i A1 1= 1A I or IA1 ! = -I A I, according as R is even OJ' odd. Proof: By Theorem M it is sufficient to carry out the proof

for the case in which .11 is generated from A by a permutation R=

(:J

of the rows. By Def. 38 we have in this case

I Al j = L'sgn Pa',Pl'" ar P!nlEn

p

,

" n

since the a"l, ... , ar"" are the n terms contained in the principal diagonal of A 1• By &pplying the permutation

R-1

=

(:t) = eJ

= S to the n factors of each summand and using the fact that sgn R sgn S = 1 (Theorem 61 [124]) we obtain

1Al ! = sgn RPinL'lEnsgn S sgn P alp

'" 8,

a"p

, 3"

where each time the column indices are generated from 1, ... , n by the permutation

(p:) =SP=Q= (q:}

Then, on the one

i

hand, sgn S sgn P = sgn Q (,rheorem 60 [123] ) j on the other hand, Q runs through the entire group @In if P does, every element once (Theorem 16 l60]). Hence we also have

131

18. Minors and Co/actors

I Al I =sgn R 1: sgn QaJ;q, ••• a'll11n =

sgn RiA

Qln @On

The two facts in Theorem 64 and Theorem 65 imply that all general theorems about determinants have a symmetric form, on the one hand, relative to the words row and column, on the other hand, (except for possibly distinct signs) relative to the individual rows as well as the individual columns. In the following we will make use of this in our proofs, just as already in th" proof to Theorem 65.

18. Minors and Cofactors. The Laplace Expansion Theorem In this section and the following we will giYe theorems which go beyond both Theorems 64 and 65 of the preYious section and which penetrate deeper into the structure of determinants. Two ob3ectives will thereby be realized. On the one hand, the way will be prepared for applying determinants to systems of linear equations; on the other hand, methods will be developed more suitable for the eyaluation of determinants than the formula used to define them. For this purpose we definf': Definition 39. Let A be at/, Cn, n)-rowed matrix, l;;;;;v;;;;;n-l. {i 1, ••• ,iv}, and {kl, ... ,kv} a combination of v- th order of its rows and its columns. 6 respectively. {iV+l' ... , in} and {kv +1' ... , k ll } the associated complementary combinations. Then we write, set up, or designate: A{i" .•. ,ip), {....... ,ky)

the (v, v)-rowed matrix generated by striking out the rows iv +1' ... , in and the columlls k. h.' .. , k" in A; A{i" ... ,i~), {l1..... ]:?}

= AU p +l,· .. ,inl. {l-.+l,···.J:nl.

6 For the sake of simplicity the rows and columns are merely indicated by their indices.

132

1. IV. Lmear Algebra with Determmants

that is, the (n-v, n-v)-I'owed matrix generated by striking out the rows i 1, ••• , i. and the columns k 1 , •• ,k. in A; a{i" •••• i,.), {k, •.•• ,k.}

= I A{i" •• .,i,.}, Ik" ... ,k,.} I

the 8ubdetermin(tnt or 'minor of v-tlt degree or (n-v)-th O1'(Zerr of A. !X{i" ... ,iv), {k, •..• ,k.}

= (_1)i,-r· '+i,.+k,+ .. ·+k9 I A{i" .. ,Iv" (k" ... ,k.) I = (_l),,+ .. +,.+k,+ ... +k. a.('.+1> ., 'n,. k.+1, .. ·,kn)

the cofact01' of (n - v)-tlt degl'ee or v-tit m'del' of A,. cmnplemen;ta'I'Y determinant, algebraic complement or coj'actm' of

For the limttmg cases v = 0, alld v = n, we ronsider' e and A

I as

the only mtnors and cofactors of O-th and n-th degree,

respectwely. Therefore, the capital letter's denote matrices; the corresponding small letters, their determmants. The Latin letters indicate that only the intersections of the rows and columns specified by their indiees are retained; the Greek letters, that these rows and columns are deleted, therefore, that only the intersections of their complements are retained. The degree indicates the number of sequences still left; the order, the number of sequences deleted. For the especially important limiting case v = 1, we simply write A"" a,k' A,A' fl,A for the A W,lJl) , ••• This is admissible for the a,le since the A,,,, and consequently also their determinants, are actually the elements a,le of A.

Furthermore we stipulate: Definition 40. Let the assumptwns of De/. 39 be valtd and the (:) combinations of v-th order of the numerals 1, ... , 1t be set somehow in a definite order. With respect to one and the same such ordering

{il , .•. , i.} zs regarded as a row index and

{k1' ... , kv} as a column index and accordmgly the (:) (:)

18. MUlors and Co/actors

133

mmors of v-th degree a{i...... i.]. {lIlY •••• k.} of A are combmed mto an ( (:), (:) )-rowed matnx AC,), ahd simIlarly the (:) cofactors of v-th order

"'(t...... i,},{I:» .... .t,l

C)

of A wto an( (:). (:)) -

rowed matrix ACv). Then ACd ~s called the v-tlt llel"il'ed 'mail'ix and A (v) the v-th oomplelnenta'l'Y matl-[X of' A or the complementary '1ntWJ-lx of A(v); the latter refers to the fact that the terms of ACv) 1cere also called the complementary determinants of the corresponding terms of ....1('). In the case '"

= 1,

where the ( ~) combinations of 1st older

are simply the n numerals 1, ... , n, let theIr natural order be taken as a basis for the formation of ACI) and ACI). Then A(l) becomes the matrix A itself. Similarly, we write for All) simply A. The formation of this 1st romplt>mentary matrix of A, or, simply, complementary matrix of A, is arrived at hy the following rule: Let every element a,J.. of A be replaced by the determinant of that (n-l,n-l)-rowed matrix A,k which is obtained by striking out the i-th row and k-th column from A, and affix to this the sign factor (-I),+k. The distribution of the sign factors 1 and ·-1 can be visuahzed by covering the square array of A with 1 and - 1, just as a chessboard with black and white squares, starting with 1 at the upper left corner (at the position of au)' For the limiting cases v = 0 and v = n, according to the convention agreed upon in Def. 39, we set A (0) A(n) (e), AC"} = A(O) A I)' Th;s means, in particular, that A (e) is the complementary matrix of a (1, I)-rowed matrix A (an)'

= (I

= = = =

These concept formations, which seem very complicated at first, are introduced in order to be able to express as simply as possible the following theorem known as the Laplace Expansion Theorem. Theorem 00. Under the assumptions of Def. 39, 40 the formulae

1. IV. Linear Algebra with Determinants

134 J;

a(i" ... ,;.). {t, ..... k,,} IXli,..... i.). {I;b .... k,,} =

I A I,

J;

a{i, ..... i ..). {k" .... ",,} lX{i, ....,i.},{k1 , ...,k.}

=

IA I

{k, .... ,k.} {iI ,· •. Jiv}

are valid, in which the summations extend over all (:) combinations {kb ... , Ie.,} and {it' ... , i.,}, l'espectively, while {ii""

iv}

and {lei"'" kv} each ittdicate a fixed combination. In words, the determinant I A I is equal to the inner product of a row (column) of the v-th derived matrix A (v) and the corresponding row (column) of the v-th complementary matrix

A(v)

of A.

This implies that the 1X{i-" ...,i.i. (I;, ..... k.) are linked to the a{i, ..... i.l, 'k, ..... ".} . This is the reason for calling the former the complementary determinant of the latter and A(V) the complementary matrix of A(vl.

=

Proof: For the limiting cases v= 0 and v n the theorem is trivial in view of the conventions agreed upon. Therefore, let 1 :;;; v :;;; n - 1, that is, in particular, n> 1. It is then sufficient to prove the first formula of the theorem. For, thE' second formula follows from the first formula formed for the matrix A' through the application of Theorem 64 [129] to the determinants on the left and right. 'l'he proof of the first formula is based on a definite grouping of the summands in the formula used to define the determinant: I A I = 1; sgn P alp,' •• a"p,,' PinS"

For this purpose we decompose the group ~1"

•• ,

St(~) relative to

a subgroup

@in

into right cosets

lLt-{i1....,ito}

of index ( nv)

determined by the combination {i1"'" iv} and then carry out the required summation J; in the grouping PillSn

1:=1:+ ... +1:. PinS"

PIn~,

Pin 2<:)

18. Minors and Co/actors

135

Each such sum, extended over a coset, is then just one of the summands from the left side of the formula to be proved. the number of which is likewise The subgroup

C).

~{il'''''ip} of

en

of index (:) to be used is the

totality of all those permutations of 1, ... , Jl in whieh the numerals i l " ' " i, (and therefore also the remaining numerals i, + l' . . • , in) are permuted only among themselves. By Theorem 20 [64] this is surely a subgroup of En. Its permutations can be written in the form

where R=(:J

(~=l, ... ,y)

and

s=(;J (£=v+1, .... n)

run through all permutations of the numerals 1. ... , v and v + 1, ... ,11, respectively, independently of one another. namely. the groups e y and $n-, (the former for the elements 1, .... 1', the latter for the elements v -t-l, ... , n). The right cosets relative to ~{i" .... i,.}. and thereby the index of ~{il' ...,ip\ are determined as follows: Let Po be any permutation in @5". We can think of it as written in the form p o

= (~ ... i~i9+1

... in)_

~ •.. k. k'+l ... kn

Then the right coset ~{i,.....,i,,}PO generated by Po consists of the . I' totality of permutations (. P.-:.GB,SPo = ~1 "'~v '1~.+1 t t l ' .•

il '"

1,.

i )

"'1' .. " .+1·"

n

~ "':r',,~+l···t

2'11

8

i. i.+ 1

. i

. ) (.

... ~.n 1 .+ 1 " , •••

in)

= ( k., ... kr • k'.+l ... ken

'

The vertieal line indieates that the left and the right parts are permutations in themselves. 7

136

1. IV. Linear Algebra with Determinants

that is, of all and only those permutations for which the numerals of {i l , ••• , i,} go over in some order into those of {kl' ... , "'} (and conseq llently the permutations of {i,+t, ... , in} into those of {lcY+I" .. , k n }). This means that each combination {kl" .. , le,} corresponds biuniquely to a right coset ~{k" .... k.} relative to(;l;{tr, ...,i.}\ therefore, in particular, the index of (;l;{il'''''~') is equal to the number (:) of combinations of v-th order of 1, ... , n.

This partition of @;n relative to er(i" ... ,i.1 is nothing but the group-theoretic form of the inferences familiar from the elements of arithmetic for determining the number (: ). In fact, by the above the order of C§:{." .... iv}. is v! (n - v)!, the well-known formula

(~) =

sO

that by

Theorem 25

[68]

v I (nn I jJ) I follows.

1,Ve now consider that part of the sum

I A I which

corresponds to such a right above this can be written in the form

2:,

representing

Plll($n coset Sf'{k" ... ,kv )'

By the

1.: = 2: sgn (OR,SPO)U"k r,··· ai,kr.aiv+lks.+l'" Qinks , P!n~:k" .• ,k.) OR,S!n(!;,i,....,i.l n Dr

by Theorem 60

2:

pm~,k, ....,k.l

=sgnPo

[123] , and since Po is fixed in this sum,

l: sgnCR' su·" k 1, ••• a·.,k1. a·'v+l k8.+1··· a··n/<$n'

Rm($•• Sm($n_.

whereby corresponding to the structure of (;l;1i" ...,i.), explained above, the summation over all CR , s is written as the summation oyer all Rand S of the kind indicated above. Now, everything depends on the computation of sgn Po and a suitable splitting up of sgn CR , s into two factors corresponding to the two parts of CR , s. Here and in the following we can assume without loss of generality that the numerals of the four combinations

{il,···, iv}, {iY+l>"" natural order.

ill}'

{kl' .... h,}, {k.;+l"'" len} are in their

18. Minors and Co/actors

137

011 thf> one hand, the ordering of the numerals of these combinations does not enter at all into the formula to be proved. For, by Def. 39, it merely dictates the order in which the rows and columns are to be deleted in forming the minors and cofactors; whereas, the rows and columns of their underlying matrices always remain in the natural order. On the other hand, (llt" .... t~l is independent of the order of the numerals il>' .. , 'i, and ,i. _ 1 ' •.. , just as the classes ~{k1, ... ,kv) are independent of the ordering of these numerals and also of the numerals ku ... ,k, and k'-'-l' ... ,kJ., and in any such class the representative Po can be chosen so that k" .... k i and k., + l ' ••• , kIt are in natural order.

i,,,

1.) ComputatlOll of sgn Po We decompose

Pu

in) ~v t.+l 2n) (1 + 1 ... n k

= (~ ... i. i.+1 ls ... k. k.+1

= (il 1

0

0

0"

0

... ... kn 0

'jI

V

0

0

0

••

1 .••

V

~

+ 1. .. 1: ) = 1-1 K

k.kv+l

0

••

lin

and then by Theorems 60, 61 [123, 124] we have Since of

i1"'"

i~

sgn Po = sgn 1-1 ~gn K = sgn 1 sgn K. and i~ +1' ... , in are in natural order, inversions can take place only between

fl

numeral of {i l , ... , i,} and one from {i'~l'"'' Ill}' These lllversions can be enumerated by counting for each of the l' numerals i l , . . . , i,/ the number of numerals i, ~1' .•. , ill with which they form inversions. It is now obvious that it leads to inversions with the i l -1 numerals 1,. i l -1' belonging to the second row and only with these; similarly, i2 with the i2 2 numerals 1, ... , i2 -1 except i l belonging to the second row and only with these, ... ; finally, i, \\ ith the L - v nume00,

S In regard to the cases it = 1, i2 = i1 + 1, . .. cf. footnote 8 (95] in Section 12 to the Theorem of Toeplitz.

1. lV. Linear Algebra with Determinants

138

rals 1, .... i, -1 ex<.:ept it' ... , i , - I helonging to the second row. Consequently, sgn 1 = (_1)i,-1+i.-2+'''+tp-. = (_1),,+···+(·-(1+·,,+,). Similarly, it follows that sgn K = (_l)k,+ ••• +l:,-(1+"'+"). We have thereby shown that sgn Po =sgn I sgnK = (_l)H,,·+i,H.+ ...+k. = (-l)(i,k), where (i, k) is an abbreviation for il + ... + i" + 10:1 + .. + lev·

2.) Splitting up of sgn OR, s Since the two parts of On, s are permutations in themselves, OR, s permits the following splitting:

I

_ -(~ '" i, i,.+1 ... AB S ' .• ,

in) .

t,• ... i r • '/,'.+1'" t'n

_ 'tr•....•. i.. I'/,,+1 i,+1 ... in) (~ ... i, Ii.+ inj = a a .. ' +

(~ -.

•.

i r,

in

. , . 1 •• , . 'I.j, ••• '/,. t •• 1 .. · i,

R

S·

By Theorem 60 [123] we then have sgnOR,s =sgn OR sgnOs; furthermore, by Theorem 62 [125 J

n)

1 ." Jl V + 1 ... sgn AR=sgn ( r1 • •• r,. 'V +••• 1n ' ) sgn As=sgn ( 1 1 ... v v + 1. .. " .. ••• V 8.+1

.,. Sn

Now, it is obvious that inversions of the permutations on the right-hand side occur only between pairs of numerals from the sequence r l' •.. , ry or pairs of numerals from the sequence 8'1-1" ••• 8 n • Hence it is further valid that

v)

=sgn R , r,. sgn As=sgn ( V+1 ... =sgn S ,

sgn AR =sgn ( 1 ...

'1' •.

8.+1

therefore,

n)

••• 8n

agnail,s =sgnR sgnS.

lB. Minors and Cofactors

139

Together with the results of 1) and 2) this now implies 1: = (-1)
Pin.!i!k, ••..• k.}

1

•

•

v.

-=EsgnR ~k• ... ai.1:r (_1)(0.k) 1:sgnS a""+lk, Bin 15,

'1

Sm 15_.

•

-'-1'"

• .

"

ai,.k. ; ..

for, the termwise expansion and muitiplication of the two r, E yields the double sum }; first written. R In is. Sin is,,_v

R in is., Sin Gn-v

since the summations over Rand S The two factors rand (-1)(', I.) 1:

are independent. appearing in the

Sin En - v

Bin €5 v

last formula are now the determinants au" ... ,.."" {k" of the formula to prayed.

... ,k.;'

and

iX{i" ••• ,iv ), {k" ... ,kv}

For, on account of the assumption regarding the ordering of the numerals of our combinations, a"k,'" aivk• and a,.+lkp-'-l'" a,,,I:,, arE' the products of the terms in the principal diagonals of the matrices A{i" ... ,ivl.{k" ... ,kvl and A,i" .... i.l,{k" ... ,k.l Furthermore, the summations extend over all permutations Rand S of the column indices k" ... ,kv and k ,+1 , •••• ,kn in these products, where in every case the proper sign factors sgn Rand sgn S are attached. Finally, the proper sign factor (-l)U k) (_l)i,-'-"·+i v+k,+ ... +k v stands before the second sum.

=

Consequently, we finally have .

r

=a{·'1.,.

pm~(k" ........}

'OJ

.. ')'l&'lf· ". •• , ktiX(' VJ ~~ •••

J

., V"J.'" ". tv;,

,., , .,tt:Vi

and this implies the formula to be proved. because by the above remarks about the cosets we have

IAI

-

I PiniS n

-

r

{k" •• .,lev} Pin§{k" .... l;~}

The first formula of Theorem 66 can also be expressed as follows: Let a fixed row combination {it> ... , iv} of the matrix A be chose,n. From these 'V rows we can then cut out, corresponding to the (~~ combinations {k1, ... ,k) of the columns, l~) (v,v)-rowed matrices A{i" ••• ,i.),lk" ... ,kvl with the determinants a{i,." ..• i.',lk" ... ,kv\' To every such matrix there corresponds a complementary A :i" ...• iv'.lk" ... ,k.J< which is cut out from the complementary row

140

1. IV. Linear Algebra wah Determmants

combination by using the complementary column combination, or can also be obtained by striking out the rows and columns of A occurring in A {tl, . ..•,tPl'l.. . 'k1)"'1 k VJ,and whose determinant, to which the sign factor (-1) h+···+i.+k,T· .. +k. is attached, is the algebraic complement IX{i" ••• ,i.}, {k, .... ,k.l to a{it, ...,;.}. {k" .•. ,k.}' Now, by running through the system of rows {iu ""\} with a{i" ...".),(k" ...,k.l and adding all products of the minors a .. , .. with their algebraic complements a .. , .. , we obtain the determinant j A I. Similarly, if the roles of rows and columns are interchanged in this rule, the second formula of Theorem 66 is obtained. These formulae are also called, in this sense, the expansions of the determinant I A j by the minors of the Rystem of rows {il , . . . , iy}, or of the system of columns {kl' ... , ky}' respectively. In the particularly important case 'lJ = 1 th€ formulae of Theorem 66 become

n

(1 )

I A 1= 1:a,lcO!.,k 1:=1 n

(2 )

I A I = 1: a,kIXil.;

(i = 1, ... , n), (k = 1, ..., n).

0=1

(Expansion of I A I by the elements of a ?'ow or colU1nn, respectively.) By (1) or (2) the evaluation of a determinant I A I of n-th degree is reduced to the evaluation of n determinants of (n -1)-th degree (say aw ... , a ln ). The implied recursive process for computing determinants can occasionally be used in applications.

19. Further Theorems on Determinants First of all we will derive some conclusion::; from the special case v= 1 of the Laplace Expansion Theorem. Namely, this theorem immediately yields the following fact, which can also be immediately read, by the way, from the formula used to define the determinant (Def. 38 [127]). Theorem 67. The determinallt 1A j of all, (/I, H)-rowed matrix A (n > 1) is linea?' and homogeneous in the elements of each row (column) of A, that is, to be more exact, it is a linear form

19. Further Theorems on Determmants

141

of the elements of anyone row (column), whose coefficIents are determined only by the elements standing ill the remainmg I'OU'S (columns).

By applying Theorem 44 [86] rule used frequently:

this immediately pelds the

Theorem 68. If the (n, n)-rowed matrices --1, AI ..... A", (n> 1) coincide in n -1 corresponding rou'S (columns), while the remaining row (column) 11 of A is the linear combination

a = E'" Cia. 0=1

of

the correspondmg tows (columlls) Ill •••. ' 11m of AI' ... , A.m. then the determinant I A I is the same lil1em' combinat!OlI m

lA 1= Ec.! Ad i=1

of

the determinants IAII, ... ,: Am I.

In particular, we frequently need the special case 'm = 1 of this theorem which says that A itself is multiplied by e if the elements of a row (column) of A are multiplied bye. Accordingly, it further follows that [cf. Section 10, c), (3')] ! eA I = ell i A I, that is, ca,k en a,,. (i,k 1, .. . ,n).

I I

I

I= I I

=

Finally, for m = 1, c1 = 0 we call special attention to the theorem (naturally also valid for n = 1) following either directly from Theorem 67 or from Theorem 68. Theorem 69. If all elements of a row (column) of A are zero, then I A 1=0. All these facts can also be read directly from the formula used to define the determinant (DeL 38 t127]). Further, in the special case v = 2 oi the Laplace Expansion Theorem we next derive the important result: Theorem 70. If two roU's (columns) of an (n, n)-rowed matrix A (n > 1) are the same, then ! A 1=0.

1. IV. Linear Algebra with Determinants

142

Proof: By the Laplace Expansion Theorem the determinant I A I is also linear and homogeneous in the minors of a pair of rows or columns. Hence it is sufficient to show that all determinants of second degree are zero if they ale formed from a pair of rows or columns which are alike. This immediately follows from the formula used to define determinants, for

according to this any determinant of the form [ :

~

l I~ ~ or

tis

equal to ab - ab = O. As a rule the proof given for Theorem 70 is based on Theorem 65 [130]. It runs as follows: By Theorem 62 [125] a permutation of 1, ... , n interchanging only two numerals, namely, one which can be written in the form (~1 ~2~3'

•• t2~1~3'"

~,,), is odd. Hence by intertn

changing the two rows (columns) that are alike it follows by Theorem 65 [130] that 1A [ =: - I A [, that is, 1A I + I AI=: 2 I A = O. In general, however, this does not imply that tAl 0, since, for example, in the field of Section 1, Example 4 (and also in any of its extension fields) e + e =: 2 e=:O but still e =F O. This simpler proof usually answers the purpose only because we restrict the ground field to numbers, for which this conclusion is admissible. The proof given above, by means of the Laplace Expansion Theorem, is theoretically more exact, since it is valid without restriction. (See, however, also Vol. 3~ Exer. 11 to Vol. 2, Section 4.) 1

=

By means of Theorems 68 and 70 we next prove the following theorem which is basic for the application of determinants to systems of linear equations: Theorem 71. If the rows or the columns of an (n, n)-1'owed matrix A are linearly dependent, then IA 1= o. Proof: For n= 1 the theorem is trivial. Let n > 1. Then, by a') in Theorem 38 [80] at least one row (column) is a linear combination of the remaining rows (columns). Hence by

19. Further Theorems on Determmants

143

Theorem 68 the determinant I A is a llllear combmation of those n -1 determinants which arise if the IO\r (column) of A in question is replaced successi\'ely by one of the remaming It -1 rows (columns). But, by Theorem iU, these 11-1 determinants are zero; therefore, their Imear combination A is also. For the practical applications (evaluation of determinants) it is convenient to state Theorem 71 also in the following form: Corollary. If the matrix B is fanned from the (n, n)-l'owed matrix A (n> 1) by adding to a row (column) of A a lineal' combination of the remaining rows (columns), then B A.. Proof: I B i is then the linear combination A I + c1 I At I + ... + C"_1 I A"_1 I, where AI"'" ( A,,_1 i designate the determinal1ts occurring in the proof of Theorem 71, which are all zero.

=

i

Finally, we apply Theorem 70 in order to prove the following extension of the Laplace Expansion Theorem: Theorem 72. Under the assumptions of Def. 39, 40 [131, 132] the formulae

q}

if {k1 , ••. , kv} ={~,., " if {klo"" k,,} =1= {k~, . , ., k~} are valid. In words: The inner product of a.row (column) of the v-th derived matrix A(v) ana a row (column) of the v- th complementary matrix A
A:

144

1. IV. Linear Algebra with Determinants

we can Furthermore, by 'rheorem 64 [129 J l133]. restrict ourselves to the Iirst formula. By the Laplace Ex· pansion Theorem (Theorem 66), already proved, the sum on the left-hand side in the first formula of Theorem 72 can be regarded as the expansion of the matrix Al by the row combination {i1•••• , i~ }, where A1 is formed from A by substituting for the n - v complementary rows i V +I " ' " in the n - v rows of A from which the coIactors iX{ir••••• i;.}.fk" ..•• k.} are formed, namely, the row combination {~:+l, .. ') t~}. However, since {iI' ... , iv } =1= {i~) ... , i~ } Implies that at least one of the ill ...• i I differs from all i;, .. , i~, namely, is equal to one of the i~ +1' ... , i~, Al contains at least two rows which are alike. Therefore, by Theorem 70 IA1 1=0, which implies the first formula. We note here the corresponding extension of the formulae Section 18, (1) and (2) relative to the special case 'P 1: t& A if i = (1 ) I alk~Ik = 0' l'f i..J... i' ,

{I I

k-1

(2 )

X~t%..c' =

t=l

'

{I ~" '

il}

=

T

if k=k'} if k=j. lei •

20. Application of the Theory of Determinants to Systems of Linear Equations in the Case m=n The theorems developed in the preceding will now be used to derive anew the complex of theorems of Chapter III about systems of linear equations and, besides this, to fill up the gaps in the linear algebra without determinants which were called to our attention in Section 15. In so doing we will, for technical reasons, make no use of the theorems or Sections 13, 14 obtained by means of the Toeplitz process. However, the developments of Sections 10, 11 preceding this process are regarded as facts

20. Applicatwn of Theory of Determinants to m=n

145

based on elementary proofs, and sll.lll seI\ e as a ba"lS followmg expositlOl1S.

fOl

the

vVe begin with the consideration of systems of linear equalions (J) and eH) with an (n, n)-rowed matrix A, since this case naturally adapts itself to the methods of determmants. The results of this sectlOn relative to this case are indispensable for the derivation of the results in the general case considered in the following two sections. This is the reverse of the treatment in Chapter III, where the case m = n could be handled only by specializing from the general case. First of all the result contained in Theorem 71 [112) immediately says: 'rheorem 73. (Theorem (52,53, 5i)a [111)). The system of equations (H) with an (n, n)-rowed matrix A and Its transpose (H') canJlot be solved if the determinant i A I =l= O. Theorem 71 [142] or 73 state the following in regard to the concept regular introduced in Section 14: Corollary. (Def. 33 [111]). If I A I =F 0, A is regular. One of the principal results of the theory of determinants is that if A is regular, then conversely A =F O. Therefore, the alternative of Section 14, which was at that time reducible only to the inapplicable, from a practical point of view, disjunction A regular 01' A singular, can be decided in a practical way by means of the equivalent disjunction A =F 0 0)" A j = O. However, to prove this converse we nE'ed the general theory of the following section.

I i

I I

I I

Furthermore, by means of the special case v = 1 of the extended Laplace Expansion Theorem ( Theorem 72 [142] ) we can prove relative to (J): Theorem 74. (Theorem (46, -18, 49)a [111J. Theorem 55 [112] and Def. 34). The system of equations (J) with an (n, n)-rowed matrix A can be solved uniquely for any vector l*

146

1. IV. Linear Algebra with Determinants

the right if I A =l= O. In such a case the system has a uniquely dptermined resolvent matrix A*, namely.

011

1

A*_I AI_1 A .D..

f

.l1

__

(lXl;i) IA 1

(.1.,

k = 1, ... , n) ,

where A is the complementary matrix of A. Proof: The theorem is trivial for n 1 (cf. the remark in connection with Def. 40 [132) ). Therefore let n> 1. a) The matrix A* specified in the theorem, which can actually be formed since IA I =1= 0, is the resolvent matrix of A. Namely, if it is used with the to form the elements

=

x;

=i:l TAl n

XI;

lX~

(k = 1, ... , n),

xt

9

then by substituting these x" in the left sides of (J) and interchanging the order of summation we obtain n

n n E O-i'l;lXii; t" 'r' 'r' lX,1; * 'r' * .0= 1 .tJ a~'kX.o =.tJ £ti'k.tJ -AI1Xi =.tJ Xt --:1-""'--'-1n

~1

n

~1

~1.l1

_1 n

By Section 19, (1), however, £ = i'

=i

.0=1

(if = 1, ... , n).

.D..

IA I or

= 0 according as

or i' =1= i. Therefore, it follows that n

E O-i7,Xk = Xt-

&=1

(if = 1, .. 0' n),

hence, (J) exists for the above [Ck. b) The solution of (J) indicated under a) is the only one. Namely, if;; is a solution of (J), therefore, n

E a,kXk =:r.t

(i = 1, ... , n),

k=1 9 Cf. footnote 5 [91] to (H') in Section 11. The way of indexing chosen here is more convenient for the following substitution than the one used in Section 14 (:1:" fCi and accordingly a.lei) , which immediately calls attention to the fact that it is the transpose matrix A' = (o.ki) with which we are operating, and not A = (Iltk)'

20, Application of Theory of Determinants to m=r.

147

then on multiplying the i-th equation by the cofactor of 1-st order (J.ll.' and summing over i we obtain

" E" tXile' E aile Xic i=l

= E"tX,Ic' xt i~l

Ie=l

(k'

= 1, ' , " n),

therefore, by interchanging the order of summation n

n

"

E Xk E a,ktXik' k=l ,=1

= ;=1 E tX,k' xi

(k' = 1" , " n),

n

But, by Section 19, (2) the

E' on the left =' A

,=1

I

or = 0

according as k = k' or k =1= k', Hence this system of equations simply says

I A I Xk' = or, since I A I =1= 0,

n

E tXile' xi

;=1

(k'

= 1, ' , " n)

(k'

= 1, , , " n),

n 1X,ic'

;~1 fAi xi

Xk'

therefore, ~ is identical with the solution under a), c) That A* is uniquely determined follows as in the proof to Theorem 55 [112], Point b) of this proof could have been settled just as in the proof to Theorem (46, 48, 49) a [111] , Namely, we could have used the fact that if A =f= 0 then by Theorem 73 the associated (H) cannot be solved and consequently by Theorem 46 [91] (J) can be solved uniquely, However, the course chosen here is significant for the actual construction of the solution 6' It amounts to the method of multiplication already mentioned in Section 17 [128], Indeed, by b) the cofactors a,k (i 1" '" n) of the elements of the k-th column of A are precisely the quantities by which the n left-hand linear forms Ii must be multiplied in order that the

I I

=

n

linear combination :E lXikfi "'" I A I xlc should contain only one inde;=1

terminate xk' Moreover, it should be pointed out that the calculation of the solution 6 by means of the method of multiplication b) does not yet say theoretically that 6 is actually a solution; instead, it

1. IV. Linear Algebra with Determinants

148

says that if a solution exists, then! is the unique solution. Point a) of the proof is therefore theoretically indispensable. 10

*

If I A [ 0, the solution of (J) always exists and is uniquely determined. In this case a general system of formulae can be established for the solution of (J) on the basis of Theorem 74. It is known as Cramer's Rule: Theorem 75. Let (J) be a system of equations with (n, n)rowed matrix A and IA I=1= O. Then for any vecior 6* on the right there exists a uniquely determined solution 6 of (J) given by the quotient of determinants

I A(k) I

xk=UT

(k=1, ... ,n),

where the matrix A (I.) arises !1·om A by replacing the k-th column of A by the vector !*. Proof: The theorem is trivial for n = 1. Therefore, If't 1t> 1. 'l.'hen by rrheorem 74 this solution is 11

k(x'""x't Xi: =,=1 IAI

(k =1, ••. , n).

By Theorem 66 (133] (Section 18, (2» the ~ appearing in • determinant this expression is simply the expansion of the IA{k) I of the matrix A(k), specified in the theorem, by the elements of its k-th column. By Theorems 73 to 75 only a half of the results of Section 14 have been regained. Our exposition still lacks the converse statements. First, we have not yet shown that for IA I = 0, on the one hand. (H) and (H ') can be solved and, on the other hand. (J) cannot be solved unconditionally and uniquely; secondly, we have not established the further statements of Theorem 55 [112] 10 Unfortunately this fact is too often overlooked or not suffieiently stressed when solving equations in elementary mathematics.

21. The Rank

0/ a Matrix

149

about the resolvent matrix A". All this will be established also from the present standpoint by the literal transfer of the proofs in question of Section 14 as soon as the converse of the above corollary has been proved, that is, the equivalence of the statements "A is regular" and" I A I =1= 0." At first it could appear that this converse, as all previous statements, 11 could be deduced from the special case v = 1 of the extended Laplace Expansion Theorem by saying: If A = 0, then the formulae Section 19, (1), (2) represent linear dependences between the rows and columns of A, and therefore A must be singular. However, this is not possible; for, as it is easy to verify by examples, the coefficients a,l-. in each formula could all be zero, so that in such a case these formulae would not yield linear dependences. The developments of the following sections will show that the converse in question lies deeper, namely, that its proof must take into account the genej'al case of the extended Laplace Expansion Theorem.

I I

21. The Rank of a Matrix The developments (and announcements) of Section 20 show that it is not the determinant A itself but only the alternative A =1= 0 or A I 0 which is significant for the solvability of systems of linear equations with the (n, n)-rowed matrbc A. Therefore, for the application to general systems of linear equations it does not seem necessary to extend the definition of determinants (Def. 38 [127]) to (m, n)-rowed matrices A. Instead, it seems to be a question of finding the proper generalization of this alternative. Now, the alternative in Section 14, in fact equivalent to it, arises from the disjunction 0 :;:;; r < n or 0 < l' = n made there, therefore, from an ., (n + 1)-ative" in which n possibilities are collected together. Consequently, here we will have to try to split up the alternative A 1=1=0 or A = 0 into an (n + l)-ative expressible for any matrix A. This is accomplished by defining the above number r in tel'ms of determinants.

I I

I

I

I i

=

I I

11 Except for the application of the case v Theorem 70 [141].

= 2 in the proof to

150

1. IV. Linear Algebra with Determinants

In order to obtain a more convenient nomenclature we first generalize Def. ~9 [131]: Definition 41. By a minoJ' of v-th degroo of an (m, n)-r'owed matrix A, where 0 < v ::s; rn and 0 < v ::s; n, we understand the deter'minant of an (v, v)-r'owed matrix obtained by striking out In - V rows and n - v columns from A. As in Section 1:.-!, Lemma 1 [96], corresponding to the existence of the maximal number l' of linearly independent rows of A we now introduce here the existence of a maximal degree Q for the minors of A different fr'om zero. Definition 42. By the rank Q of a matrix A we understand 1) the number 0 If A = OJ 2) the greatest number among the degrees of the minors of A different from zer'o if A O. In order to prove that the rank e of A so defined coincides with the number l' occurring in Chapter III, we first establish three lemmas about e. Through these a series of self-evident properties of the maximal numbers l' and 1" of linearly independent rows and columns of A will turn out to be also valid for Q. 'rhe numbers r 9.nd 1" are in fact equal to each other. Def. 41, 42 immediately imply

*

Lemma 1. For the rank

Q

of an (m, n)-rowed matrix A the

relations

o::;; e::;; m, (! ::= 0

0::;; fl < n,

is equivalent to A

= 0,

are valid. Furthermore, by applying Theorems 64, 65 [129, 130J to all minors of A we immediately obtain: Lemma 2. If the matrix A has rank e, then the transpose A', as well as all matrices deducible from A by row and column permutations, have the rank Q.

21. The Rank of a Matrix

151

Fmally we have: Lemma 3. If the (m, n)-rou·ed matrix A. arises tram the (m + 1, n)-rowed [em, n + i)-rowed] matrix .A1 by stl'lking out a row (column) lmearly dependent on the remaimng, then A and Al have the same rank. Proof: By Lemma 2 it IS sufficient to proye the theorem for the rows. Let a1• ••• , am be the rows of A and a the surplus row of A1 whose deletion gives rise to A, and which is by assumption a linear combination of a1,· •• , am. Furthermore. let Q be the rank of A, Ql that of A 1• If Q= 0, by Lemma 1 it also follows that !h = 0, smce it must then be valid that all a, = 0, and consequently a = 0. If Q> 0, A has a minor of Q-th degree different from zero. Since this is also a minor of A 1, it must also be true that Q1 ~ Q. Now, if Q1 > Q, there would then exist a (Ql' Ql)-rowed matrix .~ obtained by striking out rows and columns from A1 such that I ...11 I 0. We will show that this is impossible. Either, we have Q= n, so that the possibility of cutting out a (Ql' Ql)-rowed ~ from the (rw+ 1, n)-rowed Ai is irreconcilable with the assumption Q1 > Q (=n) (Lemma 1); Or {l < n, so that a ((>1' Ql)-rowed matrix ...11 can at any rate be cut out of A1 with Q1 > Q. In this case there are only two possibilities: a) A1 contains no part of the row a. Then Al can even be cut out of A; therefore I ~ 1=0, since it would be a minor of A

*

of degree !h > e. b) ~ contains a part of the row a. Then, by Theorem 40 [81J the part in A is a linear combination of the corre::;ponding parts of a1, ••• , am. Now, by Theorem 68 [141] .4;. i can be linearly composed from the determinants of such (Q1' (h)rowed matrices as consist of the corresponding parts of the

1. IV. Lmear Algebra wuh Determmants

152

rows of A. However, these determinants, and consequently IAll, are zero, since their matrices either have two identical rows or, if this is not the case, are minors of A of degree Ql > Q. This shows that a (Ql' Ql)-rowed Al with Ql > Q and I ~ I=1= 0 cannot exist. Therefore, Ql > Q cannot be valid, that is, Ql = Q. We now prove that Q is equal to rand r'. Theorem 76. (Theorem 53 [109]). The rank Q of a matrix A is equal to the maximal number r of linearly independent rows and equal to the maximal number r' of linearly independent columns of A. In particular, therefore, r = r'. Proof: By means of our Lemmas 1 to 3 we first reduce the statement to be proved to its bare essentials by making the following four statements: 1. It is sufficient to prove the theorem for the rows (Lemma 2). 2. It is sufficient to assume A =1= 0, therefore, r> 0, Q> 0 (Lemma 1). 3. It is sufficient to a'3sume12 that the minor of A of Q-th degree different from zero, which exists on account of 2, is the minor formed from the first Q rows and columns of A (Lemma 2). 4. It is sufficient to assume that the rows of A are linearly independent (Lemma 3). For, if AD is the matrix consisting of a maximal system of r linearly independent rows of A, which exists on account of 2 and Section 12, Lemma 1 [96], then by Section 12, Lemma 2 [97] all remaining rows of A are linearly dependent on the rows of Ao' But, by Lemma 3 (and Theorem 42 [821) the successdve deletion of the remaining rows of A leaves the rank unchanged. Hence the statement amounts to proving Q r for Ao'

=

it

This is only a matter of notation.

153

21. The Rank of a Matrix

=

Accordingly, let A (a.d be an (1', n)-rowed matrix with linearly independent rows of rank Q for which the minor IX I a.k I (i, k = 1, ... , Q) formed from the first Q rows and columns is dIfferent from zero. Then we make the first Q rows of A into an (n, n)-rowed matrix by the addition of n - Q ~ 0 (Lemma 1) rows as follows:

=

l~~ 0

ale al.a+l aee ae.e+l 0 e

0

0

::\

or e

Regardmg the determinant of ~4 we have that I X 1=*= O. For. If Q n, therefore A A, then we obviously have A = IX =*= O. However, if Q < n, then on expanding .J by the minors of the last n - Q rows it also follows that IA I= IX =*= 0, since by Theorem 69 [141] only the minor corresponding to the last n - Q columns (with the cofactor IX) is different from zero. namely

=

=

=e.

<

Now, by Lemma 1 we ha\'e at any rate Q ;:;;;; r. If Q r were true, therefore A would contain r - Q rows (all"'" a,n) (i=Q + 1, ... ,1') not in A, then by Theorem 74 [145] and [129] each of the l' -!.> systems of linear Theorem 64 equations alla; a~lxQ + OxlI +! Oa:n • a.1.

+ ... + +... + ~Qa; + ... + aoeXe + Oxe+! + ... + Ox,. ...:.. floe aI.a+la; + ... + ae•e+1 Xe + Xe+l + ... + Ox,. ...:.. lZi.e+l ~,.a; + ... + aQnxQ+ 0.1:e+1 + ... + z,. . a..,. (~=e+l, ......)

with the matrix .if could be solved, that is, each of these rows would be a linear combination or the n rows of A.

l' -

Q

1. IV. Lmear Algebra with Determinants

154

Now, in case Q = n (which is compatible with the assumption Q < r only in the case l' > n) the existence of these r - Q systems of equations would indicate that the last r - Q rows of A were linear combinations of the first Q, which contradicts the assumption that the rows of A are linearly independent. We arrive at the same conclusion for Q < n by proving that in the solutions (X I1 ' ••• , XIII) (i = (} + 1, ... ,r) of these systems of equations the last n - Q unknowns x"P + II " ' , Xln are equal to zero. In order to show this we think of the solutions (x I1" ' " xm) li = Q + I, ... ,1'), which are uniquely determined by Theorem 74 [145] and can be obtained by Cramer's rule (Theorem 75 (148]), as written in the form

I~) I a;1=

I.I~) I

1.£ 1=-lil

(~ =

(!

+ 1, ... , r).

1=1, ... ,n

The matrices A(I) , arise from A by replacing the i-th row in A by (all' .. " alII)' Now, if i is one of the indices Q + 1, ... , n in question, then M;) accordingly contains Q + 1 different rows • of A. If Al') is then expanded in terms of these Q + 1 rows and it is borne in mind that all its minors, as minors of (Q + 1)-th degree of A, are zero, it follows that IA!,) therefore $11=0 (i=e + 1, ... , rj i=Q + 1, ... , n). In accordance with what has already been said this means that the impossibility of Q< r is proved. Consequently, we have Q = r as stated. Theorem 76 iust proved immediately yields the part still missing of the results of Section 14, cited at the end of the previous section, in regard to the special system of equations with m n. Namely, we can next easily prove the following converse of Theorem 71 [142J: Theorem 77. If A is an (n, n)-1'owed matrix with I A 1=0,

1=0;

=

155

21. The Rank of a Matrix

then the ro'Ws as well as the columns of A. are linearly depf/ldent. Proof: If IA 1=0, then the rank Q < n, smce I A , is the only r.linor of n-th degree of A. Hence by Theorem 76 the 11 rows and the n columns of A are linearly dependent. Theorem 77 also says as a converse to Theorem 73 [14.5}: Theorem 78. ('l'heorem (52, 53, 54) a [111J ). The system of equations (H) with (n,n)-rowed matrix A and its transpose CH') can be solved if IA 1=0. Finally, Theorem 77 or 78 give the cont'erse of the Corolla1'y to Theorem 73 [145] mentioned in Section 20: Corollary. (De!. 33 [111) ). If A is reg1tlar, then: A I =f O. Consequently we have Theorem 79. (De!. 33 [111]). The alternatives "A regular 01' A singular" and " I A I =l= 0 or I A I = 0" are equivalent.

As already stated at the end of Section 20, the following converse to Theorem 74 [145] as well as the further statements based on it can now be regarded as proved from the present standpoint: Theol'em 80. (Theorem (46,48, 49)a l111] ). The condition I A 1*0 is also necessary in order that the system of equations (J) with (n, n)-rowed matrix A have an unconditional and unique solution. Theorem 81. ( Theorem 55 [112] ). If i A 1*0, so also is the determinant I A* i of the resolvent matrix A*= IA 1-1 A', that is, the determinant I A I = ! A In I A* 113 of the complementary matrix of A is also different from zero and (A>Io)* A..

=

This easily yields that the complementary matrix A of the complementary A of A differs from A only by a factor, namely, that 18

1 = ~ A.

The determination of this factor, that is, the

Cf. the remark after Theorem 68 [141].

156

1. IV. Lmear Algebra wtth Determinants

I I,

evaluation of A can be carried out, howev€r, only by means of the calculus of matrices. It turns out that A * is simply the inverse A of A, so that A * IA Hence for A' A A-1 we get A = A A. 1 = I A In-1 • Moreover, the determinantsl of all derived and complementary matrices A(v), A(v) are also powe!l."s or A (Cf. for this Vol. 3, Section 14, Exer. 4; Section 19, Exer. 13; Vol. 2, Section 2, Exel.'. 30 to 32.)

-1

I I i 1"1 1-

I I=

:-1.

=I I

I I.

We have thereby established from the point of view of determinants all the results of Section 14 for the special case m n. In addition to this, according to Theorem 79 we are now able to come to a decision in a practical way regarding the alternative appearing there, that is, regarding the solvability of (H) or the unconditional, unique solvability of (J). Furthermore, Theorems 74, 75 [145, 148] provide us with a simple process for determining the solution of (J) in the "rE-gular" case (I A I=l= 0). In Sections 11, 13 the decision regarding the sdivability of (H) and the delineation of the totality of solutions of (H) and (J) was made to depend only on the number r to be determined. Our Theorem 76 says for the general case that this number r can be found as the rank Q of A in finitely many steps, namely, by the evaluation of all minors of A. Moreover, this means that a maximal system of linearly independent rows (columns) of A can also be determined in finitely many steps, that is, the application of the first step of the Toeplitz process now becomes practical. Hence we can immediately deduce from Theorem 76 the following fact, which is to be applied in the following sections: Lemma 4. If A is a matrix of rank Q > 0, then every pa'ir of combinations of Q rows and Q columns of A to which corresponds a minor of Q-th degree different from zero yields a maximal system of linearly independent rows and columns of A.

=

22. Applicatton

0/ Theory of Determinants Generally

157

Proof: By Theorem 71 r142] the parts of the e rows (columns) of A under consideration going into any minor of

e-th degree are linearly independent; by Theorem 40 (81}. therefore, all the e rows (columns) are also; and by Theorem 'i6 they form in this case a maximal system of linearly independent rows (columns). Furthermore, Theorem 76 can also be thought of as establishing from a determinantal point of view Lemmasl,2 [103,104J proved in Section 13 by means of the partial application of the second step of the Toeplitz process. Consequently, this theorem also implies all inferences deduced from these lemmas in Section 13, that is, the existence of a system of fundamentril solutions of (H) (Theorem 50 [104]) and the invariance of the number of fundamental solutions (Theorem 51 [105] ). Hence. in order to completely reproduce the earlier results aU that is left for us to do is to prove from the determinantal point of view Theorems 49 [102] and 52 [106] about (J) and (H), which were deduced in Section 13 by using the second step of the Toeplitz process in its entirety. We will do this in the next section. We will then be immediately able to conclude that Theorem 54 [109] of Section 13 regarding (H) and (H'), which is the only one still not cited, is naturally also a consequence of Theorem 76.

22. Application of the Theory of Determinants to Systems of Linear Equations in the General Case The proof of Theorems 49 [102] and 52 [106] to be given using determinants will yield, besides the statements contained in these theorems, the explicit determination of the totality of solutions of (J) and (II), therefore the complete solution of both

1. IV. Linear Algebra with Determinants

158

p1'oblems J pr ) and H p .) cited at the end of Section 11, which the

developments of Chapter III were unable to do. For technical reasons it is advisable to consider (H) before (J). 1. Solution of Hpr)

The complete solution of Hpr) is obviously contained in the following theorem: Theorem 82, (Theorem 52 [106]). The system of equations (H) wzth (m, n)-rowed matrix A of rank Q possesses a system

°

of fundamental solutions of n - Q solutions. If < Q < nand, as can be assumed without loss of generality, 14 the ordering of the equations and unknowns is chosen so that the minor f01'med from the first Q rows and columns of A is different from zero, then such a system will be formed by the last n - Q rows of the complementary matrix A of the (n, n)-rowed matrix A from the proof of Theorem 76 t~l'

[152], therefore by the n -

. , ,,
(j, =

Q

vectors

e+ 1, ... , n)

from the cofactors of 1-st order of that matrix

A.

Proof: For Q= 0, namely, A = 0, the theorem is trivial (cf.

pl'oof to Theorem 52 [106]). Therefore, let Q> 0, so that A =1= o. By the assumption made in the theorem the first Q rows form in this case a maximal system of linearly independent rows of A ( Section 21, Lemma 4 [156]). As in Section 12 (first step) the system of equations (Ho) with the (Q, n)-rowed matrix Ao formed from the first Q equations of (H) is equivalent to (H), so that for the proof it is sufficient to use (Ho) instead of (H). Now, on the one hand, let Q= n, so that Ao is a (Q. n)-rowed matrix for which lAo i =1= 0 by assumption. Then by Theorem 73 14

This is only a matter of notation.

159

22. Application of Theory of Determinants Generally

[145] (Ho) cannot be solved and therefore has a system of fundamental solutions of 0 = e - Q = n - e solutions, as stated. On the other hand, let 0 < Q < n. 'rhen we have: a) The specified n - Q vectors are solutions of (Ho)' For, by the extended Laplace Expansion Theorem [Section 19, (1)], applied to the matrix A, we have n

(i =1, ... , e; i'

E Q,;,1ctxi'k =0 1'",,1

=(?

+ 1, ... , n).

b) These n - e solutions of (Ho) are linearly independent. For, by Theorem 81 [155 J , since IAi =f: 0, we also have IA I=f: O. Hence by Theorem 71 [1-i2] the n rows of A are linearly independent; therefore by Theorem 39 [81] so also are the last n - Q rows. c) Any solution ~ of (Ho) is a linear combination of these n - e solutions of (Ho). For, the system of equations

" -

~ lX,j:

.. '

(k=l, ••. ,n),

"'" --:::-X, =X1c i=1 I .A I

whose matrix is the resolvent matrix A* of A, has, since IAI =f: 0, by Theorems 74 [14.5], 81 [155J the uniquely determined solution n

:et = 1c=1 Eail;x1c

(i = 1, .•. , e)

xi= x,

(i=e+1, ... ,n)

with the matrix (..4*)* = A. Since, however, ~ should be a solution of (Ho), we have x~ , ... , xp = 0, that is, the formulae

I x~

iX.7c

i"'-e+ 1 I .A I

(= I

i=Q+l

~ iiiI:) =

I .A I

X1c

(k = 1, ... , n)

are valid. This means that ~ is a linear combination of our n - e solutions from a), b). a) to c) imply that the n - e vectors specified in the theorem form a system of fundamental solutions of (Ho), therefore also of (H).

l60

1. IV. Linear Algebra with Determinants

2. Solution of J pr)

The complete solution of J pr) is obviouSly contained in the following theorem: Theorem 83. ( Theorem 48 [92J ). Let (J) be a system of equations with (m, n)-rowed matrix A of rank Q. If the necessary solvability condition of Theorem 48 is satisfied, then (J) can be solved. If Q> 0 and the assumption of Theorem 82 is agaill made about the m'dering of the equations and unknowns, thell a solution of (J) is obtained by setting the unknowns x p +1' ••• , Xn = 0 in the system of equations (Jo) formed from the first Q equations of (J) (in case Q < n), and then determining Xl' ... , X p by solvill g the system of equations thereby obtained with a CQ, Q)-r'owed matrix whose determinant is different from zero. The solution of the latter system may be obtained by using Theorems 74, 75 [145, 148]. Proof: The theorem is trivial in case Q = 0 (d. the proof to Theorem 49 [102]). Therefore, let Q> O. By the assumption made in the theorem it again follows in this case by Section 21, Lemma 4 [156], as in Section 12 (first step), that ihe system (Jo) is equivalent to (J). That the process described in the theorem yields a solution of (Jo)' and consequently of (J), is immediately clear. By the remarks in Section 11 it follows that Theorems 82 and 83 ('ompletely solve the problem Section 5, (1) of linear algebra. In conclusion we will deduce two further rules which supplement Theorem 83 and are useful in practical applications. These will also enable us, on the one hand, to determine the totality of solutions of (J) directly, that is, without having to compute according to Theorem 46 [91] a system of fundamental solutions of (H) by Theorem 82, and, on the other hand, to come to a decision regarding the

22, Application of Theory of Determinants Generally

161

solvability of (J) directly, that is, without having to compute according to Theorem 48, Corollary 2 [93] a system of fundamental solutions of (H') by Theorem 82, Without further discussion it is clear: Corollary 1. Undel' the assumptions of Theo,'em 83 the totality of solutions of (J) in the case I? > 0 can be found as follows: In the sYBtem of equations (J o) appearing there replace the llnknMVlls xP+lI' , ., Xu by any elements ;0+1>",,1;. whatever (in case p < '11.); then by Theorems 74, 75 [145, 148] determine Xl"'" Xo by solring the resulting system of equations ' II

£~TcXTc=

1>=1

a,

+ £n (-a.Tc)~~ 15 1:=e+l

(i= 1" ,·,e)

with a (g, I?)-rowed matrix, whose determinant is different /l'01ll zero, To any arbitra1'y system ~Q+l"'" ~n there is thereby obtained one and only one solution of (J). Inciden~aIly, if CJ.ik designate the cofactors of the elements a,l. in that (I?, g)-rowed matrix (a.k) (i, k 1"", Q) and a I a,k its determinant, then by Theorem 74 [145] on interchanging the order of summation

=

Xi:

=

l!(Xik .. e"'i1: ." • =i=l £ - l l t + £ ~I £ - (- ail) = aTe + £ aklEl a l=e+1 i=l a l=e+1

(k= 1.... ,(1) becomes the solution of the system of equations in question, wmch together with xk 0 + ;k (k = I? + 1, •. " '11.) yields the general solution !J of (J). Here the solution bJ turns out to be automatically decomposed into two summands ~(~) and 68

=

according to Theorem 46

[91].

The first

~(~J

=

(a~,

... , a;,

OJ' .. ,0) is obviously the particular solution of (J), corresponding to ~e+l" .. ,~.. = 0, specified in Theorem 83; while the second !8 must accordingly represent the general solution of the associated (If). In fact the second summand seems to be a linear combination of '11. - g fundamental solutions as was the cage- in the proof to Theorem 52 [106].

=

15 For the limiting case- Q '11. cf. the statement in footnote 8 [95] to the Theorem of Toeplitz in Section 12.

162

1. IV. Linear Algebra wah Determinants

Furthermore we prove: Corollary 2. The system of equations (J) can be solved if and only if its (m, n)-1'owed matrix A has the sarne rank as the (m, n + I)-rowed matrix At arising from it by adjoining the column formed from the right side (at> ... , am) of (J). Proof: a) If (J) can be solved, then the column (a" ... , am) is linearly dependent on the columns of A. But, by Section 21, Lemma 3 [151] At has then the same rank as A. b) If (J) cannot be solved, then the column (at> ... , am) is linearly independent of the columns of A. If A = 0, then this column is different from zero, that is, the rank of At is equal to 1 while that of A is equal to O. However, if A =F 0, then by b) in Theorem 38 [80] this column together with a maximal system of Q linearly independent columns forms a system of e + 1 linearly independent columns. By Theorem 76 [152] the rank of At is therefore greater than the rank Q of A. Consequently, if A and At have the same rank, (J) can be solved.

Conclusion Dependence on the Ground Field To wind up our de\elopments we ask one further question. Ale the results of Chapter III and lV to be altered if we no longer, as has been done throughout the foregoing, require that the solutions Xi' •.• , Xn of the gi\'en system of linear equatIOns (J) belong to the field K but instead demand that ther merely uelong to any extension field K of K? Since the system of equatiOllS (J) can in this case also be regarded as a system wlth coefficients in K, our entire theory can also be larned ont using K as ground field. In doing so, the totality of solutions of (J) Is in general larger than in the case of K, since the freely disposable elements of K, occurring in the general solution of (J), can now be freely takpn from the more extensiw' K. ~ey­ ertheless, we have: Theorem 84. In regard to a system of linear equations (J) in K, the solvability or nonsolvability, the unique solvability as well as the number n - r of the elements freely disposable in the general solution are irwar'iant with respect to the passage from K to any e$tenswn f~eld K of K as ground field. Proof: If A is the matrix of (J), Ai the matrix specified in Theorem 83, Corollary 2 [162], and l' and 1'1 are the ranks of A and Ai' then by Theorem 83, Corollary :2 the solvability or

nonsolvability of (J) is equivalent to the relation l' = 1'i or r < r i , respectively, and by Theorem 83, Corollary 1 [161] the unique solvability of (J) is equivalent to the relation 1'=T1

163

=n.

164

Conclusion

But, the property of 3. determinant being zero or different from zero is independent of whether its terms are regarded as elements of K or K. Hence according to Def . 42 [150J the rank of a matrix is invariant with re-spect to the passage from K to K; therefore, on account of the self-evident invariance of n the above relations are also invariant. This means that the solvability or nonsolvability and the unique solvability of (J) as well as the number 11 - r specified in the theorem are invariant.

HIGHER

ALGEBR~~ BY

HELMUT

HASSE~

Ph.D.

Professor of Mathematics, University of Hamburg

VOLUME II Equations of Higher Degree

Translated from the third revised German edition by

THEODORE J. BENAC, Ph.D. Associate Professor of Mathematics, U.S. Nat'al Academy

FREDERICK UNGAR PUBLISHING CO. NEW YORK

Copyright 1954 by Frederick Ungar Publishing Co.

Prmted m the Umted States of America

Library of Congress Catalog Card No. 54-7418

Introduction Methodical Preliminary Observations and Survey In Vol. 1, Section 5 we formulated the basic problem of algebra guidmg our presentatlOn and called attentlOn to two especially Important subproblems. The fnst of these, the problem of solvmg a system ot Imear equatwns, was completely sol,ed in Vol. 1, Chapters III and IV. The present Vol. 2 wIll be concerned With the second of these subproblems Let K be a fIeld and l(x) = a o -t- alx an xn (all =!= 0, n ~ 1) an element of K [x] not belongtng to K. TV e seek to develop methods for obtmnwg all solutZOr1S of the al(Jeln'aic equation

+ ... +

rex) -=- 0. Since the conditions for solving the equation f(x) the same as those for solving f( x) an

-=- 0

are

-=- 0, there will be no loss of

generality if we restrict ourselves in the following to equations of the form rex}

= ao + alx + ... + an_

l

x n-

1

+ x"-=-O

(n ~ 1).

We call such elements f(x) of K[xl polynomwls (m x) in or ofl or Ot'el' K and the umquely determmed index n ~ 1 their degree l cf. VoL 1, Section 5, (2) [56]]. Furthermore, the solutions of an algebraic equatlOn t(x) -=- 0 are called, in accordance with the usual terminology, the 1'ootS of the polynomial f(x). 1 Strietly speaking, this is not correct, since the !(x) are elements of K[x]. Therefore, our terminology refers to the coefficients.

167

163

Introduction

The methods to be used to handle our present problem are basically dIfferent from those employed in VoL 1 to handle systems of linear equations. 'l'his is due to the following two closely connected facts: 1) There can exist (in contrast to Vol. 1, Chapter IV) no process formed from the lour elementary operations (briefly, ratwnal operatwns) defined in the ground field K for deciding the solvability of an algebraic equatIOn and jn the solvability case for calculating all solutions. 2) The solvability and the totality of solutions of an algebraic equation over K are (in contrast to Vol. 1, Theorem 84 [163]) dependent on the choice of the ground field, that is, on whether the solutions are to be considered as lying only in the field K or in some extension field of K. In general, algebraic equations over K can be solved only III suitable extension fields of K. To illustrate the latter general insight given in 2) we may cite the simple example of the equation X2 - 2 = O. Thue., this equution does not have a solution in the field of rational

V2

numbers but has the two solutions ± in the field of real numbers. Furthermore, 2) implies 1), for if a process, as cited in 1), were to exist, then this would be; as in Vol. 1, Theorem 84 [163] , independent of the choice of the ground field, which contradicts 2).2 Due to 1) our problem is not to be understood as one wherein the solutions of an algebraic equation shall be computed in 2 This should not be taken to mean that solution processes do not exist for special ground fields, for example, the field of rational numbers. However, such methods are no longer thought of as belonging to algebra, since besides the four elementary operations they must involve tools belonging to analysis. In this regard, cf. Section 11 [247].

lntroductwn

169

the above sense. 2) tells us what we are to striye for instead of this. In the first place, 2) imphes that for abstlact ground fields (that is, under exc1usl\re assumption of the conditions given in Vol. 1, Section 1) the extenslOn held is not at our disposal beforehand as was the case in the above example. where the real number field ,,'as assumed as known from elementary mathematics (foundations of analysis). In the second place, on the contrary, 2) implies that in the general case we have first of all no knowledge whatsoe\-er regarding the existence of extension fields which make the solutIon of an algebrall equation possible. Hence our problem amounts to that of constructing such extension fields and thereby the roots of algebraic equations. Accordingly, our presentatlOn will run as follov,s: In Chapters I and II we have to explain some preliminary facts, on the one hand, about polynomials over K formed from the left sides of algebl'aic equations and, on the other hand, the (for the time being, hypothetical) roots of algebraic equations o\"er K in extension fields; in Chapter III we will construct the root fields of algebraic equations and thereby their roots. "\Ye then can think of the above problem as solved from a practical standpoint (analogous to Vol. 1, Chapter IV: determination of solutions). From a theoretical standpoint the question. which is of very spedal interest to us here, is raised beyond this (analogous of Vol. 1, Chapter III: the structure of the totality of solutions) as to what can be said about the structure of the root field of algebraic equations, especially about its construction from components as simple as possible. This question. about which our considerations will be centered, will be handled in Chapter IV by the presentation of the so-called Galois theory, whereby the

170

Introduction

structure of these fields will be closely tied to the structure of certain finite groups, their Galois groups. Finally, in Chapter V this theory will be used to answer the question of the solvability of algeb,'aic equations by radicals, that is, the famous question: When can the roots of an algebraic equation be computed by including the operation of root e:xtraction (which, with a fixed ground field, is not defined without restriction and uniquely)?

I. The Left Sides of Algebraic Equations In Sections 1, 2 of this chapter we WIll successi....ely derive significant theorems about polynomials over K in connection with the developments of Vol. 1, Chapter 1. At first, these theorems will have nothing to do with the fact that the polynomials constitute the left sides of algebraIC equations; they will be linked to this fact only in the chapters to follow. The theorems themselves are exact analogues of the theorems in elementary number theory centered around the fundamental theorem of the unique factorization of integers into prime numbers; here they deal with the integral domain K[x] of the integral rational functions of an indeterminate x over the ground field K, whereas in elementary number theory they deal with the integral domain r of integers - just as the construction of the field K(x) of the rational functions of x over K from K[x] is exactly analogous to the construction of the field P of rational numbers from r, for in both cases the fields are constructed as quotient fields. Here we will not assume elementary number theory as known, which is to be frequently used later on. Instead, we will deduce the cited theorems for the two cases K[x] and r at the same time, that is, in terms of words and symbols bearing a twofold significance. Accordingly, in Sections 1, 2 f, g, h, ... will designate elements in K[x] or r. By means of the results of Sections 1, 2 relative to the case r we will then develop in Sections 3, 4 of this chapter some more concepts and facts about groups, integral domains and fields. These are important in the following and would already have been inserted at an earlier place (Vol. 1, Cl;tapters I and II) if we could have assumed elementary number theory as known.

171

1. The Fundamental Theorem of the Unique Decomposability into Prime Elements in K [x] and r A. Divisibility 'l'heory in an Integral Domain The fundamental theorem specified in the title presupposes for its exact formulation the concepts of the so-called divisibility theory in K[x] or r. Since this divisibility theory requires no properties of K[x] or r, other than that each is an integral domain, we will develop it for an arbitrary integral domain I. g, 11, ... shall then designate elements in I.

r,

Definition 1. 9 is said to be divisible by f or a multiple g or contained in g (notation fig,

of f, and f a divis()'J' of contl'ariwise,

f

'r

g),

if an

r exists so that 9 = rr.

It will naturally be required that T exist in I. The notation we have adopted permits us to omit this kind of additional statements, here and at similar places. However, it should be expressly emphasized that this is implicit in Def. 1. For, if the "existence" of such an element were also allowed to take place in the quotient field of I, then Def. 1 would be trivial except for the distinction between f i= 0 and f = O. Accordingly, the divisibility theory becomes meaningless if I coincides with its quotient field. This is not the case for K[x] or r.

From the basic properties of integral domains developed in Vol. 1, Section 1 we can immediately obtain the following theorems about divisibility. The proofs of these facts are so simple that we omit them 1. 1 Furthermore, we will indicate the entirely elementary proofs of a series of further theorems of Section 1 by merely making a reference to the earlier theorems that are involved.

172

1. Fundamental Theorem of UnIque DecomposabIlity

173

Theorem 1. The divisibiltty 1'elati01/s

ell, 1I1, flO f01' every f o-r f for 1=l=0 are valid.

11 g, g 1h implies hf 1hg, h =f: 0 implies fig.

1I h; fl

Theorem 2.

flf21 glg2;

Theorem 3. f 1 gl' f I g2 implzes f I glgl

gl'

+ glii2

12

g2 implies

1'01' arbitrary

iil' ii2' Definition 2. 1 is called a unit zf fie. In the following we designate units by a. b. For instance, e is a unit. Theorem 4. The units of I form a subgroup (normal divisor) of the multiplicative Abelian group of all elements =f: 0 of the quotient field of I. Pt'oof: all e. a 2 1 e implies a1 a2 e (Theorem 2): also e I e 1

(Theorem 1); a I e implies that~ belongs to I and~1 e (Def. 1). a

a'

The theorem follows from this by Vol. 1. Theorem 19, 26 [63, 69] (d. also Vol. 1, Section 6, Example 1 [61]). Definition 3. If f l' f2 are different from 0 and congruent relative to the normal divisor of units, that is, if

k=

a, then

11

and f2

are said to be associates. The residue classes relative to this normal divisor are called the cZasses of associates. This means that the class of elements associated to an element

I =!= 0 consists of all al. where a runs through aU units. For 1=0 the totality ai, that is, the single element 0, may likewise be regarded as the class of associated elements belonging to I. - In the sense of Vol. 1, Sections 7 to 9 the partition into residue classes relative to the normal divisor of the units extends not only to the integral domain I but also to its quotient field. Here, however, we consider it

174

2. I. The Left Sides of Algebraic Equations

only in the integral domain I itself. We can do this all the more as the class corresponding to an f in I belongs entirely to I.

Definitions 1 to 3 immediately imply: Theorem 5. fl and f2 are associates if and only if f1 [f 2 and

f21 fl' By Theorems 2, 5 a divisibility relation fig is equivalent to any relation f' i g', where f'is an associate of f, g' of g. Therefore, for the divisibility theory it is sufficient to consider only a representative from each class of associated elements; however, for general I it is not possible to distinguish one such representative by a universal principle (cf., however, Def. 7 [177]).

According to the preceding each 9 is divisible by all units and all elements associated to g. These divisors are called the trivial divisors of g. In order to exclude these conveniently, we formulate

Definition 4. fts said to be a pi·ope,. divisor 01' 9 if fig but t is neither a unit nor an associate of g. The fundamental theorem to be proved rests on the following definition: Definition 52, p is called a prime element if it is neither zero nor a unit and has no proper divisors. Dei. 5 does not say whether such prime elements exist. Furthermore, it cannot be decided without the addition of further assumptions regarding I. For instance, if I coincides with its quotient field, there are no prime elements.

B. Absolute Value in K[x] and

r

In order to be able to prove the fundamental theorem (which is not universally valid in integral domains) of the unique decomposition into prime elements in K[x] and r, we must have 2

Ci. also Def. 8 [178] to be added later.

175

1. Fundamental Theorem of Untque DecomposabLlity

recourse to special properties of these integral domains; namely, in r to the ordering of the integers by their absolute value, whose rules we here assume 3 as known, m K[x] to the oldering of the integral rational functlOns of x by then degree. The possibility of being able to continue handlmg both cases at the same time rests then on the fact that the ordering according to the degrees in K [x] can also be described by a more exact analogue to the absolute value in r than the degree itself. For this purpose we formulate:

Definition 6. By the absolute value K[x] u'e understand

If I of

an element

f

ill

I f I = 0 if f = 0, I f I = len, it' f is of degree 11. Here k is an arbit7'ary integer> 1 fixed once it is chosen. k could also be taken as any real number> 1; here, however, we will avoid the real numbers for technical reasons.

The following rules for absolute values, valid in the case of

r, are also valid in K[x]: (1) (2) (3)

Ifl~l,

if f=t=O,

If ± g I ~ If I+ I9 I' If' 9 i = I f 1·1 9 i .

Proof: (1) is clear by Def. 6; similarly, (2) and (3) are also

°

valid in the case f = or g = 0. However, if f =t= 0, g =1= 0, namely, f(x) =ao+···+anxn (an=l=O), If!=k n, g(x) = bo bmx m (b m =1=0), I g: = k m ,

+ ... +

3 Namely, we assume as known: 1. the relation < in r and its rules; 2. the connection existing between this relation and the arithmetical operations; 3. the definition of absolute value; 4. the connections based on 1. and 2. between the absolute value, on the one hand, and the ordering and arithmetical operations, on the other hand.

2. I. Thf' Left Sllle~ 0/ Aigebratt EquatIOns

176 then

f ± 9 contains

no higher power of .r titan

fore,

I! ± 9 1::;;:; kMa,,{(t!,m) = Max (1.:", km) :: I,'"

.l M.n(II,m).

There-

+ I, = 1f I + I 9 I. III

Accordingly, even though the relatioll

*g' r,: ; :;

Max (I! I, ,9 I) It iR valid ill K rx I· Furthermore,

(2 a) ,f is generally not correct in we have

"

'"

..

""

t(x) g(x) = ..2 a.x· . ..2 b",xl' =..2 2; a.bl , X'+IJ .=0

v 0 I'

1-'=0

n+m

~

()

).

(

'jJ =-:

0, ... , It )

=..2( ... abx - i=O .+'" t V " , It = 0"_., m = aobo + (aoVI + al bo) x + . -. + (an_1b m + anbm_ J ) xn+ m- 1 -+ {lnUm ~n+m (a"bm::j:: 0), therefore

1

t

f· 9 1 = len-I-m -= /,;11 ./em = I t ]'1

g I-

Besides (1) to (3) later on we will also have to apply repeatedly the following principle, whoHe validIty

i~

a eom,equellee of

the fact that by Del. 6 all abRolllte valueR are natnral Humbers or 0.

(4) In any non-empty subset o/, K[:r] or of smallest possible absolute vallie. The absolute value in Klx]

01'

r

r there are elements

is now relatf'd to the COll-

cepts, explained under A, of the divisihility theory ill these integral domains as follows:

Theorem 6. If

Proof: If 9

t T/

1 1-/

I~

1 /',

~ 19

I·

r

= ii', 9 =1= 0,

then we also have /' =1= 0, =1= 0;

f 1 ~ 1, I

~ 1. Since hy (:3) we further have

therefore, by (1)

/g[ =

f, g, 9 =1= 0, thell

1

r

I

it follows thnt 1 ~ If/

=:1 ~ !, t1utL is, ,f /: ; :; /g ,. =

Theorem 7. f is a unit if and only if If, 1. II ence in the case K[x] the elements a=l=O of K are the only units; l:n the case r, the integers a = ± l. Proof: a) By Theorem 6 1e implies If 1 1 since 1e 1=1.

r

=

1. Fundamental Theorem 0/ Umque Decomposab!llt}

177

b) That the f with I f I = 1, that is, the a speuhed in the theorem, are units is clear by DC± ~ [173] llll the L,lse K[x] because dIvision is unrestricted in K). By means of 0), Theorems 6, 7 yield

Theorem 8. If f1 and f2 are associates, thell Ifl I fll = If21 and fll f2' then fl and f2 are associates.

=

12 . If

The extra assumption 111'2 for the converse is not necessary in the case r; however, in the case K[x] it is necessary.

From Theorems 6 to 8 we have

Theorem 9. If fig, g =1= 0, then f is a p7'oper dIvisor of g and only If 1 < I! I< Ig I'

If

0. Formulation of the Fundamental Theorem In the special integral domains Klx] and r we can pick out a special representative for a class of associates. It is characterized by the following conventIOn:

Definition 7. f is said to be 1u:n"lnalized first, if f =1= 0 and secondly, a) in the case K[x] if the coefficient an of the highest pou'er of tc ocurrmg in f(tc) ao alltcn (all =1= 0) (briefly: the leatli'll{/ coefficient of f(x)) is equal to e. b) in the case r if f> O.

== + ... +

It is appropriate to restrict the concept of normality by the condition f =1= 0 even though 0 is also a well defined (namely single) representative of a class of associated elements. Therefore, in the case K[x] normalized is the same as the designation polynomial cited in the introduction if we disregard the single normalized element of O-th degree f e which we did not think appropliate to include in the concept polynomial. Incidentally, the terminology that we have adopted, whereby only the normalized elements in K[a:] are called polynomia18, is not generally used.

=

2. I. The Left Sides of Algebraic Equations

178

Theorem 7 immediately implies that Def. 7 actually accomplishes what we are after.

Theorem 10. In every class of associates different from the zero class there exists one and only one normalized representa-

tive. Furthermore, for normalized elements we have

Theorem 11. If f and g are normalized, so also is fg; further,

if

{J

If,

then

1 is normalized. {J

Proof: The theorem is obvious in the case r. In the case K[xJ the theorem follows by applying the multiplication formula

previously applied in the proof of (3) [175] to fg and to

19. g

For later use we formulate along with Def. 7: Definition 8. A normalized prime element is called in the case of K[x] a prime function or irreducible polynomial, in ~he case r a prime number. In Sections 1, 2 we will use, in addition, the common designation normalized prime element in order to be able to handle the cases K[x] and r at the same time.

The Fundamental Theorem to be proved can now be stated as follows:

Theorem 12. Every element f =F 0 in K[x] or r has a decomposition f=apl ... Pr 4 into r ~ 0 normalized prime elements Pi' .. " Pr and a unit factor a. This decomposition is unique except for the order of the factors, that is, a and Pi"'" Pr are uniquely determined by f. 4 We make the convention that a product PI'" Pr with l' = 0 shall mean the element e (cf. also the footnote to the Theorem of Toeplitz in Vol. 1, Section 12 [95].

1. Fundamental Theorem of Unique Decomposability

179

This theorem does not state that PH ••. , Pr are different. The uniqueness statement, however, refers as well to the frequency of the occurrence of the different prime factors.

The proof, as the statement of the theorem, will be broken up into two parts. The first, the simpler of the two, will be given under D; the second, which is deeper-lying, under F. Before giving the latter, however, a number of theorems will be derived under E. These are not only necessary to prove the second part but are also very important for reasons oyer and above this purpose. D. Possibility of the Decomposition into Normalized Prime Elements First of all we prove the lemma CDt). If f is not a unit, then f has at least one normalized p1'ime divisor. Proof: In the case f = 0 a normalized divisor of t different from units can be any normalized element =!= e; in the case f =!= 0, the normalized representative which is an associate of f. Consequently, by (4) [176] there is a normalized divisor p of f of lowest possible absolute value which is not a unit. This divisor p is a normalized prime divisor of t, for by construction it is normalized and not a unit. Furthermore, if p had a proper divisor, then its normalized representative would be a normalized divisor of f different from units (Theorem 2 [173]) of lower absolute value than p (Theorem 9 [177]), which contradicts the fact that p was chosen so as to have the minimal absolute value. In particUlar (D, ) implies the existence of p1-ime elements. For the special element f = 0 our proof in the case K[x] shows that every polynomial of I-st degree ao x is a normalized prime element; in the case r, that the number 2 is a prime number.

+

2. 1. The Left Sides of AlgebraLc Equations

180

(D l ) now yields D, that is, the theorem: (D 2 ) Evel'y f =1= 0 has a decomposition f=apl ... Pr

into r ;;;; () lw7'malized prime elements Pl"", Pr and a unit factor a. Proof: The statement is clear if f is a unit (1' = 0). If f is not a unit, then by (D l ) we can set

f= Plt1 with a normalized prime element Pl' If 11 is a unit, then this is a decomposition as stated (1' = 1). If t1 is not a unit, then by (D 1) we can set

fl = P2f2 , therefore f = P1P2f2 with a normalized prime element P2' After finitely many steps we must encounter it unit fr by this process. For, since f =1= 0 by rrheorem 9 ] it must be true, as long as 1. is not a unit, that

If I> I11 I> ... > If. I> 1, which is incompatible with an inhnite sequence of such

fi' since

all of the! fi I are whole numbers. If fr is the first unit, then

f=apl'''Pr is valid, where Pl"", Pr are normalized prime elements and a(=fr) is a unit. This proves (D 2 ) and therefore D.

E. Division with Remainder. Greatest Common Divisor The«)rem 13. If 1=1= 0 and 9 is arbitrary, then there exist uniquely determined elements rand h such that in the case K[x], lk 1<111 and h ;;,;: 0, that is, 9 = ff h and I h I I1I

+

is valid.

I

<

o : ; ; h < 11 I in the

case

r

1. Fundamental Theorem oj Unique Decomposab!iLt)'

P1'oof: a) By (4) [176] an

181

r can be chosen so that h=g-ft

has the smallest possible absolute value among all 9 us assume that I hi;;;;: i j' I' Then

f7"', Let

1) in the case K[x], if l, It are the degrees and Ct. an the leading coefflCienis of h, f' we would have l;;;;: Il, therefore f1(x)

==

~x!-"

an would be an element in K[x], Hence ff1 would have the same degree l and leading coeffiClent Cz as h. 1.'his means that h - ff'l=o-ff-ffl =g - fCf +fJ would have a lower degree than It, therefore a smaller absolute value.

2) In the case r, from

h+f=g-fT+f=g-f(J± 1) one of the two signs would give an element whose absolute value is smaller than h.

= r+

=r

But the existence of one such F f1 or ± 1 contradicts in both cases the fact that h = 9 - ff was chosen so as to have the minimal absolute value. Therefore i h I < i f I· In the case r besides the condition i It I < j f i whirl! has just been established we also wish to sho\y that h;;;;: O. For this purpose, if we let It 0 so that - I f I h O. then we only have to form

<

h1 =h

for which 0

< <

+ i tl=h ± f={J-fCf +- I))

< hl < If I is valid.

+

+

h', where hand h'satisfy the b) From 9 = ff h, 9 = ff conditions of the theorem, it follows that

fCf- t) = h' - h, therefore, f I h' - h. Now, if h' =l= h were true, then by Theorem 6 if! ~ ; h' - h I

182

2. I. The Left Sides of Algebra!c EquatIOns

would follow. According to the conditions of the theorem for h, h' this inequality would yield 1) in the case K[x] by (2 a) [176]: 2) in

1f 1~ Max (I h I, 1h' I) < 1f I, the case r: 1f 1~ (h' - h or h - h') < 1f 1- 0 = 1f I,

a contradiction in both cases. Consequently, h = h'; therefore, since f =1= 0,

r -.,. This completes the proof of Theorem 13.

Since 0 satisfies the conditions of Theorem 13 regarding h, the possibility and uniqueness of the relation appearing there yields in addition: Corollary. If f + 0 and g is arbitrary, then fig if and only if the h in Theorem 13 is equal to O. The process based on Theorem 13 for determining

T and

h from

7 the quotient and h the remainder. In order to carry out the division with remainder from a practical standpoint we assume as known (1) in the case r the process based on the decimal numeration (or any other number system) of integers; (2) in the case K[x] the process based in a corresponding manner on the normal representation (Vo!.l. Def. 9 [45J ) which follows by direct application of the indirect proof under a), 1). In particular, these processes make possible the practical determination by the Corollary whether a divisibility relation 1 I g exists. f

=1= 0 and g is called the division 01 g by

f with remainder,

By the repeated application of a similar inference we deduce from Theorem 13: Theorem 14. If f1 and f2 are not both 0, then there exists a uniquely determined normalized d such that

dlfl,dlf2' from hi fl' h 1 f2 follows hid is valid. d can be 1 epresented in the form (1) (2)

0

d=fl;, +M;·

1. Fundamental Theorem

0/ Umque Decomposabihty

183

In view of the properties (1), (2) d is called the

gl·eate.q, CO'1n-

mondiviso,/,of fl and f2 (notation d=(fllf2»)'

+

Proof: Let Tt,12 be so chosen that d = ttl1 fJ2 IS normalized and has the smallest possible absolute yalue among all norrnaiized elements of the form ftll* -t- fJ2*' This is possible by (-1) [176] ; for, if we say that 11 =FO, then the normalized representative ad1 to f1 is a normalized element of the prescribed form

CG.* = a l , T;* = 0); therefore, the set of absolute values in question is not empty. It is clear that (2) is satisfied for the d thereby determined (Theorem 3) [173J. Furthermol e, If (1) were not satisfied and, let us say, fl' then there would exist. since

d+

<

d as a normalized element is =F 0, an h = f1 - dd with I h d (Theorem 13) for which h =F 0 would be valId (Corollary). Hence, if h is normalized by the factor a, then ah = afl - add=fl(a- ad~)

+ f2(- a Ji;) +

would be a normalized element of the form fl/ fJ;'* of smaller absolute value than d. This contradicts the fact that

+

d = f1ft f2~ was ehosell so as to have a minimal absolute value. Therefore d I f1 and Similarly d I12' b) If the normalized d' also satisfies the conditions (1) and (2), then it follows from (1) for d' and from (2) for cl (with d' as h) that d' I d. Similarly, it follows conversely that d d'. This implies d = d' (Theorems 5, 10 [17-1, 178] ). The following process for the practical determination of the greatest common divisor is obtained by the repeated application of the process mentioned above for the division with remainder. It is known under the name of Eucl·idean Algorithm. Let us suppose that 124=0. Then the following divisions with remainder can be performed until a remainder I, ~ I = 0 is met:

184

2. 1. The Left Sldes of Algebraic Equations

11 == 12ul + 13 , 12 == 13u2 + 14, 1,-2 = I r-1 U,-2 + I" 1,-1 == I,Ur-l + 1,+1' Since the absolute values If,

31<[/21 1/41 < I/J 1 I I, I < I1,-1 I 1/

0

== I fT+! I < I fT

I continually

I·

decrease, we must obtain after finitely many steps f, -j 1 = O. Then the normalized representative of f, is the greatest common divisor of fl' f 2• For, by reversing these steps starting from the last l'elation it follows by Theol'em 3 [ ] that we successively have fT I f T- 1, fr I f,-2' .•• , fT: f2' fr I fl' therefore (1) is valid; and by reversing these steps starting from the next to the last l'elation, that f, can be l'epresented by the pairs f,-2' f,-1; ... ; fl,/2 in the form of the theol'em, thel'efore (2) is valid.

In addition we note the validity of the special relations (0, f) f, (f, f) = f for normalized f, (a, f) = e for any f. By elementary inferences we can now deduce the following facts from the basic Theorem 14. These will lead stepwise to the uniqueness proof that we are seeking.

=

Theorem 15. If 9 is normalized, then (fl,f2)g= (f1g, f2g)· Proof: If (ft, [2) d, (f1g, f2g) d'. then, on the one hand, dg I /lg, dg I f2g (Theorem 14, (1) for d; Theorem 2 [ ]), there-

=

=

I

fore dg I d' (Theorem 14, (2) for d'); on the other hand',

~ fl'

~ If2

~ Id

(Theorem 14, (1) for d'; Theorem 2), therefore

(Theorem 14, (2) for d), that is, d' I dg (Theorem 2). This implies d'=dg (Theorems 5, 10 [174,178]). Definition 9. f1 and f2 are said to be relatively prime or prime to each otlter if (fl' f2) = e. According to this, in particular, 0 is relatively prime to all and only the units; a unit is relatively prime to all I. Furthermore, if (f1> 12) = d and we set f1 == dUl> 12 == dU2' then Ul and U2 are rela-

1. Fundamental Theorem of Unique Decomposabdit)

Theorem 16. If f and 91 are relatively prime and

185

f

{h9~. thell

f I92' Proof: The theorem is obYious if g2 = U. If 92 =1= U and jj2 is the normalized representatiYe to 92' then by Theorem 15 it follows from (I, 91) = e that (fY2' gl(2) = a2; therefore hy Theorem 14, (2) and the assumption we ha\'e f g2' that is. f g2 is also valid.

Theorem 17. If p is a prune elemellt, then (p, 9)= e is equivalent to p -( 9, that is, p is prune to 9 if alld only if p is not a di1:isor of 9, Proof: If

p is the normalized representatiYe to p, then

can on1y be e or

p

(P.9)

(Def 4,5 [174]), Since it is a normalized

divisor of p. Now, on the one hand. if p J.. g, then (p, g) !

=p

cannot be valid, since otherwise by Theorem 11, (1) P g, and therefore p t g, would follow; consequently, (p, g) = e. On the other hand, if (p, g) = e, then p! 9 cannot be Yalid, since otherwise by Theorem 14, (2) we would have pie, contrary to Def.5; therefore p

-t g.

Theorem 18. If p is a prime element and p 9192' then p: gl or p i g2' Proof: If p 91' then p is prime to 91 (Theorem 17), therefore, on ac{;ount of the hypothesis, p I 92 (Theorem 16).

-t

Theorem 19. If p is a prime element and p I91 . , , 9" then

I

I

P gr' Proof: It follows by repeated application of Theorem 18.

p 91 01' .. '

01'

The uniqueness proof, which is now to be given. rests on the last theorem.

2. I. The Left Sides of Algebraic Equations

186

F. Uniqueness of the Decomposition into Normalized Prime Elements

Let

f=

apl" P, = bql

qs

be two decomp(}8itions of an f =1= 0 into 1';;;;; 0 and s;;;;; 0 normalized prime elements Pl"", Pr and ql"", qs, respectively, with a and b unit factors. On dividing by b it then follows by Theorem 11 [2] that : is normalized, therefore = e, that is,

a = b. Therefore Pl' Pr= ql qs· = 0, so also is s = 0; for otherwise ql Ie, contrary to Def. 5 [174]. In such a case both decompositions f = a, f = b coincide. If r > 0, so also is s > 0 by the same inference. In this case Pl I ql ... qs, therefore by Theorem 19 Pl i ql or ... or Pl I qs· Now, the qi have no proper divisors; consequently, as Pl is not a unit it must be an associate of one of the qi, therefore equal to it (Theorem 10 l178]). The ordering can be so taken that Pl = ql' It then follows that P2 ... Pr = q2' . qs· If r = 1, then, as above, so also is s = 1. In this case, therebql coincide. fore, both decompositions f = aPl' f If l' > 1, so also is s> 1. By continuing the above line of reasoning and ordering the qi in a suitable manner we successively obtain P2 = Q2' •.• , Pr = qs and l' = s. The latter statement is valid since by the above line of reasoning the Pi must be exhausted by the same number of steps as the qi' Hence both decompositions coincide except for the ordering of the factors, By D and F the fundamental theorem is now proved. If

l'

=

1. Fundamental Theorem of Unique Decomposability

187

G. Consequences of the Fundamental Theorem By having recourse to the decomposition into prime elements the concept of divisibility introduced under A and F can be looked upon from a new viewpoint. Namely, the following facts are valid: 5

Theorem 20. If g =4= 0, then fig if and only if the normalized prime factors of f OCcur amollg those of gS (Def. 1 [172], Theorem 12 [178]). The inferences, immediately following from this, in regard to the concepts unit and associate need not be cited in detail.

Theorem 21. If f 1 and f2 are different from 0, their greatest common divisor is the product of the normalized prime factors that f1 and f2 have in common (Theorems 12, 14, 20).

Theorem 22. f1 and f2' different from 0, are relatively prime if and only if they have no common prime factor. (Def. 9 [184] , Theorem 21). Theorem 22 immediately implies the following generalization of Theorem 18 [1851 in the direction of Theorem 16 [185] : Theorem 23. If f is prime to gl and g2' then it is also prime to UtU2'

H. Dependence on the Ground Field It is important for us to establish how the concepts and facts developed in the preceding for the case K[x] are affected if our

investigations are based on the integral domain R[x] over an 5 Cf. the footnote before Theorem 1 [172J. The previous theorems and definitions which substantiate our arguments are inserted in parenthesis. 6 This statement and the following are aimed at the frequency of the occurrence of the diffl'rent prime factors.

2. I. The Left S£des of Algebraic Equations

188

extension field K of K instead of K[x]. In this regard the following is valid: Theorem 24. If the elemeuts belonging to K[x] m'e ?'egarded

as elements of the integral domain R[x] over an extension field

-+

K of K, then the relatwns "f Ig, f g, h lS the rernaiuder on dividing g by f, (f1 f2) = d" m'e preserved; on the contrary, the relation "p is prime function" is not necessarily pl'eserved. Proof: a) The determination of h from f and g, and the determination of d from f1 and f2 can be performed, according to the expositions after Theorem 13 [180] and Theorem 14 [182], by calculating processes which consist in the application of the four elementary operations on the coefficients of these elements, where the coefficients belong to K. The interpretation of these coefficients as elements of K does not change any of these processes, therefore neither are the results hand d changed. Consequently, the alternative h = 0, h =1= 0, too, is preserved. By Theorem 13, Corollary [182] this yields the invariance of the alter-

=°

-+

native fig, f g in the case f =1= O. For f this invariance is trivial by Theorem 1 [173]. b) The example specified in the introduction already shows that the prime function X2 -2 in p[x] 7 acquires the decomposition (x -

V2) ex + V"2)

into proper divisors on extending P

to the field P of real numbers. b) implies, just as in the introduction under 1), that on principle no rational arithmetical processes can exist for deciding whether a specific element in K[x] is a prime function, nor is there such a process for obtaining the prime factor decomposition of a specific f 7

2 is a prime function in p[x] amounts to the irraeasily deduced from Theorem 12 for the case r,

That

X2 -

tionality of V2, which can be

[178]

2. Residue Class Rings in K[x] and

r

in K [x]. Rather, we are dependent on a trial and this purpose in each concrete case.

189 e?TOr

process for

If we have to consider, as frequently in the following, exten-

sion fields K of K besides the ground field K, we must state when llsing the deSignations prime function, irreducible (Def. 8 [178] ) whether they are used relative to K[x 1 or K[x]. IVe do this through the addition of the phrase in K or tn K, respectively (d. the first footnote in the introduction [167]). The fundamental theorem (applied in R[x]) immediately yields in addition the following theorem to be used many times in the sequel;

Theorem 25. If K

'tS

an extension field oj K, then the prime

facto)' decomposition in R[x] of an f belonging to K[x] is generated from the pnme factor decomposition in K [x] by decomposing the prime factors of f in K [x J into their prime factors in K[x].

2. Residue Class Rings in K[x] and

r

In Vol. 1, Section 2 we introduced the general concept of congruence relation in a domain. The results of Section 1 enable us to survey all possible congruence relations in the integral domains K[x] and r in a sense corresponding exactly to what was done in Vol. 1, Theorem 34, 35 [75, 75] for the congruence relations in groups. We obtain this survey from the following theorem:

Theorem 26. '1'0 a congruence relation === in K [x] or r there exists an element f, uniquely determined except for associates, such that (1) hl = h2 if and only if f Ihl - h:.

190

2. I. The Left Sides of Algebraic Equations

Conversely, for every f a congruence relation in K [x] or r is generated by (1). Proof: a) According to (1) every congruence relation = arises from an f. Let M be the set of all elements 9 =0. By Vol. 1, Section 2, (a), (B), (y), (1), (2) we have that hl=h2 is then equivalent to h j - h2= 0, that is, equivalent to saying that hl - h2 is contained in M. Now, either M consists only of the element O. In this case hl = h2 is equivalent to hi = h2 that is, our congruence relation is equivalent to equality, and the statement of the theorem is correct for f = 0 (Theorem 1 [173]). Or, instead, M contains elements different from O. In this case among the elements of M different from zero there is by Section 1, (4) [176] an f of smallest possible absolute value. Now, if 9 is any element in M and by Theorem 13 [180] we set

< if I,

{J=fT+h, 1h 1

then h = g - IT also belongs to M, since by the definition of the congruence relation in VoL 1, Section 2 we have that f= 0

°

implies fT 0 and consequently {J= implies h = {J - fT O. Therefore, since f was chosen so as to have the minimal absolute value, we must have h = 0, that is, fig. Conversely, since f= 0

(r

implies for every multiple 9 = of f that {J=== 0, M consists of all and only the multiples of f; namely, {J=== 0 is eqUivalent to fig and therefore, by the statements already made, hI = h2 is equivalent to f 1hi - h 2 • b) For any fixed f the relation (1) is a congruence relation. That the conditions Vol. 1, Section 2, (a.), (B), (y) [22] are satisfied can be seen as follows: first. flO (Theorem 1[173J); secondly, ft hI - h2 implies f 1 h2 - hi (Theorem 3 l173]); and thirdly, f i hi - h2' f 1 h2 - ha implies f I hl - hs (Theorem 3). The conditions Vol. 1. Section 2, (1), (2) (27) are satisfied;

2. Residue Class Rings in K[x] and

r

191

+

for, from f I hi - h2' f I gl - 92 we first have f' (hi 91) (h2 92) (Theorem 3), and secondly f I h l 91 - 11 2 g 1 , f 11 2 91h292 (Theorem 3), so that as aboye f: h19! - 11 292' c) f is determined uniquely by the congruence relation extept for associated elements (naturally, anyone of these can be chosen without changing the congruence relation (1)). ;\"amely. let the congruence relation hi = h2 be equi\'alent to t' i hl - 10 2

+

as well as to

Ti hi -

h 2 • Then

1': f - 0, rr r- 0 implies f= O.

r=O. As a consequence of the above assumption this also means that f fT- 0, Tf f - 0, in other words, l' anti rare nssociates (Theorem 5 [174]). On the basis of Theorem 26 we define: Definition 10. The element j' belon9ing to a cOllgruellce relation = in K [x] or r acconting to Theorem 26, aHd ulliquely deter'mined except fOl' associates, is called the 11Wdullts of this COllgr'uence relatioil. We thell 1crite in full

=

hi === h2 mod f 8 ror hi h2' that is, lor t hi - h2' and the classes ther'eby deter'mined ar'e called the l'e.sidue clas. ses mod f; the ring 9 formed by these classes, the 1'esidue class ring mod f. If f =1= 0, we assume that f is non>talized for the sake of unique-

ness. -

In the case K[x] we designate the residue class ring mod f

S Incidentally, in the case K[x] th€ addition of "mod f(x)" prevents the confusion with equality in K[x] which may arise when the element~ are written in terms of the argument x (Vol. 1, by Theorem 12 [47]). 9 Here we speak of a 1'ing in somewhat more general terms than in Vol. 1, Theorem 8 [28] ,since we are also including the case excluded there. In other words, here all elements of a ring may be congruent to one another. In such a case f would be a unit, since e = 0 mod f. Consequently, the set would contain only a single element, therefore it would no longer satisfy the postulate stated in Vol. 1, Section 1, (a) [13].

2. I. The Left Sides

192

0/ Algebraic Equations

by K[x, mod f(x)]; in the case r, by rf . Furthermore, we will write on occasIOn {h} for the residue class detet'mined by the element h relative to the modulus considered at the time.

Even though the ealculatIOns with residue dasses (Vol. 1, Theorem 8 r28] generating the residue class ring are independent of the particular lepl'esentatives that are used, it is still important for our later applications as well as for obtuining u comprehensive view of the residue classes to have a complete system of replesentatiYes ( Vol. 1, Section 2, [23] ) for the residue classes mod f which is as simple as possible. Such a system is specified in the following theorem: 'l'heorem 27. If f=O, then every element of K[x] 01' r forms by itself a residue class. If f =l= 0, a complete system oj' rep1'esentati1)eS of the residue classes mod f is formed by the elements h with the property Ih I < If I in the case K [x], 0:;;; h < f in the case r. Proof: a) For f = 0, hi = h2 mod 0 is equivalent to I hl - h 2 , that is, to hl = h 2 • b) For l' =l= 0 the existence statement of Theorem 13 [180] implies that every element is congruent to one of the specified elements mod f; and the uniqueness statement, that it is congruent to only one such element. Hence the specified elements represent all residue classes mod f, each once. The complete system of representatives mod f for f =1= can be described in more explicit form as follows: In the case K[x) : Co c1 X cn - 1 x n- 1, if f is of degree n> 0, where co' c1 , ••• , cn- 1 run through all systems of n elements in K; 0, if f is of degree (f = e); In the case r: 0, 1, ... , f - 1; here, therefore, the number of residue classes mod l' is finite, namely f.

°

°

+

+ ... +

°

2. Residue Class Rings in K[x] and

r

193

The facts of Theorem 27 motivate the designation residue classes in so far as these are formed by all elements which have one and the same remainder with respect to division by t. 10 Especially important for us is the condition for what f the residue class ring mod f is an integral domain or even a field. The following theorem gives information in this regard: TheOl'em 28. The residue class ring mod t is an integral domain if and only if f = or f is a prime element. If f is a prime element, it is actually a field. Proof: a) If f = 0, then by Theorem 27 the residue dass ring coincides with the integral domain K[x] or r. Xext, let f = p be a prime element. In this case if 9192== 0 mod p, that is, p I9192' then by Theorem 18 [185] we have p 91 or P 92- that is, 91 =0 mod p 01' 92==0 mod p. Therefore, if the product of two residue classes {91} {92 } =0, at least one of the factors {91} or {02} = 0, that is, the analogue to Vol. 1, Theorem ± [18] is valid in the residue class ring mod p. It is clear that the analogue to Vol. 1, Theorem 3 (17] is also valid, since {e} is the unity element of the residue class ring. Hence this is first of all an integral domain. Furthermore, it is actually a field. For, if 9 =1= 0 mod p, that is, p Xg, so that p is prime to 9 (Theorem 17

°

+ gg"" = e, therefore, in the case of the given h we also have ph + g'O = h. (1851), then by Theorem 14 [182] we have ph*

'l'he latter relation says, however, that g'O == h mod p, namely. that {g} {li}= {It} or {g} =

}~?

This means that the division

by residue classes mod p different from zero can be carried out 10 Besides the special integral domains K[ro] and r, the residue classes relative to congruence relations in general integral domains can also be similarly related to division (d. Vol. I, footnote to Def. 6 [27]), whereby the general designation residue classes is justified.

194

2. 1. The Left Sides of Algebraic Equations

without restriction [ Vol. 1, Section 1, (7) [16] J. Hence the proposed residue class ring is a field. b) Let f =1= 0 and not a prime element. Then, either f is a unit or there exists a decomposition f = 0102 into proper divisors 01' 02' In the first case there is only one residue class so that it cannot be an integral dOllllLin [Vol. 1, Section 1, (a) l13] ]. In the second case the relation 0192 ===' 0 mod f, that is, {OI} {g2} = says that the product of two residue classes mod f different from o is equal to 0, so that the given residue class ring is again not an integral domain.

°

In the following we designate the residue class field a prime function p(x) by K(x, mod p (x) ); the residue relative to a prime number p, by Pp.ll By Theorem 27 the residue class field P71 is a finite p elements. In regard to this field we prove the following theorem, which is to be applied later:

relative to class field field 12 of additional

Theorem 29. For every a in Pp we hc!ve aP = a; fo1' every a =l= 0, therefore, ap-l = e. Proof: This is obvious for a = O. Let a =1= 0 and 0, al' . .. a p - 1 be the p different elements of Pp' Due to the uniqueness of the division by a in Pp the p elements aO 0, aa 1, ••• , aap_l are then different from one another, therefore they must again be the totality of elements of Pp in some order, so that aa 1 , · •• , aa p-1 are identical with at, ... , ap _ 1 except for the ordering. This means on forming products that ap-1at ••• ap_l =a t ••• ap-I' However, since Ct 1 ••• a/l- 1 =1= 0, we obtain ap-l = e, a P = a as stated.

=

11 The new designations are dispensable in view of the conventions in Def. 10 [191]. They are chosen so as to be more in harmony with the nomenclature adopted in Vol. 1, Def. 9 [45], 10 [46] , Theorem 5 [19]. 12 For p = 2, P2 is the field specified in Vol. 1, Section 1, Example 4 [20] and quoted as an example at various places in Vol. 1.

2. Residue Class Rings in K[x] and

r

195

'l'heorem 28 says that division is unrestricted and unique in the full residue class ring mod f if and only if f is a prime element. However, in any residue class ring a subset can always be found in which division is so characterized. The following theorem and the adjoined definition lead to this subset. Theorem 30. All elements of a residue class mod f hare aile and the same greatest common divisor u·ith 1; consequently, it

is called the divisor of this residue class. Proof: If gl = g2 mod f, that is, gl - fh = if, then by Theorems 3 [173], 14 [182] we have (Ul' 1) I (g2 f) and (g2' f) (gll f). therefore (gl' f) = (g2' f). Definition 11. The residue classes mod f of the divisor e are called the p1-ime 1'esillue clas,r,;es moll f. Hence these represent the partition into residue classes mod f within the set of all elements prime to f (Def. 9 [184] ).

The above statement is confirmed by the following theorem: Theorem 31. The prime residue classes mod f form all Abelian group \lS, with respect to multiplication.ls Proof: Since \lS, is a subset of the residue class ring mod f, it is sufficient to show that \lSf is closed with respect to multiplication and that division in \lS, is unique and likewise without restriction. The former immediately follo,..-s from Theorem 23 [187] ; the latter, by inferences corresponding exactly to those in the proof of Theorem 28 under a) by using Theorem 16 (185] instead of Theorem 18 [185]. Theorem 31 is significant above all in the case r. The group \)Sf is then finite; its order is designated by Cf(f) (Euler's function). We have qJ(O) = 2;14 furthermore, by Theorems 17 [185],27 [1fl2] we

=

13

For f

14

Cf. footnote 13 [195] as well as Theorem 7 [1761.

0, cf., by the way, Theorem 4 [173J.

196

2. I. The Left Sides of Algebraic Equations

have cp (p) = p - 1 for prime numbers p. The general formula that can be proved without difficulty by means of the theorems of Section 1, E is r(t) =

t pit II (1 -

!). P

(f

> 0),

where p runs through the different prime numbers dividing f without a remainder. However, this formula will not be needed in the following. Likewise, we will not need the generalization of Theorem 29, which can be proved in an entirely corresponding manner, a'l' (f)= 1 mod f for every a in r prime to f, the so-called Fermat Theorem. 15 We quote these facts here only in order to round out our developments relative to r, which form an important chapter of the elementary theory of numbers.

3. Cyclic Groups In this section we will make an application to the theory of groups of the results of Sections 1, 2 relative to the case r. This application is important for later developments.

Definition 12. A group 3 is said to be cyclic if it consists of the integral powers of one of its elements A. In this case 3 is also said to be generated by A and A a primitive element of 3· For the integral powers of A we have by definition (Vol. 1 [60] the calculating rules (1)

An! An

= Am+n,

(A,,,)n

= Am",.

This implies first of all (Vol. 1, Dei. 13 [58]):

Theorem 32. Eve1'Y cyclic group is Abelian. From (1) we further infer the following basic theorem about cyclic groups. 15 Fermat himself stated it (Theorem 29) only for f was first given in this general form by Euler,

= Pi it

3. Cyclic Groups

197

Theorem 33. Let 3 be a cyclic group generated by A. Then there exists a uniquely determined integer f;;; 0 such that the correspondence (2)

Am

+-;..

{m}

maps 3 isomorphically onto the additive group iH, of the residue classes mod f.1 6 Prool': a) The convention (3) m 1 = m 2 if and only if

relation in

AmI

=

AnL,

defines a congruence

r.

For, Vol. 1, Section 2 (a), (B), ('Y) [22] are satisfied, since equality in 3 satisfies these laws. Furthermore. Yol. 1, Section 2, (1), (2) [27] are also valid; namely, if 111 1= 11l 2 • 11 1 = 11 2, therefore ATIlt = ATI", ATI. = Am, then by (1) it follows that A m.+ n, = Am, A'h = Am, An, = A""+'" Am,ft, = (Am1)'" = (Am.)", = (An.)",. = (An.)"" = Am,.."

+ = +

therefore m 1 UI iH2 11 2 , mini = 111 2 11 2 , If f is the modulus of the congruence relation (3), then (4) Am. Am, if and only if ml m 2 mod f is also valid. By (4) the correspondence (2) between 3 and ffi, is biunique. Furthermore, it is also an isomorphism. since by (1) the multiplication of the powers Am corresponds to the addition of the exponents m, therefore also to that of their residue classes {m}. Consequently, 3 c-.;) ffi, in virtue of (2). b) The ffi, belonging to different f ~ 0 are characterized by the number of their elements (Theorem 27 [192]), therefore are not isomorphic. Consequently, f is uniquely determined by 3·

=

By Theorem 33 the possible biuniquely to the non-negative these types actually exist, since seen, by the mt . If .8 is a cyclic 16

-=

types of cyclic groups correspond integers f 0, 1, 2, ... , and all they are represented, as we have group generated by A of the type

Cf. Vol. 1, Section 6, Example 1 [61]

=

2. I. The Left Sides of Algebraic Equations

198

mt , then by Theorem 27

[192] the different elements of 3 are given 0, by the totality of integral powers ... , A-2, A-1, AO::: E, AI, A2, ... ; in this case the order of 8 is infinite; b) if f> 0, let us say, by the f powers AO::: E, AI""J Af-l ; for if this system of f elements is continued in both directions it successively repeats itself in the same order.J7 This means that in this case 8 is finite of order f.

a) if I

==

In order to determine all subgroups of a cyclic group we first obselTe the following fact, which is immediately clear from Vol. 1, Theorems 19, 25 [63, 68].

Theorem 34, If A is all, element of a group ®, then the integral powers of A form a cyclic subgroup ~( of ill, the period of ...4, whose order is also called the 01'der of A. If ® is finite of 07'der n, then A is also of finite order m and min. In case A has finite order m then by Theorem 33 m can also be characterized as equivalent to Ale::: E with k mod m, that is, with m l/c; or also as the smallest of the positive exponents k lor which Alc For later ,use we prove in addition:

== °

== E.

Theorem 35. If Ai> A2 are commutative elements in @ of finite orders m l , m 2 and (m l , m 2) ::::: 1, then A1A2 has the order m l rrt2 • Proof.' If (A 1A 2 )" ::: E, then on raising this to the rrt2-th and m1-th powers, respectively, it follows in view of the assumption A2Al == AIA2 that A;:'tk= E, A:"k= E,

=

i

therefore m 1 m2k, m 2 im1k. Since (mt> m 2 ) lJ we must also have I k, m 2 i k (Theorem 16 [185]). This implies m 1m2i k (Theorems 22,20 [187] ). Conversely, since

rrtl

then (AIA2)k of AlA!.

=E

(A 1 A2 )m,m, = (A~lr' (A~'rl

if and only if

1n1rrt 2 i k,

= E,

that is,

tit l 1n 2

is the order

I

11 This is the reason for the designation cyclic. A circle degenerating to a straight line can be thought of as an image of case a).

3. Cyclic Groups

199

By applying Theorem 34 to a cyclic group obtain all subgroups of 5.

Q) = ~

we easily

Theorem 36. If B is a cyclic group of finite order n (of infinite order 18) generated by A, then to every posiU1;e divisor i of n (every positive j) there corresponds a normal divisor i2!1

of order m = ~ (of infinite ordm'), namely, the period of .-:iJ. Its 3

factor group 5/m] is again cyclic of order j. Arl subgroups (different from the identity subgroup) of B are generated in tillS way. Each and eV61'y one of these, as tL'ell as their factor groups, is therefore again cyclic. Proof: a) By 'l'heorem 34: the periods mj of the specified Al are subgroups with the orders designated. and by Yol. 1. Theorem 26 [69] they are naturally normal diYisol's of .3. b) According to the explanation of the congruence relative to m1 (Vol. 1, Def, 16 [65] and by Def. 10 [191 J Am. ===Am, em]) if and only if m1 1112 mod i. The residue classes relative to 2{j can therefore be represented by AO=E, AI, ... , AI-I. Accordingly. B/'2{J is crclic of order j, namely, is generated by the residue class of A. c) If 5' is any subgroup (therefore normal divisor) of 3. then the convention, analogous to (3) [197], m 1 =m 2 if and only if A"" Am. (5') yields a congruence relation in r. If i is its modulus, then Am is in 3' if and only if m= 0 mod i e Vol. 1, Theorem 35 [75]). that is, if i\ m; therefore 5' consists of all and only the integral powers of Ai, that is, it is the period mj of Ai. If 5 is finite of order n, then i I n, since An = E belongs to 8'. If 5 is infinite, then i = 0 corresponds to the identity subgroup. while positive

=

=

18

ing -

The facts relative to this case - not necessary for the followare enclosed in parentheses.

2. I. The Left Sides of Algebraic Equations

200

j correspond to subgroups different from the identity subgroup. Theorem 34 also makes it possible to determine aU primitive elements of a cyclic group. Theorem 37. If 5 is a cyclic group of type in, generated by A, then the primitive elements of 5 with respect to the correspondence (2) cM"l"espond to the prime residue classes mod f, that is, Am is primitive if and only if m is prime to f· Proof: Am is primitive if and only if its period is exactly 5; this is the case if and only if it contains A, that is, if an iii exists

mm

= 1 mod f. However, by Theorems 3, so that Amm = A, namely, 14 [173, 182] this is a necessary condition and by Theorem 31 [195] a sufficient condition that 'in be prime to f. According to this, if f 0 (namely, if 5 is infinite) AI, A-I are the only primitive elements; if f> 0 (namely, if S is finite of order f) there are 'fJ (I) primitive elements [196] among the f elements. In particular, if f p is a prime number, all cp(p) p-1 elements =l= E are primitive.

=

=

=

4. Prime Integral Domains. Prime Fields. Characteristic In this section we derive a basic distinction between integral domains and fields based on the results of Sections 1, 2 for the case r. For this purpose I shall designate throughout an arbitrary integral domain, K its quotient field. Since every field K can be regarded as the quotient field of an integral domain (namely, K itself), this will in no way impose a restriction on the fields K drawn into consideration. We consider the integral multiples me of the unity e:ement e of /. As explained (Vol. 1, at the end of Section 1 [19]) these satisfy the calculating rules. (1) m 1 e+m 2 e=(rn l +m2 )e, (m 1 e)(m 2 e) = (m 1 m 2 )e. The me are the integral "powers" of e, that is, the period of e

4. Prime Integral Domains. Prime Fields. Characteristic

201

in the additive Abelian group formed by the elements of I. and in this sense the formulae (1) are merely the fOllllulae (1) of Section 3 [196], though the second formula IS slightly different. As an analogue to 'l'heorem 34 [198] (but "'ith due consideration of the second rule of combination. namely. multiplication, which exists in I besides the addition which we are using as the rule of combmation of the group) we have from (1) by Vol. 1, Theorem 6 [~5]: Theorem 38. The integ/'al multiples of the Hility element of I form an integral subdomain 10 of I. Its quotient field is a subfield Ko of K. Since e and consequently also all me are contained in e\'el'y integral subdomaill of I, and Slllce the quotients of the me are contained in every subfield of K, we have 'rheorem 39. 10 is the smallest (ilttersectioll of all) il!tegJ'ai subdomaill (s) of I. Ko 1St the smallest (intersection of all) subfieldes) of K. The characterization of 10 and Ko gi\-en in Theorem 39 justifies Definition 13. 10 is called the prime integ~'al d01n(('in of I, Ko the p'l'ime field oj' K, The distinction, mentioned above, between the integral domains I and the fields K will now be expressed in terms of the type of their prime integral domains 10 and prime fields Ko' Even though the calculating operations with the me in 10 in accordance to (1) are isomorphic to the corresponding calculating operations with the m in r, this does not mean that 10 has the type r. For, as the example of Pp shows, the correspondence me -+-+ m is not necessarily one-to-one, that is, equality and distinctnes8 in 10 need not be preserved under this correspon-

2. 1. The Left Sides of Algebraic Equations

202

dence in passing over to r. On the contrary, for the possible types of 10 the situation corresponds exactly to that established in Section 3 (Theorem 33 [197] ) for the types of cyclic groups. The(}rem 40. To I there exists a uniquely determined integel'

r ~ 0 such that 10 is il)ornorphic to the residue class ring rf under the correspondence (2) rne~ {rn}. Proof: From the interpretation of 10 as the period of e in it follows by Section 3, (3) l197] that the relation (3) rn 1 === m2 if and only if m 1 e = m2 e is a congruence relation in r. If f is its modulus, namely, (4) rn 1 e = m2 e if and only if m1 1n2 mod f, then by (4) the correspondence (2) between 10 and r, is one-toone, As in the proof to Theorem 33 [197] it is by (1) also an isomorphism; this is valid not only for addition (which corresponds to the gJ;ouP operation appearing there) but also for the multiplication defined in 10 and rt. On the basis of Theorem 40 we define:

=

Definition 14, The integer f ~

°

of Theorem 40, that is, the

modulus of the congruence relation (3), is called the characteristic of I and K.

Kow, since 10 is an integral domain isomorphic to by Theorem 28 l193] :

r"

we have

°

Theorem 41. The characteristic of I and K is either or a prime number p. If it is 0, then 10 ('..) r, Ko ('..) P; if it is p, then lo=Ko

('..)Pp.

By Theorem 41 the designation characte1'istic of 1 and K is motivated by the fact that this number characterizes the type of the prime integral domain 10 and prime field Ko' respectively, That all characteristics possible by Theorem 41 actually occur, is shown

4. Prime Integral Domains. Prime Fields. Characteristic

203

by the domains r, P, PP' which are their proper prime domains. 19 By the methods of Vol. 1, Section 3, d) [36J any domain I or K can be mapped isomorphic ally on a domain which contains r or P or a Pp (that is, the integers or rational numbers, or the residue classes, relative to a prime number p, respectively) as a prime domain. Hence in the following we speak simply of the prime domains r, P, Pp •

Furthermore, the follOWing facts are evident (cf. Theorem 39) :

Theorem 42. A.ll extension domains alld subdomains of a domain have the same char-actetlstic. From rule (4) for the integral multiples of e it easily follows in virtue of the transformation m 1a-m 2 a = (m 1 - m 2 ) a = (m,-m 2 ) e a (m,e-1n 2 e) , a that a corresponding rule is valid for the integral multiples of an a=l=O:

=

Theorem 43. If a is an element of a domain of cha1'acteTistic f, then 1n 1a = m 2a if and, in case a =1= 0, also only if (m 1 =m 2 (for/=O) t 1m, m 2 mod p (fo~' / p) In particular, there/M'e, we have 'Ina = 0 i/ and, in case a =F D, also only i/ Jm = 0 (for / = 0) 1. t m 0 mod p (for f p) J This theorem shows that domains with characteristic p behave differently from those with characteristic 0 (for instance, all number domains). As a consequence all the familiar inferences from the "algebra of numbers" cannot be used in our abstract "algebra of domains," as we have stressed many times in Vol. 1 [Proofs to Theorem 12, d); Theorem 13, d ); Theorem 70 [49, 52, 141] ].

=

=

=

r

=

19 Here and in the following expositions the term domain stands for integral domain or field.

2. I. The Left Sides of Algebraic Equations

204

The following fact, to be applied later, also shows that domains with a characteristic p behave differently from the algebra of numbers: Theorem 44. In domains of characteristic p a sum can be raised to a power p by raising each summand to the p-th power:

( lak)P= 1 afc· k=l

k= 1

P1'oof: It is sufficient to prove the theorem for n = 2, since it then follows in general by induction. According to the binomial theorem (assumed as known from the elements of arithmetic) we have in this case (a1

'P- (P). p-v + a2 )P =a'P1 +.I a1 a2 +a'P2 " 9=1 v 1

where (~) designates the number of combinations of p elements taken v at a time. Now, we have already established (cf. Vol. 1, proof of Theorem 66 [133] the formula p! =p . ..(p-l) ... (p-(v-l)}. (P)= v vl(p-v)! 1·2 .. ·"

But, for 1 :;;; v :;;; p -1 the product 1 .2 ... v is prime to p (Theorem 23 [187] ) and, as the number (~) is an integer, must divide the product p. [(p -1) ... (p - (v -1»] without a remai~der; therefore, by Theorem 16 [185] it must divide the second factor [ ... J of this product without a remainder. This means that (~)=o mod p. Hence from Theorem 43 it follows that (a1

+ a2 )P = a~ + a~,

Moreover, it should be noted that a corresponding rule is also valid in the case of subtraction; for, since subtraction can be performed without restriction, (a1"+ a2 )P b;-bf= (b 1 - b 2 )P,

ai = af

implies in general

II. The Roots of Algebraic Equations In this chapter we will derive a number of theorems about the roots a of polynomials f(x) over K under the assumption that these roots belong to an extension field A of K. Here, howeyer. we will not be concerned wIth the question whether the A and a exist for a given K and fex); this point will be discussed for the first time in Chapter III. Consequently, these theorems are merely regarded as consequences of the assumptIOn that a polynomial over K has one or more roots in an extension fIeld A of K. To simplify our terminology we make the following Notational conventions fo?' Chaptm's II to IV. Capital Greek letters, except those already disposed of, i. e., M (set), B (domain), I (integral domain), r (integral domain of integers), always stand for fields even though this may not be ex· plicitly stated. In this respect K shall be the g1'ound fIeld; A any extension field (briefly, extension) of K. Other capital Greek letters that may be used will stand for extensions of K with special properties. Though these latter letters will always be used in the same sense, as in the case of P (field of rational numbers), we will always explicitly specify the attached properties for the sake of clarity. In the case of fields (later, also for groups) the property of being contained in or containing will be denoted by the symbols ;:;;;, ;;;;;, <, >. If K ;:;;; K ;:;;; A, then we say that K is a field between K and A.

We will designate elements in the ground field K by a, b, ~, 't, .... (d. Vol. 1, footnote 1 to Section 1, [15]); similarly, elements in K[fr'] by f(x), u(x), h(x), •.• ,those in A[x] by cp(x),1jJ(x), x(x), ... ITheseconventions

c, . .. , those in extensions A of K by a,

1 Whenever fractional rational functions occur, designated as such in Vol. 1, they will be represented as quotients of integral rational functions.

205

206

2. 1I. The Roots of Algebraic Equations

frequently permit us to omit explanatory additional statements regarding the field to which the elements under consideration shall belong. Likewise, we can also occasionally omit the additional statement "in K" when using the expressions "irreducible" and "decomposition into prime factors." Furthermore, in the course of our development we will introduce concepts which are defined in terms of a definite ground field. The specific reference to this ground field can also be omitted provided that in a given investigation no other ground field appears besides a fixed K. Definitions which shall use these conventions will be marked with an ".

5. Roots and Linear Factors 1) 'l'he fundamental theorem proved in Section 1 for K[xJ makes it possible, first of all, to show that the roots of a polynomial over K in an extension A of K are related to its prime factors in K. Namely, by Vol. 1, Theorem 4 [18] and the principle of substitution we have: Theorein 45. If f(x)=P1(X)"'Pr(x) 2 is a polynomial over K decomposed into its prime factors, then

every root of f(a:) in A is also a root of at least one of the Pi (a:) ; and, conversely, every root of one of the PiCa:) itt A is also a root of fex). Consequently, the roots of f(x) are under control if those of the p,(x) are under control; therefore, we could restrict ourselves to the investigation of the roots of irreducible polynomials. However, we will not impose this restriction, since it turns out to be superfluous for the theory to be developed in Chapters III, IV. This also seems desirable in view of the non-existence of a rational calculating process for bringing about the decomposition into prime factors. 2 According to the definition of polynomia.l given in the introduction, as well as by Def. 8 [178] and '!Theorem 11 [178], the unit factor appearing in Theorem 12 [178] is a = e.

5. Roots and Linear Factors

207

2) We will further prove some theorems connecting the roots of a polynomial over K with its prime factors of 1-st degree (so-called linear f a ctQl'S) in an extension /I. of K. Theorem 46. If a is a root of fex) in /I., then f(.1') is di1:isible by the linear factor x - a, that is, there exists in /I. a decomposition f(x)

=== (x -

a) qJ(x).

Convel'sely, such a decomposition implies that a is a root of f(x). Proof: a) By Theorem 13 [180] we can set f(x) = (x - a) qJ(x) + 'l'(x) with! 'l'(x) I < i x - a . Since Ix - a I= kt, then I'l'(x) 1= kO = I, so that 'l'(x)= ~ is an element of /I.. Hence, on setting x = a it also follows that ~ = 0 since f(a) = O. This means that there exists a decomposition of the type indicated. b) The converse is clear.

Theorem 47. If a 1, . · . , a, al'e different roots of f(x) in /I., then there exists in /I. a decomposition f(x)= (x -

(1)'

.

(x -

a,,) qJ(x).

Conversely, such a decomposition implies that a l "

.• ,

a, are

roots of f(x). Proof: a) By assumption the prime functions x -

(%1' • • • ,

x - a 'I are different, and by Theorem 46 each appears in the prime factor decomposition of f(x) in the field /I.. Hence, due to the uniqueness of the decomposition. its product must also be contained in f(x), that is, there exists a decomposition of the type stated, b) The converse is clear. On comparing the degrees on both sides. Theorem 47 immediately yields the important fact:

2. II. The Roots of Algebraic Equations

208 Theorem 48. The/'e is

110

extension A of K in which a

polynomial of n-th degree over K has m01'e than n diffe1'ent roots. The following theorem, to be used later, is a consequence of Theorem 48: Theorem 49. If K consists of infinitely many elements and U, (x" ... , xn), ... , Ur (x" ... , Xl') a?'e elements of K[x" ... , x n], different /I'om one anothe1', then in every infinite subset M of K there are systems of elements a" .. , an such that the elements U, (al" .. , an),"', Urea"~ ... ,all ) of K are likewise diffe?'ent from one another. r

Pyoo/: By considering the product of differences U

= II 1.,

(U, - U,)

k=l

.<1. the statement immediately reduces to one of the following two equivalent propositions: (a) If g(x" . . ,x n ) :$ 0, then there are elements a" ... , an in M for which g(a t , ••• , all) =F O. (b) If g(a ... , all) 0 for all a,,. .. , an in M, then g(x " ... , " Xn) =0.3 We will prove this by mathematical induction. For n 1, (a) is a consequence of Theorem 48. Namely, if g(x) :$ 0, then either g(x) =b =F 0 (unit) and therefore g(a) b =F 0 for all a in M, or g(x) is a polynomial except for a factor in K different from zero. In the latter case, g(a) 0 for only finitely many a in K, so that according to the assumption on K and M there exist elements a of M for which yea) =F O. Now, let us suppose that (a), and consequently (b), is already proved for n = v -1. Then we consider the polynomials g(x v) =g(xt, .. ·,xv ) over K[xl'''''x v_ 1 ] ' g'/«xV)==g(a" ••. ,av_uxv) over K, where the latter is generated from the former by replacing xl> ... , :1:: v_ l by the systems a" .. . , a v- l in M. Now, if g(a" ... , a,) 0 for

=

=

=

=

=

3 This confirms the statement made in Vol. 1, Proof to Theorem 12, d) [49] [ef. for this Vol. 1, Proof to Theorem 13, d) [52]]. For this purpose we regard K as the quotient field of the integral domain I appearing there and M as the I appearing there.

5. Roots and Linear Factors

209

=

all a I , · · · , a v in M, then by (b) (n 1) we have, first of all, that every g"(x) O. Consequently, by (b) (n v - I ) the coefficients of g (x.), that is g(x v) itself, are = 0, which means that g (x" " ., xv) == O. Hence (b) is also valid for n v, and therefore valid in general.

=

=

=

The following theorem is more precise than Theorems 47, 48 in that the roots are no longer assumed to be different.

Theorem 50. If f(x) is completely decomposed ill /I. iilto lineal' facto?' s f(x) -

then

0. 1" , . ,

an al'e roots

(x -

of

0.1)

.,

ex -

an).

f(x). In such a case

fl.,)

can have

other l'Oots neither in A nor in any extension 1\ of A. 47

Proof: a) The first part of the theorem is ob,'ious (Theorem [207J).

b) If a is a root of f(x) (in A or in an 7\) ).1 then fCa) = 0 implies that (a - 0.1) (a - an) = 0, SO that a must be equal to one of the roots a, ( Vol. 1, Theorem 4 [18] ). Let f(x) satisfy the hypothesis of Theorem 50. By the roots of f(.T) in A we will understand, from now on, the entire sequence 0. 1, ••• , an corresponding to the linear factors of f(x)) whether it contains any terms that are alike or not. Theorems 46, 47, 50 tell us how to proceed in order to construct the roots of a polynomial f(x) over K. This c(>TIstruction will be carried out in Chapter III. For this purpose we will have to extend K stepwise so that at least one linear factor will be split off of {(x) at each step. If an extension A is found in this way in which {(x) is completely decomposed into linear factors, then we can put a stop to the extension process, since by Theorem 50 no more new roots can be obtained by continuing the extension in such a case.

3) Finally, we prove some facts about the roots of irreducible polynomials.

210

2. 1/. The Roots of Algebra,c Equations

First of all, it immediately follows from Theorem 46 [207J and the concept of irreducibility: Theorem 51. An irreducible polynomial over K has a root in K if and only if it is of the first degree. This makes the fact 2) stressed in tbe introduction, that in general the ground field must be extended in order to obtain the roots of a polynomial, more evident. We must add, of CO'Ilrse, the fact, which is not to be fully discussed here, that in general there are irreducible polynomials over the ground field of degree higher than the I-st. (For special theorems in this direction, see Section 23.)

Theorem 52. Ther'e is no extension of K in which two relatively prime polyltomials have a common root. In particular, this is valid for two different irt'educible polynomials over K. Proof: If ex is a common root of J\ (x) and Mx) in A, then by Theorem 46 [207J x - ex is a common divisor of flex) and Mx) in A. By Theorem 24 1188] this means that eMx) ,

Mx» =1= e. From Theorem 52 we obtain the following, so-called F'tmdamental Theorem about irreducible polynomials: Theorem 53. Let rcx) be in'educible in K. If f(x) has a r'oot in common with any hex) oVe?' K in an extension of K, then

i

f(x) hex).

Proof: By Theorem 17 [185J f(x) would otherwise be prime to hex), which by Theorem 52 contradicts the assumption. In this theorem h(x) does not have to be a polynomial, namely, it has to be neither different from 0 and units nor normalized, In particular, the theorem is trivial for h(x) 0; for h(x) == a, it is meaningless.

==

Our construction of the roots of j'(x), to be carried out in Chapter III, depends above all on rrheorem 53,

6. llIultiple Roots. Derivative Definition 15. A root a. of f(x) decomposition

ill A is called V-10hZ

if a

*0

f(x)= (x-a.)v
v>

The designation v-fold is justified by Theorem 54. If a is a v-fold root of f(o;) in II, then there a1'e exactly v 1'oots of f(o;) in II (and in every extension of II) equal to a. Proof: By Def. 15 f(o;) then contains the prime factor x - a at least v times; moreover, since cp (a) =l= 0, it does not occur more often by Theorem 46 [207].

The multiplicity of a root of t(x) is related to the derivative rex) of f(x), a concept taken from analysis. Naturally, here we cannot define the derivative for our abstract ground field as is usually done in analysis, namely, by means of a limit process, Therefore, we give the following formal definition. Definition 16. By the de'rivative of 00

t(x)= ao we understand

+ alx + . + anxn = k=O .I aka:}:

4

co

I'(x)= a1 + 2a 2x + .,' + nanxn - 1 = .I kakxl.:-1 k=l

= k=O Z'" (k + l)aHlxk • This formal process for forming derivatives coincides with the corresponding differentiation rules of analysis. As in analysis, it satisfies the formulae (1) (f(x) ± g(x»)'==. rex) ± g'(x), (2) (f(x)g(x»,= f'(x) 9 (x) [Cx)g'(x);

+

4

Cf. Vol. 1, footnote 10 to Theorem 11

211

[39].

212

2. 11. The Roots of AlgebraLc EquatLons

in particular, therefore, (af(x))'= af'(x)) (f(x)n)' = nf(x)n-l rex).

Proof: If

therefore f'(x)

= k=l Z'" kakx"'-l = k=O Z'" (k + 1) ak+l x"',

g'(x)

= k=l Z kbkxk- 1 = Z (k + 1) bk+1 x"', "'=0 00

'"

then by Vol. 1. Section 4, (2), (3) of Section 1 [19]) we haye (I(x)

[40]

(cf. also Vol. 1, end

+ g(x))' =C~o (ak + bk) xlc)' == k~ k(alc + bk ) xk - 1 '"

= k=l Z ka",x7t- 1 +

(I(x) g(x))'

a>

Zkb",x"'-l=f'(X)+g'(x), k=l

=C! C!=k a.bf') xk)' =l

(k.+f=k a.bl') .tJc-l

=«:=1 v+j.I=k

Z'" (Z (v + p) a.bl') xk-1

co

'"

= Z ( Z va.-bp ) xk-1 +..2 ( ..2 pa.bj.l) xk-1 "'=1 .+I'=k

k=l

00

'+j.I=k co

=..2( Z (v+ l)a.+1bl') xk+ k-O'+j.!=k ..E( Z (p+l)a.bl'+l)xk k~Ov+p=k = rex) g(x) + f(x) g'(x).

Furthermore, as if to compensate for the omitted limit relation, we have: Theorem 55. If we set for an a
then

=


6. Multiple Roots. Derwative

213

from

f(x) - fCa)

+ (x -

a)
it follows by (1), (2) that

f'(x)

= t;p(x) + (x -

a) t;p' (x).

therefore f'(u) =
As a consequence of Theorem 55 the multiplicity of a root a of rCx) is related to the value of the deriYahve rca) as follows: Theorem 56. A root a of fCx) is a multIple root If and olily if f'Ca) = 0. Pr'ooj: If a is a root of fex), theTl by Theorem 46 [20'i] there exbts a decomposition 1(.1.') == (x - a)
°

Theorem 57. If K has characteristic 0, then all and only the units f(x).=a o in K[x] have the derivative f'(x)===O. If K has the characteristic p, then all and only the elements of the form (3)

00

f(x)~E aZpx 1p ,

therefore f(x)= fo(x P)

l=O

in K[x] have the derivative ['(x)=O. Proof: a) That f'(x) 0 for all of the specified f(x), is clear by Theorem 43 [203] and Def. 16 [211].

=

b) Let f(x)=.E ak,x Jc and f'(x) k==O

==.E kal.Xk-1=O, k==1

therefore

214

2. I/. The Roots of Algebraic Equations

kak=O (k=l, 2, .... ). Then by Theorem 43 if the characteristic is 0, (k = 1, 2, ... ), that is, rex) ao; whereas, if the characteristic is p, then only

==

a,,=O for Ie =1= so that iCx) has the form (3).

°

mod p,

In Theorem 57 we noted that irreducible polynomials of the form (3) behave quite differently from those which do not have this form. This difference will force us to restrict the result to be announced regarding the multiplicity of the roots of an irreducible polynomial in the case of characteristic p to the irreducible polynomials which do not have the form (3). Steinitz 3 called these irreducible polynomials of the fi?"st kind; those of the form (3), of the second kind.

Following the precedence of van der TVaerden, we will use the nomenclature given in the following definition: Definition 17. An irreducible polynomial t(x) over K is said to be separable if its derivative rex) =1= O. Hence, if K hag char'acteristic 0, all irreducible polynomials ar'e separ'able; if K has characteristic p, all the irreducible polynomials which do not have the form (3) in 'llheorem 57 are separable and only these. In other cases f(x) is said to be inseparable. The deSignation separable refers to the following fact, the consequence of Theorem 56 already announced: 5 When we refer to the name of Steinitz, here and in the following, we always mean the text E. Steinitz: Algebraische Theorie de?' Korper (Algebraic Theory of Fields), new edition, Berlin 1930. Chapters I and II of this text except for the part relating to extensions of the second kind are incorporated into our Vol. 1, Chapter I and Vol. 2, Chapters I to IV; in fact, they constitute their content. Every algebraist should a.t some time look into this basic original work on field theory.

215

6. Multiple Roots. Derivative

Theorem 5S. If an lI'l'educible polYllomial flx) is separable, then ttx) has ollly simple roots. Pl'oof: If 0. were a multiple root of fex), then rca) = 0 (Theorem 56) would be valId and therefore f(.l~) f'Cx) (Theorem 53 [210]). Due to the assumed separabilitr of f(:r), we now have f'(x) =1= 0 (DeI. 17). Hence f'Cx) has a degree. and this is smaller than that of f(x) (Def. 16 [:211J). But this contradicts the result f(x) I[,(x') (Theolem 6 [176 J ) of the assumption that f(x) has a mulliple root.

Moreover, Theorem 58 has the following converse: Theorem 59. If an lrreducible polynomial rex) has ol1e root (( which is only simple, then f(x) IS separable. Proof: In this case rca.) =F 0 (Theorem 56), therefore we certainly have rex) :$ 0, and consequently rex) is separable (Def. 17). \Ve further define in coniunction with Def. 17: *A.ddendum to Definition 17.6 An arbit1'm'1I polynomial over K is said to be sepm'able in K, if its pl'lme factors in Kart! separable; otherwise, insepm'able in K. This means that in the case of characteristic 0 every polynomial is separable. In the case of characteristic p it is easy to see that the separability of arbitrary polynomials cannot be made to depend simply on the derivative as is done in Def. 17 for irreducible polynomials. It can very well be that (x) = 0 is also valid for separable f(x) (for instance, if f(x) ===-fo<'x)l', where fo(x) is

r

separable and irreducible); furthermore,we may have f'(x) =!= 0 for inseparable f(x) (for instance, if f(x) = xfo(x), where fo(x) is inseparable and irreducible). Nor does the relationship, given in Theorems 58, 59, between separability and the multiplicity of roots, which justified the designation separable, carryover to arbitrary 6 For the significance of [206].

f,

cf. the introduction to Chapter II

216

2. 1I. The Roots of Algebraic Equations

polynomials. Nevertheless, the extension of this nomenclature to arbitrary polynomials given in the Addendum to Def. 17 is useful for later purposes.

Due to lack of space we refer to Steinitz for a method that can be used to decide the separability of arbitrary polynomials, which does not at the same time depend on the decompOSition into prime factors (in view of the remarks made in connection with 'rheorem 24 [188], this must be desirable). Here we will only note the following fact sufficient for our purposes, which immediately follows from Theorem 59 according to Def. 17, Addendum.

Corollary to Theorem 59. If an arbitrary polynomial rex) ovel' K decomposes in an extension of K into different linear factors, that ~s, the roots ai' ... , an of rex) (x - at)' (x - an) Q,re different from one anothel', then f(x) is separable (in K and every extension of K).

==

Those fields in which eve1'Y polynomial is separable are significant. They are called perfect fields. We will make some rather brief statements about perfect fields by using, in part, results which will not be deduced until later. First, the above considerations immediately yield that every field of characteristic 0 is perfect. - Furthermore, if a field K of characteristic a prime number p is to be perfect, then, in parti.cular, every polynQmial xP -

a over K must be separable. In this case the equation

==

x P a can be solved in K. For, the unsolvability in K implies by Theorem 123, b) [312], to be given later, the irreducibility of the polynomial x P -

a in K and therefore by Def. 17 its inseparability.

Moreover, the solution of x P...::.. a in K is unique. From we have that 0 at -

a2 P

by

va.

= a{ -

= 0, at =

a; = (at - a2 )P (Theorem 44

ai =

a, a;

=a

[i04]), therefore

a2 • We designate this unique solution of x P...:.. a

6. Multiple Roots. Derivative

217

Conversely, if K is a field whose characteristic is the prime number p, let us assume that the equation reP === a can be solved for every a in K. In this case, if the irreducible polynomial f (x) over K were inseparable, then by Def. 17 I(fc) would have the form

t(x) =

i

a.xvp =

.=0

i

(fPa;,)" X"P,

.=0 .

and Theorem 44 [204] would yield "

p-

\'P

f(x) = ( ~ I a.x )

contradicting the irreducibility of 1(0:). - This means that we have found that: A field K whose characteristic is a prime number pis perject il p

and only if Va is contained in K whenever a is. According to this and by Theorem 29 [194] the prime fields Ppare perfect fields. - That not every field is perfect, that is, that there are so-called imperfect fields, is shown by the example Pp(x); for, this field has the characteristic p, and the element x has no p-th

=

root in this field. Namely, if (f(X»)P x were true, that is, \g(x) (f(x»P = x(g(x»P (where we must naturally have that f(x), g (x) =1= 0) and if n, 'In were the degrees of I (x), g (x), then by Section 1, E, (3) [175J pn = 1 + pm, that is, 0 1 mod p, which is not valid.

=

III. The Fields of the Roots of Algebraic Equations In this chapter we will construct the roots of a polynomial f(x) over K according to the scheme given in Theorem 50 [209]. For this purpose we will first set up an extension ~ (stem field) in which a linear factor splits off of f(x) (Section 8); from this extension we will ascend stepwise to an extension W (root field) in which f(x) resolves completely into linear factors (Section 10). Besides this we will develop a general theory of the extensions of a field in order to prepare the way for our investigations in Chapter IV regarding the structure of these root fields. In this theory no role will be played by the fact that these extensions arise from definite polynomials (Sections 7, 9). In conclusion (Section 11) we will add a section on the so-called Fundamental Theorem of Algebra, which will be an interesting digression only from a technical and historical point of view.

7. General Theory of Extensions (1) Basic Concepts and Facts In Vol. 1, Section 4 we constructed and investigated the special extension domains I [Xi' •.. , xn] and K(x i , ... , xn) of ground domains I and K, respectively. These developments will now be pushed to a greater depth and expanded. For brevity, in the follOWing we restrict ourselves to tpe case wherein the ground field K is the only ground domain to be used. 218

7. General Theory of Extensions (1)

219

A. Adjunction. Simple and Finite Extensions Definition 18. Let A be all extension field of K alld M a subset of A. The integral domain of all integral rational functions over K, each of which consists of finitely many elements of M, is called the integ1'al snbdomain of A gelt(31'ated by the adjunction of M to K (notation K [MD. The field of all ratio1lal functions over K, each of lchich cOllsists of finitely many elements of M, is called the subfield of A generated lnJ the adj1tnction of M to K (notation K(M)). That K[M] is actually an integral subdomain, K(M) a subfield of A, follows from Vol. 1, Theorem 6 [25] and the properties of the integral rational and the rational functions. respectively, over K. Furthermore, since the elements of K, in particular, can be regarded as integral rational functions over K, K[M] and K(M) are extension domains of K. Consequently. we have K ~ K [M] ~ K(M) ~ A.

By using the concept of intersection, introduced in Yol. 1. Theorem 7 [26], K[M] and K(M) (as shown by Steinitz) can also be defined as the intersection at' all integral domains and fields, respectively, bet1ceen K aHd A which contain the subset M of A. It is true that the field A in Def. 18 only plays the role of an auxiliary field which can be replaced by any other extension 7\ of K containing M without affecting K[M] and K(M) in any way, namely, without changing the result of the adjunction. However, its existence is absolutely necessary for the process of adjunction. In this regard the word adjunction is easily misunderstood. We wish to emphasize that the adjunction of a set M to K in the two ways explained in Def. 18 has a sense only if it takes place within an extension A of K, that is, if it is, so to speak, initiated frorn above. In other words, it must be known that the elements of M can be combined with one

220

2. Ill. Fields of Roots of Algebraic Equations

another and with the elements of K by means of the four elementary operations according to the field laws. An adjunction from below, that is, without the knowledge of an extension A of K containing M, is never allowable. For, if K satisfies only the field axioms given in Vol. 1, Section 1, we have no right whatever to take for granted the existence of a set M of elements outside K which can be submitted along with the elements of K to the four elementary operations. Therefore, for instance, just because the indispensable field A occurring in Def. 18 can be identified with K(XI' ••• , xn) (or some still more extensive field), we have no right to conclude that the definition of adjunction enables us to circumvent the existence proofs of the domains K[xu' .. ,x7/] and K(xI> •.. ,xn) over K given in Vol. 1, Section 4, each of which is (trivially) generated by the adjunction of xI> •.. , x" to K in the two ways of Def. 18. A similar remark is also valid for the existence proof of the quotient field K carried out in Vol. 1, Section 3 as well as for the existence proof of stem fields and root fields to be given in the following Sections 8, 10. We note that the quotient field is generated by the adjunction of all quotients of elements of the integral domain I and that in its existence proof K plays the role of A.1

Regarding the dependence of the adjunction on the ground field K the following facts can be immediately established. For brevity they will be stated only in terms of the field adjunction K(M) as this is the only case to be used later. Tlleorem 60. K(M) = K if and only if M is a subset of K. Theorem 61. If A ;;;;; K ~ K and A=K(M), then A=K(M) is also valid.

=

Theorem 62. If A ~ K;;;;; K and A = K(M), K K(M), then A = K(M, M), where eM, M) is the union of M and M. By Theorem 62 the successive adJunction first of M and then of M is equivalent to the simultaneous adjunction of M, M. By 1 Cf. the "preliminary remarks to the existence proof" in Vol. 1, Sections 3, 4 [34, 41].

7. General Theory of Extensions (1)

221 applying the concept of compostte, introduced in Yolo 1. Def. [) r:!7] , K(M, M) can also be described as the composite at K(M) and K(M), namely, as the smallest subfield of A tontainina '" K(M) and K(M). \Ve now make a special defmition. again only for the l:ase 01 field adjunction. "'Definition 19. An extension A of K is said to be simple 0/' finite over K, if it is derived by the adjunction oj' oile 01' finitely many, respectively, of its elemellts to K, tit e)·eto/·e. il It cOllsisis of the r'ational functions over K of all eiemeltt a 0)' of fillitely many, respectively, elements al' ... , U'" 2 AllY such element a OJ' system of elements ul' ... ,an of A for ~chich A=K(a) or A = K(al' ... , a,,), respectively, is said to be a primitit'e element or primitive system of elements of A relative to K. In particular, therefore, the field K(x) of rational functions over K of an indeterminate x is simple over K, and the field K (xl> ..• , x l1 ) of rational functions over K of 11 indeterminates :C 1" •• , xI! is finite over K. Naturally, Def. 19 does not say that every simple or finite extension is of this kind, for a or Up,," Un do not have to be indeterminates. Since in the following we will be concerned almost exclusively with this latter case, we will later investigate under B more exactly the situation in which the elements are not indetM·minates.

In addition we note that the concept of adjunction introduced in DeI. 18, as well as that of Simplicity and finiteness introduced in Def. 19, are related to the concept of relati\~e isomorphism already introduced in Vol. 1, Def. 7, Addendum [30]: If the extensions A = K(M) and A' = KCM') generated by adjunction are isomorphic relative to K, and if the sets M and 2 This formulation is not essentially different from that given in Def. 18, since the rational functions over K of every subsystem of CII> ••• , un occur among those of at> ... , all"

222

2. Ill. Fields of Roots of Algebraic Equations

M' correspond to one another with respect to an isomorphism relative to K between the two extensions, then by specifying the correspondences (1)

a ~ a'l

~~

W, ...

M,)

(ai~:" .~n a, ~, ... m M

the correspondences for all remaining elements of A and A' are necessarily determined. Namely, according to the condi:tions for an isomorphism relative to K (cf. Vol. 1, Theorem 9 and Corollary, Def. 7 and Addendum l29, 30] , as well as the enclosed remarks) we must have g(a,~, ... , y) g(a', ~', ... , y') (2) ..-----.. h(a,~, ... , y) h(a', W, ... , y') • Hence to descnbe the isomorphism (2) it is sufficient to c:;pE'cify completely the correspondences (1). In the following we will frequently use this fact. To simplify our terminology we set up: Definition 20. Let A = K(M) and A' = KCM') be 'isomorphic relative to K. Let there be an iSOllW1'phism relative to K between these extensions such that the sets M and M' correspond to one another according to (1). Then the complete isomor'phism (2) is said to be generated by the correspondences (1), and A and A' are said to be isomorphic relative to K on the basis of the em'respondences (1). In particular, let A and N be simple or finite extensions of K which are isomorphic relative to K. Then an isomorphism relative to K between these extensions can be generated by a single or by a finite number, respectively, of correspondences at -<-+ a' or at -<-+ a~, ... , aT -<-+ a;, respectively, of two primitive elements a; a' or two primitive systems of elem~nts Up •• , a r ; a i .... , a;, respectively, of A and N. Naturally, it is not true that a, -<-+ a: generates an isomorphism relative to K between A and N if a, and a: are arbitrary primitive elements. For this purpose the must be chosen so as to be adapted to ai'

u:

7. General Theory of Extensions (1)

223

B. (Separable) Algebraic Elements. (Separable) Algebraic Extensions 3 *Definition 21. An element a of an extension A of K is said to be (slYparably) algebraic over K if it is a root of a (separable) polynomial over Kj if it is not algebraic over K. then it is said to be transcendental over K. By Vol. 1, Section i (cf., in particular, the explanation of Def. 9 [45] appearing there) transcendental ove?' K says the same as indeterminate over K, so that algebraic over K is the opposite of indeterminate over K. - Algebraic elements (at the time, of course, they were hypothetical) have already been used in Chapter II.

Theorem 63. Ii a is (separably) algebraic over K, thell olle and only one zrreducible polynomial f(x) eXists in K which has the root a (and it is separable). P1'00f: a) By Def. 21 a is a root of a (separable) polynomial na;) over Kj therefore, by Theorem 45 [206] at least one of the (separable - De!. 17, Addendum l215]) irreducible factors of !la;) is in K. b) By Theorem 52 [210] a cannot be a root of two different irreducible polynomials over K. Definition 22. The polynomial f(x) in Theorem 63 is said to be the irreducible polynomial over K belonging to aj its degree n, also the degree of a over K (notation: n = [a: K]). 'l'heorem 51 l210] immediately yields: Theorem 64. [a: K] = 1 if and only if ex is an element of K. Furthermore we have: Theorem 65. If A ~ K ~ K and the element ex of A is (separably) algebraic over K, then a is also (separably) algebraic J In the following, whenever sepal'able is enclosed in parentheses, the text can be read throughout without this addendum as well as with it.

224

2. III. Fields

0/

Roots of Algebraic Equations

over K, and the irreducible polynomial over K belonging to it is a divisor of the irreducible polynomial over K belongillg to this element; therefore, in particular,

[a:K];;;;;[a:KJ. Proof: The statements without "separable" immediately fol-

low from Theorem 53 [210]. If in addition a is separable over K, then by Theorem 63, 58 [215] it must be a simple root of the irreducible polynomial over K belonging to it, therefore all the more of the irreducible polynomial over K. Consequently, by Theorem 59 [215], Def. 21 it is also separable over 1<. *Definition 23. An extensi01t A oj' K is said to be (sepa'I'ably) algebraic ave)' K if each ot' its elements is (sepal'ably) algebraic over K; if it is not algebraic over K, it is said to be transce1ulental over K. The simple extension K(x) is transcendental over K, since by the above the indeterminate x is transcendental over K. A corresponding statement is also true for the finite extension K(xu' .. , x n ). In the following we will study algebraic extensions in detail.

Analogously to Theorems 61 [220], 65, we immediately obtain from Theorem 65 the following theorem, which shows the dependence of Def. 23 on the fields K and A.

K ~ K and A is (separably) algebl'aic also (separably) algebraic over K and nat-

Theorem 66. If A ~ over K, then A is urally Kover K.

We will ,be able to prove the converse (analogous to Theorem 62 [220] only later (Theorem 86 [242], Theorem 92 [255]), that is, after proving it for the special class of algebraic extensions defined under C, which follows, (Theorem 71 [228] ) and acquiring more precise knowledge regarding these (Theorem 84 [241], Theorem 91 [254J ).

7. General Theory of ExtensIOns (1)

225

C. Extensions of ]finite Degree Analogous to the concepts introduced in Y 01. 1, Def. 23, 21, [80] we first formulate *Definition 24. 1) n elements a!, ... , an of an extension A of K are called linearly dependent relative to K geneous relation

if

a linear homo-

n

..E a'f,lXk = 0 k=l exists with coefficients aJ.. in K 1-chich are not all 0; otll el'1l'ise, linem'ly independent relative to K. 2) An element a of A is called linearly dependent on a! ... " an relative to K, if there is a lineal' homogeneous representation n

()(. = k=1 2: ak()(.k with coefficients ak in K; otherwise, linearly indepeJUlent of ai' ... , an relative to K. Referring to the proofs of Theorems 38, 39,41,42 [2111-2U3] given in Vol. 1, Section 10 we deduce that these theorems can be meaningfully carried over to the concepts explained in Def. 24 if we replace the linear forms (vectors) in K, appearing there, by elements of an extension A of K. "Ve now define the special class of algebraic extensions in which we are interested:

*Definition 25. An extension A of K is said to be of finite degree over K, if there is a finite maximal system of elements of A linearly independent relative to K. AllY such maximal system a 1 " ' " (J,n is called a basis of A relative to K; the uniquely determined number n, the degree of A over K (notation: n= [A: K]). The designation basis shall mean that every element u of A is linearly dependent on u1 , ••• , a", therefore, has a basis ~'epresentation

2. III. Fields

226 a

=

at al

0/ Roots 0/ Algebraic Equations

+ ' . , + an all

with coefficients a l , ••. , all in K. By Vol. 1, Theorem 41 [811 this basis representation is unique. Conversely, by this theorem the uniqueness of the basis representation in terms of at, ... , a. 1l implies their linear independence, and the fact that every a. can be expressed in terms of these implies their maximality. Therefore, we have Addendum to Definition 25, The condition of Def. 25 can also be exp1'cssed as follows: All elements of A shall possess a unique basis representation in te/'ms of finitely many elements.

rfhe following theorem shows something that is not readily seen from Def. 25, namely, that the extensions of finite degree form a special class of algebraic extensions, and, in addition, somewhat more. Theorem 67. If A has j'inite degl'ee ove?' K, then A is algebraic over K. .1[ol'e exactly: If [A: K] = II, then every elerneltt a of A is algeb1'aic over K alld its degree is lex :K] ~ u. Proof: The n 1 elements exO e, a l fA, a 2, ' , "all are

+

=

=

then linearly dependent (Def, 25). Hence there exists a relation ao ala an(J." = 0 with coefficients al~ in K which are not all 0 (Def. 24), This says that « is a root of a polynomial f(x) over K of at most n-th degree. But the irreducible polynomial f(x) over K belonging to a also has the degree n at most, since it is a divisor of TCx). Therefore, IX is algebraic ovel' K of degree ~ 1/ (Del. 21, 22 [223, 223] ).

+ + ,.. +

The condition that an algebraic extension A of K must have finite degree over K imposes the restriction that A can contain no elements of arbitrarily high degree over K. This l'estriction, however, is not sufficient in general to ensure the finiteness of the degree of N (cf., however, Theorem 91, Corollary [254] to be given later). 4 Counterexamples are given in Steinitz: Algeb1'aische Theorie der Korper (Cf, footnote 5 to Vol. 2, Chapter 2 [214],

7. General Theory of Extensions (J)

227

That there actually exist algebraic extensions which are 1iot of finite degree is shown by the field of all algebraic numbers over P, since this field contains elements of arbItrarily high degree (d. Theorem 123 [312] to be given later).

Furthermore we prove: Theorem 68. If A has finite degree ovel' K. thell A is finite avel' K. PI'oof: If «1'" ., an is n, basis of A, then the basis representation implies that every element of A belongs to the intf'gral domain K[a1 , ••. , an]. On the other hand. since K;;;;; K[a l , .•• , all] ;;;;; K(a l . · · · , all) ;;;;; A is now valid, we have A = K [al' ... , a ll 1 = K(a t . . . . , an). therefore A is finite over K.

In the remaining theorems we will establish how the concepts explained in Def. 25 depend on the fields K and A. Theorem 69, [A: K] = 1 if and only if A = K. P1'oof: a) If A = K, then the system consisting of e alone is a basis of A relative to K, since in this case every a in A actually has the unique basis representation a = a . e. There-

fore, in this case we have lA: K] = 1. b) If [A: K] = 1, then by Theorem 67 la: K] ;;;;; 1, therefore = 1 for every a in A. By Theorem 64 [223] this means that every a in A is an element of K, that is, A = K. Theorem 70, If A ;;;;

K;;;;

K and A has finite degree over K,

then A has finite deg1'ee ove1' K; also K has finite degree over

K, alld

r

[A: K] ;;;;; A: K], [K : K] ;;;;; [A: K].

Proof: Let [A: K] = Ii. Then more than II ell'ments of A are linearly dependent relative to K, therefore a fortiori relative to K; and more than n elements of K, as they are also elements of A, are linearly dependent relative to K. Therefore, in both cases

228

2. III. Fields of Roots of Algebraic Equations

there exists a finite maximal system of at most n linearly independent elements. The followmg conyerse of Theorem 70 IS particularly important for the sequel. At the same time the above relations between the degrees will be put in a more precise form. Theorem 71. It' A ;;;;; I< ;;;;;; K and the degree of A ove}' K as well as of Kover K

~s

finite, then the degree

of

A over K is also

fillite and [A: K] = lA: K] . [I<: K]. Proof: Let [A:

KJ

= rn and a!, ' , ., am be a basis of A rela-

tive to K. Furthermore, let [K: KJ = j and a1 ,

••• ,

aj be

a basis

of K relative to K. Then we will show that the mj elements

= 1, ... , rn; J,; = 1, ... , j) form a basis of A relative to K. a) Since every a in A has a representation

a;a1. (i

m (X

with

a,

=

.I~j(X,

.-1

in 1<, and these ii, in K have in turn representations j

a. = b-l I aibiib with

a,,, in

(i= 1, .. . ,m)

K, every a in A has a representation

Consequently, every relative to K. b) Let us set

a

in A is linearly dependent on the

Then the linear independence of the a, relative to that

a.ak

K first implies (i=l, ... , m).

7. General Theory of Extensions (1)

229

This in turn due to the linear independence of the K implies that a,k=O (;=1, ... ,

Consequently, the

azUl.

Tn:

al.

relati\-e to

k=l ... .,j).

are linearly independent l'elatlye to K.

a,a,.

a) and b) imply that the mj elements form a maximal system lelative to K of linearly independent elements of A. This completes the proof of the theorem. In particular, if we take Theorem 69 into account, Theorem 71 yields: Theorem 72. If A aHd A ~

A,

then A =

A if

7\

have fillite degrees over K and

and only if [A: K] = [/\: K].

D. Conjugate Extensions. Conjugate Elements

In Vol. 1, with respect to Def. 7, Addendum [3U} • we saw that thougb extensions Al and A2 of K isomorphic relative to K were not to be di~tinguished, still a distinction between two such extensions must be made if they are contained in the same extension A. Regarding this point we define the following equh" alence relation: "Definition 26. Two fields Al aile[

A~

isomorphic relative to

K bet~lJeen K and A a7'e also called conjugate 7'elative to K.

Elements a 1 and (1.2 c07'7'esponding under an isomorphism relative to K are likewise called conjugate relative to K. Just as in Dei. 18 [219], here too A plays only the role of an auxiliary field which makes it actually possible to have the relation: •. At and A2 are conjugate," but, without affecting this relation, can be replaced by any extension A of K containing AI and A2• If, in speaking of fields At> A2 conjugate relative to K, we make no reference to such an auxiliary field A it simply means that there actually exists an extension A which contains Al as well as A 2 • As an example of conjugate extensions of K which are actually different we can cite

230

2. Ill, Fields oj Roots oj Algebraic Equations

the n fields K(x l ), Xl"'"

XII'

K(x,,) contained in K(x lI •• · , XII)' In these 1 g(x g(xn) aloe examp1es 0 f conJuga . te l ) ln genera - .. "h-()' •

••• ,

0

h(x 1 )

xn

elements.

Accordmg to the expositions to Vol. 1, Def. 7, Addendum [30J, the following fact is obviously valid for conjugate elements which are algebraic over K: Theorem 73. If 0: is an algeb1'aic element over K of an extension A of K and tea;) the irreducible polynomial ove/, K belonging to this element, then all elemellts of A conjugate to a. are alsol'oots of rex).

8. Stem Fields In this section we will carry out the first step towards the construction of the roots of a polynomial f(x) over K by proving the existence of a special extension of K in which a linear factor of f(x) splits off. This will be done by actually constructing this extension. In addition to this we will obtain a general view of all such extensions. Technically, our investigations will be entirely analogous to the existence- and uniqueness proofs in VoL 1, Section 4, 5 except that here the main task has already been performed in advance by the construction in Section 2 of the residue class field K(.'V,modj'(x» of an irreducible polynomial t'Cx). Our present existence proof is based on this field. By Theorem 45 [206] we can assume without loss of generality that [(x) is irreducible: Our basic theorem is then the following: S The existence- and uniqueness proofs for K (x) in Vol. 1, Section 4 are in fact merely the proofs of this section carried out for I(x) o.

=

8. Stem Fields

231

Theorem 74. Let f(x) be an IrreduCtble polynomwl of degree novel' K. Then thel'e exists an extensIOn ~ of K lnth the proper-ttes:

(I) f(x) has a l'Oot a in ~, thel'etor-e a decompositIOn rex)

= (x -

a) qJ(x).

(11) ~ is simple over K and a is a pmniti?;e elemellt of ~. that

is,

~=

K(a.).

(III) ~ is algebratc of fmite degree orer K: namely, .z: has a basis aO, 0.1, ..• , a,,-I, therefore (::S. K] = H. Thls also mealis that ~ = K [a]. If 2:1< is any extension of K salisl ying (I), (II), and (1.* is the root of f(x) given in (I), (II), then k* is isomorphic to k relative to K under the correspondence aX +--r a (alld therefore also has p1'ope1'ty (III». The extension type of K is therefore uniquely determined by (I), (II). a) Existence Proof The residue class field K (x, mod f(x» contains a subdomain K' which is made up of the special residue classes {a} determined by elements a of K. On the basis of the correspondence (1) {a}+--i>-a this subdomain is a field isomorphic to K.

=

Namely, since the polynomial f(x) is not a unit, a b mod a, = b. Hence (1) is a biunique correspondence between K' and K; that it also satisfies the conditions of an isomorphism follows from the rules of combination defined for residue classes. f(x) is equivalent to

Therefore, !ust as in Vol. 1, Proof to Theorem 10, d) [36] and Theorem 11, d) l43 J we can form a field :s isomorphic to K(x, mod f(x)) by replacing the elements of the subfield K' by the elements of K corresponding to them according to (1)

232

2. III. Fields of Roots of Algebraic Equations

while preserving all relations of combination. This field contains K itself as a subfield. It will next be shown that this field "Z has the properties (I) to (III). (I) II we think of the normal representation f(x) === xn a1x n- 1 an as a relation between the elements fex), x, al , · · · , an of K[x], then this implies by Vol. 1, Theorem 8 [28] the existence of (he analogous relation {f(x)}={x}n+{a 1} {x}n-l {an} between the corresponding residue classes mod f(x), therefore between elements of K(x, mod f(x)). Now, since rex) === 0 mod f(x), which means that {f(x)} is the zero of K(x, mod rex»~, then in K(x, mod f(x)) we have {x}n {a1 } {x}n-l {an} O. Let us now see what happens to this relation in passing from K(x, mod rex»~ to 2: hy (1). For this purpose let a designate the element {x} not belonging to K', and therefore to be preserved. 'l'hen in 2: the relation an a1u,,-1 an = 0, that is, f( a) = 0 is valid. The element 0.= {x} of 2: is therefore a root of f(x).

+

+ ... +

+ ... +

+

+

+ ... +

=

+ ... +

The root a which did not exist beforehand ~s created by this proof with as much conceptual pl'eciseness as pos~ioble. To be briefer but less precise, our train of thought can be expres.s'ed as follows: Since the residue class x mod f(x) satisfies the relation f(x) === 0 mod f(x), it is a root of f(x) in K(x, mod f(x».6

(H) Let ~ be an element of 2: not belonging to K. Then,

according to the construction of "Z this element is a residue class mod f(x). Let hex) be any element of this residue class, so 6 To say that the above proof is "trivial" and that it yields "nothing new" is not a valid objection, for it would arso have to be raised in the case of the construction of the integers from the natural numbers and of the rational numbers from the integers.

233

8. Stem Fields

that ~ = {h (x)}. Then, by returning to the normal representation

= .lJ Ck Xlc co

h(x) .

1..=0

in K[x] and the relation following from this

{hex)} = k~ (cd {xd' in K(a;, mod f(x)), we obtain the existence of the relation co

'"

k=O

k=O

P={h(x)}= .lJClc{X}k= ~('krxk=h((X)

in 2.:. In case ~ belongs to K, the corresponding statement is naturally valid (with an hex) of degree 0). This means that the integral subdomain K[a.] contained in 2.: exhausts the totality of elements ~ of 2.:; therefore, the ,mbfield K(a) likewise contained in E must all the more exhaust this set (Def. 19, [221}). Hence :2: = K [a] = K(a). (III) By using the reduced residue system of Theorem 27 [192] it follows from the proof for (II) that in the representations ~ = h(a) we can assume hex) as·having a unique representation in the form Co ctx C"-l Xn-l. This means that all 13 in :2: can be uniquely represented as linear homogeneous expressions in terms of aO, at, ... , a,,-l with coefficients in K. Consequently, 0.0, a 1, ... ,a,,-l is a basis of 2.: (DeL 25, Addendum [226) ) and therefore [2.:: K) = n.

+ + ... +

b) Uniqueness Proof

Let :2:* be an extension of K satisfying (I), (II) and a* the root of f(x) given in (I), (II). We will first show that the integral subdomain K (a.*] contained in 2.:* is isomorphic to K (x, mod f(x») by means of the correspondence

2. III. Fields

234

(2)

~*

+--+ {h(x)}

if

0/

Roots of Algebraic EquatIOns

W" =

h(a*).

Namely, the validity of Vol. 1, Section 2, (0), (0') follows from DeL 19 l221J. The validIty of Vol. 1, Section 2, (E), (c') can be shown as follows: First B~ = ~~ says that hI (0.';<) = h2(a*). By Theolem 53 [210J this yields that rex) I hI (x) - h2 (x) since j'(a.j.) = 0; that is, hl(x) == h2 (.t) mod f(x), therefore {ht(x)} = {h 2 (x)}. Secondly. we note that this line of reasoning can also be carried out in the opposite dIrection. Finally, the validity of Vol. 1, Section 2, (3), (4) [29J follows fl om the rules of combination defined for the residue dasses. Consequently, since K(x, mod rex») is a field, the integral domain K[o.*J is alt:io a field and therefore identical with its quotient field K(0.*) = ~*. Hence (2) stands for an isomorphism between ~ .. and K(x, mod rex»~. Now, since by a) (or by applying to ~ what has been demonstrated for ~"') the correspondence (3)

B+--+ {hex)}

if ~=h(u)

is an isomorphism between ~ and K (x, mod rex»~, ~* is isomorphic to ~ relative to K on the basis of the correspondence

B* +-+ B if WI- =

h(u*),

B=

h (a)

generated by combining (2) and (3). This correspondence actually maps the elements of K onto themselves, and it is generated by the correspondence 0.* +--+ a specified in the theorem. This completes the proof of Theorem 74. As we will see later, an extension of K can contain different fields ~ with the properties (I), (II) of 'l'heorem 74. Therefore, here, we set up the following definition in contrast to the formulation of Def. 8 to 10 in Vol. 1 [37, 45, 46] wherein the definite article was used:

9. General Theory of E'ttensions (2)

235

*Definition 27, Every extension :£ of K Leith the properties (I), (11), and therefore also (III), of Them'em 74 is called a stem field of f(x) over K, 'l'heorem 74 then yields: Theorem 75. If the il'reduc!ble polynomial I(x) ol-'el' K has a root a. in an extension A of K, then A contains a stem field oj' i(x), namely, K(a.).

Proof: K(a.) has the propeltles (I), (II) of Theorem 74. As far as the extension type IS concerned, this theorem completely describes the structure of gnyextension of Km which an irreducible f(x) over K has a root. Namely. apalt from isomorphisms relatn'e to K all and only the extensions of a stem field of f(x) ale such extensions of K. Hence the stem fields themselves represent the smallest such extension type.

9. General Theory of Extensions (2). Simple and Finite Algebraic Extensions In this section we wHl derive some important relations between the general concepts introduced in Section i. For this purpose we will use the results of Section 8. A. Simple Algebraic Extensions In the first pad of Theorem 74 (231) it was established that the stern fields over K are simple and algebraic of finite degree. Conversely, from the second part of this theorem we immediately obtain: Theorem 76. If A is simple ave)' K aHd a primitive elemeut a. of A is algebraic of degree novel' K, then A= K(n) is a stem field ror the irreducible polynomwl of n-th degree f{x) over K

2. Ill. Fields

236

0/ Roots 0/ Algebraic Equations

belongmg to 0.. This means that A is algebraic of finite degree, namely, of deg1'ee n over K. In particular, this implies the following relation between the two degrees defined independently of one another in Def. 22 f223] and Def. 25 [225] : Theorem 77. The deg1'ee of an algebJ'aic element a over K is equal to the degree of the corresponding stem field K(a) over K: [a: K] = [K(a) : K]. The statement of Theorem 76 can be sharpened by also bringing in the separability. For this purpose we fir.at derive the following criterion for separability: Theorem 78. If K has characteristic p, then an element a algebraic ovm' K is separable ovm' K if and only if K(a P) = K(a). Proof: Let a be separable. If f(x) = xn + at x n- t + ... + all is the irreducible polynomial, separable by Theorem 63 [223], belonging to a, then ft(x'J.;=;. x'" + ai xn- 1 + ... + a~ is the irreducible polynomial belonging to aP. For, in the first place, ftCaP) = f(a)p = 0 by Theorem 4i [204) . Secondly, f 1 (x) is irreducible. For the proof note that hex) I flex) implies h(xp) Ift(x p) = f(x)p (Theorem 44). Theref(x)r with fore, the irreducibility of f(x) yields h(xp ) o;;;; r ;;;; p. Now, on taking derivatives of both sides we obtain O==rf'(x) (Theorem 57 [2131; this yields 1'=0 or p - since rcx):$ 0 (Def. 17 [214]) -, therefore h(xp) = f(x)O = e or h(xp) == f(x)p fl(x P), and consequently hex) e or hex) =

=

=

=

flex). Accordingly,

[a: therefore by Theorem 77

K1 =n= [a P : K1,

9. General Theory of Eo::tensions (2)

237

[K(a) : KJ = [K(a P)

:

K]

and by Theorem 72 [229], K(a) = K(a P).

b) Let a be inseparable. Then the irreducible polynomial belonging to a is inseparable (Theorem 63 [223]); therefore. it has the form f(x) = fo(x p ) (DeL 17 [214]). This means that 'oCx) is the irreducible polynomial belonging to (J.p. For, in the first place, fo(ap) = f(a) = O. In the second place. fo(x) is irreducible, since hCx) i foCx) implies h(xp ) 'fo(.r p) = 1(x) and rcx) is irreducible by assumption. Accordingly,

[a: K] = p[uP : K] therefore by Theorem 77

> [a P : K],

lK(a) : K] > [K(ap) : K], and consequently by Theorem 72 [229] K(a) > K(a P). If K(a) = then a must be separable. We now prove the sharpened form of Theorem 76 announced above: K(a P),

Theorem 79. If K has char'acteristic p, alld a in Theorem 76 is assumed to be separable over' K, then A = K(a)ls also sepal'able over K. Proof: According to Def. 23 [224] we have to show that every element !3 of K(a) is separable. By the criterion of Theorem 78 this will follow by showing that the relation K(aP) K(a), which is valid by assumption, yields the corresponding relation K(~p) = K(j3). For this purpose let us set

=

[KGB) : K] = i, [K(,S:P) : KJ = i',

2. 1lI. Fields of Roots of Algebraic Equations

238

Then by 'rheorem 71 [228] we have mj=m'j', and by Theorem 70 [2271 m ~ m', 3' ~ j. Now, let <Jl(X) =xm

+ alxm-1 + ... + am

be the irreducible polynomial in K(~) belonging to a (cf. Theorem 77). According to Theorem 74, (III) [231] and Theorem 76 its coefficients can be represented in the form (X"

i-I = ..1: a'kpi

(k = 0, ... , 'In-I).

toO

Then, on raising the relation
== xm + a~xm-l + ... + a~

of degree m with the coefficients , (Xk

in

;-1"

= \=0 ..1: aikP

pi

(k = 0, ... , m-I)

K(~p).

By Theorem 53 [210] the irreducible polynomial in K(~p) belonging to aP is then a divisor of <Jll(X), This implies that its degree satisfies the relation (cf. Theorem 77) m'~m.

Consequently, due to the previous relation we have m' = m, therefore = j, that is, by Theorem 72 [229] we actually have K({:3P) = K(~). Theorem 76 shows that there are only two possible cases for a simple extension A of K: Either the relations of Theorem 76 are valid or every primitive element a of A, therefore also A itself, is transcendental over K. In the latter case each such a behaves as an indeterminate over K (see Def. 21 [223]), and A= K(a) has the extension type of the field K(x) of rational

r

239

9. General Theory of E"Ctensions (2)

functions of an indeterminate x over K.7 ),-ccordingly. by Theorem 76 we have (see also Theorem 79):

'fheorem 80. 'The concept stem fielfl of a (sepa/'able) il'l'efluciblepol1Jnornial is the same as that of simple (spp(('rable) algeb1'aic extension (just as the concept field of rati01wl functions of an iluletm'1J1Jinate and simple transcendental extension).

With regard to the separability we can be more precise: Every irreducible polynomial is separable u:/1ich has a simple separable algebraic extension as a stem field. Hence these concepts are only technically different, inasmuch as we think of stem lields as generated from a definite irreducible polynomial I (x), while the term simple algebraic extension is free from such a reference to a definite f(x), that is, to a definite primitive element a. The latter, more general. interpretation is useful because it appears naturally suited for investigating, not on!y the special I(x) and its root a, by which a stem field is generated, but also its other elements which are primitive. This will be done in the following,

If A is a simple algebraic extension of K and II its degree. then every element B of A is algebraic over K. and so by Theorem 67 [226J has a degree ;;;:;; n. 1\OW, by Theorem 77 [[3: K] = [K(B) : K]. Hence the application of Theorem 71 [228] and Theorem 72 [229] yields the more precise 7 The simple extensions are exact analogues of cyclic groups (different from the identity) (see Def. 12 [196] and Def. 19 [221]). Here a distinction is made between simple extensions according to whether they are algebraic, that is, are ~ K(,l;, mod I(x» with I(x) =1= 0, e and of finite degree (degree of I(x», or transcendental, that is, are QQ K(x, mod f(x» with I(x) 0 and of infinite degree. This distinction corresponds to that made in connection with Theorem 33 [197] between cyclic groups (different from the identity); namely, into those which are ~ ~ff w~th 19= 0,1 and so of finite order (absolute value of f), and those whIch are Q,l lJff with 1= 0 and so of infinite order.

==

2. III. Fields

240

0/

Roots

0/

Algebraic Equations

Theorem 81. If A is a simple algebraic extellsion of degree n ove1' K and ~ an element of A of degree j over K, then j Inand n m =---;- is the degree of A over K{~), 3

In particular, ~ is a primitive element of A, if and only if it has degr'ee n over K. We slress in addition the following important consequence of Thporem 74, (II) and (III) [231]: Theorem 82. If a is an algebraic element over K, then K [a] = K{a). This property, which is not valid for transcendental elements, is naturally very useful for operating with an algebraic element. It arises from the last conclusion in Theorem 28 (193] in virtue of the reduction, carried out in Section 8, of K(a) to the residue class field K(x, mod f (x» for the f (x) belonging to a. The proof of Theorem 28, combined with the reduction of Section 8, also tells us how to handle this property in a practical way when operating with an algebraic element (removal of all denominators which may occur not belonging to K).

B. Finite Algebraic Extensions First of all we prove an analogue to Theorem 76 [235]: Theorem 83. If A is finite over K and a pr'im'itive system of elements a 1 , •.. , aT of A is algebraic over K, then A = K(al' ... , a,) is algebraic of finite degree over K. Proof: The simultaneous adjunction of al"'" aT leading from K to A can be replaced by the successive adjunctions K=A o, AO((1.1)=A 1 , A1 (a2)=A2 , ••• , Ar-1(a,)=Ar=A (Theorem 62 [220]). Here a, is algebraic over A,_ 1 for i = 1, ... , 1" (Theorem 65 [223]), so that A, has finite degree over Ai - 1 (Theorem 76 [235]); therefore, A has finite degree over K (Theorem 71 [228]), and consequently is also algebraic over K (Theorem 67 l226]).

9. General Theory

0/ E'ttensions (2)

241

A sharpened fOlm of Theolem 83 analogous to Theorem 79 [237] will be proved later ( Theorem 90 [251] ) by including the separability. There are various types of finite extensions A = K(a!, . .. ,Cl r ). On the one hand, we may have the two extremes: ((1"'" a r algebraic over K and a 1, ••• , ((r transcendental over K; these two cases are the only possibilities if the extensions are simple. On the other hand, we may have the still further possibilities that some of the Ct, are algebraic while some are transcendental over K. Here we are only interested in the case first specified, given in Theorem 83, since it is only in this case that A can be algebraic over K, as it is by Theorem 83. From now on we apply ourselves to this case.

In Theorem 83 we have established when a fmite extension is algebraic. Conversely, we next im estigate when an algebraic extension is finite. In Theorem 80 [:239] the requirement of simplicity forced us to stress the special class of algebraic extensions of finite degree which have a basis of the spe('ial form aD, a l , .. ,an- 1 (namely, consisting of the first n powers of a single element a). Now the requirement of finiteness will make us stresf.:, however, the class of all extensions of finite degree among the algebraic extensions. Kamely, from the results of Theorems 67, 68 [226, 227] on the one hand, and from 'rheorem 83, on the other hand, we immediately obtain: Theol'em 84. The concept finite al,gw'aic extensiQf~ is the same as that of extension of finite deg}'e~. By the successive application of Theorem 82 to the chain of simple algebraic extensions appearing in the proof of Theorem 83 we further obtain the following generalization of the theorem first mentioned: Theol'em 85. For finitely many elements al.···' aT of an extension of K algebraic over K 'We have K [a p ••• , fl T] = K(al' ... , aT)'

242

2. Ill. Fields of Roots of Algebraic Equations

Moreoyer, along with Theorem 71 [228], Theorem 83 enables us to prove in addition the converse of this theorem, which was previously announced after Theorem 66 [224]:

Theorem 86. If A ;;;;; K;;;; K and A is algebraic over algebraic ove/' K, thell A is also algebraic over K.

=

8

K, K is

+

xl' al X"-1 + the irreducible polynomial over K belonging to it. Then, on the one hand, a is algebraic over K(a1' ... , aT)' therefore K(a l ,· .. , an a) has finite degree over K(u l , . . . , aT) (Theorem 76 l235]); on the other hand, al, ... ,aT are algebraic over K, therefore K(al"'" aT) has finite degree over K (Theorem 83). Con Requently, K(a l . . . , u" (1) also has finite degree over K (Theorem 71 l228]); therefore, a is algebraic over K (Theorem 67 [226]). Hence A is also algebraic over K (Def. 23 [224] ).

Proof: Let a be an element of A and
+ ... + aT

10. Root Fields In this section we will construct the roots of a polynomial j'(x) over K by proving the existence of a special extension

of K through the repeated application of the stem field constrnction of Section 8. In this extension f(x) resolves completely into linear factors. In addition we will obtain a general view of all extensions of this kind. Our basic theorem, analogous to Theorem 74 [231], is the following: Theorem 87. Let f(x) be a polynomial ove?' K. Then there exists an extension W of K with the properties: 8 Witho1.1t including the separability. account only later (Theorem 92 [255]).

This can be taken into

10. Root Fields

243

(I) f(x) resolves in W into linea?' factors tex) = (x - at) ... Cx - aT)' (II) W is finite over K and a 1, . . . . aT is a primitive system of elements, that is, W = K(a 1, ••• , aT)' (Ill) W is therefore algeb1'aic of finite degree over K and also W=K[a1 , ••• ,aT]. If W* is any extension of K satisfying (I). (II) (and thel'efore also (III» and at, ... , a~ are the roots of Hx) ill W*, then W* is isomorphic to W relative to K 1.cithl'espect to a suitable ordering of at, ... , a.~ on the basisot the correspondences

ar

a l , .. " Ct.: +-+ a,. The extension type of W is therefore uniquely determined by (I), (II). ~

a) Existence Proof

The proof can be obtained by means of the following r steps which we will only indicate schematically: 1) tCx) =Pl(X) fcx), PICX) prime factor in K. 1:1 stem field of Pl(X) over K, a l root of PICX) in 1:1, therefore tCx) = Cx - al) q?l(X) in 1:1, 1:1 = K(a1)· 2) ~)'

r) ... t(x) =

(x-~) ... (x-~~)in

2,., 2,. =

K(~,

The extension :£, = W arrived at through these the stated properties.

... ,.-xr). l'

steps has

b) Uniqueness Proof

Let W* be an extension of K satisfying (I), (II) and let ai,"" a; be the roots of f(x) in W*. We will show in the

244

2. Ill. Fields

0/ Roots 0/ Algebraic Equations

following r steps that a chain of fields exists between K and W*:

Ks.1Ji<1Jr< ... <1J:<W~ such that each field is isomorphic relative to K to the correspl)'1ding field of the chain in a) with respect to a suitable ordering of cri, ... , a~ on the basis of the correspondences of the theorem. 1) By Theorem 25 [189] the prime factor P1(X) of 1'(x) in K is a product, of certain of the prime fact.ors x x - a~ of t(x) in W*. Therefore, P1 (x) has one of the ai, ... , as a Toot. Assume the ordering as chosen so that ut is a root of Pl(X). By Theorems 74, 7'5 [231, 235] W* then contains the stem field ~r -= K(a!) of P1 (x) over K, and on the basis of the correspondence at ~ a1 this is isomorphic to ~1 relative to K. 2) In ~* there is a decomposition f(x) = (x - a*)
a.t, ... ,

a:

1 1 1

Since the coefficients of
== a; {(x) -a.*

in ai and those of

Ha;) are computed in lhe same rational way (cf. the remark after

Theorem 13 [180]) as in the case of
=.B.:L in a 1 and x-a 1

Lhose of f(x), the coefficients of .:pi (x) and .:pI (x) correspond to one another by the isomorphism a: ~ a1 between If and l:1' By the remarks to VoL 1. Def. 7, Addendum [301 the polynomial n: (x) corresponding by this isomorphism to the prime factor :1t2(X) of
a.t ' ... ,

L:

l:

10. Root Fields

245

K to the stem field };2 of :1:2(X) over ~1 on the basis of the correspondence a; -<--+ a2 together with the isomorphism ~ al between the subfields ~~ and 'S1'

ar

.............................................

r) '" W* contaim, the field ~~ = K(ui, ... , a~). and this is isomorphic to };r = K(a!' ... , aT) relative to K on the basis of the correspondences af +-+ a 1, ••• , aj: 0 ( - + ar'

Since by (II) };~ =W*, ~, =W, the statement follows from the r-th step. '1'his completes the pJloof of Theorem 87. Analogously to Def. 27 [235], here too we define with the indefll!ite article: "'Definition 28. Every extension W of K with the properties (I), (II), and therefore also (III), of Theorem 87 is called a 'l'oot ftel(l of tex) over K.

Analogously to Theorem 75 [235], Theorem 87 then yields: Theorem 88. If the polynomial rex) ovel' K resolves into linear factors in an extension A of K: f(x)

=(x -

( 1)

•••

(x - at),

then A contaills a root field of f(x), namely K(ttl"'" a,). P1'00!: K(ttl"'" aT) has the properties (I), (II) of Theo· rem 87. This enables us to survey the totality of all extensions of K in which a polynomial f(x) over K resolves into linear factors, at least as long as only 1.he extension type is taken into consideration. Namely, apart from isomorphisms relative to K, this totality consists of all and only the extensions of a root field of Therefore, the root fields themselves represent the smallest extension types of this kind. In Def. 27 [235] we were not permitted to define the stem field of an irreducible f(x) by using the definite article, because by Theorem 75 [235] a root field W= K(ttl' 00.' an) of tea:) contains the n stem fields ~1= K(al)'" 0, };n= K(an) which

rex).

246

2. Ill. Fields of Roots of Algebraic Equations

can very well be different, as our later exposition will show. In the case of root fields, howC\'el', we haye by Theorem 50 [209] : 'rheOI'em 89, Under the assumption of Theorem 88 A con-

tains no othe1' 1'oot field of fex) besides the root field K(CX1"'" U,), This means that two different root fields of f(x) cannot occur in one and the same extension of K. Consequently, (as in

the case of quotient fields - cf. Vol. 1, Theorem 10, Corollary [37] ) we can speak of the root field of rex) plain and simple, hence, also of the roots of lex) without expressly naming an extension containing them. If we sturt with a root a of f(x), namely, with a stem field ~ K(a) of one of the prime factors p(x) of f(x), then, as immediately follows from the proof to 'l'heorem 87, a) 1), the root field W K(a!' ... , a,) of can be taken so that it contains a and therefore also ~ K(a). This will frequently be done in the following.

=

rex) =

=

In our exposition we have, in principle, di'sregarded the nattt1'e of the roots. In the next section we will, for once, consider this nature, for historical re::'SOlJJS, in the special case of a ground field consisting of numbers. Except for this digression we will be concerned only with the st1'uctm'e (the extension type) of the root field under consideraVion. The following sections will be devoted to the investigation of this structure.

The constructions in Sections 8, 10 give us a complete solution of the problem of solving algebraic equations, as formulated and explained in the introduction. Accordingly, from now on the roots of a polynomial are no longer to be regarded as unknowns, that is, as elements to be determined, but as completely determined elements, given by the constructions in Sections 8, 10.

11. The So-called Fundamental Theorem of Algebra As already stressed in connection with Def. 18 [219] "we cannot get around the existence proofs of Sections 8, 10 by adjoining a root a or the roots a" ... , fl., of f(x) to the ground field K hfrom below." Only if we have somehow obtained beforehand an extension of K in which f(x) has a linear factor or resolves into linear factors, can this simpler course be followed. Now, in the mathematical literature of the past this was generally arrived at on the basis of the so-called Fundamental Theorem of Algebra. This theorem states that every polynomial over a number field resolves into linear factors in the field of complex numbers; in particular, so also does every polynomial over the smallest possible number field P. This means that the existence of the roots of all algebraic equations with numerical coefficients is proved by constructing, once and f01' all, the field of complex numbers and proving that these facts are valid in this field. Now, however, we no longer think of this theorem, first proved by Gauss, as belonging to algebra in the modern sense - , even if, as we did at the end of Vol. 1, Introduction, we takc algebra to include not only the complex of theorems dealing \\rith the solution of equations but also anything that can be deduced from the field axioms (that is, the general -rules of opel'ations of the rational numbers) or a part of them (ring, integral domain, group). - This theorem, it must be understood, cannot be proved without the tools of analysis (limit, continuity), even though, as in many of the extremely numerous proofs, these may be used to a very small extent. I). Also, the importance of the so-called Fundamental Theorem of Algebra does not extend beyond the special number fields. This fact likewise deprives it of its funda/mental role for algebra from " A simple proof of this theorem is given in K. Knopp, Fl!lIktionentheo}'ie (Theory of Functions) I, G, edition 1944, Section 28, Theorem 3, p. 112 (Sammlung Goschcn G(8), by making full use of analytic (complex-function-theoretic point of view) methods. In doing so it is presented in a natural way as a theorem in the theory of complex functions.

247

248

2. llf. Fields of Roots of Algebra,c Equations

the standpoint formulated in Vol. 1, Introduction (cf. the second and third paragraphs of Vol. 1, Introduction).1o Hence we were justified in deleting the so-called Gaussian Fundamental Theorem of Algebra from our presentation and adopting instead the existence proofs of Sections 8, 10 for the roots of algebraic equations. 'fhese proofs were devisffi by Kroneckel' and constructed by Steinitz; they rest on a foundation which is entirely abstract and consequently capable of giving much more far-reaching support than the socalled Fundamental Theorem of Algebra. In addition Steinitz proved a complex of theorems analogous to those existing between number fields and the complex number field, which is probably his main contribution in the sphere of algebra. Namely, he showed that there exist extensions A to every ground field K in which all polynomials over K simultaneously resolve into linear factors and that the smallest possible field of this kind, A, just as our stem fields and root fields, is uniquely determined by K, except for isomorphisms relative to K, namely, as the field of all algebraic elements over K. Moreover, this field A has the property that there are no proper algebraic extensions of A; therefore every polynomial over A also resolves into linear factors in A. Hence Steinitz called it algebraically closed. In the special case of the ground field P, A is the field of all algebraic numbers. Since the Steinitz existence proof for A must depend on the special exisj;.. ence proofs of Sections 8, 10 we cannot get around these proofs by starting from the existence of A. In order to prevent misunderstandings in regarding the use of the definite article, as introduced in Theorem 89, we wish to point 10 However, E. A,'tin and O. ScMeier [Algcblaischc K07!8t1Uktion reeller K(h'per (Algebraic Construction of Real Fields). Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg (Papers of the Mathematical Seminal' of Hamburg University) 5 (1926)] have extended the sphere of algebra so that the general OI'dering rule8 of rational numbers also appeal' in the circle of algebraic (axiomatic) considerations. The so-called Fundamental Theorem of Algebra as a special theorem of the complex number field is then subordinated to a corresponding theorem regarding a general class of fields and receives in this new extended form a new franchise in modern algebra.

11. The So-called Fundamental Theorem of Algebra

249

out that in the specIal case wherein K is a number field our existence proofs in Sections 8 to 10 do not YIeld the existence of the roots of a polynomial over K. Furthermore, the roots and then rational functions over K do not turn up as comple.c vumbel's but merely as abstmct calculating elements. The extent to which these E'lements can be called numbers then depends on the extension of the concept number, which is not universally settled. However to subsume it under the concept of complex nU7nber it would first have to be shown, over and above our existence proof, that these elements can be represented in the form a + b~, where i is a root of the polynomial X2 + 1 and a, b are elements of a field which is isomorphic to a subfiE'Id of the real number field. Such a proof would generally amount to the proof of the so-called Fundamental Theorem of Algebra (or its generalization mentioned in footnote 10). Only for the special case of the polynomial x2 ...,... 1 itself, in a field K of real numbers, is it immediately obvious: If this polynomial is irreducible in K and i one of its roots in a stem Held ~ over K (which, incidentally, is then, at the same time, a root field in which the decomposition X2 + 1 (x - i) (x + i) is valid), then by Theorem 74 [231] every element of :s has a unique reprE'sentation a + bi with numbers a, b in K, and the operations with these elements behave isomorphically to the known operations with the complex numbers because of the equation i2 -1Cat£chy, the founder of the concrete theory of complex functions, was the first to take this absb'act path in order to introduce the "imaginary" i starting from the real number field K~l In doing so, he laid down the cornerstone to the conceptual structure of abstract algeb?'a which was placed on a broader foundation by Kronecke1' and Steinitz and has been continuously growing to the present day.

=

=

11 Cauchy introduced the complex numbers according to the scheme of the existence proof in Section 8, therefore as residue classes mod X2 + l' in particular, i stood for the residue class x mod :1)2 + 1. (Ex;rc. d'anal. et de phys. roath. 4 (1847), p. 87).

IV. The Structure of the Root Fields of Algebraic Equations III this chapter we will first derive (Sections 12, 13) some properties of Toot fields accessible by the foregoing methods. Then (Sections 14 to 16) we will introduce as new tools certain finite groups determined by extensions of finite degree, namely, their Galois groups. Finally (Sections 17, 18) we will develop the Galois Theory on this foundation. 'l'his theory will enable us to obtain a complete grasp of the str ucture of the 1'oot field in the sense in which the main objective of this volume was stated in the introduction. In doing so we have to restrict ourselves throughout to separable polynumials and extensions. As shown by Steinitz, the theory to be developed in the following must undergo essential modifications in order to be applicable to inseparable polynomials and extensions. Due to the lack of space, here we cannot go into the points of deviation.

12. Simplicity and Separability of the Root Fields of Separable Polynomials, more Gener~ ally, of Finite Algebraic Extensions with Separable Primitive System of Elements The property of the root fields of separable polynomials specified in the title of this section is basic for the sequel. It is generally valid for finite algebraic extensions with separable primitive system of elements, to which the root fields of separable polyuumials actually belong (Theorem 87 [242]). In the proof we must restrict ourselves for methodicn.l reasons to ground fields K with infinitely many elements. 250

12. Simplicity and Separability of Root Fields

251

However, by using other methods of attack it can be ",hO,V11 that this l'estriction is actually supel'fiuous (d. end of ~ection 20 [309 ff] ). Hence we are permitted to formulate the following theurems, as well as all others depending on them. without thi" restriction. The theorem to be proyed is called the Ahelian Theorem after its discoverer. H is the following: Theorem 90. Evel'Y finite algebmic extellsion A nj' K leith sepa1'Uule prirnitwe system of elements, especially therefOl'e the root field of any sepw'able polynomial f(x) ave/, K. is simple. algebraic and separable ovel' K, thel'e!01'e a stem field of a separ'able irr'educible polynomial g(x) over K. Pr'oof: 1 Let A = K(a l , ••• , ar ), whele a 1, ••• ' U r are separable and algebraic over K, For r = 1 there is nothing morp to be proved ('l'heol'ems 76. 79 [235, 237]). a) 1'=2. Let flex), f?(x) be the irreducihle polynomials over K separable by the assumption and Theorem 63 [22:3]. belonging to a 1 , a 2 • Furthermore, let W be the root field of the pol:rnomial Mx)f2 (x) over K, and ftl

flex)

== II (X-IX1•.) , ~=1

= II (x- ~") P,

fix)

~=1

the decompositions of flex) and Mx) in W into linear factors. Then al is one of the a lv" a 2 one of the a 2., Furthermore, the alY, and the a2" are each different from one another (Theorem 58 [215]). Hence the n l n 2 linear functions 1 Hitherto this proof has nearly always been given using the Theorem on Symmetric Fmlctions (cf. Theorem 131 [329] , given later). However, the basic idea of the proof given In this text, which does not use this theorem, was used already by Galois for the same purpose.

2. IV. Structure 0/ Root Fields

252

in W l""i] are all different from one another, since any two of them differ either in the coefficients of the indeterminate x or in the coefficients of the terms not containing x. Therefore, if, as we assume here, K has mfinitely many elements, then by Theorem 49 [208J (applied to W for K, K for M) an element a::j:: 0 exists in K so that the n l n2 elements

+

it"" = it"., (a) =£1:1'1 aa.Z", are all different from one another. Now, let it a. l aa2 be the element among the it.,", corresponding to the system (a l." a21,) (al' 0: 2), Then we form the polynomial !pea;) according to 2 fl.l fil (- a)n, !p(X)."fE!:. n ({} -(~h, + ax))"fE!:. II «(<Xl aX2) - (~lv, + ax)

= +

=

~=l

n,

iSS

~=(

+

II ( {} - ax) -

~lV,)"fE!:.

t ({) -

ax).

•,=1 The fir'st representation enables us to recognize that !pea:) actually has the root a2' corresponding to the linear factor f} - (al ax). However, none of the roots a2', of rex) different from a2 can be roots of this polynomial; for, otherwise the aa2""' where equality of it = 0:1 + aa2 with a t}",", = (Xl', (alV, ' a~,) =1= (ai' (X2) would follow contrary to our construction of the '&"" .3 rfhe latter representation tells us that cp(x) is a polynomial in K(fr).

+

+

2 The prefactor (- a) 1" has the effect of making cp (x) into a polynomial, that is, of making its leading coefficient e. 3 It is easy to see that the separability of Ct.! is not necessary for this conclusion and consequently for the entire proof. An analogue to 'Ilheoretn 90 iJs therefore also valid if we only assume that all are separable except for one of the primitive elements. Here, however, we won't have to make any use of this fact (essential for the theory of inseparable extensions).

1.2. Simplicity and Separab~lity of Root Fzelds

253

Due to these two pl'opelh% of q;(x) the gleatest common divisor (
+

+

,'=2. 1'>2, b) In this case the statements follow by mathematical indnction. Let us assume that the statements are already proved up to r -1. Then K(Ul' , .. , ((I-i) = K«lr-l) for a suitable separable algebraic ar-l over K. By Theorem 62 [220J we have A K(al' ... , (.(r-1' aT) K(ar- 1 • aT)' and consequently by the proof a) A is simple algebraic and separable over K. that is, the statement is also valid for r.

=

=

2. IV. Structure of Root Fields

254

This proof easily yields the further result that a primitive element {} of A = K«(l.lO ••• , (l.r) can be found, in particular, among the linear combinations ala! + .. , + a/(I., of a primitive system of elements (1.1' ••• , aT' This fact is useful for the construction of such a {) in concrete cases.

By Def. 19 [221] a simple algebraic extension is a fOltioli finite, algebraic. Hence Theorem 90 says in connection with Theorem 80 [239] and 84 [241] ; Theorem 91. The following concepts al'e identical: Sepal'able e:den,'1iolt of finite deg1'ee, finite separable algeb1'aic extensim/, ,.nmple sepm'(lble algebraic extension, stml1 field of (t ..~eparable i1"J'ellucible polynmnial.

111 regard to the latter concept the following more precise statemellt l8 valid: EI'e1'Y irreducible polynomial is separable for whlch such an extension is a stem field. In view of the remarl; after Theorem 67 [226] we add to the above theorem: Corollary. The concept of sepa1'able alyelYJ'((ic ext('ftsion witll bounded degrees I'm' its elements is identical with the concepts of The01'em. 91; in pat'ticular, the existing "maximal degt'ee" of an element of A OVM' K for such an extension A of K is equal to the deu?'ce of A over K. Proof: Let t) be an element of A of maximal degree. If {} were not a primitive element of A, there would be an element of A which could not be represented rationally by t}. If B were such an element, then K;§;;K(&)
(Theorem 60 [220]); and since we can set K(t}, B) == K(-&,) (Theorem 90),[{): K] < [{)': K] ( Theorems 72, 77 [229, 236] ) contrary to the maximal determination of the degree of {), Hence t} is a primitive element of A; therefore A = I( (t}) is simple algebraic over K and [A: K] K]. Next, by the inclusion of separability a refinement of Theorem 86 [242] is obtained which is also valid for the case of characteristic p:

= [{} :

13. Normality 0/ Root FIelds and TheIr Primitwe Elements

255

Theorem 92. If A ;;;;; K ;;;;; K and A is sepa?"ab/e algebraic 01'er K, K separable algebraic ove1' K, then A 18 also separable algebraic over K. Proof: Let a be an element of A and f(x) the irreducible polynomial over K (existing according to Theorem 86 [242]) belonging to this element. Let s be the greatest exponent for which

_fO(Xp8) with a polynomial fo(x) over K. Then, fo(x) is irreducible due to the irreducibility of f(x) and separable due to the maximal property of s.

f(x)

By assumption a, aP • •.• , aP s- 1 are separable over K. Hence by Theorem 78 [236] we have R(ex) = K(';\P) = ... = K(ex Ps ), According to this (cf. also Theorem 82 [240]) a has a representation IX = g:(ex PS ) = h' c
K.

However, since the u, are

separable by assumption and the ups are separable as roots of fo(x) over K, Theorem 90 yields that a is also separable over K. Hence A is separable over K (Def. 23 [224]).

13. Normality of the Root Fields and their Primitive Elements. Galois Resolvents 1) A further important property of root fields (of arbitrary polynomials) is that specified in *Definition 29. An extension N of K is called 1wJ"1.ual (or Galois) over K if ellery extension N* conjugate to N relative to K is identicallmth N, that is (cr· Det. 26 [229J) if A ;;;; N;;;;: K, A ;;;;; N* ;;;; K and N" isomorphic to N relative to K always imply N* = N. As a partial analogy to Theorems 66. 70

have here:

[22-1. 227]

we

256

2. IV. Structure of Root Fields

Theorem 93. If N ~ K> K and N is normal over K, then N is also normal over K. P1'oof: According to Def. 26 [2~9) every extension N~' of K conjugate to N relative to K is also an extension of K conjugate to N relative to K. However, we require> neither, as in Theorems 66, 70 [224, 227) , that K be also normal over K, nor conversely, as in Theorems 71, 86 [228, 242] , that N be normal over K if N is normal over K and K is normal over K. Examples of this can be found by means of the Fundamental Theorem to be proved in Section 17, which at the same time gives us the deeper reason for these statements.

We now prove: Theorem 94. 'llhe root field W of a polynomial j'(x) is normal over K. PI'oof: Let t(x) = (X-a1)'" (x -a,) and W=K(al"",ar)' By Def. ~6 [229] and the lemarks to Vol. 1, Def. 7, Addendum [30] these two relations for al"'" aT remain vali.d for the elements oJ, ... ,a~ conjugate to them in passing over to a field W* conjugate to W relative to K. Therefore, W* is likewise a root field of t(x) and consequently by Theorem 89 [246] is identical with W, that is, W is normal over K. 2) In order to become better acquainted with the significance of normality £01' a simple algebraic extension A we must first of all acquire a general view of the totality of conjugates to A both for the case in which A is normal and that in which it is not normal. This is obtained by representing A as a stem field of an irreducible polynomial j'(x) (Theorem 76 [235]) and then studying it as a subfield of the root field W of t(x) (cf. the statements in connection with Theorem 89 [246]). In this respect we immediately have according to Theorems 74, 75 [231, 235] and Def. 26 [229J (d. also the statements already made before Theorem 89 [246]:

25i

13. Normality of Root F,elds and The" Pnmitive Elements

Theorem 95. Let f(x) be an irreductble polynomial

K, n

!ll

a root of I(x), 1\ = K(n) the stem field belonging to it and hen) any element of I\. Then, zf (Xl' ••• , aT are the roots of 1(x), the root field 'W = K(a1 , . . . , aT) of tez) (containing (1, therefore 1\ = K(a) contains the r stem fields ~

=

1\1 = K(al)" ." I\T = KeaT). These are conjugate to 1\ = Kea), and in these ~l = heal)"'" ~T = hear) each 1'epresents a system of elements conjugate to

~

= h(a).

In these theorems (1. designates anyone of the roots a" A the field A, determined by it and ~ the element~. corresponding to it. However, here and in similar considerations in the following (as has already been done in the proof to Theorem 90 [2511) we make no fixed agreement as to which of the a, shall be equal to a. Such an understanding would create an entirely unjustified lack of symmetry, since the (1.., An ~, of K cannot be distinguished (Vol. 1, in connection with Def. 7, Addendum (30]).4

Theorem 95 says, in partIcular, that the a l are all conjugate to a. Conversely, Theorem 73 [230] implies Theorem 96. Let the data of Theorem 95 be given. Then the ,'oots al' ... , (1, are the only conjugates to a within W or any extension of W. The conjugates to an algebraic element are therefore identical with the roots of the irreducible polynomial belonging to the element. Hence, for brevity, in the following we will also use the terminology, the conjugates to a instead of "the roots of the irreducible polynomial belonging to a."

The corresponding theorem is valid in regard to 1\,. For, by an inference analogous to that used in the proof of Theorem 94 we can show that a field (within an arbitrary extension of W) 4

In the literature we frequently find the convention

a

=

al'

258

2. IV. Structure of Root Fields

conjugate to A is likewise a stem field of l(x). Therefore it, too, is generated by the adjunction of a root of f(x), that is, of an aj (Theorem 50 [209]). Theorem 97. Let the data of Theorem 95 be givel/. Then the fields Ai' ... , A, are the only conjugates to A within W or any extension of W. The corresponding theorem cannot yet be stated for ~,. As far as our present considerations are concerned there may actually exist besides lI. i still other extensions of K within W or in extensions of W, and such extensions would lead to the existence of elements distinct from the ~, but conjugate to ~ (cf. Def. 26 [229]). We will l'eturn to this later on (Theorem 103 [262] ).

3) The facts established in 2) can be used to determine the significance of normality for simple algebraic extensions; in doing so, we can also trace, in particular, the significance of the concurrence of simplicity and normality for the root fields

of separable polynomials. It is appropriate to extend Def. 29 by the following definition based on it: *Definition 30. An element -& is called nO'l"tnal (or Galois) over K if the field K(-&) is normal over K. If an element is normal over K, it is automatically algebraic over K, since by the remark after Def. 26 [229] an element ~ which is transcendental over K has conjugates different from K(~).

By combining Def. 19 [221] and Def. 30 we immediately obtain Theorem 98. Let N be a simple algebraic extension of K. If N is normal over K, then every primitive element of N is ?tol'mal over K. Oonversely, if a primitive element of N is normal over K, then N is normal over K. From the previous standpOint this theorem is naturally tautological with Def. 29, 3D, therefo.re for the time being says

13. Normality of Root Fields and Their Primitive Elements

259

nothing new. However, by retaining its wording we can gi\"'e this theorem a new, more significant, meaning by characterizing the concept of normal element in another way_ On the basis of the facts established in 2) this is done in Theorem 99. Let {} be an algebraic element over K, q(x) the i1'reducible polynomial over K belonging to it alld ttl- _. _, itn its j'oots - the conjugates to it. If f} is normal over K, then Ire have: (1) K(f}) = K(it 1, _ •• , itll)' that is, the root fu:ld of q(x) coincides with a stem field of q(x). (II) K(,\) = ... = K(it ll ), that is, the conjugate stem fields of q(x) corresponding to the conjugates to 'it coincide. (III) it1 =gl(it) • ... ,itn=gn(it), that is, the conjugates to {t belong to K(it). Oonversely, (I) or (II) or (III) implies that {} is normal over K. Proof: a) If it is normal over K, then by DeL 29, 30 and Theorem 95 we first have that (H) is valid; this in tum yields (I) according to Theorem 60 [220] as well as (III) by Def. 18 [219] and Theorem 82 l240] . b) First of all, (HI) or (I) implies that K(it;) :;;;; K(fr); by Theorem 72 [229] this yields that (II) is valid. Therefore, it is sufficient to show that the normality of fr follows from (II). This. however, is the case by Theorem 97 and Def. 29. 30. Theorem 99 states, in particular, that if it is normal, or not normal, over K so also are all of its conjugates {ti- Hence it has a meaning to stipulate: Addendum to Definition 30. A polynomial q(x) over K is called nO'l'1naJ, (or Galois) over K, ;i1'St, if it is irreducible, a.nd secondly, i; one 0; its roots fr is normal over K. Due to the meaning already attached to the statements in Theorem 98, Theorem 99 allows us to recognize what restriction the

260

2. IV. Structure of Root Fields

requirement of normality imposes on a simple algebraie extension N of K. Namely. every primitive element -fr of N must satisfy conditions (I) to (III); conversely, for the normality of N it is actually enough that one of these conditions be valid for only one primitive element -fro Form (1) of this restriction shows that in the converse to Theorems 90, 94 (251, 256] every simple (sep,atable) normal extension of K is also a root field of a (separable) polynomial over K. In the case of separability we can be more precise: Eve~'y polynomial is separable for which such an extension is a root field. This follows from the fact that Theorem 63 [223] and Def. 17, Addendum [215] imply that a separable element can only be a root of separable polynomials. Form (II) exhibits a remarkable analogy to the like-named concepts of group theory (Vol. 1, Section 9, especially Theorem 31 [73] ). The fundamental theorem to be proved in Section 17 wH! enable us to see that this formal analogy springs from a factual connection. Form (III) is the most understandable and is appropriate to establish normality in concrete cases.

According to the statements regarding (I) the following analogy to Theorem 91 [254] is valid: Theorem 100. The following concepts a1'e identical: root field of a separable polynom,ial f( a; ), separable 1UYf'1IWl e:xtension of finite degree, finite sepa1'able normal extension, simple separable normal extension, stem field of a separable normal polynomial q (x). More exactly, eve1'Y polynomial f(x) is separable for which such an extension is a root field, and OOe1'Y normal polynomial q(x) is separable for which it is a stem field. Therefore, these concepts differ only technically from one another. (Cf. the statement in this regard under Theorem 80 (239) .) In the Galois theory to be discussed in the following sections we will be occupied with the more detailed structure of the extensions

13. Normality of Root Fields and Their Primitive Elements

261

characterized in Theorem 100. Our aim will be to formulate this theory so as not to attach singular importance either to a definite polynomial or to a definite primitive element or primitive system of elements. We will attain this objective by developing the Galois theory as the theol'Y of sepa~'able normal extensions of finite degree. Only incidentally shall we mention the way the relations to be demonstrated turn out if we take into consideration the fact that such an extension arises from a root field of a definite polynomial or from a stem field of a definite normal polynomial.

4) As a preparation for the latter investigations we will summarize iu the followiug theorem the results about the structure of the root field of a separable polynomial which were obtained in this and the preceding sections (Theorems 90, 94. 98. fl9). After this we will make a definition basic in this regard. Theorem 101. Let f(x) be a separable polynomial over K and a1, ... ,a, its roots. Then in its root field W=K(a1,···,(lr) there exists a primitive element f}=h(1J.1'···' aT) that W = K(f}) and accordingly representations Ct.1 = k 1 ({t), ... , aT = kr({t) are valid. Any such {t is normal over K. This means that if q(x) is the irreducible polynomial over K belonging to it and {t1' .•• , {tnare its roots the conjugates to {t - , then the repreSO

sentations exist. *Definition 31. Any normal polynomial q(x) (occasionally also a {t belonging to it) determined for f(x) according to Theorem 101 is called a Galois resolvent of f(:t) relative to K. The designation re80lvent descends from the older literature. It signifies that the equation f(x) 0 is to be regarded as 80lved if the resolvent q(x) == 0 is solved. For, by Theorem 101 the roots of

==

2. IV. Structure of Root Fields

262

!(re) are actually obtained by operating rationally on a root of

q(x). Underlying this is the idea of a process for solving the equation f(x) :!: O. However, we will not adopt this idea here. For, in the first place-apart from the special case wherein q(x) has the degree 1 and therefore I(re) resolves into linear factors in K -either we are no more able to solve the equation I(x) ~O than the equation q(x) :!= 0 (namely, there is no rational process lor solving either equation), or we can solve one just as well as the other (namely, by the const1"uctions in Sections 8, 10). In the second place, for reasons sunilar to those cited in the introduction regarding 1) there does not exist in principle a rational process for determining the coefficients of a Galois resolvent q(x) from those of I(x). Rather, for us the Galois resolvent has merely a theoretical significance in that its roots fr will permit the root field K (a1> ••• , (1.1) to be put in the "simple" form K(fr). Theorem 99, (1) immediately yields in addition: Theorem 102. I(x) is the Galois resolvent of itself il and only il it is a normal polynomial. For this reason the normal polynomials are also occasionally called, as mentioned in Def. 29, 30 [255, 258] , Galois polynomials. Accordingly, since normal extensions, so long as they are simple, are stem fields of Galois polynomials, they are also called Galois extensions.

5) We now return to the question of the totality of conjugates to ~ h(a), which was put aside on p. 258. Analogously to Theorems 96, 97 [257] we prove the following theorem, however, only for separable f(x): Theorem 103. Let the data of Theor'em 95 l257] be given and besides let f(x) be separable. Then the elements ~l""" ~r are the only conjugates to ~ within W or ~'1ithzn any arbztrary extension of W. Proof: Let ~* be an element conjugate to ~ (in an extension of W). By Theorem 73 [230J ~* is then a root of the same irreducible polynomial in K as ~. Hence by Theorems 74, 75

=

263

13. Normality of Root FLelds and Thezr Primiti'loe Elements

[231, 235] the stem field K(~*) belonging to it is isomoIphic to the stem field K(f)) relative to K on the basis of the correspondence ~.+--r ~.Jo. Now, according to Theorem 101 let W = K(~). Furthermore. let q(x) and cp(x) be the Irreducible polynomIals m K and K(~) belonging to i}. Then
=

=

=

On taking Theorem 96 (257] and Theorem 58 (215] into consideration, Theorem 103 yields that those ~i whieh axe different are the roots of the irreducible polynomial g(x) belonging to ~, that

2. IV. Structure of Root Fields

264

is, are the conjugates to P in the sense introduced there. Frequently, we find that the ~, collectively are also designated as tke conjugates to P; in general, these are the roots of g (x), each occurring a certain number of times. In Theorem 113 [284J we will study this more in detail.

14. The Automorphism Group of an Extension Domain In this section we prepare the way for the application of group theory to the investlgation of the structure of normal extensions of finite degree, which is to be made in the following, by mtroducing the groups in question. In order to obtain the elements of these groups we recall how we formed in Vol. 1, Section 2 the concept of isom01'ph%sm relative to domains from the concept of one-to-one correspondence, defined for sets, through the addition of the two postulates (3), (4) [29] relative to the rules of combination. In an entirely analogous manner we next create the corresponding concept of automorphism 5 relative to domains from the concept of permutation defined in Vol. 1, Section 16 for sets. Definition 32. A permutation of a domain B, that is, a one-toone correspondence of B to itself with a definite mapping rule (designated by -'.>-) is called an automorphism of B if in addition to the rules of combination defined in B it satisfies the relations: (1) a -'.>- a', b -'.>- b' imply a b -'.>- a' b', (2) a -7 a', b -,). b' imply ab -7 a'b', that is, if it satisfies the isomorphism conditions [Vol. 1, Section 2, (3), (4) [29]].

+

+

5 The concept automorphism can alSQ be introduced for groups just as isomorphism (Vol. I, Def. 17 [65]); however, we have no need of this here.

14. The AutomorphISm Croup of an Extension Domain

265

This means that the remarks about permutations made in connection with Vol. 1, Def. 35 [117] can properly be applied to automorphisms.

Besides the companson made above we can bay that the two set concepts one-to-one correspondence, permutatIOn and the two domam concepts isomorphism, automorphism bear the same relationship to one another, that is, automorphism of B means the same as zsomorphism of B to itself wtth a defwite mapping rule. Therefore, the remarks about isomorphisms made in connection with Vol. 1, Def. 7 [30] can also properly be applied to automorphisms.

From VoL 1, Theorems 56, 57 [U8) we Immediately obtain by means of VoL 1, Theorem 19 (63) Theorem 104. 'l'he automorphzsms of a domain B form a group if by the product of two automorphisms of B u'e understand thezr product as permutations, that is, the automorphism of B generated by carrying them out one after the other. The type of this group i.~ uniquely determined by the type of the domain 8. We designate by aA the element generated by applying the automorphism A to the element a. (1) and (2) can then also be put into the form (1)

(a

+ b)A = aA + bA

(2)

(ab)A

= aA bA •

The definition of the product of automorphisms further implies (3) (aA) B = aAB' If a goes over into one and the same f'lement by aU automorphisms

in a set of automorphisms 931, then we designate this fact by ai1J!. According to Theorem 1M every domain has at least the identity automorphism Il-+- a. That further. automorphisms do not have to exist is shown by the example of the prime domains r, P, Pp • Thus. as in the case of isomorphisms, since for every automorphism of an integral domain we must also have 0- 0, e-+ e (d. the statement

2. IV. Structure

266

0/ Root Fields

in Vel. 1 in cennectien with Def. 7 [30]), fer the specified demains all remaining transitiens according to' (1), (2) must of necessity yield a-+ a. In the fellowing we will study in detail examples of demains (fields) with automorphism groups different from the unity group.

Tn the study of extension domains B of a domain Bo we needed the sharper concept of isomorphism of 8 relative to Bo' Here, too, we have to have recourse to the sharper concept of automorphism of 8 relative to Bo' Addendum to Definition 32. If B is an extension domain of 8 0 , then those automm'phisms of B whereby every element of 8 0 goes over into itself are called ((uto'morphism.
~

A __

A'

a~cx.,~p,

...

(aI,~',...

in .

M)

,.

M Analogously to Def. 20 [2221 we further define in this regard: Definition 33. An automorphism of A = K(M) relative to K is said to be generatefl by the substitutions of the elements of M a,(:l, •..

1D

perfm'med under the automorphism. In particular, we can thereby have M M'; in this case every substitutien represents a permutatien of M and A is genemted by these permutations of M.

=

Finally, by means of Vol. 1, Theorem 19 diately obtain from Theorem 104:

[63]

we imme-

15. The Galois Group of a Separable Extension

267

Corollary to Theorem 104. The automorphisms of B relative to 8 0 form a subgroup of the group of all automorphisms of 8. The type of this subgroup is uniquely determined by the extension type of B. According to the above remarks this subgroup is the entire automorphism group of B if B is an integral domain (field) and Bo is its prime integral domain (prime field); on the contrary, it is the unity group if B coincides with Bo'

15. The Galois Group of a Separable Normal Extension of Finite Degree We next apply the concepts of Section 14, especially the Corollary to Theorem 104, to a normal extension N of finite degree over K, which (if separable) is by Theorem 100 [260] also simple algebraic over K. The following terminology is introduced in honor of Galois who first recognized 6 the farreaching significance of the automorphism group for this case. ~'Definition 34. :the group ® of the automorphisms relative to K of a normal extension N of finite deg1'ee over K is called the Ga70is g1'OUp of N relative to K. In particular, if ® is Abelian or cyclic, N is also called Abelum or cyclic over K. In the case of a separable N we obtain a general view of this Galois group through the following theorem: Theorem 105. Let N be a separable normal extension of degree n over K, {t a primitive element of Nand {tl' ••• , {tn its G Although not in the abstract form given here, but in the concrete representation of Section 16. - Evariste Galois was killed in a duel on the 30th of May, 1832 at the age ot 20. On the eve of his death he wrote a long letter to a friend in which he (among other things) added further important results to the first draft of his theory of algebraic equations, already presented to the Paris Academy. The letter is printed in his works.

2. IV. Structure of Root FIelds

268 conjugates. Then the Galois group

@

of N is finite of order n,

and ~ts n automorphisms are gener'ated by the n substitutions it -+ it. (~= 1, ... , It), therefore every element ~

=

h (it) of N

goes over into the n conjugates ~t = h(,'},). Pr'oof: a) By Def. 32, Addendum [266J an automorphism of N relatIve to K can also be regarded as an isomorphism of N to itself relative to K (with a defimte mapping rule). Hence by

Def. 26 [229J such an automorphism takes every element of N mto a conjugate; in particular, {t into a ttl (Theorem 96 [257J ) and then the ~=h({t) into ~,=h(I},). Consequently, there are at most the n possibilities stated in the theorem for an automorphism of N relative to K. b) Conversely, by Theorem 95 [257] each of these n possibilities gives rise to an isomorphism of N = K(iT) relative to K to one of its n conjugate stem fields K(iT,), therefore by Theorem 99, (II) [259J of N to itself. By Def. 32, Addendum this means that the isomorphism is an automorphism of N relative to K. The statements of the theorem follow from a) and b) if we add that the separability of N implies that the n conjugates iT1, ••• , it,.. to it are different from one another, since they are the roots of the irreducible polynomial belonging to {} (Theorem 58 [215J ). This means that the n automorphisms generated by the n substitutions {} -+ it, are also different. In order to obtain a better idea of what it means to operate in the Galois group @, that is, to apply successively the automorphisms gen~ated by the substitutions it -+ 'fr" we will set the elements of @ in a form which is more readily grasped. We will do so by passing over to an isomorphic group wherein these elements are no longer regarded as automorphisms or, as we say in group theory, by giving a suitable representation of @. This is brought about by

269

15. The GaloIs Group of a Separable Extenswn

Theorem 106. Let the assumptions and designatwns of Theo1'em 105 be valid; furtherm01'e, let q(x)

= (x-1't

1 ) ...

(x-1'tl1 )

be the irreducible polynomial over K belonging to 1't, and according to Theorem 99, (III) [259] (1)

it,

In the set 1)' of the

= g,(fr)

~'esidue

(2

= 1, ... ,11.).

classes mod q(x) rep1'esented by the

g,(x)

=

=

(2) {gJ X {gtJ {Uz}, if {u,(g" (x»} {Ut(:';)} then define~ an operatlOn whwh 1S unique, can be carried out without restriction a1'ld is isomorphic to G\) on the ba-sis of the con'espondences

(3)

(it-+it,)+-+{U,(x)}

(i=l, ... ,n).

Proof: First of all, let us recall the one-to-one correspondence (3) (p. 234) set up between the elements of N and the residue classes mod q (x) . a) Since U,(gk(-&» is generated from g,(iJ) by it-+ U,,(1't), therefore from i}, by iJ -+ i}1c' then by Theorem 105 It is likewise a i}z and therefore g,(Uk(-&» = gz(it),{g,(glc(x»} = {gz(x)}. Consequently, (2) defines an operation in which is without restriction and single-valued. b) Since the i}, are different due to the separability of K, the {g,} are also different. Consequently (3) is a one-to-one correspondence between @ and ~. c) Now the successive application of the automorphisms generated by it ...... i}, and it-+it", takes it first into it,=g,(it) and then this into Di(itlc ) U,(Uk(it» = Uz(it) = it l • Hence on taking them together we obtain the automorphism generated by i} -+ fr l . This means that the operation in is defined by (2) corresponds under the correspondence (3) to multiplication in @. Consequently, by (3) the isomorphism condition (Vol. 1, Theorem 23 [65] ) is also satisfied. The statements of the theorem follow from a), b), c). Let a q(x) be given (in the case of the application to the root field of a separable polynomial f(x) let the Galois resolvent of this polynomial be given) as well as the representations (1). Then the 1'epresentation of the Galois group @ as the substitution grottp lJ

a:

=

270

2. IV. Structure of Root Fields

of integral rational functions mod q(x), given in Theorem 106, is immediately accessible for practical calculations. For this purpose it is appropriate to reduce each g,(x) to its uniquely determined residue mod q(x) of degree lower than the n-th (Theorems 13, 27 [180, 192]; see in particular the remark after Theorem 13; a complete picture of the combinations within iJ (therefore also those within @) is then obtained,if the same is done for all g,(gk(x». In the prevailing literature this representation iJ of ® is usually used to define the Galois group. However. this has the disadvantage that a definite primitive element ~ is thereby singled out. It must then be proved that isomorphic groups 1:\', iJ',. " are obtained for all primitive elements i}, If, .. " Our definition depending only on the fieldEl Nand K has the advantage of bringing this fact to the fore from the beginning. In so doing it penetrates one step deeper, namely, all groups iJ, W,· " appear as representations of one and the same group ®. the automorphism group of N relative to K.

16. The Galois Group of a Separable Polynomial Let ® be the Galois group of the root field W of a separable polynomial f(x) over K. If the methods of Section 15 are to be applied in order to obtain a concrete representation of this group, then we must know a Galois resolvent q(x) of f(x), to which Theorems 105, 106 can be applied. Now, by Theorem 102 [262] f(x) is in general not a Galois resolvent of itself, and according to tlhe remarks in connection with Def. 31 [261] a Galois resolvent cannot be derived from f(x) by a rational process. Hence it is of interest to find a concrete representation of ® which does not depend on a Galois resolvent q(x) of f(x) but only on f(x) and its root..~.

We first define; *Definition 35. By the Galois (J'I'Q1tp oj' a polynomial I(x) over K we understand the Galois group ® of its root lield W over K. In particular, if ® is Abelian 01" cyclic, then I (x) is also called Abelian orc-yelie over K (as W was according to Del. 34 [267] ).

16. The GalOIs Group of a Separable Polynomial

271

We now prove the following theorem which gives the desired representation of the Galois group of W or, as we can now say, the Galois group of a separable polynomial f(x); Theorem 107. The Galois g?'O'UP ~ of a separable polynomial f(x) over K is isomorphic to the group ~ of those permutations of the diffe1'ent roots among the roots 0. 1" " , (J.T of f(x) resulting from the automorphisms of ~. These permutations can also be characterized by saying that thei1' application prese')"ves every integral rational relation f (0.1' ..• , 0.,) 0 which may exist between the roots of f(x).

=

Proof: 1) We first show that ~ can generally be represented as a permutation group ~ of the different 0. 11 " . , a r among the roots ((1' ••• ,

are

a) By Theorem 105 [267] an automorphism A relative to K of the root field W of f(x) takes each of the roots of f{x) into a conjugate element, therefore by Theorem 73 [230] again into a root of f(x). Since A is a one-to-one correspondence, it takes different elements into different elements. Consequently, through A the different 0.1"'" Or are subjected to a uniquely determined permutation P. b) Since W = K(all' .. ,00T) = K(ai, ..• , a;;), A is generated by P (Def. 33 [266]), and therefore distinct A also correspond to distinct P. Consequently, the correspondence between the Galois group G\ of Wand the set ~ of the permutations P of the 0.1' .... ' a~ produced by its automorphisms is one-to-one. c) Since in the case of automorphisms as for permutations multiplication is defined as the successive application, the isomorphism condition (Vol. 1, Theorem 23 [65] ) is also satisfied by this correspondence. By a), b), c) ~ is a permutation group isomorphic to ®. 2) We next show that the permutations of ~ are characterized by the property specified in the theorem. a} That every existing relation f(a.u ... , aT} = 0 is preserved by applying the permutations of I.l3 is clear. For, according to 1) the application of such a permutation amounts to the application of

272

2. lV. Structure of Root Fields

the automorphism of @ generated by it, and this is actually the case here by Def. 32, Addendum (266] and the expositions to Vol. 1, Def. 7, Addendum [30), b) According to Theorem 101 [261] let it be a primitive element of W, q(x) (x - fr t ) .•• (x - fr,,) the Galois resolvent of f(x) belonging to it and u, == k" (fr) (v == 1, ... , r»)

=

fr

==

h(al' . ,., ar)'

By Theorem 105 [267] the Galois group ® of W then consists of the automorphisms generated by the substitutions fr--+ 1}, (i = 1,

... ,n).

Now, if

(al, ... ,o.r)

is a permutation of the 0. 1"",0., a.71 which preserves every existing relation !(0.1"'" u t ) == 0, and the transitions for the entire sequence a l , ••• , at produced by it are designated by

~1

,."'

(at' ... ,a) ),

then the application of these tran-

Oil'· ,.. a.i r

sitions will take the special relations /Xv = k.(h(oc 1 , •• " ocr}) q(h(ex1, ••• , ex,)) = 0 into the relations exiv = kp(b(cxi1' ••. , ('
(v

== 1, ... ,1'),

(v=l, ... ,r),

Therefore these, too, are valid. Now, the latter relation implies that == fr, is one of the conjugates of 1}; consequently, by the former relations we have that (li" == k, (fr,) is generated from tX~:::: ky (iJ) by the automorphism fr -'> fri . Hence according to 1) the

h(ui ,,"" (li r )

permutation

(al' ... ,a;:) ail'··'

actually arises from an automorphism

air

and consequently belongs to \]S. This completes the proof of Theorem 107. Incidentally, from a practical standpoint it does not achieve as much as Theorem 106 [269] ,which is based on the knowledge of a Galois resolvent of f(x). For, the decision as to which permutations have the property of Theorem 107 cannot be made in general in a finite number of steps without the knowledge of a Galois resolvent of f(-x), We note in addition that Theorem 107 yields of

@

17. The Fundamental Theorem

0/

the Galois Theory

273

Theorem 108. If f(x) is a separable polynomial of degree r ove,· K and r is the number of the distinct elements among its roots, then the degree n of its root field W over K is a divisor of r! (therefore, even more so, a divisor of r!). Proof: By Theorpm 105 [267] n is at the same time the order of the Galois group Ql of W. Now, since by Theorem 107 the permutation group ~ isomorphic to Ql is a subgroup of the symmetric group 12i;: of r elements, by Vol. 1, Theorems 25, 58 [68, 121] we have n 11'1.

17. The Fundamental Theorem of Galois Theory The Galois group eM of a separable normal extension N of finite degree of K is studied because it is very useful for obtaining a more precise insight into the structure of the extension type of N. For, by means of the Galois group we are able to survey completely the building stones leading from K to N, that is, the fields lying between K and N together with their mutual relations, as soon as we know the structure of eM, especially the totality of the subgroups of eM with their mutual relations. Since eM is a finite group, the latter, at least in any concrete case, is a problem which can be handled in finitely many steps. 1) We prove the following Fundamental Theorem of the

Galois Theory: Theorem 109. Let N be a separable normal extension of degree n over K and eM its Galois group of order n. Then there exists a one-to-one correspondence between the totality of the extension fields'" of K contained in N and the totality (containing Q; 7) of the subgroups S) of eM. This correspondence is 7 As in Vol. 1, 0; designates the identity subgroup (unity group). The enclosed remarks refer to this superfluous addendum.

2. IV. Structure of Root Fields

274

completely established by each of the following two facts (I a), (1 b) valid for a /I. and S) corresponding to one another:

(1 a) ~) consists of all and only the automorphisms

01' ®

u'hich leave every element of /I. invariant. (1 aa) Hence ~ is the Galois group of N relative to /I., and therefore the order m of S) (the index of @ in S) is equal to the degree of N over /I. and the index j of S) in ® is equal to the degree of /I. over K. (I b) A consists oJ all and only the elements of N v'hich

remain invariant under every automorphism of S). ivloreover, if we say that the A aHd .\) with the same indices correspond to one another, then these correspondences also satisfy:

(II) If A is an extension field of A' of degree k over A', then S) is a subgroup of S)' of index k in S)', and conversely.

(rrI) If A, A are conjugate extension fields of K, then S",;:>, S",;:> are conjugate subgroups of ®, and conversely. To be more exact: If

A is

generated from A by the automor-

phism S, then ~ is generated from S) by transformation v;ith the element S, and conversely. (IV) If A is a normal field over K. then S) is a normal divis07' of ®, and conversely. (V) In the case (IV) (beszdes the statement in (1 aa» the factor group ®/S) is isomorphic to the Galois group of A relative to Kj for, if every automorphism of a coset of ® relative to S) i.~ applied to the elements of A one and the same automorphism of A relative to K is always obtained. The biuniqueness of the correspondence characterized by (1 a) or (1 b) implies, in particular, that the number of fields A between K and N is finite.

17. The Fundamental Theorem of the Galois Theory

275

Remarks: For the better understanding of this theorem and its proof the one-to-one correspondence (I a), (I b) is illustrated by the following Fig. 1, in which fields and groups standing at the same height shall correspond to one another. In N 0: particular, by (I a) K and @ correspond to one another; by (I b) Nand (r. 8 Therefore "in @" always corresponds to n "relative to K" and "relative to G:," to "in n N," as we tried to express as clearly as possible in the formulation of the theorem. Furthermore, by (II) to the" ascent" from K to N corresponds the "descent" from @ to Gl:, and to the relative degrees n, m, j, k Fig. 1 of the fields correspond the relative indices n, m, j, k of the corresponding groups, so that it is also appropriate to regard the orders of the groups as indices (of G: in them). As announced in the remark to Theorem 99, (II) [259], (III) and (IV) justify the dual use of the words normal and conjugate in field theory and group theory (cf. also the remark in the case of Theorem 93 (256]). In (III) to (V) (analogous to (II» the particular corresponding pair K, @ can be replaced 9 by any corresponding pair A', oIj', as immediately follows by applying the entire theorem to A' as ground field and (according to (I aa» .'0' as the Galois group of N relative to A' (for this see also Theorems 66, 70 [224, 227] ). Then (I aa) seems like a special case of (V), since in (V) @ and .\) can be replaced by .\) and (r, and accordingly K and A by A and N. In fact we actually have oIjJ(J; C'o.> ~).

B Therefore it is not N, as one might first think, which corresponds to its Galois group @. On the contrary, @ is the Galois group of N ?'elative to the K corresponding to @, just as .\) is the Galois grottp of N relative to the 1\ corresponding to oIj. 9 Fig. 1 corresponds to such a case, where this is also possible for (III), by taking A, A as conjugate even relative to I\' and oIj, ~ as conjugate even relative to oIj'.

276

2. lV. Structure of Root Flelds

Proof of Part (I)

In order to show that (I a), (I b) yield one and the same biunique correspondence between all A and all ~), it is sufficient to establish the following: (1 a) (I a) maps eve7'y A on a unique ~ (notation A-~), (1 h) (1 b) maps every ~ on a unique A (notation ~-A). (2 a) A-~ implies ~-A. (2 b) ~"""A implies A-~. For, that the unique correspondences (1 a), (1 b), which say the same thing by (2 a), (2 b), satisfy Vol. 1, Section 2, (fl), (£) [23.24] is guaranteed by (1 a), and that they satisfy Vol. 1, Section 2, (fl') , (£') [24] is guaranteed by (1 b). (1 a) This is clear, since by Vol. 1, Theorem 19 (631 and the definition of the product of automorphisms (Theorem 104 [263]) the set ~ of those automorphisms of ® which leave every element of A invariant is a subgroup of ®. This means, as established in (I aa), that Sj is the Galois group of N relative to A (Dei. 34 [267]), and therefore the order of Sj is equal to the degree of N over A (Theorem 105 l267] ), and the index of Sj in ® is equal to the degree of A over K (Theorem 71 [228] and Vol. 1, Theorem 25 [68]). (1 b) This is clear, since by Vol. 1, Theorem 6 [25] and the conditions characterizing automorphisms relative to K [Del. 32, (1), (2) [264] and Addendum [266J] the set A of those elements of N which remain invariant under all automorphisms of Sj is a subfield of N containing K. (2 a) Let A-+Sj according to (1 a). Then we form Sj-A according to (1 b) and A-~ according to (1 a). In this case the elements of A } { automorphisms of Sj •

277

17. The Fundamental Theorem of the GaloIs Thear)

. Sll1t;e

by {(I a) they are invariant with respect ~"\ } _ to cO' (I b) they leave the elements of A invariant

belong by {(I b)}, to the totality {~} of all such (Ia) ~ elements of N } { - \ { that is we haye A;;;; A , Sj ;;;; f3 automorphisms of 0) ,

f.

The first

of these relations implies that [N. A] ~ IN: A] (Theorem 70 l141] ), whereas ihe last implies [N: A) ~ [N : At since by

(1 aa) ~), Sj are the Galois groups of N relative to A, A, whose orders by Theorem 105 [267] are equal to the degrees of N over A, 7\. Consequently, [N: A] = [N : 7\], that is, A = 7\ (Theorem 72 [143]); due to the choice of A we therefore have Sj-+A according to (1 b), as stated in (2 a). (2 b) Let Sj-+A accordmg to (1 b). Then we form according to (1 a).

A-+~

~~.

This

On the one hand, it then follows as above that Sj

means that the orders m, 'iii of S'), ~ satisfy the relation m~m.

On the other hand, by means of a primitive element il' of N we form the polynomial 'Ijl(x) = (X-il'A 1)'" (X-fr.l m), where A l , ••• , Am are the automorphisms of ~'). Its coefficients are symmetric integral rational functions of the roots frAp. (!J. = 1, ... , m). Now, an automorphism A of ~) maps -!tAp. into (&A,).4 = 'frAfloA, namely, merely permutes them with one another (Vol. 1, Theorem 16 [60]). Hence the coefficients of 'Ijl(x) are invariant with respect to all automorphisms A of Sj and consequently according to (I b) belong to A. 'l'ex) is therefore a polynomial in A; it has il' frE as a root, and its degree is equal to the order m of Sj. This means that

=

2. IV. Structure of Root Fields

278

[N : A] = (K(ofr) : A] = (t\(ofr) : A] = (ofr : A] ::;;; m (Theorems 61, 77 [220, 236] Def. 22 [223], Theorem 53 [210] ). But by (I aa) and Theorem 105 l267] it follows from A-+ij that [N: AJ =

m.

Consequently we have m~m.

These two inequalities yield m = ni, that is, .~ =~; due to the choice of ~ we therefore have A-,;-~ according to (1 a), as stated in (2 b).

=

=

=

Incidentally, from m [N : A] [K('fr) : A] m together with Theorem 77 [286] and Def. 22 [228] it follows that 'Ijl(x) is the irreducible polynomial in A belonging to 'fr.

The unique correspondences (1 a), (1 b), which by (2 a), (2 b) form together one and the same biunique correspondence between all A and all lQ, are now designated by A +--+ lQ. Proof of Part (II)

a) If A +--+ lQ, A' +--+ lQ' and A ~ A', then by (Ia) the automorphisms of Sj leave, in particular, the elements of the subfield A' of A invariant, therefore by (I a) belong to the totality lQ' of all such automorphisms of ®. Consequently we have Sj ::;; lQ'. b) If A +--+ lQ, N +--+ lQ' and lQ::;;; 5;>', then by (Ib) the elements of A' are left invariant, in particular, with respect to the automorphisms of the subgroup lQ of lQ', therefore by (Ib) belong to the totality A of all such elements of N. Consequently we have A ~ A'. It immediately follows from (laa) that in both cases the degree of A over A' is equal to the index of lQ in lQ' if we set there A', lQ' instead of K, ®.

17. The Fundamental Theorem of the Galois Theory

279

Proof of Part (Ill)

For this purpose we first note that an automorphism S of N relative to K takes a field A between K and N into a field As between K and N isomorphic to it relative to K, namely, into a conjugate field. This immediately follows from Def. 32, Addendum [266] and Del. 26 [229] Now, if A ~ S), As ~S)s and ~ is an element of A, ~s the element of As corresponding to it, then ~.p = ~ implies

(~S)S-l.pS= ~SS-l.pS= P.ps= ~s; therefore ~s is invariant with respect to the subgroup S-l S)S of 0) conjugate to S). Consequently S-l S)S ~ S)s. But S-l takes As again into A since SS-1 = E. Hence it likewise follows that Ss;;, s8-1 ~ S), or also S) s ~ 8-1 S)8. Consequently ~) s = 8-1 S)S, that is, As ~ 8-1 S)S. In order to complete the proof of (III) we only have to establish in addition due to the biuniqueness of our correspondence that all conjugates to S) in 0) or A in N are represented by 8-1 S)8 or Ag respectively, if S runs through the group 0). a) For S) this is immediately clear by Vol. 1, Def. 21 L71] b) For A it follows thus: If we set A = K(f3), then As = K(~s). Now, if 8 runs through the group 0), then f3s runs through all conjugates to ~ (Theorem 105 [267] ), therefore As runs through all conjugates to A (Theorem 97 [258]). Proof of Part

CIV)

By (III) and the biuniqueness of our correspondence we have that if the conjugates of A coincide so also do the conjugates of S) for a corresponding pair A ~ 3), and conversely. (IV) immediately follows, on the one hand, from Theorem 99, (II) [259] and, on the other hand, from Vol. 1, Theorem 31 [73].

280

2. IV. Structure 01 Root F,elds

Proof of Part (V)

Let A ~ Sj and accordmg to (rV) let A be a normal fIeld over K, Sj a normal divisor of ®. Then, due to As = A (cf. proof to part (III» every automorphism S of ® produces an automorphism P of A relative to K; and since every element of A is invariant with respect to Sj, all automorphisms of a residue class SjS produce one and the same automorphism P of A. Conversely, every automorphism P of A relative to K is generated in this way from an automorphism S of ®. For, if ~ is a primitive element of A, then by Theorem 105 [267] ~p is one of the conjugates to fl, and therefore there exists, again by Theorem 105, an automorphism S of ® which takes ~ into ~p and consequently produces the automorphism P for the elements of A = K(~). Accordingly, to the totali1.y of co sets of ® relativQ to Sj, that is, to the totality of elements of the factor group ®/~, there uniquely corresponds the totality of elements of the Galois group of A relative to K. This correspondence is also one-to-one in this case, since by (Iaa) the order of ®ISj, that is, the index of ~, is equal to the degree of A, therefore by Theorem 105 is equal to the order of the Galois group of A. Finally, the definition of coset multiplication (Vol. 1, Theorem 22, [65]) implies that the correspondence under consideration also satisnes the isomorphism condition (Vol. 1, Theorem 23 [65]). Consequently, the Galois group of A relative to K is isomorphic to the factor group ®/i;J and is generated by it in the manner specified in (V). This completes the proof of the Fundamental Theorem. Now, under the one-to-one correspondence specified in the theorem by (II) all containing relations and therefore also maximal and minimal properties relative to such relations likewise correspond

17. The Fundamental Theorem of the GaloIs Theory

281

to each other lU inverted order. On takmg this fact into consideration we obtain the following additional ploperties of that correspondence: 'flleOl'em 110. If in the sense

of

Theorem 109

Ai +---+ ~i' " ., Ar +---+ S";?r and [ .. J designates the mtersectzon, {... } the composite of the fields and groups,lO then

[Ai' ... , Ar]

+---+ {~l'

... , ~r},

{Al' .... ,A r }+--+ [~l'···'S"d)] is valid.

In particular, by the remark made with respect to Theorem 62 [220J we have

Corollary. If al"'" aT are elements of N and in the sense of Theorem 109 then K(al' ... , aT) +--+ [~l' ... , ~r J. 2) The Fundamental Theorem enables us to draw conclusion:;; about the structure of N over K from the structure of the Galois group ® of N relative to K. According to the introductory remarks we think of the latter as known 11 in applying the Fundamental Theorem to the investigation of the structure of the extension N of K. In order to break up the step from K to N to be investigated into a sequence of simpler steps by inserting a chain of intermediate fields

K=Ao
11

[270J

Ci. Theorem 106 f269] as well as the remark before Def. 35 and after Theorem 107 [271].

282

2. IV. Structure of Root Fields

we have to determine, starting from a chain of subgroups @ = S)O > 3)1 > ... > ,))8 = (t, the A, as the intermediate fields corresponding to the subgroups ~,. If we regard the intermediate links A, themselves as new ground fields, the successive expansion from K to N is then tied to it successive reduction of the Galois group @ of N relative to K to the subgroups S)" which according to (1aa) arc actually the Galois groups of N relative to the A,. The entire step from K to N is covered when @ is reduced completely, that is, to (t. In particular, if ~)i is chosen so that &),+1 is a norma.l divisor of S)" then by (rV) A'+1 is a normal field over Au and the Galois group corresponding to this step is ,)),1S),+l' In order to reduce the Galois group @ of N to a subgroup 3) in the stated sense we have to determine the subfield A of N corresponding to S). This can certainly be done by the correspondence rule (I b) ; however, this does not enable us to control the field A as completely as if we were to give a primitive element ~ of A. For such a primitive element we introduce the follOWing definition: Definition 36. If A --- S) in the sense of Theorem 109 and A = K(M, that is, if ~ is a primitive element of the A C01"f'esponding to &), then ~ is called an element of N belonging t.o &).

Here we will not go into the question of how such a {1 is determined from a primitive element i} of N (cf. Vol. 3, Section 17, Exer. 4). In the following we will only derive a series of facts, which are of theoretical signiiieance in this regard. Conversely, it is naturally valid that every element {1 of N also belongs to a certain subgroup Sj of @, namely, the subgroup (uniquely) determined according to K(P) A +-+ Sj. Hence these facts, in addition, extend, and yield a deeper group-theoretie basis for the statements made earlier about the conjugates of an algebraic element (Theorems 95, 96, 103 [257, 257, 262] ).

=

283

17. The Fundamental Theorem of the Galois Theory

Theorem 111. Under the assumptions of Theorem 109 let ~ be an element of N belonging to the subgroup S) of 0), and j the index oj' S); furthermm'e let S)Sl + ... + .'1)8] be the right decompo.sition of 0) relative to .'I). Theil, on applying the automorphisms 8 of 0) one and the same cOlljugtrte ~.5)S of ~ is always generated by all automorphisms of a coset S)8. Fur0) =

thermore, the conJugates ~~S" ••• , ~.!!>s corresponding to the j -0 J eosets ~;;k"lu"" ~ 1 are different from olle another. Or, in short, the element ~ is znvariant with respect to S) and j-valued with respect to 0). In particular, therefore

's

(1) g(x) = (x - ~?Sl~ ... (x - ~~s,) is the irreducible polynomial in K belonging to ~. P1'00{: According to Def. 36 the elements of A =

K(~)

remain invariant with respect to the automorphisms of S). This means, in particular, that ~.? = i3 and therefore i3~s = ~s for any S of 0). Now the !3s represent all the conjugates to !3 (Theorems 103, 105 [262, 267] ). Furthermore, by Theorem 109 A = K(!3), so that ~ has the degree j. Hence there are generally exactly j different conjugates to i3 (Theorems 96, 58 [257, 215] ). But, in view of what has already been shown, there are among the !3s at most the j elements i3.?s ('V = 1, .. " j) different from one • another. Therefore none of these can be equal. The statement of the theorem follows from this. Theorem 111 can also be reversed: Theorem 112. Under the assumptions of Theorem 109 let ~ be an element of N invariant with respect to the subgroup S) of ® of index j and j-valued with respect to ®. Then !3 is an element belonging to S).

2. IV. Structure 0/ Root Fields

284

Proof: If K(~) =A -<'-+ Sy, that is, ~ is an element belonging to S)', then by (I a) we first have S) ;:;;;; S)', since by the assumption the elements of A = K(~) are surely invariant with respect to :g. Furthermore, by Theorem 111 ~ is i'-valued with respect to @, where l' designates the index of S)'. Therefore, according to the assumption l' = j, that is, S)' = S), and consequently ~ is an element belonging to :g, as stated. The two extreme cases of Theorems 111, 112 are worthy of note: Corollary. Under the assumptions of Theorem 109 an element ~ of N is a primitive element of N if and only if it is n-valued with respect to ®, and an element of K if and only if it is onevalued (invariant) with respect to @.12 The results expressed in Theorem 111 can be applied (by Theorem 94 [256] ) to the data taken as a basis in Theorem 95 [257] , if the irreducible polynomial f(x), whose stem field A= K(<x) and root field W= Kea!, ... , ar ) are to be studied, is again (as was already done in Theorem 103 [262]) assumed to be separable. As a supplement to 'I'heorem 95 and Theorem 103 we prove in ihis regard: Theorem 118. Let the data of Theorem 95 [257] be given. and in addition let [(x) be separable. If we set [K(a): K(~)] =k, [K(M: K]=i, therefore kj= [K(a): K]

=r,

12 These facts follow most simply directly from the fact that the correspondences (1 a) and (1 b) coincide, as established in the Fundamental Theorem. For this purpose we only have to state that K and @ correspond, not only (as already stressed in the remarks included there, trivially) according to (1 a), but also according to (1 b); and similarly that N and ~ not only correspond according to (1 b) but also according to (1 a).

17. The Fundamental Theorem of the Galois Theory

285

then the elements ~1' ••• , ~, conjugate to ~ fall into j different sets each consisting of k elements equal to one another. In particular, therefore, g(x) == (x - ~1) ... (x- ~r) is a polynomial in K which is related to the irreducible polynomial g(x) in K belonging to ~ as follows: g(x) == g(x)k. Proof: We apply to a, and with this to ~=h(a), all automorphisms of the Galois group ® of W. (Besides the degree relations already introduced in the theorem for the fields W ~ K(a) ~ K(~) ~ K) let 1=[W:K(a)], m=[W:K(~)], n=[W:K], therefore lk=m, mj=n. On the one hand, Ly Theorem 111 the seL of the r different al' ... , ar conjugate to a is now generated l times from a by applying ® (Theorem 58 [215]), and therefore the set of conjugates ~1"'" ~r is generated, likewise l times, from ~ by applying ®. On the other hand, however, the set of the j different conjugates to ~ is generated m times from ~ by applying ® a('cording to Theorem 111. Therefore, the set ~1' ••• , ~r (written only once) must represent these different conjugates exactly k

= 7times,

as stated.

In analogy to the Corollary to Theorems 111, 112, it further follows from Theorem 113 that: Corollary. Let the data of Theorem 95 [257] be given and in addition let f(x) be separable. Then an element ~ of A is a primitive element of A if and only if the conjugates Pl"'" ~r are all different from one another, and it is an element of K if and only il these conjugates are all equal to one another.

286

2. IV. Structure of Root

F~elds

In connection with rrheorem 111 we further establish: Theorem 114. Let A ~.'b in the sense of Theorem 109 [273]. Furthej'more, let 'It be an element belonging to ~

A

~

and

an element of N belonging to &), therefore N = K(&) and Finally, let A l , ••• , Am be elements of &) and

= K (~).

~81'

.... &)81 right casets of ® relative to &). Then the j con~.ps, each belong to the j conjugate subgroups

jugate elements S;-l~Si

Furthermore,

=

(2) 4h(X) (X-1'tA)"'(X-1'tA 't' 1 m) is the zrreducible polynomial over A belonging to 1't. The irreduczble polynomial q(x) over K belonging to 1't has the decamposztion (3) u:here (4)

q(x)

=

'IjIo.s (x) ... tjJ.ps (x), "" 1

l

,ho. (x)=(x-1't AS )···(X-1'tA ms) (v=l, ... ,j). 't'''b'' sv 1 ~ 'J Here the factors 'IjI.psJx) are the conjugates to 'IjI(x); therefore they too are irreducible polynomials in the conjugate fields

A.ps.= K(~.ps) to A. In particular, if ~ is a normal divisor of ®, so that A is a normal field over K, then (3) accordingly represents the decomposilion of q(x) into its irreducible factors (4) in the field A. Theref01'e, these are all of the same degree and consist of the linear factors of q(a:) corresponding to the decomposition of ® relative to 0. Proof: The statement regarding the ~.pSy follows from (III). The statement about 'IjI(x) was already shown in the proof of part tI) to Theorem 109 under (2b). On applying the automorphisms S., to AI) = A, 'IjI1> (x) 'IjI(x) it finally follows that the 1/J.psv(x) are polynomials over the A.psv'

=

17. The Fundamental Theorem 0/ the Galois Theory

287

If N is the root field of a separable polynomial I(x) over K, hence q(x) a Galois resolvent of I(x), then in the older literature the relations described in Theorems 111, 114 were expressed as follows (here too remarks similar to those made in connection with Def. 31 [261] are appropriate) : The Galois group @ of the polynomial I(x) over K will reduce to ~ by the adjunction of an irrationality fJ of N belonging to the

subgroup ~ (index j, order m), that is, with respect to A = K(,8) as ground field. This adjunction is made possible by solving the resolvent of j-th degree (1) and produces a decompo.s.ition (3) of the Galois resolvent q(x) into irreducible factors of m-th degree (4) which belong to the j conjugate fields to A K((J). After the adjunction of ,8, in order to determine a root it of the Galois resolvent q(x) we must still solve the resolvent of m-th degree (2), the Galois resolvent of I(x) relative to A, which is obtained from those of the factors (4) of q(x) corresponding to the field A. In particular, if ~ is a normal divisor of @, then A K({J) is a normal field over K and the resolvent (1) is the Galois resolvent of itself with the Galois group @/~. The conjugates {JS'JS~ of {J are irrationalities over N belonging to the conjugate subgroups 8:;1 R;!8v of~. Consequently, by Theorem 110, Corollary [281] the simultaneous adjunction of all conjugates (JS'JS~' that is, of all roots of the auxiliary equation (1), reduces the Galois group @ to the intersection [S11 &,)S1"" ,Sf1 ~Si] of all subgroups conjugate to &'). By Vol. 1, Theorem 33 [74] as well as by Theorem 94 [256] and Theorem 109, (IV) [274] this means that this intersection is a normal divisor of @.

=

=

3) Finally, It should be noted that the Fundamental Theorem also can be used to investigate the structure of an arbitrary (not nf'cessarily normal) separable extension /\ of finite degree over K. For. If /\ = K(a) and a!, .. " ar are the coniugates to a, then 1\ is a subfield of the normal fieldN= K(a 1, ••• , ar) over K. In this case, if ® is its Galois group, s;, the subgroup corresponding to /I., then a one-to-one correspondence with the properties of Theorem 109 can be set up mapping the groups

288

2. IV. Structure of Root Fields

between OJ and ~ on the fields between K and A. The proof of Theorem 113 [284J is an example of this kind of approach. A. Loewy 13 has actually shown much more by carrying out the entire train of ideas of the Galois theory from the beginning for an arbitrary finite separable algebraic extension A = K(a l ,· •• , aT) (where fJt" •• , aT are, therefore, not necessarily the roots of a single polynomial) rather than restricting himself to a normal extension. Instead of the automorphisms and permutations he used isomorphisms and so-called transmutations (one-to-one correspondE'nces with a definite mapping rule of the a l , · . · , aT to a system of conjugates al"'1 , ... , aT<,J'T ,cf. the proof of Theorem 90 [251]) which no longer form a group but a so-called groupoid. Incidentally in this way Loewy was liKewise able to establish the Galois theory independently of the symmetric functions (cf. footnote 1 to Theorem 90).

18. Dependence on the Ground }'ield The Fundamental Theorem of the Galois theory and the expositions related to it enabled us to determine how the structure of a separable normal extension N of finite degree over a ground field K is ai'fected if an extension A of K contained in N is used as ground field instead of K. In addition we now wish to see what happens to the relations if the ground field K is leplaced by an arbttl'ary extension K of K. 'l'his question arises, as the considerations at the end of Section 17, from the effort to attain a given extension N of K of the stated kind by steps as simple as possible or - and this is the point which represents 13 Neue elementa're Begrilndung und Erweiterung der Galoissehen Theorie (New Elementary Foundation and Extension of the Galois Theory), Sitzungsberichte del' Heidelberger Akademie del' Wissen.schaften. Mathematisch-naturwissenschaftlicher Klub, 1925 (Records of the Heidelberg Academy of Sciences, Department of Mathematics and Natural Science, 1925).

18. Dependence on the Ground F!eld

289

a generalization of ~ection 17 - merely to mclude it. Above all we intend to impose on these '>teps some general sllnphcity condition, which has nothing to do with the gh'en N; for instance, as will be dOlle in Chapter V, the condition that the n

extension take place through the adjunction of roots special sense of the word. 1) If N is the root field W of a polynomial rex)

... ex -

Va

in the

= ex -

at)

at) over K, then on passing over to an extension K of

K as ground field the root field W = K(Ut, ... , aT) of f(x) O\'e1' K is also replaced by the extended root field

VI = K(al' .... aT)

of f(x) over K, and by Theorem 60 [~20) VI = W if and only

if K;;:;; W, namely, if we have the same situation as in the previous section. Since by Theolem 100 [26(J] any extension N of K of the stated kind can be represented as a root field W of a polynomial f(x) over K, we can also explain in this way what is to be understood by N 1.vith respect to an extension

R of

K as ground field, namely, the passage to the extension

N= VII of K. This explanation seems at first dependent on the choice of the polynomial f(x). However, we prove: Theorem 115. If N is a separable normal extension of finite degree of K and

K an arbitrary extension of

K, then the root

fields over K of all polynomials over K, for 1.vhich N is the root field over K, establish altogethe1' one and the same separable normal extension N of finite degree over K. Proof: Let f(x) = (x - (1) ... (x - ar), f*(x) = (x-a't)· .. (x-a;.) be two polynomials over K for which N is the root field over K,

290

2. IV. Structure oj Root Fields

and let N, N* be the root fields of !,(x), f*(x) over K. In this case since

N" =

K(at,· ... a;.) ~ K(a1, .... (J,~.) = N = K(a1' ... , ar),

N'" contains K, on the one hand, and

a1,

••• ,

aT'

on the other

hand. Therefore, we also have K(a1"'" ar) = N, that is,

N"!"

~

N. Likewise, N ~ N*. Consequently N= N-".

The extension N of K, independent, according to this, of the choice of the polynomial ftx). is naturally separable and normal of finite degree over

K (Theorem 100 [:260] ).

The extension hi of K in Theorem 115 is determined uniquely 14 by the property of being a root field over K for all polynomials over K for which N is a root field over K, in tho same sellse as the root field of a polynomial is determined according to Theorems 87, 89 [:2-1:2, 246J. This means that any two snch extensions of K are isomorphic relative to J{, and no extension of K contains two different such extensions. We can therefore define: Definition 37. The extension N of K in Them'em 115 deter!/tined uniquely by K, N, K is called the extension N of K with 1'eSlJe,(:t to

K ((S

f/l'Qwul fiel(l.

It is important tor us to characterize this extension in another way:

N still

14 This uniqueness depends essentially on the normality of N. For arbitrary extensions of finite degree A = K(a l " " , ar), where al' ... ,a, are not necessarily the roots of a single polynomial over K, the conjugates of f.. = K(a1 •• , ' . a,) would necessarily have to be characterized with respect to K,

18. Dependence on the Ground Field

291

Theorem 116. If, under the assumptlOl! of Theorem 115, N is til e e.1:tension N of K with respect to K as gl'oancI field. thell 1)

N contains N,

2) no field between K and N contains N except N itself.

N is determined umquely by 1), 2). Proof: That 1), 2) are valid for N follows immediately from the property used to !.lefine N as well as the minimal property of root fields (Theorem 88 [243 J). b) The extension N" of K has the properties 1), 2). In this case by 1) N* contains the roots al •... ' a r of every polynomial f(x) over K for which N is the root field over K. therefore it also ('ontains the field N of Theorem 115. Hence N is a field

K and N* which contains N, ancI therefore N= N"', since we have assumed that N* satisfies 2). between

The properLies 1), 2) say that N is the smallest subfield of

N containing Nand K; therefore, it is the composite {N, K}. Here N itself is to be regarded as the auxiliary field (field K of Vol. 1, Def. 5

[27J ) containing both Nand

K, whose existence

is required in order to be able to form the composite {N, K} = N.

N cannot be defined from the beginning as the composite {N, 'K} On the contrary, the representation of N as the composite {N, K} is only made possible by the construction of N carried out Hence

according to Theorem 115, and the use of the definite expression "the composite {N, K}" is based on the fact that N is determined uniquely by Nand

K

alone as proved in Theorem 115. Before the

composite is formed according to Theorem 115 the fields Nand K are" free" from one another, that is, are not contained in a common extension. We express this logical relation as follows, in connection with Def. 37:

2. IV. Structure of Root Fields

292

Addendum to Definition 37.'1'he extension N of K with ?'espect

to K as ground field, which exists in the sense of Theorem 115 and Def. 37, is also called the fi'ee composite of Nand (notation {N, K}).

K

I t may help to clarify the point under consideration if we make the following additional statement: Steinitz proved that the free composite can be defined from the beginning as the ordinary composite of Nand

K if we make use of the existence and uniqueness

of the field A of all algebraic elements over K (see Se<:tion 11). Then this field A, which also contains the field A of all algebraic elements over K and therefore, in particular, the field N, can be used as an auxiliary field for forming the composite of Nand K. 15

In Theorem 116 we characterized the extension N of K

with respect to K as the free composite {N, R}. This characterization is important because it shows that the ground field K entering in Theorem 115 and Def. 37 essentially plays only the role of an auxiliary field. Namely, since in Theorem 116, 1), 2) no mention of K is made, we have Theorem 117. If N is a separable normal extension of finite de{Jree over K, K an extension of K aHd N the extension N of

K 1.,vifh respect to K, then N is also the extension N of K* with

respect to K if K* designates any common subfield of Nand K over which N is sepal'able and normal of finite de{Jree.

15 This is also valid for an arbitrary algebraic extension A of K, and even uniquely. Th.is follows from the fact that the conjugates of /.. {A, i<} which are not distinguishable "from below," that is, with respect to free composition by a process generalizing Theorem 115, are, from the beginning, distinct "from above," that is, inside A.

=

18. Dependence on the Ground Field

293

Therefore, in ],Jartieular, the largest common subfield of N and K, that is, the intersection [N, KJ, can also be regarded as ground field, since it satisfies the required conditions (Theorems 70, 92, 93 [227, 255, 256] ). In order to be in harmony in the future with the designations of Section 17 we now write

X instead of K, so that N = {N, A}; furthermore, we set N A -= [N, X] and illustrate the fields under consideration and their relations to one another by the adjoining Fig. 2, which is an extension of the graphic illmtration used in Fig. 1 for Theorem 109 l275]. By Theo-

N

K Fig. 2.

rem 1 t7 N also has in this case the property of Theorem 115 for A as ground field. Furthermore, due to the symmetry of the composite

{N, X} in N and A we have: Theorem 118. If N and

A are separable normal extensions

oj' fillite degl'ee over' K, then the free composite

N=

{N.

A}

is

the extension N of K with respect to 7\ as well as the extension

A of

K with respect to N. Consequently,

normal

of

N is separable and

finite degree over A as well as over N.

2) We now prove the following basic theorem, which completely answers the question asked at the beginning: Theorem 119. Let N be a separable 1101'mal extension of fillite

degr'ee over K and ill the Galois group of N relative to K. [i'urthermore, let 7\ be any extension of K and

N = {N, X}, A= (N, 7\]. Finally, let ~) be the subgroup of ® corresponding to the field A between K and N. Then

294

2. IV. Structure of Root Fields

(I) The Galois group ~ of N relative to Ii. zs isomorphic to the Galois group Sj of N relative to A. Namely, the automorphisms of Sj and §,? can be generated:

a) By the same substitutions of any pr'imitive element 'It of N relative to A (therefore, in particular, by such a one from N relative to K). b) By the same permutations of the t'oots (.(1"'" aT of any polynomial cp(x) over A for which N is the 1'00t field over A (therefore, in particular, by such a one over K for whieh N is the root field over K).

III particular, according to this (Fig. 3)

[N : li.J = allll,i!

A has

IN : AJ,

finite degree over A, also

[N : Nl = [Ii. : A). (II) If the fields A~' between A and N, and the fields between

Ii.*

A and N correspond to oue another on the basis of the

isomorphism betu'eell ~) and S5 described under' (I) and according to Theorem 109 [273), then this is a one-to-one correspondence between the A* alld

71.*,

ill which the relatiolls

"contamed, conjugate, normal" correspond tv one another' as

N N

N N

rn m

I .\

7\ A

A

A

K

I{

Fig. S

Fig. 4

18. Dependence on the Ground Field

295

long as they refer to fields correspondiilg to aile another and gr01ln(1 fields of the stated kind. With /'espect to fhis correspondence we have further (Fig. 4):

N = {N, k}, A = [N, Aj, A" -= {N. A}. (III) If, in particular, A is normal over K, then A A*

=

lN, A"],

is also

normal over K (the converse, hon'ever, need not be w/id). Proof of Part (I) a)1r. Let

{t

be a primitive element of N relative to A, and

'ljl(x) and 1Jj(x) tho irreducible polynomials over A and 7\. l'espectively, belonging to this element. Then 'li(.r) is first of all a divisor of 'ljl(x) (Theorem 53 [H)!:)]). Fmthermole, since 'ljl(x) resolves into linear factors in N (Theorems 93 l:256] , 98 [258], 99, (Ill) 125H]), 1Jj(x) is a ploduct of certain of these linear factors; therefore it, too, is a polynomial in N. Consequently, 1Jj(x) actually belongs to the intersection A = [N. 7\]: Due to the irreducibility of'ljl(x) weihereforehave1jJ(x) ='ljl(x}. From N={N,A}={A(fr),A}=A(tT) it further follows that fr is also a primitive element of N relative to i\. This result and the fact 1Jj( x) 'ljl(x) previously proved

=

yield the statement (I) a) by Theorem 105 [267] . b) Let N be the root field of cp(x)

=

(x·-

((1)

•.•

(x -

((T) over

A, therefore N the root field of cp(x) over A (Theorems 115, 117). Then by Theorem 107 [271] the automorphisms of the Galois 16 For this proof we must realize that the elements of N belong to N (Theorem 116, 1 [ [2911), therefore can be replaced by those of A in calculating relations. Without the existence of a common extension of N and A this would not be permissible. In such a case, for instance, the polynomial 'ijJ(x) could not be defined as in the text.

296

2. IV. Structure of Root Fields

groups S) and .i5 are generated by those permutations of the different elements among the ai' ... ,ar, whose application takes any existing integral rational relation X (a 1,· •• , a r ) = 0 and X(u1' ... , a r ) = 0 with coefficients in A and A, respectively, over into a relation which is also correct. The groups '.j3 and

1$

of these permutations,

which are isomorphic to :Q and.iS, respectively, are also isomorphic to one another due to the isomorphism of S) and S5 proved by a); therefore, in particular, they have the same finite order. Now, the permutations in 'l3 have the specified property relative to the field A a fortiori for the subfield A, because the relations X (ai' ... , ar ) = 0 occur among the relations X(ai' .. " at) = O. Hence ~;;;;; \j3 and consequently by the above ~ = \fl, as stated under (I) b). The degree relations specified under (I) immediately follow from Theorem 109, (I aa), [274] and the double application of Theorem 71 [142] according to Fig. 3.

Proof of Part (II) The first part of statement (II) is clear by (1) and Theorem 109 [273]. Furthermore, let N, 7\* be a corresponding pair of fields. To complete the proof use the elementary properties of intersections and composites: First, A* = [N, A") . To prove this we let ~*, &,5* be the subgroups of Sj, ~ corresponding to A*, A*, and {} be a primitive element of N relative to A, therefore by the proof for (I) a)

N relative

7\. Then by our correspondence rule in (II) and by Theorem 109, (Ib) [274] A* and A* consist of all rational functions over A and X, respectively; of {} which also of

to

are invariant with respect to !Q* and

Therefore,

A*;;;' [N, A*]. Conversely, the elements of the intersection [N, A*] are, as elements of N, rational functions of {} over A; and as elements of A* they by (I) a)

A*;;;; A*,

&5*, respectively.

and consequently also

297

18. Dependence on the Ground Field

are invariant with respect to variant with respect to

~)*,

55".

Hence by (I) a) they are in-

that is. we also have [N, A#] ;;;;; A*.

Taken together this means that A* = [N, Secondly,

A*], as slated.

N = {N, A"'} follows trivially from

N~ {N,A*} ~ {N,A}=N.

A]

Thirdly, A = [A \ A -;:;. lA*,

AJ ;;;;;

[N,

A]

follows trivially from

=A.

A* = {A\ A}. Namely, from the relation A" ;;;;; A* previously proved it follows, at any rate. that A* ~ {A *, A} ~ow, suppose that A* > {A*, A} were valid. Then if the oneFourthly,

to-one correspondence established in the first part of (II) were canied out using the intersection mechanism already proved as valid under "first," it would also follow that

A*= [N,A*] > [N,{A*,A}], whil€' Ai';;;;; [N, {A *, A}] is trivially true. The latter proof justifies the technique of drawing the cross connection between A'" and

71.*

in Fig. 4.

PrOOf of Part (III)

If, besides N, A also is normal over K, then by Theorems 98 [258] ; 99, (III) [259]; 103 [262J the conjugates relative

to K of a primitive element of the intersection A = [N, A] are contained in N as well as in A, therefore also in the intersection A. By these theorems, therefore, A is then normal over K. This completes the proof of Theorem 119. 3) The result expressed in 'rheorem 119 under (I) will now be interpreted in terms of the expositions in Section 17, 2) [281]. From this point of view it says that in the transition

2. IV. Structure of Root Fzelds

298

to an arbitrary extension 7\ of K as ground field the Galois group G) of N relati\Oe to K is reduced just as in the tl ansition to the intersection A = [N, AJ, that is, to the part A of

A

contained in N. Accordingly, the adjunction of elements to K not contained in N furthers the construction of N no more than the adjunction of suitable elements to K contained in N. Hence, following K1'onecker, we call the former (so far as they are algebraic over K) accesso1'Y irrationals, the latter natural i~'?'ationals for the extension N of K. That the accessory irrationals must be taken into account with respect to certain investigations in spite of the results contained in Theorem 119, (I) goes back to the fact that the adjunction of an accessory irrational may very well satisfy a prescribed simplicity condition, whereas this is not the case for the adjunction of an equivalent natural irrational according to Theorem 119, (I).

The result expressed in Theorem 119, (II) says that the

A to N still over to A as

step from

to be made for the inclusion of N after

passing ground field [cf. Theorem 116, 2)] is equivalent to the step from A to N in any consideration that may concern us, Moreover, it says that any pair of fields A"and

A*

corresponding to one another in the sense of Theorem

119, (n) can take on the roles of A and A. Now, to generalize the expositions in Section 17,2) [281] take an arbitrary chain of extensions (1)

as a basis. Then there corresponds to this a chain

(2) K=Ao;:;::; Al ;:;::;~;;;;; ... ;;;;; Ar;:;::; N of fields between K and N, the development of which can be illustrated by the following Fig. 5.

299

18. Dependence on the Ground Field

Fig. 5

By the successive construction we thereby have (3)

N, = (N'-l' A,},

A~i-!) = CN'-b A,] , A~'-2), "."' A;I), A, are the subfields of

A;'-I>, N,-2"'" N1, N corresponding

and the fields

N,-I' Theorem 119, (II).

to

A;'-1) in the sense of

Theorem 119, (III) yields that of these A;,-I) is normal over Ai_I' therefore by Theorem 119, (II), A, is also normal over Ai - 1 if A, is normal over A'_l (but the converse need not be true). By the successive application of Theorem 119, (II) it easily follows in view of the minimal property of composites (Vol. 1, after Def. 5 (27} or Vol. 2, Theorem 116, 2)) and the maximal property of intersections (Vol. 1, after Def. 5) that in any parallelogram of our schematic figure (it may be

300

2. IV. Structure of Root Fields

composed of a "ground mesh" or of many "ground meshes") the lower left field is the intersection and the upper right field i" the composite of the two fields standing in the' upper left and lower right positions. In particular, therefore, besides the recursive representations (3) there also eXIst the representations skipping over all intermediate steps (4) Az= IN,A,]. N,={N,A,}, By Theorem 119, (I) this implies that the successive expansion of the ground fields through the fields of the cham (1) is tied to the successive reduction of the Galois group ® of N relative to K through the chain of subgloups (3) @ = &)0 ;:;;; ~1 ;:;;; Sj2 ;:;;; ..• ;:;;; -Pr;:;;; Q; corresponding to the chain (2) by the Fundamental Theolem. By (4) and the properties of composites and intersections

AT ;:;;;

N is valid, that is, N, as visualized in the end, is included

in chain (1) if and only if AT = N and therefore AT =N" that is, if, according to (5), the Galois group G) is reduced to &), = Cit.

v.

Solvability of Algebraic Equations by Radicals

The theory developed in Chapter IV owes its origin to the celebrated question already mentioned at the beginning of Section 18, namely, under 'what conditions is an algebraic equation solvable by radicals. Accordmgly, this theory forms the foundation for the handling of this question. The last chapter to be given is devoted to answering this question for ground fields of characteristic O. For this purpose we will first state the question in a more precise form by defining what we mean by solvable by radicals (Section 19). Then we will develop as a necessary tool the theory of cyclotomic fields (Section 20) as well as the theory of pure and cyclic extensions of prime degree (Section 21). From this by applying the Galois theory considered in Chapter IV we will deduce a group-theoretic criterion for the solvability by radicals (Section 22). Finally, we will sketch in addition the proof based on the Galois theory for the non-solvability by radicals of the generiC algebraic equation of degree higher than the four-th, which was first established by Abel in another way, (Section 23). In Section 20 we add a short sketch of the theory of finite Iields and thereby, in particular, fill up the gap still remaining in the proof of Theorem 90 [251] regarding such fields.

19. Definition of Solvability by Radicals In this section we give an exact formulation for what is to be understood by the expression solvable by radicals. The 301

2. V. Solvabthty of Algebratc Equattons by Radtcals

302 n

conC'ept l'a familiar from the elements of arithmetic, where a is an element of a field K and n is a natural number, is explained there, as is well known, as the solution of the n

equation x t , - a ~ o. Since Va is in general a many-valued function we will not use this notation. Instead we will use the equation belonging to the radical rather than the radical itself. Definition 88. A polYJIQmial of the form xn - a is called l)lII'e.

In order that the question posed at the beginning of this chapter be not trivial, we naturally have to admit into the circle of the allowable operations not only the specified operation of root extraction but also the four elementary operations, subordinate to the former from this standpoint.1 Along with a root a of a pure polynomial we then consider all its rational functions over the ground field K as known, that is, all elements of K(a). However, the significance of our question goes still further: It would be unsystematic if we were to hold the operation of root extraction in check by one such step. On the contrary, it is reasonable to regard not only the roots of pure polynomials as known but also those of polynomials belonging to the extended field K(a), etc. Our question amounts then to this: What are the conditions under which we can reach or introduce the roots, that is, the root field of a polynomial f(tr) over K or in general any extension A of K, by the successive 2 adjunl'tion of roots of pure polynomials starting from K. This immediately enables us to curtail the type of radicals to be drawn into consideration: Namely, if xn - a is a pure polynomial over K of compound degree n n 1 ns and a one of its roots, then an, al is a root of the pure polynomial xn , - a over K and further a is a root of the pure polynomial ron. -al over K(a1)' Consequently, we

=

1

=

Otherwise, only pure equations could be solved by radicals.

2 Not only by simultaneous adjunction. This actually says (in contrast to the case of Theorem 62 [220]) more than this, for a can very well be a root of a pure polynomial over an extension K of K without still being a root of a pure polynomial over K.

20. Cyclotomic Fzelds. FZnlte Fields

303

can restrict ourselves to the successive adjunction of roots of pure polynomials of prime degree. According to one's wish the restriction can also be imposed here that these polynomials shall or shall not be irreducible in the field reached at any given time. Since the irreducible polynomials are the simplest building stones for the construction of algebraic extensions, it seems theoretically more proper to adopt this restriction.3 Accordingly, we define:

.. Definition 39. An extens'ion A of K K

if

1S

called pW'(' ovel'

tt can be derived by the adjunctwn of a I'oot of all

irreducible pure polynomwl over K.

* Definition

40. An

called salt-able by

extension

1'(ldical.~

K=Ao
0, [uute degree A of K is

over K zj' a chain of extensions

<···
exists in which A, is pure and of prtme degree over

A'-l'

A polynomial f (x) over K is said to be solt'able by 1·(ulic((1.~ over K if its root field over K is also.

20. Cyclotomic Fields. Finite Fields In order to be able to handle the question of the solvability by radlcals we have to make some preliminary statements about the theory of the special pure polynomial x" - e. We remark that if the roots of this equation in the case of the rational ground field P are represented as complex numbers in accordance with the so-called Fundamental Theorem of Algebra, they will effect the division of the circumference of the unit circle into It equal parLs. In view of our applications we will I Actually the answer to our question (for ground fields of characteristic 0) given in Theorem 127 [319] is independent of this restriction, as easily follows from Theorems 123, 126 [312, 316] given later. Out of r€gard to Theorem 126 it appears to me however more proper to demand the irreducibility, since the "c;uder" statement of the question entirely overlooks the algebraically interesting "finer" structure of cyclotomic field".

2. V. Solvability oj AlgebraIc Equatwns by RadIcals

304

not restrict ourselves here to the special case P. Instead, we will alIo\\- more general ground fields K, though in general we will stiU call the equation xn - e ....:-.0, due to its association with this special case, the cyclotomic equati01t and its root field Tn the cyclotomic field for novel' K. Regarding the roots of the cyclotomic equation for novel' K, the so-called n-th 1'OQtS of 'unity over K, we first of all prove the following theorem: Theorem 120. Let K be a field whose characteristic is 0'1'

°

a prime number which is not an exact divisor of n. Then the n-th roots of unity over K form a cyclic g1'OUp .5 of order 11 with respect to multiplicatlO'l1. Therefo1'e, there eXlst n different n-th roots of unity over K which can be represented as the powers ~\ ••. , ~n-J

of one among them, a so-called p'ri1nitive n-th 1'oot of wnity b' Proof: Let ~1' ••• , (;" be the roots of f"Cx) xl! - e. Then

=

the derivative is f'(t)=nl;n-l=!=O n -t 'L

(i=l, ... , n),

sinl:e naturally ~,* 0, which by the assumption regarding the characteristic of K in this case also implies n(;,:,-1 0 , (Theorem 43 [:203]). Therefore, by Theorem 56 [213] the n roots (;, are different from one another. Furthermore, since b~=e, (;k=e implies (~,l;k)"=e, the n different n-th roots of unity form an Abelian group .5 of order n (VoL 1, Theorem 20 [64] applied to the multiplicative group of the elements 0 of the cyclotomic field Tn). By Theorem 34 [198] the order of each element (;, of .8 is then a particular divisor m, of n. Now, let ~ be an element of 8 of highest possible order m. We wish to show that m=n,

*

*

305

20. Cyclotomic Fields. Finite Fields

for this result will imply that the n powers to. t 1, ••. , t n- 1 are different and consequently exhaust the group 3. To prove this equality let p be an arbitrary prime number and set (according to Theorems 12, 22 [178, 11)7].

m=pllm, m,=pll'ml with (m,p) =1, (iii" p)=1. Then, the orders of ~m{, ~pJ.l. are obviously pIJ.', m, therefore by I

Theorem 35 [198] the order of ~ m,~pIJ. is pIJ.;m,. The maximal I choice of m implies pIJ.Im,;;;; p!L m, that is, !!I ;;;; !!. Therefore, m, contains any prIme number p at most to the power to which p occurs in m, that is, m, I m (Theorem 20 l187]) and conse-

r =

quently ,ll e. The n different n-th roots of unity ~I are therefore, all of them, roots of the polynomial of moth degree ;z;m - e. This implies that m ~ n (Theorem 48 [208J), which together with min gives m = n. This completes the proof of the theorem. By Theorem 37 [200] (cf. also the statement to Theorem 31 [195]) we immediately have in addition: Corollary. If ~ is a primitive n-th root of unity over K, then all and only the powers ~m, which rorrespond to the
g,,(a;) = H (a;-C m ) ",=0 (m.n)-l

whose roots are the
2. V. Solvabtlity oj Algebraic EquatIOns by Radicals

306

is the irreducible (separable by Theorem 59 [215], normal by Theorem 99, (III)

[259]) factor of gn(x) belonging to

~, then

the iii represent a subgroup ~n of the prime residue class group \j3n mod u. The Galois group ®,. of the (separable, normal) cyclotomic field Tn is then isomorphic to this group ~n on the basis of the correspondence of the automorphism generated by ~ ~ ~m to the residue class

m mod

01

Tn

n.

In particular, Tn is Abelian (Def. 34 [267]), and furthermore the degree of T" over K is a divisor of cp(n) (Theorem 105 [267] ). Proof: a) By Theorem 107 [271] an automorphism of T" relative to K, on the one hand, permutes the n different roots ~i of xn - e only among themselves and, on the other hand, leaves their power representations ~I ~'-1 invariant. Hence such an automorphism takes ~ again into a primitive n-th root of unity, so that the cp(n) different primitive n-th roots of unity are again permuted only among themselves by such automorphisms. Accordingly, the coefficients of gn(x) are invariant under all automorphisms of Tn relative to K and consequently belong to K (Theorem 112, Corollary [284]). b) TII=K(~o, ... ,t7l-1)=K(~), therefore a primitive n-th root of unity over K is at the same time also a primitive element of Tn relative to K. This implies that the automorphisms of Tn relative to K can (Theorem 105 [267 J ) be written in terms of the substitutions of ~ corresponding to them. Therefore, if jjn(a;) has the significance specified in the theorem, then the Galois group ®n of Tn relative to K is represented by the sub-

=

kltiiutions

t ~ ~m,

and their elements correspond biuniquely

through this to the set ~n of the prime residue classes mod n represented by the

m.

Now, since t ~ t m, and ~ -.. ~;;-2 carried

307

20. Cyclotomic Fields. Finite Fields

out ont' after the oiher yield ~ -+(~m2)m, = ~m,m, under this correspondence, the multiplication in ®n amounts to the multiplication of the residue classes in ~n' Hence this correspondence is an isomorphism and \l3n is a subgroup of ~" isomorphic to ®no In investigating the solvability by radicals the roots of unity of prime order n p play, according to Def. 40 [303], a special role. For this case we prove an extension of Theorem 121: Theorem 122. Let p be a prime number and K a field with characteristic different from p. Then the cyclotomic field Tp is cyclic over K and its degree is an exact divisor of p -1. Proof: By Theorem 121 the degree of Tp over K is a divisor of
=

Pp different from zero (Theorem 17 l185]) and as such are roots of the cyclotomic equation xp-J. - e ...:... 0 (Theorem 29 [194]), therefore are the totality of (p -1)-th roots of unity over Pp • Consequently, by Theorem 120 [304] they form a cyclic group with respect to multiplication. Hence I.j3p is cyclic, so that by Theorem 36 [189J the subgroup ~p, that is, ®p, is also. Therefore, Tp is cyclic over K (Def. 34 [267]) and its degree is an exact divjsor of p-1. Moreover, Theorem 44 [204] trivially implies:

= (oo-e)P,

Corollary. If K has characteristic p, then XI'-6 therefore e is the single p-th j'oot of unity ove?' K, and Tp

=

K.

308

2. V. Solvability of Algebraic Equatzons by Radicals

It is worth notlllg how the abstract theory of fields yields in a very simple way the conclusion that the prime residue class group ~p is cyclic. For this purpose we extend, without further difficulty, Theorem 120 to Pp; in number theory this theorem is usually proved only for the ground field P. In number theory we describe this conclusion by saying that there exists a primitive root mod p, namely, there is an integer r such that for every integer m prime to p there exists the power representation m = rlJ. mod p (fA. = 0, .... p - 2). We add a further remark about the special case of the cyclotomic field Tp over the rational ground field P. By number-theoretic methods (Theorem of Eisenstein-Schonemann, see Vol. 3, Section 20, Exer. 6) we can show that the polynomial x1'-1 1 2 P

gpCx)=: x - I =:xP- +x

-

+,··+x+l

of Theorem 121 is irreducible in P, therefore that Tp has the degree rp(p) = p - l over p, Now, if p - l =2" ('I';;;; 0) is a power of 2, then by Theorem 109 [273] Tp can be reached from P by the successive adjunction of quadratic irrationals. This follows from the fact that the Galois group ®p of Tp relative to P, which is cyclic by Theorem 122, has the order 2", and consequently by Theorem 36 [199] there is a chain of subgroups ®p = S)o > S)1 > ' .. > 5), = a; such that 5), is a subgroup of S)'-l of index 2. Conversely, if Tp can be reached from P by successive adjunction of quadratic irrationals (or even only ineluded), then by the expositions in Section 17, 2) (281] and Section 18, 3) [297] the group ® contains a chain of subgroups of the kind 4 just deseribed, and consequently its order p -1 is then a power of 2. This implies the famous Result of Gauss. The 'l"egular p-gon for a prime number p can be constructed by ruler and compass if and only if p is a prime mtrnber of the f01'm 2' 1.

+

4 For the general case of inclusion, d. the detailed proof to Theorem 127, part a), footnote 6 [320] given later.

20. Cyclotomic Fields. Fmite Fields

309

We do not know, even today, whether or not the sequence beginning with p = 2, 3,5,17,257,65537 of prime numbers of this form breaks off (see, for this, also Vol. 3, Section 20, Exer. 14, 15). In the next section we will analogously deduce the solvability of Tp by radicals over definite ground fields K. This is the main reason for the digression of this section. On the basis of Theorem 120 [304] we can now easily give: Brief Sketch of the Theory of Finite Fields A. We have already met finite fields, that is, fields which contain only a finite number of elements. For instance, for any prime number p the prime field Pp (residue class field mod p) is a finite field having exactly p elements (SectIon 4). Next, let E be an arbitrary finite field. Then the prime field contained in E is also finite, therefore not isomorphic to the rational number field. Hence we have (Theorem 41 [202] ): (I) The characteristic of E is a pl"i'lne nuntbe1" p. By the statements in connection with Theorem 41, E can then be regarded as an extension of the prime field Pp' It is trivial by this that E has finite degree over Pp (Def. 25 [225]). From the unique representation Il = aill i + ... + amu m of the elements CL of E in terms of a basi,s Ill>"" Ilm of E relative to Pp with coefficients ai' ... , am in Pp it then follows: (II) If [E: P p] m, then E has exactly pm elements. N ext, we generalize to E the inferences applied to the prime field Pp itself in the proof to Theorem 122. The multiplicative group of the elements of E different from zero (Vol. 1, Section 6, Example 1 [61] ) has by (II) the order pm - 1. Hence these pm _ 1 elements different from zero satisfy the equation

=

:,vPm-l _ e"'::'" 0 (Theorem 34 [198] ), therefore are the totality of

(pm -1)-th roots of unity over Pp • Consequently, the

g"L'OUP

formed

from these elements is cyclic (Theorem 120): (III) If [E: Pp] = m, then E is the cyclotomic field Tpm_l over Pp • The elements of E different from zero are the roots of the

equation a;pm-l_ e"'::'" 0; therefore the totality of elements of E are the roots of the equation xpm - x --=- o.

310

2. V. SolvabtlLty of Algebrazc Equatzons by Radzcals

There exists in E a primitive element e such that the pm - 1 elements of E dtfferent ft'om zero can be represented as the powers QO = e, el , ••• , epm-2. Conversely we have: (IV) For arbitrary m the cyclotomic field Tpm_1 over Pp is a finite field with [Tpm_l : Pp] = m. For, Tpm_1 is itself a finite field (Def. 25, Addendum [226]) as it is an extension of finite degree of the finite field P11 (Theorem 83 [240]). This has exactly pm elements. Its elements are already exhausted by zero and the pm - 1 roots of xpm-l_ e, that is, by the pm roots of xpm - x. For these pm roots already form a field, since al'm = a, ~pm = ~ not only (as in the proof to Theorem 120) implies that

(a~)pm=a~

and (in case

~=l=O)(~ym=~.

but also by Theorem 44 [204] that (a. ± ~)pm = a ±~. Hence by (II) we have prrplll_l,ppl = pm, that is, [Tpm_l: Pp] = m. Since the characteristic p and the degree m are determined uniquely by the number of elements pm, by (III) and (IV) we have: (V) For any number of elements of the form pm there is exactly one finite field type, namely, the cyclotomic field Tpm_l ove?' Pp' Furthermore we have; (VI) Tpl1l-1 has only the totality of fields Tpl'-1 wUh It I m as subfields and thereby

,.

[Tpm_l : Tp P-_1] =~.

For, on the one hand, if TpP--l ;:;;; T~m_1> by Theorem 71 [228] we have that I.t = [TpP--1 : Pp] [Tpm_1 : pp] = m and

I

,.

[Tpm-1 : Tp!'--a =~. On the other hand, if

111 m and we according-

= ,.,.'. then pm -1 = pl'1-" -1 = (pi' -1) (pI'(p'-l) + ... + pi" + 1), therefore pi" -11 pm - 1. Consequently T7'1'-1 ;;;;;; Tpm_I' since the (pi' -1 )-th roots of unity occur in this case among the (pm -1 )-th roots of unity. (V) and (VI) yield a complete survey of all finite field types and there mutual relations. ly set m

20. Cyclotomic Ftelds. Fmite Ftelds

311

B. Next. let E = T))711_1 be a finite ground field and H a finite extension of E. First of all it is trivial that H has in this case finite degree n over E (Def. 25 [225]) and therefore is again a finite field (Def. 25. Addendum [226]). which by (VI) has the form H = Tpmn_l' Then, if e is a primitive element of H in the sense of (III). Q is all the more a primitive element of H relative to that sub field in the sense of Def. 19 [221]. Therefore: (VII) H is simple ove?' E. This fills the gap still remaining in the proof of Theorem 90 [251] . Since the characteristic p of H is not an exact divisor of the order pmn - 1 of the roots of unity. which form H. it is further valid by the remarks to Theorem 121 [305]: (VIII) H is separable over E. Finally, by Theorem 94 [256] we have: (IX) H is normal over E. Hence the theorems of the Galois theory can be applied to the extension H of E. Although we have already obtained a general view of the fields between Hand E by (VI) without using the Galois theory - they are the totality of Tpmv_l with v I n - , it is still of interest from a theoretical point of view to establish: (X) H is cyclic ove?' E. Namely. the Galois group of H relative to E consists of the powcws of the automorphism

A: a_ a pm for eve?'Y C!. in H with An = E. that is, of the n automorphisms A:a-+fl,mv for every C!. in H(v=O.1 •... ,n-1). Namely. by Theorem 44 ['204] [see as well the deductions in the proof to (IV)] these are actually automorphisms of H which leave every element of E invariant, since each element is a root of xpm - x. Hence these automorphisms are automorphisms of H relative to E. Furthermore. these n automorphisms of H relative to E are different from one another. since for a primitive element e of H (in the sense of (III» the totality of powers Qt (i 1•...• pmn -1) are different

=

from one another, therefore, in particular, so also are the n powers

ePmv

(v=O,I, ...• n-l). Hence they are all n=[H:E] auto-

2. V. Solvability of Algebraic EquatIOns by Radicals

312

morphisms of the Galois group of H relative to E (Theorem 105 [267] ).

21. Pure and Cyclic Extensions of Prime Degree In order to be able to handle the question of the solvability by radicals we still have to supplement the special developments of the previous sections by the theory of irreducible pure polynomials of prime degree on which the Def. 40 [303] of the solvability by radicals rests. We first prove the following theorem regarding the irreducibility of a pure polynomial of prime degree: Theorem 123. Let p be a prime number and xl' - a a pure polynomial with a =l= 0 in K. Then the root field W of this polynomial contains the cyclotomic field Tp over K, and only the following two cases are possible: a) XV - a has a root in K, that is, a is a p-th power in K. Then xl' - a is reducible in K and W = Tp. b) Xl' - a has no root in K, that is, a is not a p-th power in K. Then xp- a is irreducible in K and even in Tp. Moreover, it is normal over Tp, and therefore W is pure of degree p over Tp. Proof: Let Ul"'" Up be the roots of xl' - a and u one of these. Since a =l= 0, it then follows from uP = a that u =l= 0, and

( x)p _e=xP-<xP _ X - I l l .

,

,x-u1' : : : ( . : . _ ~) • • • ( . : . _

aP

"1').

" a (X "" "" By the principle of substitution (Vol. 1, Theorem 12 [ 47] , applied to I[x'] with I=W[x] and the substitution x'=ax) ax can be substituted for x in this, so that

_( x-~ <Xl) ... ( x--Uap) xP-e=

21. Pure and Cyclic Extensions of Prime Degree

313

"I are therefore the p-th roots of unity that is, W ~ Tp." Furthermore, if t is a primitive p-th

follows. The quotients

over K, root of unity (Theorem 120 [304]; - in case K has characteristic p, t = e [Theorem 122, Corollary] ). then by a suitable ordering we have a l = t, a (L = 1, ... , p). a) Now, if a hes m K, then according to this the a l lie in Tp that is, W ~ Tp and consequently by the above W = Tp' b) If, however, none of the a l lie in K and if, in this case,

an irreducible factor of x P - a in K, then ± ao would have a representation as a product of certain v factors a" namely, ± ao= '1;1'-,,'. If by Theorems 14, 17 [182. 185] we were to set vv' = 1 kp, then since a P = a it would follow that (± ao)" =tl'-"aak , IS

+

so that, since a =l= 0, the root ""-,, = ~"a= (+ ~oV would still lie a

in K. Hence $11 - a is in this case irreducible in K and consequently K(a) has degree p over K. Now, if in the above line of reasoning we had made the assumption that h($) was to be an irreducible factor of $11 - a in Tp, then it would follow that <x, and consequently K(n), would be contained in Tp. Hence the degree of Tp over K would be a multiple of p, whereas by 'l'heoreru 121 lB05] this degree is at the same time an exact divisor of p -1. Consequently $p-a is in this case also irreducible in Tp and by Theorem 99, (III) [259] normal over Tp as well; therefore by Theorem 99, (I) W = TI1 (a) and consequently pure of degree p over T1) (Def.39 [303]).

314

2. V. Solvabihtyof Algebraic Equations by Radicals

For the question of the solvability by radicals we are naturally especially interested in the case b) of Theorem 123. If K has characteristic p, then xP - a is an inseparable irreducible polynomial (Def. 17 [214] ). Its single root a is a p-fold root. In the sense of our prevailing restriction to separable extensions we exclude this possibility in the following by assuming in the case of the consideration of pure extensions of prime degree p that the characteristic of K is different from p. Then XV - a (and generally any irreducible polynomial over K of degree p) is a fortiori separable (Def. 17). Furthermore, in the case b) xP - a is in general not normal over K though it is over Tp, so that it seems appropriate for the application that we have in mind to adjoin at any given time first a primitive p-th root of unity t to K before the adjunction of a root of a pure polynomial xP - a of prime degree p. This means that we should first go over to the extended ground field

K=

K(~) = Tp which coincides with the field

Tp = {Tp, K) roots of unity over K.

(DeI. 37, Addendum [292] ) of the p-th In this regard the following theorem, which is an immediate consequence of Theorem 123, is of interest to us (in which K, so to speak, is to be identified with the

K just specified):

Theorem 124. If p is a prime number and K a field with characteristic different from p which contains the p-th roots of unity over K, then every pure extension of p-th de(Jree A of K is normal (separable and cyclic) over K. That A is cyclic over K is trivial, since it is a separable normal extension of prime degree p. For, by Theorem 105 [267] its Galois group relative to K has the prime order p and by Theorem 34 [198] must therefore coincide with the period of any of its elements different from E.

315

21. Pure and Cyclic Extensions oj Pnme Degree

As a converse to Theorem 124 we now prove the theorem upon which our apphcation is based: Theorem 125. If p is a prime numbel' and K a field with charactel'istic different from p which contains the p-th roots o/' unity over K, then any normal extension of p-th degl'ee 1\ of K is (a fortiori separable, cyclic and) pure over K. Pl'oof: Let A be a primitive element of the cyclic Galois

*

group of 1\ relative to K, a primitive element of 1\ relative to K and ~ a primitive p-th root of unity over K. Then we form the so-called Lagrange resolyent of tt: 0.

='h

+

~-1

lTA

+.,. +

~-(p-l)lT.#-I,

If this element a of A is different from zero, we proceed as follows:

Let A be applied to a. Due to Ap = E, ~p = e and the invariance of the element ~ of K with respect to A we generate (); A

=

=

+ ,-10..4,8 + . , . + ,-<1>-1)0.AP We + ,-10.A + .. ,+ C-
0.A

o;C;

therefore by the repeated application of A

o.A =

a~·.

This means that the elements 0., one another due to the assumption relative to K, that is, g(x)

=(x-a)

o.~, 0.

... , o.~ P -1 different from 0 are the conjugates to a

*

(x-a~)'"

(x-a~p-l)

is the irreducible polynomial over K belonging to a, and 0. is a primitive element of A (Theorems 111, 112 [283, 283] together with Corollary). Hence xP - e (x- e) (x-~)", (X_~p-l), implies, as in the proof to Theorem 123, that g(x) xP- o.P and consequently

=

=

316

2. V. SolvabUuy of Algebraic .equations by Radicals

g(:c) =:CP- a with a = c.P in K, therefore c. is a root of the irreducible pure polynomial :cP - a over K. Hence A K(c.) is in this ease actually pure over K (Def. 39 [303]).

=

We now show that by a suitable dhoice of the primitive p-th root of unity t we can obtain an a =f O. Namely, if for each of the p _ 1 primitive p-th roots of unity tv (v 1, ... , p - 1) the Lagrange resolvent belonging to it were a., = 0, then the system of equations p-1 Ct. = I W)-II {}All = 0 (71= 1, .. .,p-1)

=

1'-0

would exist. If we were to multiply the v-th equation by ~ VII-' and sum over v, then by interchanging the order of summation it would follow that Now,

~1 (Cl"-llt = {- e for p' $ p mod p .. _1 pe-8 for p'a p mod p, because in the former case tv---II-' is a primitive p-th root of unity, therefore a root of

ItP-8 --D

~-"'.

=

ItP-l

+ 1tP"-2 + ... + It + e,

whereas

in the latter case e stands (p -1) times as the summand. From this we obtain the relations (p' = 0,1, ... , P -1).

Since K does not have the characteristic p, this would mean that all 80AII-' would be equal to one another, which by Theorem 112, Corollary [284] is not the case for a primitive element t of A.

In conclusion we will make further use of the preceding results in order to prove the solvability of Tp by radicals over fixed ground fields K: Theorem 126. Let p be a prime number and K a field whose characteristic is 0 or a prime number > p. Then the cyclotomic field Tl1 is solvable by radicals over K, and besides there actually e:cists a chain of fields

21. Pure and Cychc Extensions of Pnme Degree

K= in which

A,

Ao < Ai < ... < A, with

317 AT ;;;;; Tp l

is not only ( accordmg to Def. 40 [303] ) pure

of prime degree but also normal over A'-1' proof: Use mathematic-al induction, assuming for this purpose the statements as already proved for all prime numbers < p (and all ground fields allowable according to the formulation of the theorem). Now, let d be the degree of Tp over K which by Theorem 122 [307] is an exact divisor of p-1, and let d = P1 ... P be the factorization of d into (not necesv sariIy different) prime numbers Pk. Now, by assumption the characteristic of K, if =l= 0, is also greater than each of these prime numbers. Hence by the induction assumption there first of all exists in this case a chain of fields

K= Ao < A1 < ... < AT! with AT1 ~ Tp1 , in which

A,

is pure and normal of prime degree over 1\'-1'

Secondly, (starting next from there is a chain of fields

A..,

~nstead of K as ground field)

Al' < Al' + 1 < ... < Al' with Al' :2: Tp, Tp Ii in which A,," +, is pure and normal of prime degree over A + l' etc. Therefore taken together there is a chain of fields K = Ao
rl

1

2 -

2;

1

%

I

1.-

A, is pure and normal of prime degree over Ai-I- Now, let Ip be the cyclotomic field for p over AT, and Ii its degree over A, , which by Theorem 119 [2!)3] is an exact divisor of d. Then" by inferences entirely similar to those in the in which every

case of the Result

of Gauss

[308 J

6 The cyclotomic field TP2 for p. over cyclotomic field Tp, for P2 over K.

AT,

(,Theorems 36, 109, naturally contains the

2. V. Solvability oj Algebraic Equations by Radicals

318

122 l199, 273, 307] ) there exists a chain of fields

Ar
in which

Ar, +i

is normal of prime degree over A r,+i_1' The

successive degrees in the last chain are, as divisors of d, certain of the prime numbers Pk. Now, since by construction the Plc-th roots of unity are contained in

A,v +" AT v +, is pure over

I\r,+;_l by Theorem 125 which can be applied in this case (cf. also Theorem ,12 [203J ), due to the assumption regarding the characteristic of K. Consequently the complete chain K=

Ao < ... < AT =Tp ,

if we further bear in mind that Tp ~ Tp , has all properties required for the statements of the theorem. Since the theorem is trivially valid for the smallest prime number p = 2 because T2 = K, this completes the proof of the theorem by mathematical induction.

22. Criterion for the Solvability by Radicals In this section we will derive the criterion for the solvability by radicals. In order to be able to express this criterion we make the following definition: * Definition 41. A separable normal extension N of finite degree of a field K igo called metacyclic over K if a chain of intermediate fields K=Ao or

- which by the Fundamental Theorem of the Galois theory amounts io the same - if the Galois group ® of N relative to K contains a chain of subgroups

22. Criterion for the Solvability by Radicals

319

® = S)o > )))1 > ... > S"dr = ~ such that S)i is a normal divisor of prime index of &1.

'''-1'

A separable PQZyn01ni(tl f(x) over K is called metacyclic ove?' K if its root field is metacyclic over K.

Therefore, the expression metacyclic says that the individual steps Ai over A i -l and ~'-l/~' are cyclic, Incidentally, a gl'OUp ~ of the kind specified in Def. 41 is likewise caned metaeyclic.

We will now state and prove a criterion for the solvability by radicals of normal extensions of finite degree. Here we restrict ourselves to ground fields of characteristic 0 due to the complications arising in the previous sections in the case of prime characteristics. This means, in particular, that the assumption of separability is always trivially satisfied. Theorem 127. A normal extension N of finite degree over a field K of characteristic 0 is solvable by radicals if and only

if

it is meta cyclic.

A polynomial f(:J.·) over K is solvable by radicals if and only if it is metacyclic.

P1'oof: a) Let N be solvable by radicals over K. According

to Def. 40

[303J

there exists in this case a chain of fields

K=A'o
with

A'?' , . -N'

in which A~ is pure of prime degree Pi over Ri_l' By Theorem 126 there exists (just as in the proof of that theorem) a chain of fields K= in which

Ai

Ao < Al < ... < As

with

As;;:;; TP1 , ••• , Tpr ,

is (pure and) normal of prime degree over

Ai _ 1-

Now, let C1.; be a sequence of elements of A~ such that a.i is a

Ai-i'

there-

A~ = K(Ul"'" (1.,). Either x P1 -

a1 is

root of a pure irreducible polynomial x Pi - ai over fore A~ =

Ai_l (a.i) and

also irreducible in As; then by Theorem 123

l:312]

and due

2. V. Solvability

320

0/ Algebraic Equations by Radicals

to TIll ;;;; As its root field A8 +lover

As

i:..; (pure and) normal of

As and in addition As+ 1 = As(al)' Or a/ a 1 IS reducible in As; then by the same theorem As + 1 = As and in addition As + 1 = AS(Ul)' Similarly, we conclude that the root field As + 2 of X1'2 - a2 over As + 1 is either (pure and) prime degree Pi over 1 -

A8 + 1 or = As+ 1 and in both cases in addition to this we have As + 2 = As + 1 (0.2)' etc. By

normal of prime degree P2 over

continuing on to As+r and counting only once fields successively appearing many times we obtain the chain of fields

< Al < ...
K=Ao

(the latter

= A' ;;;: N), in which every Ai is (pure and) normal of prime T degree over Ai_I' By the expositions in Section 18, 3) (cf. Fig. 5 (299]) to this corresponds a chain of intermediate fields K = Ao ;;;; Al ;;;; ... ;;;; Ai = N, in which Ai is normal over A.-I and, more specifically, of prime degree unless Ai = Ai - 1 (which means that A, could be omitted). For, the degree [Ai: A,-d, as a divisor of the prime degree

(A. : Ai_d,

is either 1 or this very prime number itself.6 According to Def. 41 N is then metacyclic over K. b) Let N be metacyclic over K. According to Def. 41 there exists in this case a chain of intermediate fields K=Ao
321

22. Cmerion for the Solvability by Radicals

As with As ~ Tp, ... , Tpr'

K=Ao
-

1

in which A. IS pure (and normal) of prime degree over Ai_I' Now', let 11, be a sequence of elements of A, such that 'l'l', is a root of a normal polynomial g,(x) of prime degree p, over Ai-I'

therefore

A j =A,_I({}')

and

N=A r =K({}l"",{}r)'

Either UI(X) is irreducible and consequently also normal in then due to

As over As

Tp,

A.+1=A.;(f}1)

;;;;

by rfheorem 125

[315]

As;

its root field

is pure (and normal) of prime degree

As. Or U1 (x) is reducible in As; thenAs+ 1 = As (111) = A., [A + 1 : As] is in this case, on the one hand,
PI over

since 8 account of the normality of UI (x) in Ao and actually account of the reducibility in (by Theorem 119

of Pl

A=

As,

N= A8 + 1),

As);

<

on

on the other hand, a divisor

[293], applied to K = Ao, N = 11. 1,

therefore

[A + 1 : As] = 8

1.

Similarly,

we conclude that the root field As + 2 =As + 1(f}2) of U2(X) over

As + 1

is either pure (and normal) of prime degree

As + l' etc. By continuing on to As + rand counting only once fields appearing successively many times,

P2 over As

+ J or =

we obtain a chain of fields

Ao < Ai < ... < Ai' with Ar ~ N (the latter due to Ar = As (/}1' ... , -Or) ~ K(111, ..• , {}r) = N), in which every Ai is pure (and normal) of prime degree over K=

Ai-I'

According to Def. 40

[303]

N is then solvable by

radicals over K. As is evident from each of the two parts of the proof, a refinement similar to that already obtained in the special Theorem 126 [316] is also generally valid: Corollary, If N is solvable by radicals over K under the assumptions of Theorem 127, then there actually exists a chain of fields

322

2. V. Solvabihty of Algebraic Equatzons by Radzcals

K:::: 1\0
in which

A,

with AT;;;; N, is not only (according to Def. 40) pure of prime degree

but also nOi'mal over A'_1' Theorem 127, together with the statements already made in Def. 41, answers the question asked at the beginning of this section, namely: Under what conditions is an algebraic equation f(x) = 0 in a ground field K of characteristic 0 solvable by radicals,' that is, when can its roots be represented in terms of calculating expressions which are formed by means of the four elementary operations and the operation of root extraction? There is another qUeJstion, somewhat different from the above, which could be asked: When can one root of an irreducible polynomial f(x) over K be represented in this manner? It amounts to the question of the solvability by radicals of an arbitrary extension A of finite degree of K, which conversely is not more general either since any such extension A can be regarded as the stem field of an irreducible polynomial f(x). In this regard we cite that the conditions for the latter question are exactly the same as those for the question considered above. Namely, an arbitrary extension of finite degree A of K is solvable by radicals if and only if this is the case for the normal extension N belonging to it (the root field of any irreducible polynomial I(x) for which A is a stem field). It is clear that if N is solvable by radicals so also is A. Conversely, by passing over according to Def. 41 from a chain of fields for A to their conjugates we can show that if A is solvable by radicals so also are all conjugate extensions. This easily yields that N i.s solvable by radicals. Examples of Algebraic Equations over Ground Fields of Characteristic 0 Which Are Solvable by Radicals

1) All equations of second, third and fourth degree. Namely, their Galois groups are isomorphic to subgroups of the symmetric groups $2' ®3' ®. (Theorem 107 [271]), and it is easy : Strictly speaking, however, this does not answer the question, for It merely reduces it to the statement and investigation of the Galois group_

23. EXLstence

0/ Equatwns not Solvable by Radzcals

323

to prove that the latter (together with their subgroups) are metacyclic. 2) All cycl~c and, more generally, Abelian equatwns; in particular, by Theorem 121 [305] the generic cyclotomic equation xn-1~ O. Namely, by Theorem 36 [199] any finite cyclic group is metacyclic. It is especially easy to show from the general theory of finite Abelian groups that these groups are all metacyclic. Here, however, we cannot give more details. s

23. Existence of Algebraic Equations not Solvable by Radicals The existence of equations not solvable by radicals was discovered by Abel, who first proved the zmpossibility of solving by radicals the generic equatIOn of degree higher than the fourth. In this last section we will sketch a modern proof of this Abelian Theorem. First we define: Definition 42. If Kn = K(x i , •.. , xn) is the f~eld of rational Xi"'" Xn over K, then the

functions of n indeterminates polynomial (1)

over Kn is called the generic polynomial of n-th deg1'ee over K. 'fhis generic polynomial of n-th degree over K can be re-

garded as an "indeterminate" condensation of all special polynomials of n-th degree over K, which, conversely, can be obtained from it by substituting in the generic polynomial any system of elements a i , . . . , an of K for the indeterminates Xi' .•. ,

X'n

over K.

~ For this we refer to Vol. 3, Section 3, Exer. 9 to 20. In regard to further theorems about metacyclic groups cf. the text of A. Speiser Theorie der Gruppen von endlicher Ordnung (Theory of Groups of Finite Order) 3rd ed., Berlin 1937.

2. V. Solvability of Algebraic Equations by Radicals

324

We now consider the decomposition into linear factors (2)

InCx)

=xl! +

Xl X,,-l

+ .. ,+ Xn =(x -

~1)

'"

(x -- ~n)

of the generic polynomial of n-th degree over K in its root field Wn K" (;1' ... , §n). If we think of the n hnear factors on the right as multiplied out, then by Vol. 1, Theorem 11 [39] the coefficients of the same powers of x on the left and the right must be equal. Thus, we obtain the system of formulae 9

=

~

{

(3)

= - (~l + ... + ~ft)

~ ~ !~ ~~ ~ :::~ ~~~ ~~

Xn = (-1)" ~l'" ~n or on collecting them together :l;.

=

(-l)'(il,.~i,./il ... ~i,.

(v

= 1, ... , n),

where the sum on the right extends over all (:) combinations taken v at a time (Vol. 1, Section 16 [l~lJ ) of the numerals 1, .... n. According to these formulae we have, in particular, (4)

Wn

=

Kn (~1' ... , ~n)

= K(x1! ... , .e,,;

~l"'" ~n)

= K(~l"'" Sn). Besides the foregoing interpretation leading from (1) over (2) to (3) and (4), we can conversely also give another. For this purpose we start from (4), that is, the field W n= KCSI"'" Snl of rational functions of S1"" ~n over K now taken to be n indeterminates; then we define Xl"'" xn by the formulae (3) as elements of this field Wn and define fnCx) by the formula (2) as a polynomial in the subfield Kn K (Xl' ••• , xn) of Wn = K(Sl' ... , Sn), whereby Wn is again the root field to

=

9 Here, deviating from the convention in Vol. 1, p. 50, we designate equality in Kn:::: K(x j , • • • , :en) and in the algebraic extension W" Kn(~j"'" ~n) only by =, in order to reserve for equality in the case of the addition of further indeterminates (say x in (1) and (2) as well as X;, ... , Xii in the proof of the following Theorem 128).

=

=

23. Existence of Equations not Solvable by

Rad~cals

325

In(x) over Kn and has the representations (4). Of course, in the case of this latter interpretation the question remains unanswE'red whether the polynomial f,,(x) so formed is the generic polynomial of n-th degree over K. Since the specified second interpretation is handier for the application which we have in mind, it is important to answer this question affirmatively:

Theorem 128, If W n = K(~l"'" ~n) is the field of the rational functions of n indeterminates ~1' ••• , ~" over K, then the elements Xl' •.. , Xn of W 11 defined by formulae (3) likewise ha11e the character of n indetermhwtes over K,10 that is. the subfield Kn = K (Xl' ... , xn) of W" = K(~1' ... , ~n) is likeu'ise of the e.'ctension type of the field of the rational functions of n indeterminates over K. In particular, the polynomial f,,(x) ove1' K1I defined by (2) is in this case the generic polynomial of n-tl1 degree over K. Proof: We have to show that the normal representations ( Vol. 1, Def. 9 [45] ) of the elements of the integral domain K [Xl' ••• ,xn ] in terms of x 1, ••• ,.'tn are unique. For this purpose it is sufficient to prove that a relation

=

(5) g(Xl' ••• , x,,) 0, where g(x1 , ••• , xn) is an integral rational function of n indeterminates Xl' ... , tin over K, implies the relation (6) g(x1 •••• , xn) == O. We will prove this by double mathematical induction,l1 first on the number n of indeterminates, secondly on the degree v,~ of g in ?in. 10 Cf. Vol. 1, Def. 9 explanation.

[45]

together with the accompanying

11 For the idea of applying double mathematical induction in this proof, I am indebted to a communicaHon by letter from Ph. F1.£rtwangler.

2. V. Solvability of Algebraic Equations by Radicals

326

For n = 1, Xl = - ;1' therefore the theorem is obviously valid on the basis of the assumed indeterminate character of Sl' Let us assume that the theorem has been proved up to n - 1. Then, let (7)

g(Xl' ... , i~"'n) =

~n

..J: x" g. (Xl' ... , iin-1)

k-O 'n

Ie

be the rE:'presentation of 9 (following from the normal representation by combining as an integral rational function of 33" over K [Xl' ... , xn-d, therefore, in particular, (8) 9(X1 , ••• ,;1;,,_1,0)=gO(x1,· .. , X,,_1)' Now, if we set ~,,= 0 in (5), then by the principle of substitution ( Vol. 1, Theorem 12 [47] ), which can be used due to the indeterminate character of ~m the relation (9) g(xl, ... , X~_I' 0) = 0 is generated, where xl' ... , X~-l arise from xl" .. , X n-1 by the substitution ~n = 0 in (3); if we then set (Xl"'" Xn_l) = (Xb •. ,X~_l) in (8), by the principle of substitution we further obtain from (9) the relation (10) go(xi, . .. , X~_I) = O. Now, since the significance of xi. •.. ", X~_I according to their definition in terms of S1"'" Sn-l corresponds to that of Xl"'" x" in terms of S1"'" Sn. i! follows from (10) according to the induction assumption first made that the relation go('X1' · · · ' Xn- 1) is valid: therefore by (7) the relation

== 0

~n-1

(11)

g(x1 ,·

•. ,

xn)= xn ~o ii~gk+l (Xl' ... , $n-l)

=

~g(l)(~,

.. . , ~),

is further valid, where g(1) ($1' " •• , :en) is again an integral rational function of $1 •••• ' Xn over K, which (in case Vn > 0) has the degree Vn -1 in x".

23. Existence of Equations not Solvable by Radicals

327

Now, if the degree is vn=O, then g(l)(Xl"'" xn) == 0 and therefore the statement (6) is true by (11). Let us assume that it has been proved up to the degree Vn -1 (for fixed n). Then by the substitution (Xl"'" X,.) = (Xl' •..• x,,) we obtain from (11) the relation g(x1• •••• X,.) = xng(l) (Xl' ...• X,,) and by (5) and the fact that X,. =!= 0 we further obtain from this the relation g(1) (Xl' ...• x,,) = O. Hence by this second induction assumption we have g(1)(x1••••• X,.) == 0. which by (11) implies statement (6). This completes the proof of the statement of the theorem by double mathematical induction. Incidentally, Theorem 128 easily yields the converse, namely, the fact that the elements ;1""';" defined by (2), starting from the indeterminates Xl"" X,., have the character of indeterminates over K. However, here we have no need of this result.

We now prove: Theorem 129. The generic polynomial of n-th degree over K is separable and its Galois group relative to Kn is isomorphic to the symmetric group @:i,.. Proof: According to Theorem 128 we think of the generic polynomial of n-th degree f,,(x) over K in terms of the second interpretation discussed before Theorem 128, that is. as formed through the formulae (3). (2) starting from the root field W,. = K(;l' ... , ;n}.First, that 1,.(x) is separable immediately follows in this case from the distinctness of its roots ;1"'" ;1& chosen as n indeterminates over K (Theorem 59. Corollary [216]). Furthermore, let

(.1. .. tn~) ~l'"

be an arbitrary permutation

328

2. Y. Solvability of Algebraic Equations by Radicals

in e>1I' By Vol. 1, Theorems 10, 11 [33, 39] , K(;'., ... , ;'n) is then isomorphIc to K(;l"'" ;11) relative to K on the basis of the correspondences ;1 +---+ ;", ••. , ;n ~ ;'11' However. since the formulae (3) are symmetIic in ;1""';11' the elements Xl"'" X n, and consequently all elements of KII = K(Xl' ... , x n ), are mapped onto themselves under this correspondence, so that the specified isomorphism is valid even rela.tive to Kn. Since W n =K(;l"'" ;,a) = K (;." ... , ;'11)' any permutation in $11 generates an automorphism of Wn relative to KII, and conversely is therefore obtained by such an automorphism in the sense of Theorem 107 l271]. On taking into consideration the distinctness of the roots ;1' ... , ;11> pointed out before, it follows from this theorem that the Galois group of W n, that is, the Galois group of f,,(x) relative to KII • is isomorphic to the symmetric group @SII' Moreover, Theorem 129 yields in particular: Theorem 130. The generic polynomial of n-tk degree over K is irredtl.cible in KII• Proof: If fn(a:) is the irreducible polynomialm KII belonging to a root ~ of fll(a:), then, on the one hand, fll(a:)lfn(a:) (Theorem 53 [210] ). on the other hand, 1,,(;,) == 0 for every root ;, of fn(a:) (Theorems 73 [230],129). Therefore, since the roots 1;, are distinct,

I

it also follows that f,.(a:) III (a:) (Theorem 47 [207]). These two fn(a:) , as stated. facts taken together yield fn(a:) We will dwell a moment on formulae (3). again from the second point of view discussed before Theorem 128 [325]. In this case we call the elements a:1O •••• a:n of K(~1> ••• , ;11) the elt!tmeni;a'l'Y wymim,ef;ric funcfJlon8 of the indeterminates ~1"'" ;,.. Furthermore, we generally call a rational function over K of the indeterminates ~l""';1I symmetric in ;1' ...• ;11' if it goes into

=

23.

Ex~tence

of EquatJ.ons not Solvable by Rad,cals

329

itself by all permutations of the 11. elements ;1"'" ;n' By applying ThE'orems 107, 112 [271, 283] along with Corollary, Theorem 129 immediately Ylelds: Theorem 131. A rational function over K of the indeterminates ;1"",;n [that is, an element of K(;1>"" ;1/)]' is symmetric if and only if it ill a rational function ove1' K of the elementary symmetric functions a: 1" . " a:n of ;1>".,1;11' that is, an element of the subfield K(a: 1, · · ·,a:n) of K(;I"'" ;n)' The deeper-lymg statement of thlS theorem, namely, the statement "only if," which says that every symmetric rational function over K of ;1>"',;n is a rational function over K of XU"" xn> is a substatement of the theorem known under the name of the Theorem on Symmetric FunctiQns, which in the past was nearly always taken as a basis for the Galois theory (cf. footnote 1 to the proof of Theorem 90 [251]). This theorem goes beyond the statement of Theorem 131, in as much as it states that: 1) Any integral rational symmetric function over K of ;1> .• ',;n is an integral rational function of XV"'' x n. 2) The latter is also valid even if an integral domain I is used instead of the field K. However, in contrast to Theorem 131, these further statements cannot be inferred from the Galois theory. 12

We now return to the proper problem of this section, which we can next attack on the basis of Theorem 129. Since the symmetric group @in for 11. > 1 always has the normal divisor m,. of index 2 ( Vol. 1, Theorem 63 l126]), we can reduce the root field Wn Kn (;1' . , ., ;n) of degree n! over Kn to a field

=

of degree

~!

over a field V n generated from

Kn

by the ad-

junction of a square root: 12 Here we cannot give a proof communicated to me by Ph. Furtwangler - of the statements 1), 2) by means of double mathematical induction, which is entirely analogous to th-e proof of • Theorem 128. See, Vo1. 3, Section 23, Exer. 3.

330

2. V. SolvabIlity of Algebraic Equatwns by Radicals

=

Theorem 132. The root field Wn Kn CS1' ... , Sn) of the generic polynomial of n-th degree 1) over K has a subfield Vn 0/ degree 2 over Kn. This is obtained, in case K does not have the characteristic 2, by the adjunction of the element 1:,..2 l:n-l 1 '>1 ~1"'''1

en>

(j

= ........... .

...

1 ;n ~! ;:-1 to Km which is the root of a pure polynomial x 2 -

d of second

degree over Kn.1S Proof: That Vn = Kn(b) is the field between Kn and Wn corresponding to l!!n follows according to Theorems 112, 129

[283, 327] from two facts: first, b is unchanged by the even permutations of S1" .. , Sn, but changes its sign under the odd permutations ( Vol. 1, Theorem 65 [130]); secondly, 5 =1= 0 (see Vol. 3, part 1, Section 19, Exer. 4), so that the assumption about the characteristic implies b =1= - b. Furthermore, this means that 52 = d is unchanged by all permutations of S1"'" Sn; therefore it must be an element of Kn (Theorem 112, Corollary l284] ). The element d = 52 is called the di'lcri'ff~inant of f n (x). Naturally, it even belongs to K[Sl'"'' ;n] and therefore is an integral rational function over K of the roots ;11···. ;/1'

Now, in the theory of groups it is proved that the alternating g1'OUp mn for n =1= 4 has no proper normal divisor,14 and that l!!n is the only normal divisor of @3n. 15 Since

~!

is not a prime

I:} Regarding the case where K has the characteristic 2, see Vol. 3, Section 23, Exer. 20. 14 Spfti,ser, 1. c. (cf. footnote 8 of this Chapter), Theorem 94. See also Vol. 3, Section 23, Exer. 13, 14.

15 This is a consequence of the so-called Jordan Theorem (Speiser, same place, Theorem 27) together with the obvious nonexistence of normal divisors of @in of order 2. See also Vol. 3, Section 23, Exer. 16.

23. Ex£stence of Equations not Solvable by Radzcals

331

number f01' n ;;;;; 4, there can therefore exist for n> 4 no chain of subgroups of en or the kind specified in Def. 41 [318] ,so that en is not meta cyclic in this case. Theorem 127 [319] therefore implies: Result of Abel. The generic polynomial of n-th degree ove1' a field K of characterzstic 0 tS not solvable by radicals for n>4, This theorem insures the existence of equations not solvable by radicals, first of all, only for the particular ground field Kn of Def, 42 [323]. Another question which arises is then the following: In a gi1:en ground field K are there special (that is, situated in K itself) equations of any degree n> 4 not solvable by radicals? This question IS answered affirmatively for the special case of the rational ground field P by the Irreducibility Them'em of Hilbert,16 If g(x; Xl' • , ., xn) is an integl'al rational function of the indeterminates X; Xl' ••• , Xn over p, which is a polynomial in X irreducible o'!.'er Pn =P(x1, ... , x n ), (hen there are infinitely many systems of elements a i , · . · , an of P such that g(x; ai' ... , an) is irreducible in P. This theorem gives the following answer to the question asked above regarding the ground field P: If ;1' .. ',;n are the roots of the generic polynomial of n-th degree f,,(x) = x" X 1X n - 1 Xn over P, then by Theorem 112, Corollary

+

+

+ ' .. +

[284] and Theorem 129 [327] it cl Sl n is a primitive element of the root field Wn = PnCs l , ••• , Sn) relative to Pn = P(X1' ... , xn) provided that the coefficients c. are

=

+ ... + ens

16 D. Hilbert, tJber die Irreduzibilitat ganzer rationaler Funktionen mit ganzzahligen Koeffizienten (On the Irreducibility of Integral Rational Functions with Integral Coefficients), Crelle 110, 1892.

2. V. Solvability 0/ Algebraic Equations by Radicals

332

chosen from Pn so that all permutations

(~) of

the

~

v

yield

different conjugates

+ ... +

it, = Ci~', cn~'n' We think of the Cv as chosen in this way; in domg so we can even take them, as seen in Theorem 49 [208], as elements of the integral domain r n = P [Xi' ... , X,..]. Then n!

gCx; Xl"

•• ,

Xn)

= II (x ,=1

{},)

is a Galois resolvent of Wn relative to Pn and satisfies the assumptions of Hilbert's Irreducibility Theorem. Therefore, there are infinitely many systems of elements ai , ••• , an in P such that gCa;; ai' ... , an) is irreducible in P. The root fields W over P of the special fCx) corresponding to these systems a1 , ••• , an then have the highest possible degree n! over P (Theorem 108 [273]), since they each contain an element {} of degree n! and, consequently, have a Galois group isomorphic to Sn itself (Theorem 107 [271]). Hence by the expositions of 4. these sections these !Cx) are not solvable by radicals for n We therefore have: Corollary. For every degree n there are in P infinitely many algebraic equations whose Galois group is isomorphic to @;n (so-called equations without affect; in particular, therefore, for every degree n > 4 there are infinitely many algebraic equations not solvable by radicals. Whether this result is alse valid for general ground fields K, as well as for any subgroups of @;n as prescribed Galois groups, is undecided even today except for simple cases.

>

AUTHOR AND SUBJECT INDEX (The numbers refer mainly to the pages on 1chich the terms appear for the first time.) Abel 301, 323 result of 331 Abehan extensIOn 267 group 58 polynomial 270 Theorem 251 Absolute value 175 Accessory llTatlOnals 298 AddltJ.on 14 Adjunction 219 from above 219 from below 220 simultaneous 220 succeSSlVe 220 Affect 332 Algebra 9 linear 79 problem of 10, 11, 55 Algebraic complement 132 element 223 equation 56, 167 Algebraic extension 224 firute 240 normal 255 ofthe first kind 214 of the second kmd 214 simple 235 with bounded degrees 254 Algebraically closed 248 independent elements 46 Alternating group 126 Alternative 112 Analysis, in the sense of 38 Apply a permutation 117 Artin 248 Associates 173 classes of 173 ASSOCiative law 14, 57 Automorphism 264 relative to 266

BaSIS 225 BlUmque correspondence 24 Cantor 24 Cantor's diagonal method 25 Cardinal number 24 Carry out a permutation 117 Cauchy 249 CharactenstJ.c 202 Classes 21 of associates 173 of conjugate subsets, elements, subgroups 71

Coefficients 45 Cofactor 132 Columns of a matrix 87 Combination 121 complementary 121 linear 80 order of 121 Commutative law 14, 58 Complement, algebraic 132 Complementary combination 121 determinant 132 matrix 133 Complete system of representatives 23 right and left residue system 67 Complex numbers 19, 249 Composed from field 27 group 65 Composite 27, 65 free 292 Congruence relation 27, 65 Conjugate 229 subsets, elements, subgroups 71

Correspondence blUmque 24 one-to-one 24 Conset 67 Countable 24 Cramer 128 Cramer's Rule 148 Cychc extension 267 group 196 pol)nomial270 Cyelotonuc equation 304 field 304 Dedekmd 24 Degree of a suhdeterminant and mmor 132 of an element 223 of an equatJ.on 56 of an extension 225 DerivatJ.ve 211 Derived matnx 133 Determinant 127 expansion of 140 Diagonal method, Cantor's 25 Difference 14 Discnmmant 330 Distinctness 13 Distnhutive law 14 Divisible 172 Divisibility theory 172 Division 16 left and right 57 With remainder 180 Divisor 172 greatest common 183 proper 174 Domain 18 Eisenstein-SchonelDllDll" Theorem of 308 Elementary operations 18 symmetric functions 328

INDEX Elements 13 algebraically independent 46 conjugate 71 Element"ise addition 28 multlplication 28, 70 Empty set 21 Equality 13, 50, 53 Equation 53 algebraic 56 Equations system of 55 linear 56 Equipotent 24 Equivalence relation 22 right and left 67 Equivalent systems of equations 94 Euclidean algorithm 183 Euler's function 195 Even permutation 122 Expansion of a determinant 140

Extension 205 Abelian 267

algebraically c10sed 248

by automorphisms 267 cyclic 267 finite 221 Galois 255 integral domain 25 meta cyclic 318 normal 255 of finite degree 225 of first kind 214 of ground field 290 of second kind 214 over Dew ground fields 290

pure ~03 (separably} Jllgebraic 224simple '221 sailwable by radicals 303 transcend.ental224 ExtimsiOJ1 domain 25 prqper32 true ll:2 Extension field 25 Extension:rlng 25 Extension type 230, 231

Factor group 76 Fermat Theorem 196 Field 17 imperfect 217 perfect 216 Finite extension 221 Form 79 linear 79 Free composite 292 Function 38, 49 in the sense of analysis 38,47,51 integral rational 38, 45, rational 46 [47 value 47, 49, 51 Fundamental solutions, system of 92 theorem of algebra, socalled 247 Furtwiingler 325, 329 Galois 251, 267 Galois element 258 Galois extension 255 Galois group 267 as a permutation group

271

Ideal 27 Identity 17, 58 group 61, 66 logical 23 solution 90 subgroup 66 Imperfect field 217 Indeterminates 39, 46, Index 68 [47,221 Inner product 85 Inseparable polynomial 214,215

Integers 19 Integral domain 18 multiples 19 powers 19, 60 subdomain 25 Intersection 21 field 26 group 64 Invariant subgroup 69 Inverse 59 Inversion 122 Irrationals 298 Irreaucible polynomial 178

as a substitution group 269

of a polynomial 270 of an extension 267 reduction of 287 Galois polynomial 259 Galois resolvent 261 Gauss 247, 308 Generic polynomial 323 Greatest common divisor 183

Ground field 79, 205 extension 290 Group 57 Abelian 58 alternating 126 composed from 65 identity 61, 66 symmetric 119 unity 61 Groupoid 228 Bilbert 331 Irreducibility Theorem of 331 homogeneous 79 equations, system of 89

334

Isomorphic 30, 65 relative to 30 Isomorphism 30, 65 Jordan's Theorem 330 Knopp 247 Kronecker 248, 249, 298 Lagrange resolvant 315 Laplace expansion theorem 133 Leading coefficient 177 Left coset 67 division 57 equivalence 67 partition 67 residue class 67 residue system 67 Leibniz 128 Length of a linear form 94 Linear combination of 80 equations, system of 56 factor 207 form 79

INDEX Linearly dependent 80 independent 80 Loewy 288 Logical identity 23 Matrix 87 calculus 89 complementary 133 derived 133 null 88 of a system of equations 90 product 89 rank of a ISO regular III resolvent 113 singular III transpose 88 Maximal number and maximal system of linearly independent linear forms 96 Metacyclic extension 318 group 319 polynomial 319 Minor 132, 150 Modulus 191 Multiple 172 roots 211 Multiplication 14, 57 Mutually exclusive 21 Natural irrationals 298 v-fold root 211 Noether 27 Nonhomogeneous equations system of 89 Normal divisor 69 element 258 extension 255 polynomial 259 representation 45 Normalized 177 linear form 94 n-termed vectors 84 Null element 15 matrix 88 set 21 vector 84

Number 24 cardinal 24 Numbers, ratIonal, real, complex 19 Odd permutation 122 One-to-one correspondence 24 Operation 13 Operations, elementary 18 Order of combination 121 group 58 group element 198 subdetermmant and minor 132 Partition 21 right and left 67 Perfect fields 216 Penod 198 Permutation 117, 120 even and odd 122 Polynomial 167 Abelian 270 cycl1c 27() Galois 2~ generic 323 inseparable 214 Irreducible 178 metacyclic 319 normal 259 of the first kind 214 of the second kind 214 pure 302 roots of a 167 separable 214 solvable by radicals 303 Power 24 Powers, integral 19, 60 Prime element 174 field 201 function 178 integral domain 201 number 178 relatively 184 residue classes 195 to each other 184 Primitive element in the case of groups 196 (system) in the case of extensions 221

335

Primiti"e roots of unity 304 Pnnclpal diagonal 127 Pnnclple of substitution 47,51 Product 14, 57 mner 85 Proper diVIsor 174 subdomain and extension domain 32 subgroup 66 subset 21 Pure extensIOn 303 polynomial 302 Quotient 16 field 37 RadIcals, soh'able by 303 Rank of a matnx 150 Rational function 39, 47, 51 numbers 19 operatIOns 168 Real numbers, 19 Rearrangement 117 Reduction of Galois group 287 Reflexivity, law of 22 Regular matrix III Relatively prime 184 Remainder 182 division with 182 Representation, normal 45 Representatives, complete system of 23 Residue class 67 group 65 ring 28, 191 Residue system, right and left 67 Resolvent 261 Galois 261 Lagrange 315 matrix 113 Result of Abel 331 Right, see left Ring 15 Root field 245 primitive 308

INDEX Roots multiple 211 of a polynomial 167 of uruty 304 v-fold 211 Rows of a matrix 87 Rule of ComhinatlOn 13, 57 Schreier 248 Separahle algehraic element 223 algehraic extensIOn 224 polynomlal 214, 215 Set 13, 21 empty 21 null 21 sgn 122 Simple extension 221 Simultaneous adjunction 220 Singular matrix 111 Solvable bv radicals 303

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Solving equations 148 Speiser 323, 330 Steinitz 46, 214, 216, 219, 248, 249, 250, 292 Stem field 235 Subdeterminant 132 Subdomam 32 Subfield 25 Subgroup 63 conjugate 71

Subgroup (cont.) identity 66 invariant 69 proper 66 true 66 Subring 25 Subset 21 conjugate 71 proper 21 true 21 Substitution, principle of 47,51 Subtraction 14 SuccessIve adjunction

220 Sum 14 Summation interchange of order of 82 symbol 39 Symmetric functlOns 328 group 119 Symmetry in :t't, ... , %11 45 law of 22 System of equations 55 of fundamental solutlOns 92 of homogeneous and non· homogeneous equations 89 of linear equations 56

336

Toeplitz 94 process 94-102 Theorem of 95 Transcendental elements 46, 223 extension 224 Transformation of set 71 Transitivlty, law of 22 Transmutation 288 Transpose matrix 88 system of homogeneous equations 90 True extension domain 32 subdomain 32 subgroup 66 subset 21 Types 30, 65 Union 21 Unit 173 vector 84 Unity element 17, 58 group 61 Unknowns 246 Variables 38 Vector 84 n-termed 84 Waerden, van der 214 Zero 15