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' . " 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 l1)
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
' .. ,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 auo-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=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
xJolvability ( 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
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
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 ., '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"" 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 ... , 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 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.
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 -
. , ,, 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 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
.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 •.. ,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 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.';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 - (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 , ... , aTteps 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 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)0.AP We + ,-10.A + .. ,+ C- 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 ... > 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 "" ;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