VH-G(A)=IIVHGF(S)"J=1.

Size: px

Start display at page:

Download "VH-G(A)=IIVHGF(S)"J=1."

Abigail Simmons
5 years ago
Views:

1 ON A GENERALIZED PRINCIPAL IDEAL THEOREM FUMIYUKI TERADA (Received May 27, 1954) 1. Introduction. The author proved several years ago following theorem1), which is a generalization of the Hilbert's principal ideal theorem. THEOREM. Let K be the absolute class field of a number field k, and 12 be an intermediate field of K/k such that fl/k is cyclic. Then each ambignus ideal in 12 is principal when it is considered in K. By the Artin's law of reciprocity, this theorem can be translated into a group theoretical one. Let G be a finite group whose commutator subgroup G' is abelian. Let H be an invariant subgroup with cyclic factor group G/H. Let us denote S (=So) a representative of a generator of the cyclic group G/ H, and also denote Si,..., Sm representatives of generators of the abelian group Hl G', with orders mod G' e1,...em, respectively. We shall assume also that S1,..., S,n generate the group H; this is accomplished by adding to them, if necessary, certain elements in G' with e=1. Now the theorem is translated into the following THEOREM 1. If an element A=S1...S then VH-G(A)=IIVHGF(S)"J=1. j=1 7zn of H satisfies SASA-E H, Author's proof of this theorem was rather complicated, and an alternative simplified proof was given by Prof. T. Tannaka2. The aim of this note is to give another proof transforming it into a problem concerning a group of linear transformations as it was done by Magnus3>, and we avoided the computations concerning determinants as much as possible. 2. A group of linear transformations. Let us consider a group generated by the following m+1 linear transformations; Sj:z'=t1z+at(i=0,1,...,m) where m is the number of S in 1, and t,, a are supposed to be algebraically independent with respect to the rational integral domain Z. We can show easily that Sai...Sanz=Tz+A=to'...tanz+A, 1) F. TERADA, On a generalization of the principal ideal theorem, this, journal, 2nd Ser., Vol. 1 (1949). 2) T. TANNAKA, An alternative proof of a generalized principal ideal theorem, Proc. Japan Academy, vol. 25 (1949). 3) W. MAGNUS, Ueber den Beweis des Hauptidealsatzes, Crelle's Journal 170 (1934).

2 96 F. TERADA where A is a linear firm of a1 with rational functions of t as coefficients.4) More precisely, expanding 1-T as (2) A=S1a a zn and 8,(1)=a,. Moreover following relations are also verified easily. (4) S:z=Tz+A, S':z'=z+C-SS'S: z=z+tc (5) S:z'=z+C, S':z'=z+C'-SS': z'-z+c+c We now introduce m relations tit=1(i=1,...m)7>into the coefficients of the above transformations, et being the order of S, mod G'. Let us denote by the group obtained by this manner, and also denote 0 the subgroup of consisting of the elements of the from S:z'=z+C(i. e. T 1). Then S;i(i=1,..., m) is contained in o as it follows from the relation (6) Sit:z'=z+(1+t+...tt-1)=z+fa(i=1,...,m), where f=1+tt+...+ti-5). It follows from (3)-15) that G0 is an abelian normal subgroup of J with abelian factor group J/0. To avoid confusion, we shall describe an element S:z=z+C of simply by C, and the group operation will be denoted additively. The elements S7i (i=1,...,m) of G are contained in G', and there is m relations between these elements and commutators. These will be written as k(i1,m),(7) Sit=II[Sk,S1]Prz> where the sign [x,y] means the commutator xyx-ly-1 and Pk/is an element if the group ring [G/G'] and the powers mean the usual symbolic cower. In the following we shall confine ourself with a fixed representation (7) among the possible representations. Replacing all s; by tf in P, we have a function which will be denoted by the same symbol Pki). Now, let us introduce the relation (7) into the group (i and denote the group obtained by These relations may be denoted additively as 4) The denominator of this coefficient is a monomial of t0, ti,...ti. All the rational functions of ti which will be appear in the followings are of this type, and we shall denote h1, gi. Pki, etc., without notice there. We shall call the ti-degree of a function the ti-degree of the numerator of this function in its incommensurable form. 5) This symbol will be used till the end of this paper. 6) The coefficient 8j is just the derivation at which is defined in the free group generated by t0,...,tm. Cf. R. H. Fox, Differential calculus in free groups, Ann. of Math., vol. 57(1953). 7) Notice that we introduce no relations for t0, which is corresponded to S=So in G, and is treated distinctively from the other elements t1,..., tm in the following.

3 PRINCIPAL IDEAL THEOREM 97 The subgroup of (fi corresponding to (will be denoted by (o. Then the correspdndence Sz-S1 defines a homomorphism r of (f3 onto G (c. f. 4). 3. Proof of the theorem. An inverse image S...Sm1z in our Theorem by the homomorphism i is expressed as z=tz+a,t=t..a..tan, A=laz,+...+n as,. Then an inverse image of SAS-1A-1 is an element of (f j expressed, from (2), as and this will be rewritten as n i (D0a piao). But also, an inverse image of VHr'(S,3Y'(IISi...SmSr'...S;'m)ai is,...fm a j a,=fi... and therefore, f...fm ry aj is an inverse image of Vg.,a(A). Now let us prove the following PROPOSITION. If there is a relation in the group then there is a rational function D of to,..., tm such that Each element of H' has an inverse image of the formand the relation (8) is a general form of the inverse image of the assumption SAS-lA-1 E H' of our theorem, where C satisfies the relation J'(C)=1. From this proposition, we have T7J, G'(A)=(fl...ryca)=*(DC), and it follows from (4) that A(DC) is a conjugate of i/r(c)=1, and this shows our main theorem. PROOF OF THE PROPOSITIONS. From (7*) we have we have 8) It can be assumed that the functions q, f j, Pa),...in this proof are polynomials of ti, although it is not necessary for our purpose.

4 98 F. TERADA L t us denote these determinants by Do and D. respectively. Then we have- (11) DIak=D.a1(k,l=0,1,...,m). For l=0, this is the identity (10) itself:and for k0, l=0. after transposing, in the equality (9), the term of a1 in the left-hand side to the right and also the term R, a0 in the right-hand side to the left (i. e, exchanging the term of aj and R;a0 with negative sign), we have (11) by a similar method. As the above equality-poi) (a41-aok)a=qtkak is an identity, we may put z into ak, and we have Q 74k=- =Rii0. Also, by the definition, 0JJ=0. Therefore, after multiplying the first row of the determinant D0 by 0,..., the last row of Do by 0m, we have the following identities by adding them to the k-th row: f1+ql1...jqlkk...q1n Qrn,...Qmb...fn+Qmm and id=0 by comparing the t0-degree of the both side of the identity. Moreover, the last formula L1D=0 shows that D' is divisible by each f (i=1,...m), and D' is expressed as D'=t...fnD" where D" is a functionn of t,..t and therefore it may be considered as a constant because t, f, f=f..fn(i=1,..,m). Thus we have Do=0Dfi. and putting 1 into all ti (i=0,...,m) of this identity, we have D0(1)=e1.. en D". It is shown easily from the definition of Do, D0(1)=c,em, and this shows D'=1. Therefore we have

5 which is our proposition. q, e. d. 4. Remarks. a) We shall prove that i/r is a homomorphism of the group (3 onto the group G. Let us consider a free group generated by m+1 elements F0,...,F and prove that the correspondence q: FL-S defines an isomorphism o of the group..., F3 onto the group i. It is easy to see that our purpose follows from this immediately. Moreover, it is enough to prove that if there is a relation we have Firstly, rewriting F as F=F0..Fm (mod), we have (F)=So Snit=1 mod (i, and this shows tam=1, and it follows/3 =0, $-0 mod ec for i>1. Therefore F is expressed as where the powers mean the symbolic power. In this expression, we may assume that g, is polynomial of..., F, and especially such that 1) the Fe-degree of g is less than e, for all a>1, 2) the Ft-and F,-degree of gi is less than e,-1 and e-1 for k, l>1, 3) the F'-degree of gk1 is zero for j<l<k, 4) the Ft-degree of ryti is zero for all 7. For, 1) follows from [1', 3'] co, 2) follows from [F, +1 =F1Fay, which is combined with F k, k into a factor, 3) follows from [F, Fa]1-Fj=[Fa, Fj]"[Fk,Fj]I-FZ which are combined with [Fl, Fj]9 and [F,F]9,j and finally4) follows from F4ai=1. Now we have from (14) and (16) where ry and gz are polynomials of tz obtained from ry and gkz in (16) by replacing all Fz by tti. Expressing this condition by means of ai, and recalling the algebraic independence of a1, we have Comparing the ti-degree, we have yi=0 from the normality of y and g. Moreover, comparing the tj-degree, we have U o = 0, and so on. Thus we have ry(t)=0, g(t)=0, and this shows y(f)=0, g(f)=0; that is F-1 mod o, as it was desired. b) In our group f of linear transformations, let us denote an invariant subgroup generated by S,..., S,n and Vp(=(W). Then the factor group

6 100 F. TERADA is a cyclic group with generator S, and/is an abelian group of the type (e,...,e). It will be shown easily that we have our main theoremm concerning the group (which is an infinite group. But also, we have the inverse of this theorem concerning this group J ; that is, we have THEOREM Z. A necessary and sufficient condition for anelement A S to satisfy (A)=1 is that A is an ambigous element, that is, A satisfies SAS-IA-'EY. PROOF. The commutator subgroup' is generated by the following elements with symbolic power and the group (o=(w is generated by these elements and, a0-i0af(p= 1,...,m). As it was shown in 3, V,t(A) and SAS1A-1 are expressed as where X1 has no terms of t1,...,t. Then, as it was proved in the preceding proposition, But XL has no terms of t,...,t and therefore, a necessary and sufficient condition for fi...f n rya=0 is X=0(i=1,...,m), that is, SAS-IA-i is contained in This theorem suggests us that the condition SAS-'A-1 E H' will be necessary in general for the validity of the main theorem, though for individual groups some special condition will guarantee a generation. MATHEMATICAL INSTITUTE, TOnoKU UNIVERSITY

Chapter 3. Rings. The basic commutative rings in mathematics are the integers Z, the. Examples

Chapter 3. Rings. The basic commutative rings in mathematics are the integers Z, the. Examples Chapter 3 Rings Rings are additive abelian groups with a second operation called multiplication. The connection between the two operations is provided by the distributive law. Assuming the results of Chapter