ON A CERTAIN GROUP CONCERNING TIIE p-adic NUMBER FIELD.

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1 ON A CERTAIN GROUP CONCERNING TIIE p-adic NUMBER FIELD. Hideo By Kuniyoshi.*) In the local class field theory, we consider the norm group of a finite extension field of a p-adic number field k. An abelian extension K of k is uniquely determined by this subgroup of k*, where k* is the multiplicative group of all non zero elements of k. We denote this norm group of K by VK, c. Then the galois group of K/k is isomorphic to the factor group We may consider, in some sense dually to the above fact, a subgroup G(k/K) of K which consists of all the elements of K* whose norms to k are unity. It is likely that G(k/K) has close connections with the subfield K, When K/k is cyclic, the structure of G(k/K) was determined by Hilbert. When K/k is abelian, a certain property of G(k/K) was given by Prof. T. Tannaka,1) who gave also another theorem which is analogous to the ordering theorem of local class field theory. The former property was extended to non-abelian cases, by Mr. T. Nakayama and Mr. Y. Matsushima. In this paper, restricting to the abelian case, I shall give a detailed structure of G(k/K), and add a certain remark to a particular non-abelian case. 1. The striletilre of G (k/3)). Let k be a p-adic number field, and K be a finite extension of k. We denote the multiplicaitve groups of their non zero elements by k*, K*, respectively, and norm group of K/k, by N. The elements of K whose norm to k are unity, form a subgroup of K* and we denote this by G(k/K). When K is a normal extension of k with its galois group G, we mean by a factor set of K/k a system of elements a, T (o, T E G) of K satisfying (1) saa7 p=a, at, P. *) Received Aug. 1st, ) T. Tannaka L8J. 2) T. Nakayama and Y. Matsushima (4), T. Nakayama C7J.

2 p-adic NUMBER FIELD 187 Further, we shall denote by KGA the group generated by 81 8EK, a E G. One of Tannaka's results runs as follows: Theorem 1.3) Let Q be a mite abelian extension field of k with its galois group A, and (aa, T) be a factor set of Q/k whose exponent is equal to the degree of extension Q/k. Then G(k/fl) is generated by 12 and aa, 7/a7, a. where o,- run over A: (2) G(k/f) avtf11 R a7 Let (3) (n1n)nz-fiin be an invariant system of A, then A decomposes directly in cyclic groups Z of order nz: (4) A=Z1xZxxZ, This decomposition is up to isomorphism unique. Let o be a generator of Z, and we shall fix it throughout this section. In the Theorem 1, it is not necessary to take all the elements of A, but sufficient to do with o of (4). We show this fact in next Leiiuna 1. (5) G(k/Q)=(ai,fZ4x We prove this by induction. Let N=Z1xx Zand M be the corresponding intermediate field. We assume the lemma for the extension fl/n1: Then we take an element B of G(k/Q), Nok8=-1. As N/k is cyclic, it follows from Hilbert's lemma that (6) Nnif8=n1, nem. Furthermore, as Q/M is abelian extension with its galois group N, there exists4) an element where Then a of N such that n=a, H mod N15) a=ila-ci. (7) n=ia, N mod N/H. 4 3) We refer this theorem to (8). 4) T. Nakayama (6) and Y. Akizuki C1). 5) We denote a product IIaa, T by a7, H, and in a similar way IIa, a by aq p. TE1TEN

3 188 HTDEO KUNIYOSHI Next, we calculate a:, using the relation (1), (8) It follows from (6), (7) and (8) that therefore From the assumption of the induction, we obtain G(k/fI)=(aSip) q.e.d. invariant Then Let K be an abelian extension field of k, whose galois group H has system (n1, n2) n1 n1. (9) H=HixH, Hz={T} where K, are the cyclic groups of order n, and TZ their fixed generaters. Let (b) 1e a factor set of K/k whose exponent is equal to the degree of K/k, From the lemma 1 G(k/K)=bTZKAb T2, T1 Concerning the order of bt1, T2/bT2, Ti mcd KA, Lemma 2. If (bt2, Ti/bT2, Tj)z belongs to KID X, then we obtain next proof Let Kz be the intermediate field which corresponds to Hi. From the assumption of the lemma and (9), we have (10) TIaT2=1-Tl Tn, Tr1Z 81-T2 J02.EKb Taking the norm with respect to K2, the left-hand side of the equation (10) becomes Tl)T2 -- bt2 Tlj 2, H2 NIc)K2 8)1-Tj(b1-TlH2)=x-(b2, T1 and the right-hand side

4 p-adic NUMBER FIELD 189 therefore, In this relation, Tx, tt2=(nkjk28)1-z1ek. b72x, H2 and NKJK2 0 belong to the field KW, S and as the galois group H, of K/k is generated by Tl, it follows that (11) bt2, H2X0NK;K20 where a belongs to the field k. On the other hand we have (12) where if we regard a as an element of K2, for implies (12), owing to the "verschiebungssatz" of the local class field theory. From (11) and (12) follows and from this using the Nakayama's theorems we get Again we return to the extension Q/k, and use the same notations as in the Theorem 1 and the Lemma 1. (13) Lemma 3, Prod Let L5 be an intermediate field which corresponds to the subgroup Zg of A, then Q/L1 is a cyclic extension with its galols group Zj. From this and (S) we get Hence, Now, we point out a relation between the galois gloup A of Q/k and the group G(k/12). Theorem 2. 7) This proof is given by prof. T. Tannaka. Our original proof was much longer and considerably complicated.

5 190 VIDEO KUNIYOSHI where AzZixZi+lxxZr and Z2 are the cyclic groups of (4). Proof We assume a relation between and i. e, (14) We choose Zi, Zj, i<j arbitrary, and let N' be a direct factor excluding GzxZ, and Z be the corresponding intecme liate field, then Z/k is a normal extension with its galois group ZzxZ5. We take norm of (14) with respect to Z, then a simple calculation will show that hence (14) changes to the form (15) From Chevalley's lemma8,) (av) is also a factor set of Z/k whose exponent is equal to (Z:k). Thus we can regard (aq) and Z as (bffq), and K respectively in the lemma 2, hence (16) This shows that in the relation (14) no can really appear, and there exists essentially only relations of the form (15) with (16). It follows ail, a 62 mod Q4A forms a cyclic subgroup Zj,i of degree n, and Then putting we obtain the desired theorem. q.e.d. result, From this, as an immediate consequence, we obtain the Matsushima's namely: Theorem 3, Let k be a p-adic number field and SZ be a finite abelian extension field. then Q/k If is a cyclic extension. This theorem is not true for a nonabelian extension K/k. that For example, let K/k be a nonabelian extension with galois group G. And we assume 8) C. Chevalley (3 or E, witt (9J.

6 p-adto NUMBER FIELD 191 that G/G' and G' are both cyclic groups, G' being the commutator subgroup of G. Then after a slight calculation we get 2. Connections with the class field theory. Let Q and k denote the local fields as in the section 1. There exists the maximum abelian extension field Q of k, and obiously I Q/12. Let A be an infinite abelian extension field of k and we put (10) where A is any intermediate field of A/k of finite degree over k. For the infinite abelian extension A of k, we are able to constitute similar theory with finite abelian extension fields by using H(A/k) instead of N*. Now, we shall show that G (k/f) is closely connected with the maximum abelian extension field of k. Lennna 4. (17) Proof. Let aeh (Q/k), and we put a in the from where P is a fixed prime element of k, and e an unit element, If E=0, we denote the group of all the units by E, and construct a subgroup H of k* generated by E and F3, B =0(2 C ). Then H1 has finite index in k* hence from the existence theorem of the local class field theory, there exists a finite abelian extension Al of k such that Furthermore from (16) (18) On the other hand, from the construction of H1, it is obvious that and this contradicts with (18). Therefore E=0, a is a unit. If a=1, there exists a natural number n such that and we denote by Eta the group of all the element em of k* congruent with unity modulus pn: e1m1 mod p. From Ez and P, we construct a subgroup of k* which has a finite group index in k*.

7 192 HIDEO KUNIYOSHI Analogcusly to the above discussicn, we get an atelian extension An of k such And that similarly From (19) and (20) obviously Thus we lead to a contradiction, arid the lemma is proved. Theorem 4. Let K/k be any finite extension, then (22) Proof. Obviously Cenvenely, we take an elment Q from H (IU/K) and put We assume B=1 and lead to a contradiction. If B=1 there exists an abelian extension A of k such that (23) From (16) follows Therefore, using the Versehiebungssafz we get This contradicts with (23), hence we have As an immediate consequence of this theorem, using the ordering theorem of the local class field theory, we get one of Chevalley's results (2J Corollary. Let k be a p-adie number field and K be its finite extension field. When use take a finite abelian extension A of K, then A/k is abelian, if and only if References 1. Y. AKIZUKI; Eine homom. orphe Zuordnung der; Elernenten der galoisschen Gruppe zu den Elementen einer Untergruppe der Normklassengruppe, Math, Ann. 112 (1936) , C. CHEVALLEY; Sur 1a theorie du corps de classes darts les corps finis et

8 p-adic NUMBER FIELD 193 les corps locaux, Journ. Fac. Sci. Tokyo II (1933) C. CHEVALLEY; La theorie du symbole de restes normiques, Journ. fur Math. 169 (1932) Y. MATSUSHIMA; A remark on Tannaka's "Hauptgeschlechtssatz im minimalen" (in Japanese) Zenkoku-Shijo-Sugaku-Danwakai, 252 (1943). 5. T. NAKAYAMA and Y. MA. TSUSHIMA; Uber die multiplikative Gruppe einer -adischen Divisionsalgebra. Proc. Imp. Acad. Tokyo, vol. XIX (1934). 6. T. NAKAYAMA; Uber die Beziehungen Zweischen den Faktorensystemen and der Normklassengruppe eines galoisschen Erweiterungkorpers, Math. Ann. 112 (1935) T. NAKAYAMA; A remark on Tannaka's "Hauptgeschlechtssatz im Minimalen" (in Jap.) Zenkoku-Shijo-Sugaku-Danwakai 247 (1942). 8. T. TANNAKA; Some remarks on p-adic number fields. (in Jap.) ibid. 236 (1942). 9. E. WITT; Zwei Regeln uber verschrankte Prod ukte, Journ, fur Math. 173 (1935) Mathematical Institute, Tohoku University, Sendai.