JOURNAL OF NUMBER THEQRY 32. 203-219 (1989) On the Norm Groups of Global Fields LEONID STERN * Department of Mathematics, Towson Slate University, Baltimore, Maryland 21204 Communicaled by P. Roquelfe Received May 11, 1988 INTRODUCTION Let Klk be a finite extension of fields. We define N,,K* = (K&W E K* >, where K* is the multiplicative group of K, and N,,(a) is the norm of CI from K to k. The subgroup N,, K* of k* is called a norm group of the field k. One of the fundamental objects of study in local class field theory is the family of norm groups of local fields (i.e., fields complete for a discrete valuation with finite residue field). It is proved in local class field theory that the mapping KH N,,,K* from the set of finite Abelian extensions of a local field k into the set of norm groups of k is a bijection. It is also shown that N,, K* = Nr,, K’* for any finite separable extension K of a local field k, where K’ is the maximal Abelian extension of k contained in K [ 121. In this paper we show that the mapping KH N,,K* from the set of finite extensions of a global field k (i.e., an algebraic number field or a function field in one variable over a finite field) into the set of its norm groups is essentially different from that in the local case. Our result (Theorem 1) says that KS L iff N,,L* s N,,K* for any finite Galois extensions K and L of a global field k. In particular, this implies K = L iff N,, K* = N,,, L*. In connection with our result it is worth mentioning the following result of Neukirch [S]. Let a be the algebraic closure of the field of rational numbers Q. Let G, denote the absolute Galois group of an algebraic num- ber field K, i.e., GK= G(Q/K). Neukirch [S] proved that for any finite Galois extensions K and L of Q, if GK and GL are isomorphic as topological groups with the protinite topology, then K= L. This result has been subsequently generalized independently by Ikeda, Iwasawa, and Uchida [16]: for any finite extensions K and L of Q, if G, and G, are isomorphic as topological groups, then K z L (i.e., K and L are conjugate * Supported in part by the Faculty Research Committee of Towson State University. 203 0022-3 14X/89 $3.00 Copyrght ‘Y’. 1989 by Academic Press. Inc. All rights of reproductmn m any form reserved 204 LEONID STERN over Q). Two other interesting notions of equivalence of algebraic number fields have been investigated in recent years. Two algebraic number fields K and L are arithmetically equivalent iff they have identical zeta functions: ~Js) = CL(s). Gassmann [4] and more recently Perlis [lo] have shown that arithmetically equivalent fields have the same normal closure over Q, but are not necessarily isomorphic. Sonn [ 131 introduced the following notion of equivalence of algebraic number fields. A finite extension K of a field k is k-adequate if there is a finite dimensional k-central division algebra containing K as a maximal subfield. A finite group G is k-admissible if there is a Galois extension K/k with G E G(K/k), and K is k-adequate. Both notions were introduced by Schacher [ 1 l] and investigated by him and others. Sonn [ 131 proved that if K and L are algebraic number fields such that for every finite group G, G is K-admissible iff G is L-admissible, then K and L have the same normal closure and the same degree over Q. In this paper we introduce a notion of equivalence of global fields, whose relation to the above equivalence relations is not clear. It is well known that if finite separable extensions K and L of a global field k are conjugate over k (k-isomorphic), then N,,K* = N,/,L*. Our result (Theorem 1) shows that the converse is true if K and L are Galois over k. Finite separable extensions K and L of a global field k are k-equivalent if N,K* = N,,L*. We are thus led to the following question: does k-equivalence of K and L imply that K and L are conjugate over k? Let K and L be finite Galois extensions of a global field k. Let F be the compositum of K and L (F= KL). We prove (Theorem 1) that the index of N,,F* in N,,,K* n N,,, L* is finite. By local class field theory the corresponding index is one whenever K and L are finite Abelian extensions of a local field k. An example will be given to show that in general the index is different from one in the global case, even if F is the compositum of Abelian extensions K/k and L/k. The question of whether or not an equality N,,F* = N,,K* A NLlkL* holds in the global case is shown to be related to the so-called Hasse norm principle (HNP). We say that a finite Galois extension E/k has Property I if for any Galois extensions kc KLr E of k the equality N,, F* = N,,k K* r\ N,!,L* holds. By local class field theory every finite Abelian extension E/k of a local held k has Property I. We prove in Theorem 7 that a finite Abelian extension E/k of a global field k has Property I iff HNP holds for E/k. Our result for non- Abelian Galois extensions is Proposition 5. In the second section we apply Theorem 1 to obtain the following result. There is a subfamily of irreducible over a global field k decomposable forms such that f(x, , . x,,) and g( y,, . .. y,) are equivalent (over k) if and only if {f(a,, . a,)la,Ek) = {Ah,, .. b,)/b,Ek}. (For details see Theorem 8.) ONTHENORMGROUPSOFGLOBAL FIELDS 205 1. THE NORM GROUPS OF GLOBAL FIELDS In this section we will use class field theory. The notation is fairly standard but is summarized below for the convenience of the reader. Let G be a finite group, written multiplicatively. Let A be a G-module, written additively or multiplicatively: AG= {a~A/aa=aforalla~G} N,= 1 OEZ[G] .YtG I, = the ideal in Z[G] generated by { 0 - l/a E G> NGA= {aEA/N&=o} N,A = {N,a/aEA}. The Tate cohomology is used: H”(G,A1 if n>l AC/N, A if n=O @(G, A) = Nd/IJ if n=-1 I H -n-l(GA) if n< -2. Let K/k be a finite Galois extensions of a global field k with the Galois group G. The subgroup N,,K* = N,K* = {N,,(a)/aE K*} of the multiplicative group k* of k is a norm group of k, where N,(a) = N,a = IL G a(a) for UE K*. Let J, and C, be the idele group and the idele class group of K, respectively. The short exact sequence of G-modules l+K*+J,+C,+l yields the exact sequence of cohomology groups . - I?(G, JK)- I?‘(G, C,)-% fi’(G, K*) 4 ri’(G, JK) - I?“(G, C,) ----+ 1. It is well known in global class field theory that the groups &‘(G, C,) and fi-‘(G, Z) are isomorphic. Since fiP3(G, Z) is dual to the Schur mul- tiplier (of G) H3(G, Z) 2 H2(G, Q/Z), and the Schur multiplier of G is a finite group, it follows that fi-‘(G, C,) is finite. Hence the kernel of f, &l/32/2-6 206 LEONID STERN being equal to Im g, is a finite group. The kernel off has been the subject of investigation in connection with the Hasse norm principle (see [6] and references therein). For each prime u of the field k we fix a k-embedding of the separable closure it of k into the separable closure it, of the completion k, of k at u. Note that a k-embedding of E into i;, determines a unique extension of v to k. If M is a finite separable extension of k, then the compositum Mk, is the completion of M at the extension of v to M corresponding to the k-embedding of /? into /?,. We denote this completion of M by M,. We will now find the kernel off: By global class field theory h*(G, J,c) = LI I?“( G( KU/k,), Kg), where v ranges over all primes of k. Hence the homomophism f is equivalent to the canonical homomorphism (v ranges over all primes of k). We set N(K/k) = f-j (k* n N,c,,$Z), where u ranges over the set of primes of k. For the kernel off we have the following equality Ker f = N(K/k)/N,, K*. Let i(K/k) denote the order of the factor group of N(K,Jk) by N,K*. The Hasse norm theorem asserts that if K/k is a cyclic extension, then i(K/k) = 1. We say, more generally, that the Hasse norm principle holds for K/k if i( K/k) = 1. THEOREM 1. Let K and L be finite Gal&s extensions of a global field k. Let F = KL be the compositum of K and L. Then (a) N,,F* is a subgroup offinite index in N,K* C-INLIkLL*.