Determination of quadratic extensions of linearly compact fields by norm groups

Abstract

It is shown that quadratic extensions of a field not of characteristic two, which is linearly compact at a valuation, are determined by their groups of norms, provided the residue field has a unique quadratic extension and is perfect if of characteristic two. It is indicated that Henselian can replace linearly compact in some cases. Necessity of the condition on the residue field is shown. 1. In this brief paper we shall apply the techniques and results of the paper Quadratic extensions of linearly compact fields by Ron Brown and myself (referred to below as [BW]) to prove the following result: THEOREM 1. Let F be a field of characteristic char(F)$2. Let v be a (nonarchimedean) valuation on F with arbitrary value group rF and residue field kF; assume only that kF is perfect if char(kF)=2. Suppose that F is linearly compact at v and that kF has a unique quadratic extension. Then for K1 and K2 quadratic extensions of F, K1 K2 if and only if N1K1 = N2K2. (Here Ni denotes the norm map Ki-*F.) For definition and properties of linear compactness, see [BW] or [Bour]. All the hypotheses of Theorem 1 are satisfied by any classical local field of characteristic not two. Indeed Theorem 1 is a generalization of a special case of the local class field theorem which says that an abelian extension of a local field is determined by its group of norms. The conclusion of Theorem 1 is equivalent to the assertion that a binary quadratic form over F is determined up to equivalence by the elements of F which it represents. A straightforward application of the Global Squares Theorem extends this result to the global case, obtaining the well-known result that over any local or global field of characteristic not two, binary quadratic forms are equivalent if and only if they represent the same elements. Received by the editors July 9, 1971. AMS 1970 subject classifications. Primary 12B10, 12J10, 12B25; Secondary 10C05, 12A25, 12J20.