Stability: index and order in the Brauer group

Abstract

A field is stable if for every division algebra A in its Brauer group order of A = index of A. Index and order in the BraLer group of a field F with discrete valuation and perfect residue class field K are calculated. Division algebras with specified order and index are constructed. For F complete, necessary and sufficient conditions for the stability of F are given in terms of the Brauer group of K. These results follow. A finite extension of a stable field need not be stable. The power series field K((x)) is stable for K a local field. K((x)) and K(x) are not stable for K a global field. Introduction. A field F is stable if for every division algebra A in the Brauer group Br(F), order eF(A) = index SF(A). In general order divides index and they have the same prime factors. Any local or global field is stable by class field theory. Many results concerning subfields and subalgebras of division algebras depend on stability. See, for example, [Sc] and [R]. For a profinite group G, the character group G = continuous Hom(G, Q/Z). For a field F, G(F) = Gal(FS/F) with Fs a separable algebraic closure of F. If L is a finite Galois extension of F and f is a character on Gal (L/F), then f inflates to a character on G(F). These topics are exposed in [S]. Lemma 1. Let F be a field and G = G(F). If f e G, H = Ker f, and L = fixed field of H, then L is a finite cyclic extension of F with Gal (LIF) = G/H and order f = index [ G: H] = [ L: F]. If L is a finite Galois extension of F, then L is cyclic over F iff there is a character on Gal (L/F) of order [L: F]. Proof. Immediate from Galois theory and the fact that every finite subgroup of Q/Z is cyclic. For M a field extension of F and f e G(F), fM = the restriction of f to G(M). This definition assumes Fs C Ms. If fM = ?, M is said to split f. For Received by the editors March 26, 1974 and, in revised form, May 3, 1974. AMS (MOS) subject classifications (1970). Primary 16A40, 12B20, 12G05, 12J10; Secondary 13A20, lOMIO. Copyright ? 1975, American Mathematical Society 33 This content downloaded from 157.55.39.186 on Sun, 09 Oct 2016 04:23:44 UTC All use subject to http://about.jstor.org/terms