On fields with finite Brauer groups

Abstract

Let K be a field of characteristic = 2, let Br(K) 2 be the 2-primary part of its Brauer group, and let G K (2) = Gal(K(2)/K) be the maximal pro-2 Galois group of K. We show that Br(k) 2 is a finite elementary abelian 2-group (Z/2Z) r , r ∈ N, if and only if G K (2) is a free pro-2 product of a closed subgroup H which is generated by involutions and of a free pro-2 group. Thus, the fixed field of H in K (2) is pythagorean. The rank r is in this case determined by the behaviour of the orderings of K. E.g., it is shown that if r ≤ 6 then K has precisely r orderings, and if r < ∞ then K has only finitely many orderings.

Open problems on central simple algebras

“…We remark that Fein and Schacher's main result in [FS76] was that "many" of the possibilities predicted by Brumer-Rosen actually occur: every countable abelian torsion group of the form D V for D divisible and V a vector space over F 2 occurs as the Brauer group of some algebraic extension of Q. Efrat [Efr97] gives general criteria on the field F for ÔBr F Õ 2 to be finite and nonzero.…”

Distinguishing division algebras by finite splitting fields

“…If k is a higher local field, for example, an iterated Laurent series field k 0 ((t 1 )) · · · ((t m )) where k 0 is finite, local, or algebraically closed, then one may show that the p-torsion part of the Brauer group, Br(k) [p], is finite for any p, and in particular, the hypotheses of Theorem 2.2(1) will automatically hold for N = | Br(k)[p]|. A description of when Br(k)[2] is finite is given in [Efr97] in the case the characteristic of k is not 2. We note also that the weaker conditions of Theorem 2.2(2) hold in the case that k is a global field.…”

Relative Brauer group and pro-p Galois group of pre-p-henselian fields

“…Then M = L( p √ y). Thus, if w is a extension of v to L, then w has exactly p extensions to M (see (2) of Proposition 3.1 and Remark 3.1). By the same reference, x also is in (1 + m w )L p , which is a contradiction with the choice of xL p .…”

“…It consists of the classes of central simple F -algebras in p Br(F ) which splits over F h . According to [2,Lemma 1.3], there are index sets J and J 2 such that p Br(F h ) is isomorphic to…”