On fields with finite Brauer groups

Ido Efrat

Vol. 177 (1997), No. 1, 33–46

DOI: 10.2140/pjm.1997.177.33

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.

Milestones

Received: 24 March 1995

Revised: 27 June 1995

Published: 1 January 1997