On fields with finite Brauer groups

Abstract

Let K be a field of characteristic = 2, let Br(K) 2 be the 2primary 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.

Locations

• Pacific Journal of Mathematics - View - PDF