On Fields with Finite Brauer Groups

Let K be a field of characteristic 6= 2, let Br(K)2 be the 2primary part of its Brauer group, and let GK(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 ∈ N, if and only if GK(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.