On fields with finite Brauer groups

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.