On fields with finite Brauer groups (Q1816518)

Let \(K\) be a field of characteristic \(\neq 2\), let \(\text{Br} (K)_2\) be the 2-primary part of its Brauer group, and let \(G_K (2)= \text{Gal} (K(2)/ K)\) be the maximal pro-2 Galois group of \(K\). We show that \(\text{Br} (K)_2\) is a finite elementary abelian 2-group \((\mathbb{Z}/ 2\mathbb{Z})^r\), \(r\in \mathbb{N}\), 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 \(r\leq 6\) then \(K\) has precisely \(r\) orderings, and if \(r< \infty\) then \(K\) has only finitely many orderings.