Divisible subgroups of Brauer groups and trace forms of central simple algebras

Abstract

Let $F$ be a field of characteristic different from 2 and assume that $F$ satisfies the strong approximation theorem on orderings ($F$ is a SAP field) and that $I^3(F)$ is torsion-free. We prove that the 2-primary component of the torsion subgroup of the Brauer group of $F$ is a divisible group and we prove a structure theorem on the 2-primary component of the Brauer group of $F$. This result generalizes well-known results for algebraic number fields. We apply these results to characterize the trace form of a central simple algebra over such a field in terms of its determinant and signatures.