Divisible Subgroups of Brauer Groups and Trace Forms of Central Simple Algebras

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.

2000 Mathematics Subject Classification: 16K50, 11E81, 11E04

Keywords and Phrases: Central Simple Algebras, Trace Forms, Brauer Groups

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