On the Brauer groups of quasilocal fields and the norm groups of their finite Galois extensions

TL;DR

This paper characterizes the Brauer groups of quasilocal fields and explores norm groups in finite Galois extensions, providing new realizability results and conditions for norm group equalities.

Contribution

It demonstrates that divisible abelian torsion groups can be realized as Brauer groups of quasilocal fields and analyzes norm group relations in finite Galois extensions.

Findings

Divisible abelian torsion groups are realizable as Brauer groups of quasilocal fields.

Provides conditions for norm group equality in finite Galois extensions.

Describes the structure of Brauer groups for certain Henselian valued fields.

Abstract

This paper shows that divisible abelian torsion groups are realizable as Brauer groups of quasilocal fields. It describes the isomorphism classes of Brauer groups of primarily quasilocal fields and solves the analogous problem concerning the reduced components of the Brauer groups of two basic types of Henselian valued absolutely stable fields. For a quasilocal field E and a finite separable extension R/E, we find two sufficient conditions for validity of the norm group equality $N(R/E) = N(R_{0}/E)$, where R_{0} is the maximal abelian extension of E in R. This is used for deriving information on the arising specific relations between Galois groups and norm groups of finite Galois extensions of E.