Galois subfields of tame division algebras

TL;DR

This paper establishes a criterion linking the existence of Galois maximal subfields in tame division algebras over Henselian fields to their residue division algebras, extending known constructions and aiding the identification of noncrossed products.

Contribution

It generalizes the crossed product criterion to all tame division algebras, including those over residue fields of characteristic zero, and describes the placement of noncrossed products in the Brauer group.

Findings

Galois maximal subfields in tame division algebras correspond to those in residue algebras.

The criterion applies to all tame division algebras, especially over residue fields of characteristic zero.

It helps locate noncrossed products within the Brauer group.

Abstract

We show that a finite-dimensional tame division algebra D over a Henselian field F has a maximal subfield Galois over F if and only if its residue division algebra has a maximal subfield Galois over the residue field of F. This generalizes the mechanism behind several known noncrossed product constructions to a crossed product criterion for all tame division algebras, and in particular for all division algebras if the residue characteristic is 0. If the residue field is a global field, the criterion leads to a description of the location of noncrossed products among tame division algebras, and their discovery in new parts of the Brauer group.