Galois subfields of inertially split division algebras

TL;DR

This paper proves that in valued division algebras with unramified splitting fields, the Galois property of maximal subfields is preserved in the residue division algebra, aiding in understanding noncrossed products.

Contribution

It establishes a link between Galois maximal subfields in a division algebra and its residue algebra, extending noncommutative valuation techniques.

Findings

Residue division algebra contains Galois maximal subfield if original does

The result applies to noncommutative valuation contexts

Facilitates explicit constructions of division algebras

Abstract

Let D be a valued division algebra, finite-dimensional over its center F. Assume D has an unramified splitting field. The paper shows that if D contains a maximal subfield which is Galois over F (i.e. D is a crossed product) then the residue division algebra of D contains a maximal subfield which is Galois over the residue field of F. This theorem captures an essential argument of previously known noncrossed product proofs in the more general language of noncommutative valuations. The result is particularly useful in connection with explicit constructions.