Embedding problems of division algebras

TL;DR

This paper extends the concept of embedding problems from fields to division algebras, providing new insights into the admissibility of groups over function fields of curves and solving related algebraic problems.

Contribution

It introduces a generalized notion of embedding problems for division algebras and demonstrates solutions for admissible groups over specific function fields.

Findings

Embedding problems of division algebras can be solved for certain admissible groups.

The work extends classical Galois embedding problems to division algebras.

Results apply to function fields of curves over complete discretely valued fields.

Abstract

A finite group G is called admissible over a given field if there exists a central division algebra that contains a G-Galois field extension as a maximal subfield. We give a definition of embedding problems of division algebras that extends both the notion of embedding problems of fields as in classical Galois theory, and the question which finite groups are admissible over a field. In a recent work by Harbater, Hartmann and Krashen, all admissible groups over function fields of curves over complete discretely valued fields with algebraically closed residue field of characteristic zero have been characterized. We show that also certain embedding problems of division algebras over such a field can be solved for admissible groups.