Patching subfields of division algebras

TL;DR

This paper characterizes which finite groups can be Galois groups of maximal subfields of division algebras over certain function fields, using patching techniques to extend previous partial results.

Contribution

It provides a complete characterization of such Galois groups over function fields of curves over complete discretely valued fields with algebraically closed residue fields, employing patching methods.

Findings

Complete classification of Galois groups over specified function fields

Extension of Schacher's partial results to new cases

Application of patching techniques in division algebra context

Abstract

Given a field F, one may ask which finite groups are Galois groups of field extensions E/F such that E is a maximal subfield of a division algebra with center F. This question was originally posed by Schacher, who gave partial results over the field of rational numbers. Using patching, we give a complete characterization of such groups in the case that F is the function field of a curve over a complete discretely valued field with algebraically closed residue field of characteristic zero, as well as results in related cases.