Patching subfields of division algebras

Abstract

Given a field F F , one may ask which finite groups are Galois groups of field extensions E / F E/F such that E E is a maximal subfield of a division algebra with center F F . This question was originally posed by Schacher, who gave partial results in the case F = Q F = \mathbb Q . Using patching, we give a complete characterization of such groups in the case that F 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.