Admissible groups over number fields

TL;DR

This paper classifies which finite groups can be realized as Galois groups of maximal subfields of division algebras over number fields, extending previous results and providing new characterizations for tame and wild ramification cases.

Contribution

It provides a complete classification for solvable Sylow-metacyclic groups over number fields in the tame ramification case and offers new characterizations for admissible groups in the wild ramification case.

Findings

Complete classification for tame ramification case.

Characterizations for admissible groups in wild ramification case.

Partial results for various classes of number fields.

Abstract

Given a field K, one may ask which finite groups are Galois groups of field extensions L/K such that L is a maximal subfield of a division algebra with center K. This connection between inverse Galois theory and division algebras was first explored by Schacher in the 1960s. In this manuscript we consider this problem when K is a number field. For the case when L/K is assumed to be tamely ramified, we give a complete classification of number fields for which every solvable Sylow-metacyclic group is admissible, extending J. Sonn's result over the field of rational numbers. For the case when L/K is allowed to be wildly ramified, we give a characterization of admissible groups over several classes of number fields, and partial results in other cases.