Tamely ramified subfields of division algebras

TL;DR

This paper investigates which finite solvable groups can be realized as Galois groups of tame maximal subfields within division algebras over arbitrary number fields, extending existing theorems and conjectures.

Contribution

It generalizes Liedahl's theorem and Sonn's solution of the Q-admissibility conjecture to all number fields, characterizing solvable groups as Galois groups of tame maximal subfields.

Findings

Characterization of solvable groups as Galois groups of tame maximal subfields

Extension of Neukirch's embedding theorem with local constraints

Generalization of Liedahl's and Sonn's theorems to arbitrary number fields

Abstract

For any number field K, it is unknown which finite groups appear as Galois groups of extensions L/K such that L is a maximal subfield of a division algebra with center K (a K-division algebra). For K=Q, the answer is described by the long standing Q-admissibility conjecture. We extend a theorem of Neukirch on embedding problems with local constraints in order to determine for every number field K, what finite solvable groups G appear as Galois groups of tame maximal subfields of K-division algebras, generalizing Liedahl's theorem for metacyclic G and Sonn's solution of the Q-admissibility conjecture for solvable groups.