On Galois extensions with prescribed decomposition groups

TL;DR

This paper explores whether all finite groups can be realized as Galois groups over the rationals with only cyclic or abelian decomposition groups, providing new criteria and infinite families for nonsolvable groups.

Contribution

It introduces general criteria using specialization of function fields and constructs the first infinite families of Galois extensions with nonsolvable groups and cyclic decomposition groups.

Findings

Established criteria for realizing groups with prescribed local conditions.

Constructed infinite families of Galois extensions with nonsolvable groups and cyclic decomposition groups.

Extended analysis to global function fields.

Abstract

We study the inverse Galois problem with local conditions. In particular, we ask whether every finite group occurs as the Galois group of a Galois extension of $\mathbb{Q}$ all of whose decomposition groups are cyclic (resp., abelian). This property is known for all solvable groups due to Shafarevich's solution of the inverse Galois problem for those groups. It is however completely open for nonsolvable groups. In this paper, we provide general criteria to attack such questions via specialization of function field extensions, and in particular give the first infinite families of Galois realizations with only cyclic decomposition groups and with nonsolvable Galois group. We also investigate the analogous problem over global function fields.