Decomposition Configuration Types in Minimally Tamely Ramified Extensions of $\mathbb{Q}$

TL;DR

This paper investigates the realization of finite groups as Galois groups over $Q$ with minimal tame ramification, focusing on controlling inertia groups and ramification decomposition.

Contribution

It introduces methods to realize finite groups as Galois groups of minimally tamely ramified extensions with specified inertia and decomposition groups.

Findings

Conditions for realizing groups as Galois groups with minimal tame ramification.

Explicit constructions of such extensions with prescribed ramification behavior.

Insights into the structure of ramified primes in these extensions.

Abstract

We examine whether it is possible to realize finite groups $G$ as Galois groups of minimally tamely ramified extensions of $\mathbb{Q}$ and also specify both the inertia groups and the further decomposition of the ramified primes.