The Inverse Galois Problem for p-adic fields

TL;DR

This paper develops a method to count Galois extensions of p-adic fields with a given group, providing new insights and conditions for realizing groups as Galois groups over these fields.

Contribution

It introduces a counting method based on the absolute Galois group description and establishes new necessary and sufficient conditions for Galois realizability over ield.

Findings

Counted extensions for groups up to order 2000 (excluding multiples of 512) for p=3,5,7,11,13.

Identified minimal groups not realizable as Galois groups over ield.

Proved new theorems giving conditions for Galois realizability over ield.

Abstract

We describe a method for counting the number of extensions of $\mathbb{Q}_p$ with a given Galois group $G$, founded upon the description of the absolute Galois group of $\mathbb{Q}_p$ due to Jannsen and Wingberg. Because this description is only known for odd $p$, our results do not apply to $\mathbb{Q}_2$. We report on the results of counting such extensions for $G$ of order up to $2000$ (except those divisible by $512$), for $p=3,5,7,11,13$. In particular, we highlight a relatively short list of minimal $G$ that do not arise as Galois groups. Motivated by this list, we prove two theorems about the inverse Galois problem for $\mathbb{Q}_p$: one giving a necessary condition for $G$ to be realizable over $\mathbb{Q}_p$ and the other giving a sufficient condition.