The Inverse Galois Problem for p-adic fields

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.

Presented at ANTS 13 (2018)