#### Vol. 2, 2019

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The inverse Galois problem for $p$-adic fields

### David Roe

Vol. 2 (2019), No. 1, 393–409
##### Abstract

We describe a method for counting the number of extensions of ${ℚ}_{p}$ with a given Galois group $G$, founded upon the description of the absolute Galois group of ${ℚ}_{p}$ due to Jannsen and Wingberg. Because this description is only known for odd $p$, our results do not apply to ${ℚ}_{2}$. 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 ${ℚ}_{p}$: one giving a necessary condition for $G$ to be realizable over ${ℚ}_{p}$ and the other giving a sufficient condition.

##### Keywords
p-adic extensions, inverse Galois theory, profinite groups
##### Mathematical Subject Classification 2010
Primary: 12F12
Secondary: 11S15, 11Y40, 12Y05, 20C40