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- W2893670668 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." @default.
- W2893670668 created "2018-10-05" @default.
- W2893670668 creator A5051295900 @default.
- W2893670668 date "2019-01-28" @default.
- W2893670668 modified "2023-10-17" @default.
- W2893670668 title "The inverse Galois problem for p-adic fields" @default.
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- W2893670668 doi "https://doi.org/10.2140/obs.2019.2.393" @default.
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