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 p with a given Galois group G, founded upon the description of the absolute Galois group of p due to Jannsen and Wingberg. Because this description is only known for odd p, our results do not apply to 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 p: one giving a necessary condition for G to be realizable over 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
Milestones
Received: 2 March 2018
Revised: 17 June 2018
Accepted: 23 September 2018
Published: 13 February 2019
Authors
David Roe
Department of Mathematics
MIT
Cambridge, MA
United States