Thirteenth Algorithmic Number Theory Symposium ANTS-XIII University of Wisconsin, Madison July 16 – 20, 2018 Thirteenth Algorithmic Number Theory Symposium (ANTS-XIII) July 16 – 20, 2018

The inverse Galois problem for p-adic fields

David Roe

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 Q₂. 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.

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