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Séminaire de Théorie Algorithmique des Nombres

The inverse Galois problem for p-adic fields

David Roe

( MIT )

Salle 2

10 septembre 2019 à 09:30

We describe a method for counting the number of extensions of Qp\mathbb{Q}_p with a given Galois group GG, founded upon the description of the absolute Galois group of Qp\mathbb{Q}_p due to Jannsen and Wingberg. Because this description is only known for odd pp, our results do not apply to Q2\mathbb{Q}_2. We report on the results of counting such extensions for GG of order up to 20002000 (except those divisible by 512), for p=3p = 3, 5, 7, 11, 13. In particular, we highlight a relatively short list of minimal GG that do not arise as Galois groups. Motivated by this list, we prove two theorems about the inverse Galois problem for Qp\mathbb{Q}_p: one giving a necessary condition for G to be realizable over Qp\mathbb{Q}_p and the other giving a sufficient condition.