Salle 2
10 septembre 2019 à 09:30
We describe a method for counting the number of extensions of
with a given Galois group
, founded upon the description of the absolute Galois group of
due to Jannsen and Wingberg. Because this description is only known for odd
, our results do not apply to
. We report on the results of counting such extensions for
of order up to
(except those divisible by 512), for
, 5, 7, 11, 13. In particular, we highlight a relatively short list of minimal
that do not arise as Galois groups. Motivated by this list, we prove two theorems about the inverse Galois problem for
: one giving a necessary condition for G to be realizable over
and the other giving a sufficient condition.