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
),
for
.
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
to
be realizable over
and the other giving a sufficient condition.
Keywords
p-adic extensions, inverse Galois theory, profinite groups
