I'm having trouble with this question: Find the splitting field of
over
and find all the intermediate fields of the extension.
The splitting field is
where
. Now this extension has degree
since
and
. By the isomorphis extension theorem there are
automorphisms such that
and
and
from this we know that
. Now the fixed field of
is
and the fixed field of
, but I'm having trouble with the other subgroups
,
,
,
. I know they must be degree
extensions of
, but using the trace I get either trivial fixed elements or ones that I don't know how to check their degree over
.
Tnaks in advance