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