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