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
Okay, I think I got it:
The fixed field of . This is easily seen to be fixed by and since it cannot be a root of a degree polynomial over , it's minimal polynomial must have degree (since otherwise it would generate the whole extension which cannot be).
Proceeding in almost the same way I get:
Fixed field of and .
Is this right?