Here is the problem I have:
Let E be a splitting field of f = x^4 - 4 in Q[x]
1. determine the galois group of E/Q (up to isomorphism)
2. Determine all intermediate fields between E and Q. Which of them are Galois over Q?
so far I have this:
letting a = 2^(1/2) then the roots are +/-a and +/- ai
I know Q[a] < E and |Q[a]:Q| = 2 (minimal polynomal is x^2 - 2) and ai is not in Q[a].
Letting E = Q[a, i] we have all roots of f in E and |Q[a, i]: Q[a]| = 2 since minimal polynomial is x^2 + 1.
thus |E| = |Q[a, i]: Q[a]||Q[a]:Q| = 4
from here I start drawing a big blank. Any help is appreciated.