Galois extension of a field with Characteristic 0

[bham10246]

This is something I've been trying to work on on my own for the past few days but I'm not sure how to approach it.

My Question:
a. Let E be a Galois extension of a field F with characteristic 0. Prove that there is a unique smallest subfield K such that F \subseteq K \subseteq E, K is normal over F and E is subradical over K.
[One needs the following result: Let A and B be solvable subgroups of a group G and suppose that A is normal in G. Then AB is solvable.]

b. Let f be an irreducible polynomial over \mathbb{Q} which has degree 5 and at least two complex roots. Prove that Gal(f) has order 10, 20, 60, or 120.

Thanks in advance for any kind of direction that you can provide me with...

 

[mathwonk]

what does smallest mean? the intersectiuoin of all such thigns? or smallest degree?

and are your extensions of finite degree?

so i guess you are claiming there is a unique largest normal solvable subgroup of a group. your description majkes it fairly obvious how to proceed.