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 [itex] F \subseteq K \subseteq E [/itex], K is normal over F and E is subradical over K.
[One needs the following result: Let [itex]A[/itex] and [itex]B[/itex] be solvable subgroups of a group [itex]G[/itex] and suppose that [itex]A[/itex] is normal in [itex]G[/itex]. Then [itex]AB[/itex] is solvable.]

b. Let f be an irreducible polynomial over [itex]\mathbb{Q}[/itex] which has degree 5 and at least two complex roots. Prove that [itex]Gal(f)[/itex] 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.