# Galois extension of a field with Characteristic 0

1. Aug 6, 2007

### 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...

2. Aug 7, 2007

### 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.

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