This is my question: Suppose the $K/F$ is a Galois extension with an abelian Galois group $G$. Prove that every intermediate field $L: F \subseteq L \subseteq K$ is the splitting field (over $F$) of some separable polynomial $f \in F[x]$.
I would like help in identifying which facts or theorems to use. Namely, I was thinking that I could prove that $L/F$ is Galois, thus a splitting field. I would use the FTGT in order to do so. Such $L$ (should it exist) must correspond to an abelian subgroup because $G$ is abelian. But, beyond that, I am not sure how abelianity helps at all (especially with the separability of $f$), or how even best to organize these thoughts into a coherent proof. Even an ordered list of properties and consequences would be nice.