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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.

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(Below, "extension" means "finite extension".)

(1) Yes, $L/F$ is Galois. Intermediary extensions of a Galois extension correspond to subgroups of the Galois group, and those sub-extensions which are themselves Galois over the base $F$ correspond to normal subgroups. In an abelian group, every subgroup is normal.

(2) An extension is Galois if and only if it is normal and separable. In particular, $L/F$ is separable. By the primitive element theorem, there exists an $\alpha\in L$ such that $L=F(\alpha)$.

(3) Show that $L$ is the splitting field of the minimal polynomial of $\alpha$ over $F$.

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  • $\begingroup$ In (2), $L/K$ or $K/L$, or $L/F$? $\endgroup$ – kevin Apr 2 '14 at 0:19
  • $\begingroup$ @kevin Thanks, that was a typo. Fixed. $\endgroup$ – Bruno Joyal Apr 2 '14 at 0:21
  • $\begingroup$ Ah, the big thing that I was missing was the normalcy of the subgroups. That all was very helpful! Thank you so much. $\endgroup$ – kevin Apr 2 '14 at 0:29
  • $\begingroup$ @kevin My pleasure! $\endgroup$ – Bruno Joyal Apr 2 '14 at 0:30

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