What degrees are possible for minimal polynomials? My friend is working on a question of the form:

Suppose $E/F$ is a Galois extension with Galois group isomorphic to $G$ (a finite group). Find the set of possible degrees of elements of $E$ over $F$. (The field $F$ and the group $G$ are given, but $E$ are unknown. The extension has characteristic $0$.)

He's not really sure how to approach it, and neither am I. If I understand correctly, a degree of an element $e \in E$ over $F$ is the same as the degree of the minimal polynomial of $e$ over $F$. If so, we're trying to put some contraints on the possible minimal polynomials of a Galois extension based on the structure of the Galois group of that extension. I'm not really sure how to do this however, and we'd appreciate some help.

Question. How does one approach questions like this in general?

 A: My previous answer was very far off. Here's another answer, which is a more detailed version of Marc Paul's comment to my previous answer.
By the fundamental theorem of Galois theory there is a bijective correspondence between the set of intermediate fields of the extension $E/F$ and the set of subgroups of $\operatorname{Gal}(E/F)$. It is given explicitly by mapping an intermediate field $F\subset L\subset E$ to the subset of elements of $E$ that fix every element of $L$, i.e. to $\operatorname{Aut}(E/L)$, and conversely by mapping a subgroup $H\subset\operatorname{Gal}(E/F)$ to the set of elements of $E$ that are fixed by all elements of $H$, denoted $E^H$.
It is then a routine exercise to verify that for any subgroup $H\subset\operatorname{Gal}(E/F)$ the intermediate field $F\subset E^H\subset E$ satisfies $[E:E^H]=|H|$. By the tower law it follows that
$$[E^H:F]=\frac{[E:F]}{E:E^H}=\frac{|\operatorname{Gal}(E/F)|}{|H|},$$
where the latter is the index of $H$ in $\operatorname{Gal}(E/F)$. Hence the degrees of the intermediate fields of $E/F$ are precisely the indices of the subgroups of $\operatorname{Gal}(E/F)$. By the primitive element theorem, every intermediate field of $E/F$ is of the form $F(\alpha)$ for some $\alpha\in E$, and the set of degrees over $F$ of elements of $E$ is also precisely the set of indices of subgroups of $\operatorname{Gal}(E/F)$.
