I'm trying to find a source for the proposition from the title: if $L : K : F$ is a tower of field extensions, and furthermore both $L$ and $K$ are finitely generated over $F$, then $L$ is finitely generated over $K$.
This statement does not seem to be in Dummit and Foote, and I have had no luck finding a proof online. Does anybody have a reference, source, or proof for it?
Well, I figured out a proof. It turns out it's way simpler that I thought it would be, and I didn't think about it enough before posting the question.
If $L : F$ is finitely generated, then by definition that means that there are some $\alpha_1, \ldots, \alpha_s \in L$ such that $L$ is the smallest field (by inclusion) containing $F$ all the $\alpha_i$. Now consider some intermediate field $K$ (so $L : K : F$ is a tower of extensions). Certainly $L$ contains $K$, and $L$ contains all the $\alpha_i$, and if $L$ was not the smallest field containing $K$ and the $\alpha_i$, then it would also not be the smallest field containing $F$ and the $\alpha_i$, since $F \leq K$, which is absurd since $L$ is finitely generated over $F$. Therefore $L$ is finitely generated over $K$.