Let $K/E/F$ be a tower of fields such that $K/F$ is finite and $E/F$ is Galois. I want to prove that if $K/E$ is radical, then there exists an extension $L/K$ so that $L/F$ is Galois and $L/E$ is radical.
I have previously seen that, if our initial assumption is that $K/E$ is simple radical, then the conclusion holds.
My thought is that I should be able to use this result as a base case and proceed by induction on the length of the radical tower from $K$ down to $F$ -- a radical tower $K = K_m / K_{m-1} / \cdots / K_1 / K_0 = F$ is a tower of fields where $K_{i+1}/ K_{i}$ is simple radical for each $i$.
Unfortunately, I can't figure out how to put everything together. Does this idea work? How do the pieces come together?