Let $E|K$ and $E'|K$ be splitting fields of a polynomial $f \in K[X]$. Prove that if $E$ is contained in a radical extension of $K$, then so is $E'$.
My "proof":
$E \cong K(R(f))$, where $R(f)$ denotes the set of roots in some closure. Splitting fields are unique up to isomorphism. Therefore also $E' \cong K(R(f))$. By assumption $K(R(f)) \subset F$ for some radical extension $F|K$. And we're done.
Clearly there must be something wrong and I feel very confused myself by my solution.
I get very confused by questions about splitting fields and I'm unsure if that's the right way to think of them.
I'd really appreciate any help.