I am trying to prove the following exercise:
If $E/k$ and $E'/k$ are splitting fields of $f(x)\in k[x]$ and there is a radical extension $K_t/k$ with $E\subset K_t$, prove that there is a radical extension $K_r'/k$ with $E'\subset K_r'$.
Since splitting fields are isomorphic, I know I need to use an isomorphism between $E$ and $E'$ to construct a radical extension for $E'$ from the radical extension of $E$. But, I have no idea how to proceed. Any hints?