Exercise from Cox's book Galois Theory.
Let $F \subset L$ be the splitting field of $f(x) \in F[x]$, and let $K$ be a field such that $F \subset K \subset L$. Prove that $L$ is the splitting field of some polynomial in $K[x]$.
So let $\alpha_i$ be the roots of $f(x)$ in $L$. Then I'm guessing I need to show that $L$ has the form $K(\alpha_{i_1}, \dots, \alpha_{i_m})$, and then conclude that $L$ is the splitting field of a polynomial like $g(x)= c(x-\alpha_{i_1}) \dots (x- \alpha_{i_m})$. But how can I show that $L$ has the form I want, and also how can I be sure that $g(x)$ has coefficients in $K$?