If $[L:F]$ is prime, show that either $M\supseteq L$ or $M\cap L=F$.
$M \cap L$ is a field between $F$ and $L$, and since $[L : F]$ is prime, $M \cap L$ has to be either $F$ or $L$. But saying $M \cap L = L$ is the same thing as saying $L \subseteq M$.