Let $E/F$ be a finite extension and $L \supseteq E$ an splitting field of a polynomial $g \in F[x]$ such that every irreducible factor of $g$ in $F[X]$ has a root in $E$. I need to prove $L/F$ is a norma closure (doesn't exists normal intermediate extensions).
Since $L$ is splitting field of a polynomial en $F[x]$ it follows that $L/F$ is normal.
Now if $F \subseteq K\subseteq L$ is a field such that $K/F$ is normal I want to prove $K=L$ but I don't know how to, any suggetion would be appreciated. Thanks