Let field extensions $K \subset L \subset F$ such that $F/L$ is normal extension, and $L/K$ is purely inseparable extension. Show that $F/K$ is normal extension.
My strategy: let $f(x)\in K[x]$ is irreducible polynomial, and $\alpha \in F$ is solution of $f(x)=0$. So if $\alpha \in L$ then $f(x)=0$ has only solution $\alpha$. Thus $f(x)$ split on $F$,then $F/K$ is normal.
PS: I don't what should I do when $\alpha \notin L$.