If $F/L$ is normal and $L/K$ is purely inseparable, then $F/K$ is normal

Question

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 $\alpha \notin L$.

Answer 1

Hint. Take a set of polynomials $f_i\in L[x]$ such that $F$ is generated by the roots of these polynomials. Since $L/K$ is purely inseparable, how can you make the $f_i$ into polynomials in $K[x]$ without affecting their roots?

You might find the fact that if $\text{char}(K)=p>0$, then $x\mapsto x^p$ is an endomorphism of $K$ (called the Frobenius endomorphism) useful.

Answer 2

Take a $K$ - homomorphism $T$ from the $F$ to the algebraic closure $M$ of $F$, we want to show that $T(F)=F$,(check this equivalence of normality in morandi's book) take an element "a" in $L$, so take then an irreducible polynomial in $K$ that solves "a" we say $f(x)$, then $f(T("a))=O.$ and since $L/K$ is Purele Ins. T("a")="a", in conclusion T(b)=b for every b in L, so T is really an L - homomorphism, since F/L is normal T(F)=F.