Let $K$ be a field with characteristic $p \neq 0$. Let $L:K$ be a cyclic extension s.t $[L:K]=p^s, s>1$. Show that there exists a field $F$ s.t $K \subset F \subset L$, $[F:K]=p^{s-1}$ and $F:K$ is cyclic.
$\textbf{My attempt:}$
Since $L:K$ is galois (because it is cyclic) we have $|Gal(L/K)|=p^s$. Now, by Cauchy's theorem, there is a subgrup H of $Gal(L/K)$ s.t $|H|= p$. Take $F = L^H$, since $L:K$ is galois we have $L:F$ galois and $[L:F] = p$. Therefore, $[F:K]=p^{s-1}$ and since $Gal(L/K)$ is cyclic, we have $Gal(F/K)$ cyclic.
My only question is, how can I prove that $F:K$ is normal??