# Fundamental theorem of Galois Theory problem

Let $E/F$ be a Galois extension of degree $p^k$. Prove that there exists an intermediate field $K$ with $[E:K] = p$ and $K/F$ Galois of degree $p ^{k−1}$.

I think I know how to prove the former but I don't think I can write it down as a formal argument. And using tower law we have the degree of $K/F$. However, I couldn't see why it has to be Galois (separability is obvious but why is it normal?)? Could anyone help with the proof please? Thanks.

• Translate to a statement about groups using the fundamental theorem of galois theory. This is really a statement about p groups. May 20, 2016 at 17:03
• The Galois group is a finite $\;p\,-$ group, and such groups have normal subgroups of any order dividing the group's. May 20, 2016 at 17:04

Let $G=\mathrm{Gal}(E/F)$. $G$ is a $p$-group, hence has a non-trivial center. Since the center $Z(G)$ is a non-trivial $p$-group, we can choose a subgroup $H\leq Z(G)$ of order $p$.
Let $K=E^H$ be the fixed field of $H$. Then $[E:K]=|H|=p$. Moreover, $H$ is a normal subgroup of $G$ since it is contained in $Z(G)$, so $K/F$ is a Galois extension with $\mathrm{Gal}(K/F)\simeq G/H$.