# Galois closure of a $p$-extension is also a $p$-extension

I'm working on a problem in Dummit & Foote and I'm quite stumped. The problem reads:

Let $p$ be a prime and let $F$ be a field. Let $K$ be a Galois extension of $F$ whose Galois group is a $p$-group (i.e., the degree $[K:F]$ is a power of $p$). Such an extension is called a $p$-extension (note that $p$-extensions are Galois by definition).

Let $L$ be a $p$-extension of $K$. Prove that the Galois closure of $L$ over $F$ is a $p$-extension of $F$.

This is what I've done so far:

Using the tower law we can readily show that $L$ is a $p$ extension of $F$ so we have $[L:F]=p^{\ell}$ for some integer $\ell$. Then if $M$ is the Galois closure of $L$ over $F$ then $$[M:F]=[M:L][L:F]$$ and therefore $[M:F]=p^{\ell}n$ for some integer $n$ that is not divisible by $p$. So $[M:L]=n$.

From here it seems like I want to show that either $n=1$ or that $n$ is in fact a power of $p$. I just don't see how to proceed. I've considered using the Sylow Theorems, but I'm not sure how that would really work. I also realize that this statement depends on $K$ being Galois over $F$ but I can't figure out how to take advantage of that.

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I don't want to spoil it for you, but if you have some self-restraint, there's a brief solution to this problem on this old CIT coursepage. –  Buble Apr 20 '12 at 2:56
Using the Galois correspondence, this problem translates into a group theory problem. Let $1 < A \lhd B \lhd C$ be groups such that $B/A$ and $C/B$ are finite $p$-groups for some prime $p$. Then the core of $A$ in $C$ (that is, the intersection of the $C$-conjugates of $A$) has index a power of $p$ in $B$. This is true because the core is the intersection of finitely many normal subgroups of $B$, each having index a power of $p$ in $B$. –  Derek Holt Apr 20 '12 at 12:36
@DerekHolt, You should post your comment as an answer. –  JSchlather Jan 25 '13 at 7:00