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.