# Subfield of a compositum of all $p$-extension is also a $p$-extension

Definition: A $p$ extension $K/F$ is a finite Galois extension whose Galois group is of order $p$-powers, $p^a$

Fix a prime $p$, fix a algebraic closure of $F$, call it $\overline{F}$

Let $L$ be the compositum of all $p$-extension of $F$

Show that any sub-field of $L$ of finite degree and galois is a $p$-extension.

I already showed 2 key lemma

1.) Composite of 2 $p$-extension is a $p$-extension

2.) Any $\alpha \in\ L$ is contained in some $p$-extension

But I'm having some difficulties showing the desired conclusion.

Any help or insight is deeply appreciated.

• Are you familiar with the primitive element theorem? – user263732 Oct 30 '17 at 14:58

## 1 Answer

By primitive element theorem for every finite Galois extension $L/F$; there is an element $\alpha\in L$ such that $L=F(\alpha)$. You already proved that $L$ is inside a $p$-extension. Now, by Galois Correspondence $L$ must be a $p$-extension (since subgroup of a $p$-group is a $p$-group).