Let p be a prime and let $\displaystyle F$ be a field.
Let $\displaystyle K/F, L/K$ be p-extensions.
Then the Galois closure of $\displaystyle L$ over $\displaystyle F$ is a p-extension of $\displaystyle F$.
($\displaystyle A$ is called a p-extension of $\displaystyle B$ if $\displaystyle A$ is a Galois extension of $\displaystyle B$ whose Galois group is a p-group.)