Why p-cyclic extension iff $p^m$-cyclic extension $\forall m$

Question

A theorem is stated as follows. For a field $F$ of characteristic $p$, $F$ has a $p$-cyclic extension if and only if for every positive integer $m$, $F$ has a $p^m$-cyclic extension. I wonder if there is an elementary proof of it, without too much discussion on Abelian extension theories. All I am acquainted with is basic Galois group and group theory.