Let $F<E$ be a finite extension that is not separable.
Show that for each $n\geq 1$, there exists a subfield $E_n$ of $E$ for which $E_n<E$ is purely inseparable and $[E:E_n]_i=p^n$ ($[...]_i$ means inseparable degree).
I know that we can divide the extension into two part: separable and purely inseparable part. We need to form subfields $E_1,...,E_n$ such that
$[E:E_1]_i=p$, $[E:E_2]_i=p^2$,....
But how?