# existence of purely inseparable extension

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?

• If $E/F$ is a finite extension, then I don't see how there can be one subfield $E_n \subset E$ for each $n \geq 1$ such that $[E_n : E] = p^n$. Unless you mean that the $E_n$'s are not necessarily intermediate fields? – Brahadeesh Jun 1 at 17:04