Finite Purely Inseparable Extension

Question

Let $F\subseteq E$ be purely inseparable with $|E:F|=n$. If $\alpha\in E$ show that $\alpha ^n\in F$ ?

HINT

If $ α∉F $ show that $|F[α]:F[α^p ]|=p$, where $\mathrm{char}(F)=p$.

We know that $α^{p^m }\in F$ for some integer $m≥0$....and we know that $n$ is a power of $p$, but how I can show that $p^m=n$?

Thanks.

Answer 1

Suppose $k\geqslant 1$ is such that $\alpha^{p^k}\not\in F$. Then you can apply your hint to each extension $F[\alpha^{p^i}]/F[\alpha^{p^{i+1}}]$ with $i<k$, which gives $[F[\alpha^{p^i}]:F[\alpha^{p^{i+1}}]]=p$, so by induction $[F[\alpha]:F[\alpha^{p^k}]]=p^k$.

You should be able to conclude from there.