exercise 19.13 in Algebra (Issac)

Question

Let $F\subset E$ be purely inseparable with $|E:F|=n<\infty$. If $\alpha \in E$, show that ${\alpha}^n \in F$.

Answer 1

Hint. If $\text{char}(F)=p>0$, then $n=p^k$ for some $k$.