TITLE: Suppose that characteristic $F$ is $p$. If $K/F$ is separable then $K = F(K^{p})$ where $K^{p} = \{ x^{p} \, |\, x\in K\}$.
QUESTION [6 upvotes]: I am having difficulty finishing this problem.  So far I have this:  
Want to show $K \subset F(K^{p})$.  Since $K/F$ is separable then $K/F$ is algebraic.  In particular, $\alpha\in K$ is separable over $F$ hence separable over $F(\alpha^{p})$.  So $\alpha\in F(\alpha^{p})$ by an exercise already done.  But then I do not know how to conclude $\alpha\in F(K^{p})$. Any suggestions?  Or is my approach completely wrong?

REPLY [3 votes]: Let $L=F(K^p)$. Clearly, we have $F\subseteq L\subseteq K$. It is also clear that for every $\alpha\in K$, we have $\alpha^p\in L$. This shows that the extension $K/L$ is purely inseparable (see characterization 2 here). But $K/F$ separable implies that $K/L$ is separable (see the second bullet here) and because $K/L$ is both separable and purely inseparable, we have $K=L$.