I'm aware that this question has been answered here. However, I wanna try and prove this directly:
Let $L/K$ be a finite field extension, suppose $L$ is perfect and $\text{char}(K) = p > 0$. Show that $L/K$ is separable. (Hint: Show that there exists an $n \in \mathbb{N}$ s.t $x^{p^n} \in K_s = \{a \in L\; |\; a\; \text{is separable over}\; K\}$ for every $x \in L$.)
Since $L$ is perfect, every $x \in L$ has a $p$-th root in $L$. I don't see how to prove the hint though. Any help is greatly appreciated!
Best Answer
For $K(a)/K$ a finite extension in characteristic $p$ let $$f(x)=\prod_j (x-a_j)^{p^{e_j}}\in K[x]$$ be its minimal polynomial and $F$ the splitting field. There is an automorphism $\sigma_j \in Aut(F/K)$ sending $a_1\to a_j$ thus $e_1=e_j$ so that $$f(x)=g(x)^{p^{e_1}}, \quad g(x)=\prod_j (x-a_j)=\sum_{n=0}^d c_n x^n$$ $$f(x)=\sum_{n=0}^d c_n^{p^{e_1}} x^{p^{e_1}n}, \qquad \sum_{n=0}^d c_n^{p^{e_1}} x^{n}=\prod_j (x-a_j^{p^{e_1}}) \in K[x]$$ Whence $K(a^{p^{e_1}})/K$ is separable.
Going back to your extension, if $n= [L:K]$ then $K(L^{p^n})/K$ is separable.
But $L=L^{p^n}$ so $K(L^{p^n})=L$ and $L/K$ is separable.