Let $K/F $ be a separable extension with char $F =p > 0$. Prove that for any given $\alpha \in K , \alpha \in F (K^p) $.
I am not sure how to approach this problem. I think Viete's formula is too comolicated, if not useless; this problem is from field theory anyway. If $K $ is finite then $K=K^p $ and we're done.
EDIT: Here is my solution: let $\alpha \in K $. Then $x^p-\alpha^p = (x-\alpha)^p$ by the binomial formula and the fact that $p $ choose $k $ is divisible by $p$ for $k = 1,2,...,p-1$. Hence the minimal polynomial of $\alpha$, denoted $f \in F [x]$, divides $(x-\alpha)^p$ , and since the extension is separable $f $ must be $x-\alpha $. As $f \in F [x] $ it follows that $\alpha \in F \subset F (K^p) $.
Here is now my updated question: The same proof shows that $K \subset F $, and so $K=F $ if $K/F $ is a separable extension and also $F=K^p $. But I don't think this is true. Is something wrong in my proof?