Let $k,F$ be two fields with $char\ k = p >0$. Prove that an algebraic element $u \in F$ is separable over $k$ iff $k(u) = k(u^{p^n})$ $\forall n \in \mathbb{N}$.
Again, still studying for my algebra final. After this problem, I think I will be prepared adequately. I've tried to find something similar in my book but I've had no luck. Thanks for the help in advance.
Let $u$ be separable. Then the extension $k(u)/k(u^{p^n})$ is separable AND purely inseparable, hence trivial.
On the other hand let $k(u)=k(u^{p^n})$ for all $n$, in particular $k(u)=k(u^p)$. If $u$ is not separable, its minimal polynomial over $k$ is of the form $f(x^p)$. But then $f(x) \in k[x]$ is a polynomial, which annihilates $u^p$ and has smaller degree then $f(x^p)$. This is a contradiction to $k(u)=k(u^p)$.