Let $K$ be a field oh characteristic $p$. Let's take $\sigma \in \operatorname{Aut}(K(x),K)$ where $x$ is trascendental over $K$, where $\sigma(x)=x+1$. Find a primitive element of the fixed field of $ \left\langle {\sigma} \right\rangle $.
I have no idea how to attack this problem. Maybe one step it's to note that $ \left\langle {\sigma} \right\rangle $ is finite, in fact has order p (the characteristic of the field). I was trying with particular cases to note something general, but I could not find even one element fixed by the automorphism...