Suppose $k$ is a field, and $k(t)$ is the rational function field. If $f(t)=P(t)/Q(t)$ for some polynomials $P(t)$ and $Q(t)\neq 0$, then the map $t\mapsto t+1$ sends $f(t)$ to $f(t+1)$.
So the fixed field for this map the set of rational functions $P(t)/Q(t)$ such that $ \frac{P(t)}{Q(t)}=\frac{P(t+1)}{Q(t+1)}\text{ or }P(t)Q(t+1)=P(t+1)Q(t). $ This does not seem awfully descriptive. Is there a more explicit description of what the fixed field looks like, (perhaps in terms of the form of $P$ and $Q$?), in order to get a better handle on it?
Later: With the wonderful help of Professor Suárez-Alvarez, I understand that the $\sigma$-invariant polynomials in $k[t]$ have all factors of form $\phi_\alpha$ where $\phi_\alpha=\prod_{i=1}^{p-1} (t-\alpha-i)$ for $\alpha$ a root of $f$ in $k$ when $k$ is algebraically closed.
What happens when $k$ is not algebraically closed? I think you can still factor out polynomials of form $\phi_\alpha$ for $\alpha$ roots of $f$ in $k$. But if $f$ does not split completely then there will be some $\sigma$-invariant factor with not roots in $k$ in the factorization of $f$. Is there anything more one can say to describe this remaining part? Or is that about the best we can do? Thank you for your time.