TITLE: finding fixed field of automorphism
QUESTION [6 upvotes]: Let $F$ be a field and let $g:F(x) \to F(x)$ be the automorphism which maps $x$ to $x+1$.
I need to find the fixed field of this automorphism.
So far I know $g$ fixes $F$. I want to use Galois applications but the group generated by $g$ is infinite.
Can I have a hint how to start?
Thanks.

REPLY [3 votes]: Hint: a standard way to show that a rational function over a field is constant is to show it has either has infinitely many zeroes (in its algebraic closure), or that it has no zeroes and no poles (in its algebraic closure).
I'm assuming you're in a field of characteristic zero? Otherwise, your automorphism group is finite.