Let $q$ be a power of a prime $p$ and $K=\mathbb F_q\left(\left(\frac1T\right)\right)$ and consider the following automorphism group $G$ of $K$ defined by $G=\{K\to K;\,T\mapsto T+\xi\mid \xi\in\mathbb F_q\}$. I try to determine the subfield of $K$ invariant by $G$. For that, I consider a formal power series $F(T)=\sum_{n\le m}a_nT^n$ with $m\in\mathbb N$ and the $a_n$'s in $\mathbb F_q$. Then, one has $$\sum_{n\le m}a_n(T+\xi)^n=\sum_{n\le m}a_nT^n$$ for every $\xi\in\mathbb F_q$. One deduces that $a_{n-1}=0$ or $n=p$. But equations become harder and harder. So I am stuck
Can anyone know $K^G$?