Can one characterize the $a\in\mathbb F\_q\left(\left(\frac1T\right)\right)$ such that $a(T+1)=a(T)$? Although this seems elementary, I did not manage to find a answer.
Thanks in advance for any help.



Equality in $\mathbb F_q\left(\left(\frac1T\right)\right)$
This is a field automorphism of order $p$, since $T+p=T$, so it fixes an index $p$ subfield. $\mathbb F\_q (( \frac{1}{T^p - T } ))$ is an index $p$ subfield and is fixed, hence is the fixed field. This is the simplest example of a wildly ramified extension of local fields in equal characteristic, and is in particular an Artin-Schreier extension, generated over $\mathbb F\_q((u))$ by the equation $T^p-T = u^{-1}$ where $u = \frac{1}{T^p - T}$.


