I was wondering how one would determine the galois group of the extension $\mathbb{F}_p(t,s)$ of $\mathbb{F}_p(t)$, where $s^6 = t + 1$? I certainly know (and Magma reassures) that this depends on the prime $p$ being congruent to 1 modulo 6 or not. If $p \equiv 1 \mod 6$, then $\mathbb{F}_p$ contains a sixth root of unity, so we for example have the automorphism sending s $\to$ $\xi_6s$ and t $\to$ t right? Does this generate the entire galois group?
I am quite rusty with Galois theory over function fields, so any hints\help would be greatly appreciated!