Let $F = C(T), R = C[T]$ ie, $F$ is the fraction field of the ring, $R$. $C$ is the complex numbers. Let p be prime.
Let $y$ be a non-trivial p-th root of unity. Choose $a$ and $b$ in the algebraic closure of $F$ such that $ap = T$ and $bp = (1-T)$. Let $E = F(a,b)$ e.x joining the binary set of roots to the field.
How do you determine the isomorphism class of the Galois group of $E/F$?