Finding Galois group of function field extension

Question

Let $C((T))$ be the field of formal Laurent series in the variable $T$ over an algebraically closed field $C$ of characteristic $0$.

I need to find to the Galois group of the splitting field for the following polynomial:

$F(X,T) = f(X)-T$

for $f(X) = \prod_{i=1}^r(X- \alpha_i)^{e_i}$

$\alpha_i \in C$

$e_i >= 1$

I've read somewhere that the extension degree is $lcm(e_1, \dots, e_r) $ and the Galois group is cyclic but I can't find any proof.

Thank's :)