Let $C(T)$ be a function field in the variable $T$ over an algebraically closed field $C$ of characteristic $0$.
Consider L as 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 know that the inertia group of primes of L above $(T)$ is cyclic with order $lcm(e_1, \dots, e_r)$
My question is:
Is that true for $C$ with $char(C) != 0$ ?
If it doesn't, please help me find some counter example.
Thank's!