Can Someone help me solve the following problem?
Let C((T)) be the field of formal Laurent series in the variable T over an algebraically closed field C of characteristic $0$.
(1) Prove that the algebraic closure of C((T)) is $\bigcup_{n=1}^{\infty}$ C(($T^{1/n}$)) and deduce that for any $n \ge 1$, the field C((T)) has a unique extension of degree $n$.
(2) Let $f(X) \in C[X]$ be a polynomial such that $f(X) = \prod_{i=1}^{r}(X-\alpha_{i})^{e^{i}}$ with $\alpha_1, \dots , \alpha_r \in C$ distinct and $e_1, \dots , e_r \ge 1$. Prove that the inertia group of $F(T, X) = f(X)−T$ over C(T) at the prime T is generated by a permutation $\sigma$ of cycle type $(e_1, \dots , e_r)$.
(3) Give a counterexample to (2) when C is of positive characteristic $p$ (one must have $p \mid e_i$ for some $i$).
Edit:
This is my outcome so far
(1) Found a proof here
(2) I think it somehow related to the fact that the decomposition group is isomorphic to the Galois group of the completion: $D(Q/P) \cong Gal(L_Q | K_P)$. And that the $D(Q/P) = I(Q/P)$ since $\bar{K_P}$ is algebraically closed.
(3) Haven't found anything yet