Let the fraction field of the ring of formal power series over $\mathbb C$ be $\mathbb C((X))$. Let $D$ be the splitting field of $Y^n - X \in \mathbb C((X))[Y]$. Why is it also the splitting field of $Y^n - (X^2+X)$?
I am thinking of proving that $Y^n - (X+1)$ has a root in $D$, then we can show that it is the splitting field for $Y^n - (X^2+X)$.
It's enough to show that $\mathbb{C}((X))$ has an $n$-th root of $1+X$.
One way to do this is using the binomial theorem to expand $(1+X)^{1/n}$.