$n$th root of a polynomial

Question

Let $n\geq 2$ be an integer, and $D=\mathbb Z[1/n]$. Consider the polynomial $S=(1+T)^n-1\in D[T]$. How can I show that $D[[S]]=D[[T]]$ and there exists an $f(S)\in SD[[S]]$ such that $1+S=(1+f(S))^n$?

Qing Liu: Algebraic Geometry and Arithmetic Curves ex. 1.3.9 a.

Answer 1

For $D[[S]] = D[[T]]$ you should notice $S = nT + \mathcal O(T^2) = T(n+\mathcal O(T))$. Since $n$ is invertible in $D$, the power series $n+\mathcal O(T)$ is invertible, hence $S$ is the product of $T$ and an invertible power series.

For the second assertion, consider $(1+S)=(1+T)^n$. By the first assertion, $T$ can be written as power series in $S$.