Let $n\geq 2$ be an integer, $D=\mathbb Z[1/n]$, and $A$ a complete commutative ring with unit for the $I$-adic topology, where $I$ is an ideal of $A$. Suppose that $n$ is invertible in $A$. Let $x\in I$. How can I show that there exists a unique continuous homomorphism $\phi:D[[S]]\to A$ such that $\phi(S)=x$? Why do this implies that there exists a $y\in I$ such that $1+x=(1+y)^n$?
Qing Liu: Algebraic Geometry and Arithmetic Curves ex. 1.3.9 b.