Suppose I have the following problem
Where $A=\mathbb{C}[[z]]$ and $I^n=z^nA$, and $S_n=A/I^n$. $f$ is a monic polynomial of $\deg(f)=d$ and furthermore $\bar{f}=ev_*(f)$ is a product of polynomials $\bar{g},\bar{h} \in \mathbb{C}[x]$ which are coprime with $\deg(\bar{g})=r$ and $\deg(\bar{h})=d-r$ where $ev_*: A[x] \rightarrow \mathbb{C}[x]$ is the map coming from the map $ev: A \rightarrow \mathbb{C}$, sending a formal power series $a(z)$ to $a_0$ $ev(a(z))=a_0$.
The first part in (2) of showing it for $n=0$ is easy enough, but the inductive step I feel is rather unwieldy since I don't know how to define the $a_n(x),b_n(x)$. I know the maximal ideal of $S_n$ is $I/I^n$ for all $n$, but I can't see where I am supposed to use that fact.
Errata: in the above attached picture $S_{n+1}$ should be $S_{n+1}[x]$.
I solved the problem myself by using more or less the method described here