If $K$ is a nonarchimedian field with integer ring $O_K$ whose maximal ideal is $\mathfrak{m}$ and residue field $k$. Let $\pi \in \mathfrak{m}$ (not necessarily a uniformizer as we are not assuming $O_K$ is a DVR) and $f, g \in O_K[X]$ are such that there exists $a, b \in O_K[X]$ $af+bg-1 \in \mathfrak{m}[x]$, why is $(O_K/\pi)[X]/(f,g)$ finitely generated as an $O_K/\pi$-module?
I'm really trying to understand this proof (from notes by Richard Crew on local class field theory) so any explanation or alternative proof about it is welcome.
If you meant that $K$ has a real valued non-archimedian valuation $v$ and $O_K=\{s\in K,v(s)\ge 0\}$ then let $w(\sum_{n=0}^d c_n x^n)=\inf_n v(c_n)$, note that $w(uv)\ge w(u)+w(v)$ (in fact it is an equality).
$af+bg-1 \in \mathfrak{m}[x]$ gives that $w(af+bg-1)>0$ so there is some $r$ such that $w((af+bg-1)^r)\ge v(\pi)$ ie. $(af+bg-1)^r\in (\pi)$.
Thus $$O_K/(\pi)[x]/(f,g)=O_K/(\pi)[x]/(f,g,(af+bg-1)^r)=O_K/(\pi)[x]/(f,g,1)=0$$