Let $n\in\mathbb{N}$ and $f\in\mathbb{Z}_n[x]$ a non-constant, univariate, integer polynomial $\mod n$. Let $(f)$ be the ideal generated by $f$ in $\mathbb{Z}_n[x]$. Then, the quotient ring $\mathbb{Z}_n[x]\big/(f)$ is a finite ring.
I was wondering, how can we count the number of its elements?
Take for example, $\mathbb{Z}_7[x]$ and $f(x)=3x^2+1$. How many elements does the ring $\mathbb{Z}_7[x]\big/(3x^2+1)$ have?
P.S.: there is a relevant question here