I am interested in finding the number of prime ideals in $\mathbb{Z}[x]/(12,x^2+1)$.
Here is what I think. Modding out by $x^2+1$, we get $\mathbb{Z}[i]/(12)$. Factoring $12$ in the Gaussian integers, we get $3(1+i)^2(1-i)^2$. Then we have four primes $(0)$, $(1+i)$, $(1-i)$ and $(3)$. Is this about right?