Let $d \in \mathbb{Z}$ be a square free integer. $R=\mathbb{Z}[\sqrt d]$ =$\{a+b\sqrt d | a,b \in \mathbb{Z} \}$. Overall, I'm trying to show that every prime ideal $P \subset R$ is a maximal ideal.
So far I showed that $I \subset R$ is finitely generated. $I=\{(x,s+y \sqrt d)\}$ And now I'm trying to show that R/P, for some prime ideal is a finite ring with no zero divisors. From there it would follow that R/P is a field and any prime ideal is maximal.
I know R could also be written as $R= \mathbb{Z}[x]/(x^2-d)$. How could I show that R/P is a quotient of $\mathbb{Z}/n\mathbb{Z}[x]/(x^2-d)$?