## Compute ZZ[sqrt(-5)]/(3)

Compute $\mathbb{Z}[\sqrt{-5}]/(3)$.

By Corollary 9.11, $|\mathbb{Z}[\sqrt{-5}]/(3)| = N((3)) = N(3) = 9$.

Suppose $a+b\sqrt{-5} \in \mathbb{Z}[\sqrt{-5}]$, and suppose further that $a \equiv a_0$ and $b \equiv b_0$ mod 3, where $a_0,b_0 \in \{0,1,2\}$. Then $a+b\sqrt{-5} \equiv a_0+b_0\sqrt{-5}$ mod $(3)$. That is, $\mathbb{Z}[\sqrt{-5}]/(3) = \{\overline{a+b\sqrt{-5}} \ |\ a,b \in \{0,1,2\}\}$. Since we know that this ring has 9 elements, none of these cosets are equal.