## Compute a quotient ring

Let $K = \mathbb{Q}(\sqrt{-5})$ with ring of integers $\mathbb{Z}[\sqrt{-5}]$. Consider the ideals $A = (3,1+\sqrt{-5})$ and $B = (2)$. Compute $\mathcal{O}/AB$.

We showed in this previous exercise that $\mathcal{O}/A = \{\overline{0}, \overline{1}, \overline{2}\}$ and that $\mathcal{O}/B = \{\overline{0}, \overline{1}, \overline{\sqrt{-5}}, \overline{1+\sqrt{-5}}\}$.

Clearly $((2), AB) = B$. By the proof of Theorem 9.14, $\mathcal{O}/AB = \{\overline{2\alpha+\beta} \ |\ \alpha \in \{0,1,2\}, \beta \in \{0,1,\sqrt{-5},1+\sqrt{-5}\}\}$, and these cosets are distinct.