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.

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