Compute a quotient ring

Let K = \mathbb{Q}(\sqrt{-37}) with ring of integers \mathcal{O} = \mathbb{Z}[\sqrt{-37}]. Consider the ideal A = (2,1+\sqrt{-37}). Show that \mathcal{O}/A \cong \mathbb{Z}/(2).


Let a+b\sqrt{-37} \in \mathcal{O}, and suppose a-b \equiv k mod 2 where k \in \{0,1\}. Now a+b\sqrt{-37} \equiv k mod A. In particular, \mathcal{O}/A = \{\overline{0}, \overline{1}\}.

Suppose 1 \in A. Then we have 1 = 2(a+b\sqrt{-37}) + (1+\sqrt{-37})(h+k\sqrt{-37}) for some integers a,b,h,k. Comparing coefficients, we have 2a+h-37k = 1 and 2b + h + k = 0. Mod 2, we have h+k \equiv 1 and h+k \equiv 0, a contradiction. So 1 \not\equiv 0 mod A.

So \mathcal{O}/A \cong \mathbb{Z}/(2).

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