What is the order of $\mathbb{Z}[\sqrt{-5}] /(2)$

Question

I've been trying to figure this out. The hint is that it isomorphic to some group that is not a field. I keep on getting that $\mathbb{Z}[\sqrt{-5}] /(2)$ is isomorphic to $\mathbb{Z_2}$

Answer 1

Hint: no two of $0,1,\sqrt{-5},1+\sqrt{-5}$ are congruent modulo 2.

Answer 2

You may obtain an explicit description of $\mathbb{Z}[\sqrt{-5}]/(2)$ by looking at $\mathbb{Z}[\sqrt{-5}]/(2) \cong \mathbb{Z}[X]/(2,X^2+5) \cong \mathbb{F}_2[X]/(X^2+1)$. Also, the quotient ring may be realised as the following matrix ring: $\mathbb{Z}[\sqrt{-5}]/(2) \cong \left\{ \begin{pmatrix}a&b \\ b&a \end{pmatrix} ~|~ a,b \in \mathbb{F}_2 \right\}$.