Let $I : = (3,1+\sqrt{5}i)$ and $R : = \mathbb{Z}[\sqrt{5}i]$. Show that $I$ is maximal. My goal is to show that $R/I$ is a field. So far I have the following:
\begin{align*} \mathbb{Z}[\sqrt{5}i]/(3,1+\sqrt{5}i) & \cong (\mathbb{Z}[X]/(x^2+5))/((3,1+X,X^2+5)/(X^2+5))\\ & \cong \mathbb{Z}[X]/(3,1+X,X^2+5)\\ & \cong \mathbb{Z}/3\mathbb{Z}[X]/(1+X,X^2-1)\\ & = \mathbb{Z}/3\mathbb{Z}[X]/(1+X) & \text{(since $X^2-1 \in (1+X)$)}\\ & \cong \mathbb{Z}{3\mathbb{Z}}. \end{align*} Hence, $\mathbb{Z}[\sqrt{5}i]/(3,1+\sqrt{5}i)$ is a field which happens exactly when $(3,1+\sqrt{5}i)$ is maximal.