Suppose I have some ideal $I$ in $\mathbb{Z}[\sqrt{m}]$ (usually, $ m < 0 $). What, in general, would be a course of action to prove this ideal is maximal or prime? I know the former implies the latter, and an ideal $I$ being prime or maximal is equivalent to $R/I$ being a domain or field, respectively.
An example I have been given is $I = (3,1-\sqrt{-23})$ in $\mathbb{Z}[\sqrt{-23}]$. I proved this ideal is not principal as I thought that might be useful.
However, when trying to establish $R/I$ in these $\mathbb{Z}[\sqrt{m}]$ I fail to understand the course of action.
Any links to useful material would also be much appreciated.
In rings of algebraic integers, non-zero prime or maximal is the same thing.
Here, note $\mathbf Z[\sqrt{-23}]\simeq \mathbf Z[X]/(X^2+23)$, and $X^2+23=(X+1)(X-1)+8\cdot 3\in (1-X,3)$, so \begin{align*} \mathbf Z[\sqrt{-23}]/(3,1-\sqrt{-23})&\simeq \bigl(\mathbf Z[X]/(X^2+23)\bigr)/\bigl((3, 1-X)/(X^2+23)\bigr)\\ &\simeq\mathbf Z[X]/(3,1-X)\simeq \mathbf Z/3\mathbf Z, \end{align*} by the third isomorphism theorem.