Finding an isomorphism between $\mathbf Z[\sqrt{-5}]\big/(3,1-\sqrt{-5})$ and $\mathbf Z\big/3$

Question

Finding an isomorphism between $\mathbf Z[\sqrt{-5}]\big/(3,1-\sqrt{-5})$ and $\mathbf Z\big/3$ ?

In general how does the elements of the ideal $(3,1-\sqrt{-5})$ look like,

in the form $(3)\cup(1-\sqrt{-5})$ ?

Answer 1

As $\;\mathbf Z[\sqrt{-5}]\simeq\mathbf Z[X]/(X^2+5)$, \begin{align*}\mathbf Z[\sqrt{-5}]/(3,1-\sqrt{-5})&\simeq\mathbf Z[X]/(X^2+5)\boldsymbol/(3,1-X)\cdot\mathbf Z[X]/(X^2+5)\\ &\simeq\mathbf Z[X]/(X^2+5)\boldsymbol/(3,1-X,X^2+5)/(X^2+5)\\ &\simeq\mathbf Z[X]/(3,1-X,X^2+5)\simeq\mathbf Z/3\mathbf Z[X]/(1-X,X^2+5) \\ &\simeq\mathbf Z/3\mathbf Z[X]/(1-X,X^2-1). \end{align*} As $X^2-1\in(1-X)$, the ideal $(1-X,X^2-1)$ is simply $(1-X)$, so this is finally the same as $$\mathbf Z/3\mathbf Z[X]/(1-X)\simeq\mathbf Z/3\mathbf Z.$$

Explicitly, the isomorphism maps an element $a+b\sqrt{-5}\;$ to $\;a+b\bmod3$.