Finding an isomorphism between $\mathbb{Z}[i]/(7)$ and $\mathbb{Z}[\sqrt{-2}]/(7)$ I was having some trouble finding an explicit isomorphism between $\mathbb{Z}[i]/(7)$ and $\mathbb{Z}[\sqrt{-2}]/(7)$.
$\textbf{What I have noticed is}:$


*

*7 is a prime element in $\mathbb{Z}[i]$ so $(7)$ is a maximal ideal in $\mathbb{Z}[i]$ and $\mathbb{Z}[i]/(7)$ is  a field. 

*7 is also a prime element in $\mathbb{Z}[\sqrt{-2}]$


$\textbf{What I have been trying to do is this}$


*

*Find a surjective homomorphism between $\mathbb{Z}[i]/(7) \rightarrow \mathbb{Z}[\sqrt{-2}]/(7)$. Since $\mathbb{Z}[i]/(7)$ was a field, the kernel of this homomorphism will be either the whole ring or just $0$. In the latter case it would be a isomorphism. 


I am having trouble finding this surjective homomorphism:
I have noticed that $\bar{{i}}^{2}=-1$ so the image of $\bar{{i}}$ must be sent to something whose square is $-1$. Any help would be appreciated. I may be missing some obvious insight.

Hint $\ $ We get an image of $\,\Bbb Z[\sqrt{\color{#c00}{-9}}]\,$ in $\,\Bbb Z[\sqrt{-1}]\,$ by mapping $\,\sqrt{-9}\,\mapsto\, 3\sqrt{-1}$
$\!\bmod 7\!:\ \color{#c00}{{-}9\equiv -2}\,$ and the map is onto, since $3$ is a unit so $\,3^{-1}\sqrt{-9}\,\mapsto \sqrt{-1}$
