How do I go about calculating what $R = \mathbb Z [\sqrt{-5} ] / \langle 1 + \sqrt{-5} \rangle $ actually is (i.e. how do I find a simpler ring isomorphic to $R$)?
I can see that $\langle \sqrt{-5} \rangle \subsetneq \langle 1 + \sqrt{-5} \rangle$, so $R \subsetneq \mathbb Z[\sqrt{-5}] / \langle \sqrt{-5} \rangle \cong \mathbb Z$. So it's isomorphic to a proper subring of $\mathbb Z$. I can't see how to get any further with this line of reasoning, though.
I can also see that $a + b \sqrt{-5} \equiv a - b \ (\mathrm{mod} \langle 1 + \sqrt{-5} \rangle ) $, and since $4 \in \langle 1 + \sqrt{-5} \rangle $ any element in $R$ is congruent to $0,1,2$ or $3$. How do I proceed?
Thanks