Does there exist a surjective ring homomorphism $\mathbb{Z}[\sqrt{-5}]$ to $\mathbb{Z}$?

Question

Does there exist a surjective ring homomorphism $\mathbb{Z}[\sqrt{-5}]$ to $\mathbb{Z}$?

Obviously projecting to the real/square-root part does not work, and neither does the norm $N:\mathbb{Z}[\sqrt{-5}]\rightarrow \mathbb{Z}$, $a+b\sqrt{-5} \mapsto a^2-5b^2$.

Am i missing something fairly obvious?

Answer 1

There is no ring homomorphism at all from ${\mathbb Z}[\sqrt{-5}]$ to $\mathbb Z$. This is because $\sqrt{-5}$ should go to an element whose square is $-5$ and that doesn't exist.