$\mathbb Z[\sqrt {-5}]$ is Noetherian I'm trying to prove that $\mathbb Z[\sqrt {-5}]$ is Noetherian. I already know that $\mathbb Z[X]$ is Noetherian and I'm trying to find a surjective map 
$$\varphi: \mathbb Z[X]\to \mathbb Z[\sqrt{-5}]$$
with $\ker\varphi=(X^2+5)$.
If I could find this map I could prove that $\mathbb Z[\sqrt{-5}]\cong \mathbb Z[X]/(X^2+5)$ and then $\mathbb Z[\sqrt{-5}]$ is Noetherian.
Thanks

Define $\varphi:\mathbb Z[X]\to\mathbb Z[\sqrt{-5}]$ by $\varphi(f)=f(\sqrt{-5})$.  It's that simple.
