# Is $(11)$ a prime ideal of $\mathbb{Z}[\sqrt{-5}]$?

Is $(11)$ a prime ideal of $\mathbb{Z}[\sqrt{-5}]$? I know that $11$ is an irreducible element in $\mathbb{Z}[\sqrt{-5}]$. Now to determine whether it is prime we can say $\mathbb{Z}[\sqrt{-5}]$ isomorphic to $\mathbb{Z}[x]/(x^2 + 5)$. So we get an isomorphism $$\mathbb{Z}[\sqrt{-5}]/(11) \;\;\simeq\;\; \mathbb{Z}_{11}[x]/(x^2 + 5) \,.$$

Since $\mathbb{Z}_{11}$ is a field, $\mathbb{Z}_{11}[x]$ is a PID, and since $(x^2 + 5)$ is irreducible over $\mathbb{Z}_{11}[x]$, the ring $\mathbb{Z}_{11}[x]/(x^2 + 5)$ is a field. Hence $(11)$ can be treated as a maximal ideal as well as a prime ideal in the ring $\mathbb{Z}[\sqrt{-5}]$.

• Your method is correct and answer too. Refer this link for more methods. – Error 404 Jan 30 '17 at 8:07

As indicated in the comments, yes, you are correct. It might be wise to justify why $(x^2+5)$ is irreducible over $\mathbb{Z}_{11}[x]$ though.