Prove that the ideal $I = \left( 3, 2 + \sqrt{-5} \right)$ is a prime ideal in $\mathbb{Z}\left[ \sqrt{-5} \right]$.

Question

Prove that the ideal $I = \left( 3, 2 + \sqrt{-5} \right)$ is a prime ideal in $R = \mathbb{Z}\left[ \sqrt{-5} \right]$.

The book recommends observing that $$ R/I \cong \left( R/(3) \right)/\left( I/(3) \right). $$

My trouble is breaking down the RHS of the isomorphism.

I believe I am trying to reduce it down to what is clearly an integral domain and then I can use the following proposition.

An ideal $P$ is a prime $\iff$ $R/P$ is an integral domain.

How would I go about understanding what the RHS looks like?

Answer 1

Hint: \begin{align*} \frac{\mathbb{Z}\left[\sqrt{-5}\right]}{\left(3, 2 + \sqrt{-5}\right)} &\cong \frac{\mathbb{Z}[x]/(x^2 + 5)}{(3, 2 + x, x^2 + 5)/(x^2 + 5)} \cong \frac{\mathbb{Z}[x]}{(3, 2+x, x^2 + 5)} \end{align*}