Show that $\sqrt{-n}$ and $\sqrt{-n} +1$ are not prime in $\mathbb{Z}[\sqrt{-n} ]$

Question

What I have done is that tried writing $\sqrt{-n} =(a+b\sqrt{-n} )(c+d\sqrt{-n} ) $ .

And also $-n =(a^2+b^2(-n)) (c^2+d^2(-n)) $. (Using the definition of norms.) This however didn't yield me something useful. And also how do you attempt such problems in general?

Answer 1

Start with $\mathbb Z[\sqrt{-n}]\simeq\mathbb Z[X]/(X^2+n)$ and show:

1. $\mathbb Z[\sqrt{-n}]/(\sqrt{-n})\simeq \mathbb Z/n\mathbb Z$

2. $\mathbb Z[\sqrt{-n}]/(\sqrt{-n}+1)\simeq \mathbb Z/(n+1)\mathbb Z$