If z is a common divisor of x and y then z is a unit.

Question

In the ring $Z[\sqrt{-5}]$, I would like to show that if $z$ is a common divisor of $x=3$ and $y=2+\sqrt{-5}$, then $z$ is a unit. I know that I will have to prove that $N(z)=1$, am I right in thinking that this implies that $z=1$ or $-1$? If so, how do I show this? Thank you!

Answer 1

Hint: Note that $N(3)=N(2+\sqrt{-5})=9$. Show that no element in $\mathbb{Z}[\sqrt{-5}]$ has norm $3$, so that both $3$ and $2+\sqrt{-5}$ are irreducible. Then you can finish the argument by showing that $3$ does not divide $2+\sqrt{-5}$.