How do I know that $3$ and $2 + \sqrt{-5}$ are not associates in $\mathbb Z[\sqrt{-5}]?$

Question

How do I know that in $\mathbb Z[\sqrt{-5}],$ the elements $3$ and $2 + \sqrt{-5}$ are not associates? Thanks a lot.

Answer 1

Use the norm of these elements (9), since they have the same norm we investigate if there is an unit $u$ in the ring that realize the equation $3=u(2+\sqrt{-5})$. An unit must have norm=1 so we can search for such elements: these elements are the solutions over intergers of $a^2+5b^2=1$. The only solutions are 1 and -1 so they are not associates.

Answer 2

The elements $3$ and $2+\sqrt{-5}$ are not associates in $\mathbb{Z}[\sqrt{-5}]$ because $\frac{2+\sqrt{-5}}{3}=\frac{2}{3}+\frac{1}{3}\sqrt{-5} \notin \mathbb{Z}[\sqrt{-5}]$, so $3 \not\mid 2+\sqrt{-5}$.