I need to find out whether the number $\ 4+\sqrt{-5}\ $ is a prime element of the ring $\ \mathbb{Z}[\sqrt{-5}]$.
To be honest, I just don't have a clue how I can start the solution. So, if someone could just give me a hint, I would be very greatful.
In the ring $\mathbb{Z}[\sqrt{-5}]$, you can define the norm of an element $a+b\sqrt{-5}$ as $$|a+b\sqrt{-5}|=a^2+5b^2.$$ One can show that this norm is multiplicative (i.e., $|xy|=|x|\cdot|y|$ whenever $x,y\in\mathbb{Z}[\sqrt{-5}]$). Also, the only units of $\mathbb{Z}[\sqrt{-5}]$ are $\pm 1$. More precisely, an element is a unit iff it has norm $1$. Note that $|4+\sqrt{-5}|=21$ so if $4+\sqrt{-5}$ were not a prime then there must exist $x,y\in\mathbb{Z}[\sqrt{-5}]$ such that $4+\sqrt{-5}=xy$ and $|x|=3$ and $|y|=7$ which is clearly impossible.