The problem is in the title. I have spent hours to guess the answer (using norm $N(a + b\sqrt{5}) = a^2 - 5b^2$ for checking irreducibility, of course). I also proved that $a^2 - 5b^2 \neq 1$ unless $b=0$, but I don’t know if I am going to need it.
I have seen something similar in some book long time ago. There was a ring $\mathbb{Z}[\sqrt{-5}]$, though. And the author didn’t explain how did he find it. He rather just gave and example.
Edit: althoug my question has already arised here, there wasn’t explained why $1 + \sqrt{5}$ is irreducible. I failed to check it.
