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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.

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    $\begingroup$ I am sorry, I just found out that it’s a duplicate. I couldn’t find it when I was making my own question, though. $\endgroup$ – Vladislav Oct 18 at 16:16
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    $\begingroup$ It's OK. Perhaps you could attach the link of the question that already existed. $\endgroup$ – WhatsUp Oct 18 at 16:57

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