# Finding an irreducible element which is not prime in $\mathbb{Z}[\sqrt{5}]$.

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.

• 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. – Vladislav Oct 18 at 16:16
• It's OK. Perhaps you could attach the link of the question that already existed. – WhatsUp Oct 18 at 16:57