In the ring $\mathbb{Z}[i]\ $ $\ \ 2|(1+i)(1-i)=2$ but $2\nmid (1\pm i)$
So is $2$ not prime (or is it prime but irreducible) in $\mathbb{Z}[i]$ or is $\mathbb{Z}[i]$ not a unique factorization domain?
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Yes, $2$ is not prime in $\mathbb{Z}[i]$, as you have just demonstrated. No, $\mathbb{Z}[i]$ IS a unique factorization domain, in fact it is a Euclidean domain. – Lee Mosher Apr 08 '18 at 18:28
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But it is irreducible right? – John Cataldo Apr 08 '18 at 18:29
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@StanislasHildebrandt No, it is not irreducible, because neither $i+1$ nor $1-i=-i(1+i)$ are units. – Apr 08 '18 at 18:34
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Of course $\mathbb{Z}[i]$ is a UFD, see here. – Dietrich Burde Apr 08 '18 at 19:01