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Let $R=\mathbb{Z}[\sqrt{-d}]$. I know that $R$ is not a UFD if $d>2$ is odd. We can also factorize $1+d=2.\frac{1+d}{2}=(1+i\sqrt{d})(1-i\sqrt{d})$. But how can we factorize when $d>2$ is even?


marked as duplicate by user26857 abstract-algebra Feb 19 '16 at 9:14

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  • $\begingroup$ $R$ is a euclidean domain (hence a UFD) for $14$ odd values of $d$. And there exists a conjecture of Gauß that it is a PID for an infinite number of vales of $d$. $\endgroup$ – Bernard Feb 18 '16 at 23:58
  • $\begingroup$ @Bernard... I don't understand your comment.."for14 odd values of d" $\endgroup$ – UserAb Feb 19 '16 at 0:13
  • $\begingroup$ You wrote $R$ is not a UFD if $d$ is odd. I'm saying it's not exact, since it is, not only a UFD, but a Euclidean domain for fourteen odd values of $d$. $\endgroup$ – Bernard Feb 19 '16 at 0:25
  • $\begingroup$ @Bernard... I see Thanks. But how can we factorize it when d is even? $\endgroup$ – UserAb Feb 19 '16 at 0:35
  • 2
    $\begingroup$ @Bernard: Gauss' conjecture concerns real quadratic fields ($d<0$), while OP is asking about complex quadratic fields ($d>0$). For complex quadratic fields, there are only finitely many fields where the ring of integers is a UFD. These are the ones with $d \in \{1,2,3,7,11,19,43,67,163\}$. However UserAb is not asking about the ring of integers, but about $\mathbb Z[\sqrt{-d}]$. For the cases listed above, $\mathbb Z[\sqrt{-d}]$ is the ring of integers of $\mathbb Q(\sqrt{-d})$ if and only if $d \in \{1,2\}$. Thus, indeed, $\mathbb Z[-\sqrt{d}]$ is not a UFD if $d > 2$, as the OP claims. $\endgroup$ – moonlight Feb 19 '16 at 7:19

If $d$ is even then $d=2n$ for some integer $n$, so we have:

$$ 4+d=2 \cdot (2+n) = (2+\sqrt{-d})(2-\sqrt{-d}) $$

  • 1
    $\begingroup$ Is it clear that these elements are irreducible? $\endgroup$ – moonlight Feb 19 '16 at 8:02
  • $\begingroup$ Why doesn't this argument apply when $d=2$? $\endgroup$ – Gerry Myerson Feb 19 '16 at 8:08
  • $\begingroup$ @GerryMyerson when d=2, these elements are further reducible ($2=-\sqrt{-2}^2$ and $3=(1+\sqrt{-2})(1-\sqrt{-2})$) and from there you can see that these factorizations are indeed the same $\endgroup$ – ASKASK Feb 19 '16 at 12:17
  • $\begingroup$ @moonlight to be honest I'm not sure if then even always are irreducible for $d>2$ $\endgroup$ – ASKASK Feb 19 '16 at 12:18

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