It is possible to see euclidean rings with lattices. For instance

enter image description here

from : https://en.wikipedia.org/wiki/Gaussian_integer


enter image description here

from : https://en.wikipedia.org/wiki/Eisenstein_integer

my question is the following :

We love euclidean rings. They have lattices regular enough to create an euclidean function. When we lower hypothesis, we can easily fall and lose the property to be factorial. For instance, $ \mathbb Z [i \sqrt 5 ] $ isn't factorial. Would it be possible to plot the set of those integers and "see" why specific integers are irreducible but not prime ? Which is a necessary condition to be factorial.

I don't know how to plot graphs like the one I put above, and I'm even worse to interpret them. That's why I asking you if you could give a hand.

thanks a lot!


Let $\alpha\in\Bbb{C}$ be some quadratic algebraic integer and consider the ring $\Bbb{Z}[\alpha]$, and let $N$ denote the norm function on $\Bbb{Z}[\alpha]$. Then $\Bbb{Z}[\alpha]$ is Euclidean w.r.t. $N$ if and only if for all $x,y\in\Bbb{Z}[\alpha]$ with $y\neq0$ there exist $q,r\in\Bbb{Z}[\alpha]$ with $N(r)<N(y)$ such that $x=qy+r$. In $\Bbb{Q}(\alpha)$ we can express this as $$\frac xy-q=\frac ry,$$ where $N(\tfrac ry)<1$. This shows that $\Bbb{Z}[\alpha]$ is Euclidean if and only if for all $z\in\Bbb{Q}(\alpha)$ there exists $q\in\Bbb{Z}[\alpha]$ such that $N(z-q)<1$. Geometrically speaking, if and only if the unit discs centered at the lattice points of $\Bbb{Z}[\alpha]$ cover the plane.

And indeed, drawing unit discs around the lattice points in your pictures confirms that $\Bbb{Z}[i]$ and $\Bbb{Z}[\omega]$ are Euclidean domains. And for $\Bbb{Z}[\sqrt{-5}]$ you will see that the lattice points are too far apart in the vertical direction; drawing unit discs around the lattice points leaves narrow horizontal bands that are not covered. This means that the norm map is not a Euclidean function on $\Bbb{Z}[\sqrt{-5}]$.

Note that this does not imply that $\Bbb{Z}[\sqrt{-5}]$ is not Euclidean; there may still be some other Euclidean function on $\Bbb{Z}[\sqrt{-5}]$. It certainly does not imply that $\Bbb{Z}[\sqrt{-5}]$ is not factorial. It turns out that this isn't a factorial ring, but I doubt there is any geometrical way to see this.

The reason for my doubt is that for $\Bbb{Z}[\tfrac12(1+\sqrt{-19})]$ the same argument as for $\Bbb{Z}[\sqrt{-5}]$ shows that the norm is not a Euclidean function on this ring; note that $\tfrac12\sqrt{19}$ and $\sqrt{5}$ are very close to eachother. But $\Bbb{Z}[\tfrac12(1+\sqrt{-19})]$ is factorial!

  • $\begingroup$ Your answer is really instructive but you re not dealing with the core of my question, irreducibility and primality. Am I missing something? $\endgroup$ – Marine Galantin Apr 6 at 16:23
  • 1
    $\begingroup$ My answer is not conclusive. If I understand correctly, you ask for a geometrical way to see whether a ring $\Bbb{Z}[\sqrt{-d}]$ is factorial or not from such a picture, and perhaps even to visually identify elements that are irreducible but not prime. My last two paragraphs suggest that this is not possible; the pictures for $\Bbb{Z}[\sqrt{-19}]$ and $\Bbb{Z}[\sqrt{-5}]$ are very similar, but one ring is factorial and the other is not. $\endgroup$ – Inactive - Objecting Extremism Apr 6 at 17:16
  • $\begingroup$ In fact, there are only nine integers $d>0$ for which $\Bbb{Z}[\sqrt{-d}]$ is factorial; five of them are Euclidean, four of them are not. I doubt any picture is helpful. $\endgroup$ – Inactive - Objecting Extremism Apr 6 at 17:18

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