# Visualizing euclidean rings

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

or

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!

• For drawing the pictures I can recommend Geogebra, or TikZ if you want to draw them in LaTeX directly. – Inactive - Objecting Extremism Apr 6 at 13:17
• – Jean Marie May 17 at 22:11

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 $$N(r) such that $$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!
• 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. – Inactive - Objecting Extremism Apr 6 at 17:16
• 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. – Inactive - Objecting Extremism Apr 6 at 17:18