# Visualization of quadratic rings $\mathbb{Z}[\sqrt{d}]$

The extensions $$\mathbb{Z}[\sqrt{d}]$$ of $$\mathbb{Z}$$ by the root $$\sqrt{d}$$ of the quadratic polynomial $$X^2 - d$$, $$d \in \mathbb{Z}$$ square-free, have degree $$2$$ and all have the same additive structure:

$$(x_0 + y_0\sqrt{d}) + (x_1 + y_1\sqrt{d}) = (x_0 + x_1) + (y_0 + y_1)\sqrt{d}$$

and on $$\mathbb{Z}$$ they have the same multiplicative structure:

$$(x_0 + 0\sqrt{d}) (x_1 + 0\sqrt{d}) = x_0x_1 + 0\sqrt{d}$$

The general product in $$\mathbb{Z}[\sqrt{d}]$$ is

$$(x_0 + y_0\sqrt{d}) (x_1 + y_1\sqrt{d}) = (x_0x_1 + dy_0y_1) + (x_0y_1 + y_0x_1)\sqrt{d}$$

It's mainly due to the factor $$d$$ in $$x_0x_1 + dy_0y_1$$ that the $$\mathbb{Z}[\sqrt{d}]$$ are pairwise non-isomorphic.

I asked myself how one could visualize the multiplicative structure of $$\mathbb{Z}[\sqrt{d}]$$, making it apparent that

• they are non-isomorphic if $$d_1 \neq d_2$$

• they somehow resemble each other if $$d_1$$ and $$d_2$$ are both positive or negative

• they are "really" different if $$d_1$$ and $$d_2$$ have different signs.

It's important to note, that the multiplicative structure is completely determined by the value of $$f(z)=z^2$$ for $$\sqrt{d}$$ which is just $$d$$.

Since any graph of the full multiplication function $$f(z_0,z_1): \mathbb{Z}[\sqrt{d}]\times \mathbb{Z}[\sqrt{d}] \rightarrow \mathbb{Z}[\sqrt{d}]$$ with $$f(z_0,z_1) = z_0z_1$$ is hard to visualize (because it would require a more than three-dimensional space), I restricted myself to the "reduced" functions $$f(z) = z^2$$, $$f(z) = z\sqrt{d}$$ and $$f(z) = z(1+\sqrt{d})$$ which can be depicted in two-dimensional space.

I came up with these pictures in which for some or all points $$z = n + m\sqrt{d}$$ in the upper right quadrant a line is drawn to $$f(z)$$. This gives characteristic patterns.

I wonder if these pictures might have some educational value? Are they helpful, and do they reveal something interesting?

## $$d = -1$$, $$f(z) = z\sqrt{d}$$, $$f(z) = z(1+ \sqrt{d})$$, $$f(z) = z(2 + \sqrt{d}),\dots$$

to be considered as times tables