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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?


$d = 2$

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$d = - 2$

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Overview

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With arguments in all quadrants

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$d=-1$

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Comparison for $d=2,3,5,6$

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Comparison for $d=-2,-3,-5,-6$

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

to be considered as times tables

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