Describing the elements of quotient ring of $\mathbb{Z}[\sqrt{D}]$.

Question

For Gaussian integer ring $\mathbb{Z}[i]$, there is a method describing distinct elements of certain quotient ring of $\mathbb{Z}[i]$ using 'the visualization'.

The images(in K. Conrad's note) below are examples of the method using visualization.

So, i wonder about that:

Is there a similar method for quotient ring of $\mathbb{Z}[\sqrt{D}]$, where $D$ is square-free?

I guess it is possible in imaginary quadratic field $\mathbb{Q}[\sqrt{-d}]$, where $d>0$.

Give some advice! Thank you!

Answer 1

I'll make a try...

Let's say we have $\mathbb{Z}[i\sqrt{k}]=\{a+bi\sqrt{k}:a,b\in\mathbb{Z}\}$ where $k\in\mathbb{N} $ is square-free.

So for the multiples of $(1+2i)$ we see that $$(1+2i)(a+bi\sqrt{k})=a(1+2i)+b(-2\sqrt{k}+\sqrt{k}i)$$

and $(1+2i),(-2\sqrt{k}+\sqrt{k}i)$ correspond to perpendicular vectors, so you can proceed as before (now you will not have squares but rectangles).