Plotting $\mathbb{Z}[\sqrt[]{-N}]$ as a lattice in $\mathbb{C}$

Question

$\mathbb{Z}[\sqrt{-N}]=\left\{a+b\sqrt[]{-N}:a,b\in \mathbb{Z}, N\geq 1\right\}$

(note that $N$ is a fixed natural number)

I was curious how one would plot this as a lattice in $\mathbb{C}$

For example, if $N=1$, I know we get the Gaussian integers and we form a bunch of dots at all the integer coordinates

My question is when $N\neq 1$. Would we change the imaginary axis?

For example, I am not sure how one would plot $2+\sqrt[]{-3}$

Any help is appreciated

Answer 1

$2+\sqrt{-3}$ can be corresponded to $2+\sqrt{3}i$ in the complex plane. So we can identify $\mathbb{Z}[\sqrt{-3}]$ to be $\{a+b\sqrt{3}i \mid a,b\in \mathbb{Z}\}$. When we plot these points on the complex plane, we obtain a rectangular lattice, with a basis given by $\{1,\sqrt{3}i\}$.

Similar things can be done for $\mathbb{Z}[-N]$, the lattice will be rectangular unless $N=1$.

Answer 2

Consider the identification: $$a+bi=\left(\begin{array}{cc} a&-b\\ b&a \end{array}\right) $$

Then this gives you a natural change of $\mathbb{Z}$-basis, and you can view the plot of your lattice in $\mathbb{C}$ that way.