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I was trying to plot all the points of $\mathbb{Z}[\zeta_5]$ and see if there is a nice lattice structure.

It is easy for the Gaussian Integers: $\mathbb{Z}[\zeta_4] = \mathbb{Z}[i]$, which is a square lattice.

And it is easy for the Eisenstein Integers: $\mathbb{Z}[\zeta_3] = \mathbb{Z}[\omega]$, which is a triangular lattice.

But I was wondering if there is a intuitive geometry for $\mathbb{Z}[\zeta_5]$.

I have some suspicion that $\mathbb{Z}[\zeta_5]$ has rank 4 as a free $\mathbb{Z}$-module since I computed that $\zeta_5^3$ cannot be written as a $\mathbb{Z}$-linear combination of $1,\zeta_5,\zeta_5^2$, but since each of the fifth roots of unity can be written as a linear combination of the other 4 due to rotational symmetry.

So, I tried plotting a good amount of them on Wolfram Mathematica via this code:

ComplexListPlot[
 Flatten[Table[
   n + Exp[(2 Pi I)/5] m + Exp[(4 Pi I)/5] k + 
    Exp[(6 Pi I)/5] j, {n, -4, 4}, {m, -4, 4}, {k, -4, 4}, {j, -4, 4}
]]]

and this seems like it is orders of magnitude more "dense" over $\mathbb{C}$ than the Gaussian and Eisenstein integers.

I am wondering if there is an understandable geometrical structure of this ring?

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  • $\begingroup$ You may be interested in the Minkowski embedding. In particular, $\mathbb{Z}[\zeta_5]$ can be embedded as a (discrete) lattice $\mathbb{C}^2$ in the same way that the Gaussian and Eisenstein integers can be embedded as a lattice in $\mathbb{C}$. $\endgroup$ Sep 4, 2023 at 6:46

1 Answer 1

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For most cyclotomic rings you generate what are known as quasilattices.

A quasilattice is a linear projection of a higher-dimensional lattice into a lower-dimensional domain, such as projecting a four-dimensional lattice into two dimensions. With the right projection parameters you can create symmetries that ordinarily would not be present in a lattice. For example, the four-dimensional lattice can be projected into a plane to give eightfold, tenfold, or twelvefold rotational symmetry, and a six-dimensional lattice can be projected into three-dimensional space to give icosahedral symmetry.

In the case of the cyclotomic ring $\mathbb {Z}[\zeta_n]$, you project a lattice having $\phi(n)$ dimensions into the plane, where $\phi$ represents the Euler totient function and cases where $\phi(n)=2$ correspond to regular lattices. The general projection is given by

$\sum a_k\zeta_n^k$

where $a_k\in\mathbb{Z}$, $\zeta_n$ is any primitive $n$th root of unity, and $k$ belongs to an appropriately chosen set of $\phi(n)$ whole numbers. Below is listed one choice of $k$ values for each distinct cyclotomic ring from $\mathbb{Z}[\zeta_3]$ to $\mathbb{Z}[\zeta_{15}]$.

$\mathbb{Z}[\zeta_3]: k=1,2$

$\mathbb{Z}[\zeta_4]: k=1,2$

$\mathbb{Z}[\zeta_5]: k=1,2,3,4$

$\mathbb{Z}[\zeta_7]: k=1,2,3,4,5,6$

$\mathbb{Z}[\zeta_8]: k=1,2,3,4$

$\mathbb{Z}[\zeta_9]: k=1,2,3,4,5,6$

$\mathbb{Z}[\zeta_{11}]: k=1,2,3,4,5,6,7,8,9,10$

$\mathbb{Z}[\zeta_{12}]: k=1,2,5,10$ (note the nonconsecutive values)

$\mathbb{Z}[\zeta_{13}]: k=1,2,3,4,5,6,7,8,9,10,11,12$

$\mathbb{Z}[\zeta_{15}]: k=1,2,4,7,8,11,13,14$

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