# Lattice Geometry of $\mathbb{Z}[\zeta_5]$

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?

• 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 $\mathbb{C}$. Sep 4, 2023 at 6:46

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)

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

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