Adam P. Goucher discusses lattices constructed from cyclotomic fields. In particular, one possible lattice that can be constructed is the Leech lattice. As the Leech lattice has a notorious lack of simple constructions, it’s worth considering this in more detail.

If $\zeta_n$ is a primitive $n$th root of unity, the $n$th cyclotomic ring $\mathbb Z[\zeta_n]$ is the set of numbers of the form $a+b\zeta_n+c\zeta_n^2+\cdots$, where $a$, $b$, $c$, and so on are integers. Elements of $\mathbb Z[\zeta_n]$ can be added and multiplied. The cyclotomic field is similar, but allows the coefficients to be rationals.

Each number in $\mathbb Z[\zeta_n]$ has a minimal polynomial. This is a polynomial with integer coefficients, leading coefficient 1, and having the number as one of the roots. If the minimal polynomial has second coefficient $a$ and degree $m$, then the trace is defined to be $-\frac{a}{m}$. Equivalently, the trace is the average of all the roots of the minimal polynomial.

## Inner products

Lattices of the same dimension are distinguished by their inner product. The inner product is essentially a way to compute lengths and angles. For lattices in Euclidean space, the inner product is the Euclidean dot product.

Over $\mathbb Z[\zeta_n]$, an inner product can be defined by $\langle u,v\rangle=\operatorname{tr}(\alpha u v^*)$ where $\alpha$ is a fixed number and $v^*$ is the conjugate. This has the property of symmetry: $\langle \zeta_n u,\zeta_n v\rangle=\langle u,v\rangle$. Thus, any lattice constructed in this way will have quite a lot of symmetry. For example, using $\mathbb Z[\zeta_{84}]$, the lattice has only 24 dimensions, yet has an order 84 symmetry.

For this to actually define an inner product, $\langle u,v\rangle=\langle v,u\rangle$. This happens if and only if $\alpha$ is real. Additionally, $\langle u,u\rangle>0$ for all nonzero numbers $u$.

One simple example is $\alpha$ being a positive integer. Are there other examples?

There’s a large class of examples we don’t need to worry about. For any $u$, $uu^*>0$, and also $\operatorname{tr}(uu^*)>0$ as well. The condition $uu^*$ ends up as a linear condition with irrational coefficients on the components of $uu^*$ unless $n=3$ or $6$, so we do not need to care about it. If we want $\operatorname{tr}(\alpha uu^*)>0$, this ends up as a linear condition on the components of $uu^*$. If it has to be always satisfied, then either there’s another linear constraint on $uu^*$, or the condition $\operatorname{tr}(\alpha uu^*)>0$ is exactly the same as $\operatorname{tr}(uu^*)>0$. (In hindsight, this was kind of tautological.) Since both are linear in $uu^*$, this implies $\operatorname{tr}(\alpha uu^*)=\operatorname{tr}(\alpha)\operatorname{tr}(uu^*)$. By polarization, this means $\operatorname{tr}(\alpha uv^*)=\operatorname{tr}(\alpha)\operatorname{tr}(uv^*)$. Therefore, we just get a scalar multiple of the inner product $\langle u,v\rangle=\operatorname{tr}(uv^*)$. Thus, we will restrict attention to this specific inner product.

## Some lattices

We will restrict ourselves to the cyclotomic rings of even order, since the cyclotomic ring of order $n$ for $n$ odd is the same as the cyclotomic ring of order $2n$. ($\zeta_{2n}=-\zeta_n^{\frac{n+1}{2}}$)

For 4, we have the basis $\{1,i\}$ with realization $1=(1,0),i=(0,1)$.This is the ordinary square lattice. For 6, we get the hexagonal lattice. For 8, we have a 4-dimensional lattice. We can pick the basis $\{1,\zeta_8,i,\zeta_8^3\}$. These basis vectors are all at right angles to each other. For 10, we get another 4D lattice, with 4 basis vectors all at $\cos^{-1}\left(-\frac{1}{4}\right)$ angles to each other. Interestingly, these together with the negative of their sum form a regular simplex. 12 yields another 4D lattice. Picking the basis $\{1,\zeta_{12},\zeta_{12}^2,i\}$, we find the pairwise angles as follows: $$\begin{matrix} – & 90 ^ \circ & 60 ^ \circ & 90 ^ \circ \\ 90 ^ \circ & – & 90 ^ \circ & 120 ^ \circ \\ 60 ^ \circ & 90 ^ \circ & – & 90 ^ \circ \\ 90 ^ \circ & 120 ^ \circ & 90 ^ \circ & – \end{matrix}$$ By drawing a Dynkin diagram, we identify this lattice as $A_2^2$, i.e. two copies of the hexagonal lattice.