TITLE: Lattice of Gauss and Eisenstein Integers
QUESTION [10 upvotes]: Z is a 1D lattice
Gaussian and Eisenstein integers are 2D lattices
But the golden integers (for example) are dense on the real line.

Are there rings of integers which have 3D, 4D, ... lattices?

Here is a plot of $(a + \tfrac{1}{2}(1+\sqrt{5})b,a + \tfrac{1}{2}(1-\sqrt{5})b)$ for $-10\le a,b \le 10$.

which is the lattice corresponding to the golden integers, if I understand correctly. The green points represent rational integers and the blue points represent multiples of $\varphi$.

REPLY [12 votes]: The example of the golden integers (and more generally rings of integers in quadratic number fields with positive discriminant) shows that the single embedding into $\mathbb{R}$ is inadequate. Instead, if $K = \mathbb{Q}(\sqrt{d}), d > 0$ then the appropriate way to embed $K$ as a lattice in the plane is to look at both embeddings $\sigma_1, \sigma_2 : K \to \mathbb{R}$. The first one sends $\sqrt{d}$ to $\sqrt{d}$ and the second one sends $\sqrt{d}$ to $-\sqrt{d}$. Together they give an embedding $(\sigma_1, \sigma_2)$ of $K$ into $\mathbb{R}^2$, and relative to this embedding the ring of integers $\mathcal{O}_K$ really is a lattice.
More generally, if $K$ is a number field of degree $n$, then there are $n = r + 2s$ embeddings $\sigma_i : K \to \mathbb{C}$, $r$ of which have image in $\mathbb{R}$ and $2s$ of which have image outside of $\mathbb{R}$, which come in complex conjugate pairs. Here the appropriate generalization of the above embedding is to use all of the real embeddings $\sigma_1, ... \sigma_r$ and one representative of each complex conjugate pair of complex embeddings $\sigma_{r+1}, ... \sigma_{r+s}$. This gives an embedding $K \to \mathbb{R}^r \times \mathbb{C}^s$, and embedding $\mathbb{C}$ into $\mathbb{R}^2$ gives an embedding $K \to \mathbb{R}^n$.
Relative to this embedding, it's a standard exercise that $\mathcal{O}_K$ is a lattice in $\mathbb{R}^n$ of rank $n$. This is the standard construction used to prove the finiteness of the class group and Dirichlet's unit theorem, and details can be found in any book on algebraic number theory.