An order in a number field \(K\) is a unital subring \(\mathcal{O} \subseteq K\) that is a finitely generated \(\mathbb{Z}\)-module under addition such that \(\mathbb{Q} \cdot \mathcal{O} = K\). The ring of integers \(\mathcal{O}_{K}\) is the unique maximal order in \(K\). To construct more, you can take any integer \(N \geq 1\) and consider \(\mathcal{O} = \mathbb{Z} + N \mathcal{O}_{K}\).

Multiplicative group

The group \(\mathcal{O}^{\times}\) satisfies Dirichlet’s unit theorem.

Cubic orders

There is a nice classification of orders in a cubic field~[1].

References