orders in a number field

If $\mu_1,\,\ldots,\,\mu_m$ are elements of an algebraic number field $K$ , then the subset $$M = \{n_1\mu_1+\ldots+n_m\mu_m\in K\,\vdots\;\; n_i\in\mathbb{Z}\;\;\forall i\}$$ of $K$ is a $\mathbb{Z}$ -module, called a module in $K$ . If the module contains as many over $\mathbb{Z}$ linearly independent elements as is the degree of $K$ over $\mathbb{Q}$ , then the module is complete.

If a complete module in $K$ contains the unity 1 of $K$ and is a ring, it is called an order (in German: Ordnung) in the field $K$ .

A number $\alpha$ of the algebraic number field $K$ is called a coefficient of the module $M$ , if $\alpha M \subseteq M$ .

Theorem 1. The set $\mathcal{L}_M$ of all coefficients of a complete module $M$ is an order in the field. Conversely, every order $\mathcal{L}$ in the number field $K$ is a coefficient ring of some module.

Theorem 2. If $\alpha$ belongs to an order in the field, then the coefficients of the characteristic equation of $\alpha$ and thus the coefficients of the minimal polynomial of $\alpha$ are rational integers.

Theorem 2 means that any order is contained in the ring of integers of the algebraic number field $K$ . Thus this ring $\mathcal{O}_K$ , being itself an order, is the greatest order; $\mathcal{O}_K$ is called the maximal order or the principal order (in German: Hauptordnung). The set of the orders is partially ordered by the set inclusion.

Example. In the field $\mathbb{Q}(\sqrt{2})$ , the coefficient ring of the module $M$ generated by $2$ and $\frac{\sqrt{2}}{2}$ is the module $\mathcal{L}_M$ generated by $1$ and $2\sqrt{2}$ . The maximal order of the field is generated by $1$ and $\sqrt{2}$ .

Bibliography

1

S. BOREWICZ & I. SAFAREVIC: Zahlentheorie. Birkhäuser Verlag. Basel und Stuttgart (1966).