orders in a number field

If  ${\mu }_1,\mathrm{\dots },{\mu }_m$  are elements of an algebraic number field $K$, then the subset

$$M=\{n_1{\mu }_1+\mathrm{\dots }+n_m{\mu }_m\in K\mathrm{⋮}{n}_i\in \mathbb{Z}\forall i\}$$

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 (http://planetmath.org/NumberField) of $K$ over $\mathbb{Q}$, then the module is complete.

If a complete module in $K$ 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 ${ℒ}_M$ of all coefficients of a complete module $M$ is an order in the field.  Conversely, every order $ℒ$ 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 (http://planetmath.org/CharacteristicEquation) 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 ${𝒪}_K$, being itself an order, is the greatest order; ${𝒪}_K$ 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 ${ℒ}_M$ generated by $1$ and $2\sqrt{2}$.  The maximal order of the field is generated by $1$ and $\sqrt{2}$.