# 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 (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 $\mathcal{L}_{M}$ 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 (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 $\mathcal{O}_{K}$, 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 $\sqrt{2}$.

## References

• 1 S. Borewicz & I. Safarevic: Zahlentheorie.  Birkhäuser Verlag. Basel und Stuttgart (1966).
 Title orders in a number field Canonical name OrdersInANumberField Date of creation 2013-03-22 16:52:46 Last modified on 2013-03-22 16:52:46 Owner pahio (2872) Last modified by pahio (2872) Numerical id 17 Author pahio (2872) Entry type Topic Classification msc 12F05 Classification msc 11R04 Classification msc 06B10 Related topic Module Defines module Defines complete Defines order of a number field Defines principal order Defines maximal order