orders in a number field

If  μ1,,μm  are elements of an algebraic number fieldMathworldPlanetmath K, then the subset


of K is a -module, called a module in K.  If the module contains as many over linearly independentMathworldPlanetmath elements as is the degree (http://planetmath.org/NumberField) of K over , 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 α of the algebraic number field K is called a coefficient of the module M, if  αMM.

Theorem 1.  The set M of all coefficients of a complete module M is an order in the field.  Conversely, every order in the number field K is a coefficient ring of some module.

Theorem 2.  If α belongs to an order in the field, then the coefficients of the characteristic equationMathworldPlanetmathPlanetmath (http://planetmath.org/CharacteristicEquation) of α and thus the coefficients of the minimal polynomialPlanetmathPlanetmath of α are rational integers.

Theorem 2 means that any order is contained in the ring of integersMathworldPlanetmath of the algebraic number field K.  Thus this ring 𝒪K, being itself an order, is the greatest order; 𝒪K 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 (2), the coefficient ring of the module M generated by 2 and 22 is the module M generated by 1 and 22.  The maximal order of the field is generated by 1 and 2.


  • 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