orders in a number field
If are elements of an algebraic number field , then the subset
If a complete module in the unity 1 of and is a ring, it is called an order (in German: Ordnung) in the field .
A number of the algebraic number field is called a coefficient of the module , if .
Theorem 2. If belongs to an order in the field, then the coefficients of the characteristic equation (http://planetmath.org/CharacteristicEquation) of and thus the coefficients of the minimal polynomial of are rational integers.
Theorem 2 means that any order is contained in the ring of integers of the algebraic number field . Thus this ring , being itself an order, is the greatest order; 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 , the coefficient ring of the module generated by and is the module generated by and . The maximal order of the field is generated by and .
- 1 S. Borewicz & I. Safarevic: Zahlentheorie. Birkhäuser Verlag. Basel und Stuttgart (1966).
|Title||orders in a number field|
|Date of creation||2013-03-22 16:52:46|
|Last modified on||2013-03-22 16:52:46|
|Last modified by||pahio (2872)|
|Defines||order of a number field|