orders in a number field
If μ1,…,μm are elements of an algebraic number field K, then the subset
M={n1μ1+…+nmμm∈K⋮ni∈ℤ∀i} |
of K is a ℤ-module, called a module in K. If the module contains as many over ℤ linearly independent elements as is the degree (http://planetmath.org/NumberField) of K over 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 α of the algebraic number field K is called a coefficient of the module M, if αM⊆M.
Theorem 1. The set ℒM 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 α belongs to an order in the field, then the coefficients of the characteristic equation (http://planetmath.org/CharacteristicEquation) of α and thus the coefficients of the minimal polynomial
of α 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 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 ℚ(√2), the coefficient ring of the module M generated by 2 and √22 is the module ℒM generated by 1 and 2√2. The maximal order of the field is generated by 1 and √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 |