canonical basis

Let ϑ be an algebraic integerMathworldPlanetmath of degree ( n.  The algebraic number fieldMathworldPlanetmath (ϑ) has always an integral basis of the form


where the aij’s and di’s are rational integers such that



didi+1i=2, 3,,n-1.

The integral basis  ω1,ω2,,ωn is called a canonical basis of the number field.

Remark.  The integers aij can be reduced so that for all i and j,


Then one may speak of an adjusted canonical basis.  In the case of a quadratic number field (d) with  d1(mod 4)  we have (see the examples of ring of integers of a number field)


The discriminantPlanetmathPlanetmathPlanetmath of this basis is d.

Title canonical basis
Canonical name CanonicalBasis
Date of creation 2015-02-06 13:12:19
Last modified on 2015-02-06 13:12:19
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 14
Author pahio (2872)
Entry type Theorem
Classification msc 11R04
Related topic MinimalityOfIntegralBasis
Related topic ExamplesOfRingOfIntegersOfANumberField
Related topic ConditionForPowerBasis
Related topic IntegralBasisOfQuadraticField
Related topic CanonicalFormOfElementOfNumberField
Defines canonical basis
Defines canonical basis of a number field
Defines adjusted canonical basis