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单词 CanonicalBasis
释义

canonical basis


Let ϑ be an algebraic integerMathworldPlanetmath of degree (http://planetmath.org/ExtensionField) n.  The algebraic number fieldMathworldPlanetmath (ϑ) has always an integral basis of the form

ω1=1,
ω2=a21+ϑd2,
ω3=a31+a32ϑ+ϑ2d3,
      
ωn=an1+an2ϑ++an,n-1ϑn-2+ϑn-1dn,

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

d2d3d4dn,

i.e.

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,

-di2<aijdi2.

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)

ω1=1,ω2=1+d2.

The discriminantPlanetmathPlanetmathPlanetmath of this basis is d.

Titlecanonical basis
Canonical nameCanonicalBasis
Date of creation2015-02-06 13:12:19
Last modified on2015-02-06 13:12:19
Ownerpahio (2872)
Last modified bypahio (2872)
Numerical id14
Authorpahio (2872)
Entry typeTheorem
Classificationmsc 11R04
Related topicMinimalityOfIntegralBasis
Related topicExamplesOfRingOfIntegersOfANumberField
Related topicConditionForPowerBasis
Related topicIntegralBasisOfQuadraticField
Related topicCanonicalFormOfElementOfNumberField
Definescanonical basis
Definescanonical basis of a number field
Definesadjusted canonical basis
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