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单词 NormEuclideanNumberField
释义

norm-Euclidean number field


Definition.  An algebraic number field K is a norm-Euclidean number field, if for every pair  (α,β)  of the integers (http://planetmath.org/AlgebraicInteger) of K, where  β0,  there exist ϰ and ϱ of the field such that

α=ϰβ+ϱ,|N(ϱ)|<|N(β)|.

Here N means the norm function in K.

Theorem 1.  A field K is norm-Euclidean if and only if each number γ of K is in the form

γ=ϰ+δ(1)

where ϰ is an of the field and |N(δ)|<1.

Proof.  First assume the condition (1).  Let α and β be integers of K,  β0.  Then there are the numbers  ϰ,δK  such that ϰ is integer and

αβ=ϰ+δ,|N(δ)|< 1.

Thus we have

α=ϰβ+βδ=ϰβ+ϱ.

Here  ϱ=βδ  is integer, since α and ϰβ are integers.  We also have

|N(ϱ)|=|N(β)||N(δ)|<|N(β)|1=|N(β)|.

Accordingly, K is a norm-Euclidean number field.Secondly assume that K is norm-Euclidean.  Let γ be an arbitrary element of the field.  We can determine (http://planetmath.org/MultiplesOfAnAlgebraicNumber) a rational integer m(0) such that mγ is an algebraic integerMathworldPlanetmath of K.  The assumptionPlanetmathPlanetmath guarantees the integers ϰ, ϱ of K such that

mγ=ϰm+ϱ,N(ϱ)<N(m).

Thus

γ=mγm=ϰ+ϱm,|N(ϱm)|=|N(ϱ)||N(m)|<1,

Q.E.D.

Theorem 2.  In a norm-Euclidean number field, any two non-zero have a greatest common divisorMathworldPlanetmathPlanetmath.

Proof.  We recall that the greatest common divisor of two elements of a commutative ring means such a common divisorMathworldPlanetmathPlanetmath of the elements that it is divisible by each common divisor of the elements.  Let now ϱ0 and ϱ1 be two algebraic integers of a norm-Euclidean number field K.  According the definition there are the integers ϰi and ϱi of K such that

{ϱ0=ϰ2ϱ1+ϱ2,|N(ϱ2)|<|N(ϱ1)|ϱ1=ϰ3ϱ2+ϱ3,|N(ϱ3)|<|N(ϱ2)|ϱ2=ϰ4ϱ3+ϱ4,|N(ϱ4)|<|N(ϱ3)|ϱn-2=ϰnϱn-1+ϱn,|N(ϱn)|<|N(ϱn-1)|ϱn-1=ϰn+1ϱn+0,

The ends to the remainder 0, because the numbers |N(ϱi)| form a descending sequenceMathworldPlanetmath of non-negative rational integers — see the entry norm and trace of algebraic number.  As in the Euclid’s algorithm in , one sees that the last divisor ϱn is one greatest common divisor of ϱ0 and ϱ1.  N.B. that ϱ0 and ϱ1 may have an infiniteMathworldPlanetmath amount of their greatest common divisors, depending the amount of the units in K.

Remark.  The ring of integersMathworldPlanetmath of any norm-Euclidean number field is a unique factorization domainMathworldPlanetmath and thus all ideals of the ring are principal idealsMathworldPlanetmathPlanetmathPlanetmathPlanetmath.  But not all algebraic number fields with ring of integers a UFD (http://planetmath.org/UFD) are norm-Euclidean, e.g. (14).

Theorem 3.  The only norm-Euclidean quadratic fields (d) are those with

d{-11,-7,-3,-2,-1, 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73}.
Titlenorm-Euclidean number field
Canonical nameNormEuclideanNumberField
Date of creation2013-03-22 16:52:26
Last modified on2013-03-22 16:52:26
Ownerpahio (2872)
Last modified bypahio (2872)
Numerical id17
Authorpahio (2872)
Entry typeTopic
Classificationmsc 13F07
Classificationmsc 11R21
Classificationmsc 11R04
Related topicEuclideanValuation
Related topicQuadraticImaginaryEuclideanNumberFields
Related topicListOfAllImaginaryQuadraticPIDs
Related topicEuclideanField
Related topicAlgebraicNumberTheory
Related topicMixedFraction
Definesnorm-Euclidean
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更新时间:2025/5/4 7:06:22