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单词 QuadraticImaginaryNormEuclideanNumberFields
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quadratic imaginary norm-Euclidean number fields


Theorem 1.  The imaginary quadratic fieldsMathworldPlanetmath (d) with  d=-1,-2,-3,-7,-11  are norm-Euclidean number fields.

Proof.1.d1(mod 4),  i.e.  d=-1  or  d=-2.  Any element γ of the field (d) has the canonical form  γ=c0+c1d  where  c0,c1.  We may write  γ=(p+r)+(q+s)d, where p is the rational integer nearest to c0 and q the one nearest to c1.  So  |r|12,  |s|12.  Thus we may write

γ=(p+qd)ϰ+(r+sd)δ,

where ϰ is an integer of the field.  We then can

0N(δ)=(r+sd)(r-sd)=r2-s2d=r2+s2|d|(12)2+2(12)2=34<1,

and therefore  |N(δ)|<1.  According to the theorem 1 in the parent entry (http://planetmath.org/EuclideanNumberField), (-1) and (-2) are norm-Euclidean number fields.

2.d1(mod 4),  i.e.  d{-3,-7,-11}.  The algebraic integersMathworldPlanetmath of (d) have now the canonical form a+bd2 with  2|a-b.  Let  γ=c0+c1d  where  c0,c1  be an arbitrary element of the field.  Choose the rational integer q such that q2 is as close to c1 as possible, i.e.  c1=q2+s  with  |s|14,  and the rational integer t such that q2+t is as close to c0 as possible; then  c0=q+2t2+r=p2+r  with  |r|12.  Then we can write

γ=p2+r+(q2+s)d=p+qd2ϰ+(r+sd)δ.

The number ϰ is an integer of the field, since  p-q=2t0(mod 2).  We get the estimation

0N(δ)=r2+s2|d|(12)2+11(14)2=1516<1,

so  |N(δ)|<1.  Thus the fields in question are norm-Euclidean number fields.

Theorem 2.  The only quadratic imaginary norm-Euclidean number fields (d) are the ones in which  d=-1,-2,-3,-7,-11.

Proof.  Let d be any other negative (square-free) rational integer than the above mentioned ones.

1.d1(mod 4).  The integers of (d) are  a+bd where  a,b.  We show that there is a number γ that can not be expressed in the form γ=ϰ+δ  with ϰ an integer of the field and  |N(δ)|<1.  Assume that  γ:=12d=ϰ+δ  where  ϰ=a+bd  is an integer of the field (a,b).  Then  δ=γ-ϰ=-a+(12-b)d  and  N(δ)=|a|2+|d||12-b|2.  Because b cannot be 0, we have |12-b|12  and thus

|N(δ)|0+|d|(12)2=|d|454>1.

Therefore (d) can not be a norm-Euclidean number field(d=-5,-6,-10  and so on).

2.d1(mod 4).  Now  |d|15.  The integers of (d) have the formϰ=a+bd2 with  2|a-b.  Suppose that γ=14+14d=ϰ+δ.  Then δ=γ-ϰ=(14-a2)+(14-b2)d  and

|N(δ)||14-a2|2+|d||14-b2|2(14)2+15(14)2=1.

So also these fields (d) are not norm-Euclidean number fields.

Remark.  The rings of integersMathworldPlanetmath of the imaginary quadratic fields of the above theorems are thus PID’s.  There are, in addition, four other imaginary quadratic fields which are not norm-Euclidean but anyway their rings of integers are PID’s (see lemma for imaginary quadratic fields, class numbers of imaginary quadratic fields, unique factorization and ideals in ring of integers, divisor theoryMathworldPlanetmath).

Titlequadratic imaginary norm-Euclidean number fields
Canonical nameQuadraticImaginaryNormEuclideanNumberFields
Date of creation2013-03-22 16:52:32
Last modified on2013-03-22 16:52:32
Ownerpahio (2872)
Last modified bypahio (2872)
Numerical id14
Authorpahio (2872)
Entry typeTheorem
Classificationmsc 13F07
Classificationmsc 11R21
Classificationmsc 11R04
Synonymimaginary quadratic Euclidean number fields
Synonymimaginary Euclidean quadratic fields
Related topicEuclideanNumberField
Related topicImaginaryQuadraticField
Related topicClassNumbersOfImaginaryQuadraticFields
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更新时间:2025/5/4 2:53:25