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 formγ=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


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


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


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


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


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 form ϰ=a+bd2 with  2|a-b.  Suppose that  γ=14+14d=ϰ+δ.  Then  δ=γ-ϰ=(14-a2)+(14-b2)d  and


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).

Title quadratic imaginary norm-Euclidean number fields
Canonical name QuadraticImaginaryNormEuclideanNumberFields
Date of creation 2013-03-22 16:52:32
Last modified on 2013-03-22 16:52:32
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 14
Author pahio (2872)
Entry type Theorem
Classification msc 13F07
Classification msc 11R21
Classification msc 11R04
Synonym imaginary quadratic Euclidean number fields
Synonym imaginary Euclidean quadratic fields
Related topic EuclideanNumberField
Related topic ImaginaryQuadraticField
Related topic ClassNumbersOfImaginaryQuadraticFields