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 ( of K, where  β0,  there exist ϰ and ϱ of the field such that


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


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


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 ( 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





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


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 ( 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}.
Title norm-Euclidean number field
Canonical name NormEuclideanNumberField
Date of creation 2013-03-22 16:52:26
Last modified on 2013-03-22 16:52:26
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 17
Author pahio (2872)
Entry type Topic
Classification msc 13F07
Classification msc 11R21
Classification msc 11R04
Related topic EuclideanValuation
Related topic QuadraticImaginaryEuclideanNumberFields
Related topic ListOfAllImaginaryQuadraticPIDs
Related topic EuclideanField
Related topic AlgebraicNumberTheory
Related topic MixedFraction
Defines norm-Euclidean