number field that is not norm-Euclidean

Proposition.  The real quadratic fieldMathworldPlanetmath (14) is not norm-Euclidean.

Proof.  We take the number  γ=12+1214  which is not integer of the field (142(mod4)).  Antithesis:  γ=ϰ+δ  where  ϰ=a+b14  is an integer of the field (a,b) and


Thus we would have


And since  (2a-1)2=4(a-1)a+11(mod8),  it follows  E1-1413(mod8),  i.e.  E=3.  So we must have

(2a-1)2(2a-1)2-14(2b-1)23(mod7). (1)

But  {0,±1,±2,±3}  is a complete residue systemMathworldPlanetmath modulo 7, giving  the set  {1, 2, 4}  of possible quadratic residuesMathworldPlanetmath modulo 7.  Therefore (1) is impossible.  The antithesis is wrong, whence the theorem 1 of the parent entry ( says that the number fieldMathworldPlanetmath is not norm-Euclidean.

Note.  The function N used in the proof is the usual


defined in the field (14).  The notion of norm-Euclidean number field is based on the norm (  There exists a fainter function, the so-called Euclidean valuation, which can be defined in the maximal ordersMathworldPlanetmath of some algebraic number fields (; such a maximal order, i.e. the ring of integersMathworldPlanetmath of the number field, is then a Euclidean domainMathworldPlanetmath.  The existence of a Euclidean valuation guarantees that the maximal order is a UFD and thus a PID.  Recently it has been shown the existence of the Euclidean domain [1+692] in the field (69) but the field is not norm-Euclidean.

The maximal order [14] of (14) has also been proven to be a Euclidean domain (Malcolm Harper 2004 in Canadian Journal of Mathematics).

Title number field that is not norm-Euclidean
Canonical name NumberFieldThatIsNotNormEuclidean
Date of creation 2013-03-22 16:56:56
Last modified on 2013-03-22 16:56:56
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 15
Author pahio (2872)
Entry type Example
Classification msc 13F07
Classification msc 11R21
Classification msc 11R04
Related topic UniqueFactorizationAndIdealsInRingOfIntegers