number field that is not norm-Euclidean
Proof. We take the number which is not integer of the field (). Antithesis: where is an integer of the field () and
Thus we would have
And since , it follows , i.e. . So we must have
But is a complete residue system modulo 7, giving the set of possible quadratic residues modulo 7. Therefore (1) is impossible. The antithesis is wrong, whence the theorem 1 of the parent entry (http://planetmath.org/EuclideanNumberField) says that the number field is not norm-Euclidean.
Note. The function N used in the proof is the usual
defined in the field . The notion of norm-Euclidean number field is based on the norm (http://planetmath.org/NormAndTraceOfAlgebraicNumber). There exists a fainter function, the so-called Euclidean valuation, which can be defined in the maximal orders of some algebraic number fields (http://planetmath.org/NumberField); such a maximal order, i.e. the ring of integers of the number field, is then a Euclidean domain. 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 in the field but the field is not norm-Euclidean.
The maximal order of 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|
|Date of creation||2013-03-22 16:56:56|
|Last modified on||2013-03-22 16:56:56|
|Last modified by||pahio (2872)|