Norm-Euclidean ring of integers
This article defines a property that can be evaluated for a ring of integers in a number field
Definition
Definition in terms of Euclidean norms
The ring of integers of a number field is termed norm-Euclidean if the absolute value of the algebraic norm is a Euclidean norm.
Since the norm in a ring of integers is multiplicative, norm-Euclidean rings possess multiplicative Euclidean norms.
Definition in terms of norms in the field of fractions
The ring of integers of a number field is termed norm-Euclidean if for any , there exists such that , where denotes the algebraic norm in a number field.
Equivalence of definitions
For full proof, refer: Equivalence of definitions of norm-Euclidean ring of integers
Relation with other properties
Weaker properties
- Euclidean ring of integers: For proof of the implication, refer Norm-Euclidean implies Euclidean and for proof of its strictness (i.e. the reverse implication being false) refer Euclidean not implies norm-Euclidean