Norm-Euclidean ring of integers

This article defines a property that can be evaluated for a ring of integers in a number field

Definition

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.

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