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 ${\mathcal {O}}$ of a number field $K$ is termed norm-Euclidean if for any $x\in K$, there exists $y\in {\mathcal {O}}$ such that $N(x-y)<1$, where $N$ 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