CGAL 5.5.1 - Algebraic Foundations
EuclideanRing Concept Reference

## Definition

A model of EuclideanRing represents an euclidean ring (or Euclidean domain). It is an UniqueFactorizationDomain that affords a suitable notion of minimality of remainders such that given $$x$$ and $$y \neq 0$$ we obtain an (almost) unique solution to $$x = qy + r$$ by demanding that a solution $$(q,r)$$ is chosen to minimize $$r$$. In particular, $$r$$ is chosen to be $$0$$ if possible.

Moreover, CGAL::Algebraic_structure_traits< EuclideanRing > is a model of AlgebraicStructureTraits providing:

Remarks

The most prominent example of a Euclidean ring are the integers. Whenever both $$x$$ and $$y$$ are positive, then it is conventional to choose the smallest positive remainder $$r$$.

Refines:
UniqueFactorizationDomain
IntegralDomainWithoutDivision
IntegralDomain
UniqueFactorizationDomain
EuclideanRing
Field
FieldWithSqrt
FieldWithKthRoot
FieldWithRootOf
AlgebraicStructureTraits