CGAL 6.1 - Algebraic Foundations

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UniqueFactorizationDomain Concept Reference
Reference Manual » Concepts

## Definition

A model of `UniqueFactorizationDomain` is an `IntegralDomain` with the additional property that the ring it represents is a unique factorization domain (a.k.a. UFD or factorial ring), meaning that every non-zero non-unit element has a factorization into irreducible elements that is unique up to order and up to multiplication by invertible elements (units). (An irreducible element is a non-unit ring element that cannot be factored further into two non-unit elements. In a UFD, the irreducible elements are precisely the prime elements.)

In a UFD, any two elements, not both zero, possess a greatest common divisor (gcd).

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

- `CGAL::Algebraic_structure_traits< UniqueFactorizationDomain >::Algebraic_category` derived from `CGAL::Unique_factorization_domain_tag`
- `CGAL::Algebraic_structure_traits< UniqueFactorizationDomain >::Gcd` which is a model of `AlgebraicStructureTraits_::Gcd`

Refines
  `IntegralDomain`

See also
  `IntegralDomainWithoutDivision`
  `IntegralDomain`
  `UniqueFactorizationDomain`
  `EuclideanRing`
  `Field`
  `FieldWithSqrt`
  `FieldWithKthRoot`
  `FieldWithRootOf`
  `AlgebraicStructureTraits`