# Unique factorization domains¶

class sage.categories.unique_factorization_domains.UniqueFactorizationDomains(s=None)

The category of unique factorization domains constructive unique factorization domains, i.e. where one can constructively factor members into a product of a finite number of irreducible elements

EXAMPLES:

sage: UniqueFactorizationDomains()
Category of unique factorization domains
sage: UniqueFactorizationDomains().super_categories()
[Category of gcd domains]

class ElementMethods
radical(*args, **kwds)

Return the radical of this element, i.e. the product of its irreducible factors.

This default implementation calls squarefree_decomposition if available, and factor otherwise.

EXAMPLES:

sage: Pol.<x> = QQ[]
x^2 - x
sage: pol = 37 * (x-1)^3 * (x-2)^2 * (x-1/3)^7 * (x-3/7)
37*x^4 - 2923/21*x^3 + 1147/7*x^2 - 1517/21*x + 74/7

10
10
Traceback (most recent call last):
...
ArithmeticError: Radical of 0 not defined.


The next example shows how to compute the radical of a number, assuming no prime > 100000 has exponent > 1 in the factorization:

sage: n = 2^1000-1; n / radical(n, limit=100000)
125

squarefree_part()

Return the square-free part of this element, i.e. the product of its irreducible factors appearing with odd multiplicity.

This default implementation calls squarefree_decomposition.

EXAMPLES:

sage: Pol.<x> = QQ[]
sage: (x^2*(x-1)^3).squarefree_part()
x - 1
sage: pol = 37 * (x-1)^3 * (x-2)^2 * (x-1/3)^7 * (x-3/7)
sage: pol.squarefree_part()
37*x^3 - 1369/21*x^2 + 703/21*x - 37/7

class ParentMethods
is_unique_factorization_domain(proof=True)

Return True, since this in an object of the category of unique factorization domains.

EXAMPLES:

sage: Parent(QQ,category=UniqueFactorizationDomains()).is_unique_factorization_domain()
True

additional_structure()

Return whether self is a structure category.

The category of unique factorization domains does not define additional structure: a ring morphism between unique factorization domains is a unique factorization domain morphism.

EXAMPLES:

sage: UniqueFactorizationDomains().additional_structure()

super_categories()

EXAMPLES:

sage: UniqueFactorizationDomains().super_categories()
[Category of gcd domains]