Unique factorization domains¶

class sage.categories.unique_factorization_domains.UniqueFactorizationDomains[source]¶

Bases: Category_singleton

The category of (constructive) unique factorization domains.

In a constructive unique factorization domain we can constructively factor members into a product of a finite number of irreducible elements.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> UniqueFactorizationDomains()
Category of unique factorization domains
>>> UniqueFactorizationDomains().super_categories()
[Category of gcd domains]

class ElementMethods[source]¶

Bases: object

radical(*args, **kwds)[source]¶

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.

See also

squarefree_part()

EXAMPLES:

Sage

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

sage: Integer(10).radical()
10
sage: Integer(-100).radical()
10
sage: Integer(0).radical()
Traceback (most recent call last):
...
ArithmeticError: radical of 0 is not defined

Python

>>> from sage.all import *
>>> Pol = QQ['x']; (x,) = Pol._first_ngens(1)
>>> (x**Integer(2)*(x-Integer(1))**Integer(3)).radical()
x^2 - x
>>> pol = Integer(37) * (x-Integer(1))**Integer(3) * (x-Integer(2))**Integer(2) * (x-Integer(1)/Integer(3))**Integer(7) * (x-Integer(3)/Integer(7))
>>> pol.radical()
37*x^4 - 2923/21*x^3 + 1147/7*x^2 - 1517/21*x + 74/7

>>> Integer(Integer(10)).radical()
10
>>> Integer(-Integer(100)).radical()
10
>>> Integer(Integer(0)).radical()
Traceback (most recent call last):
...
ArithmeticError: radical of 0 is 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

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

Python

>>> from sage.all import *
>>> n = Integer(2)**Integer(1000)-Integer(1); n / radical(n, limit=Integer(100000))
125

squarefree_part()[source]¶

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.

See also

radical()

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> Pol = QQ['x']; (x,) = Pol._first_ngens(1)
>>> (x**Integer(2)*(x-Integer(1))**Integer(3)).squarefree_part()
x - 1
>>> pol = Integer(37) * (x-Integer(1))**Integer(3) * (x-Integer(2))**Integer(2) * (x-Integer(1)/Integer(3))**Integer(7) * (x-Integer(3)/Integer(7))
>>> pol.squarefree_part()
37*x^3 - 1369/21*x^2 + 703/21*x - 37/7

class ParentMethods[source]¶

Bases: object

is_unique_factorization_domain(proof=True)[source]¶

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

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> UFD = UniqueFactorizationDomains()
>>> Parent(QQ, category=UFD).is_unique_factorization_domain()
True

additional_structure()[source]¶

Return whether self is a structure category.