Unique factorization domains#
- class sage.categories.unique_factorization_domains.UniqueFactorizationDomains(s=None)#
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: UniqueFactorizationDomains() Category of unique factorization domains sage: UniqueFactorizationDomains().super_categories() [Category of gcd domains]
- class ElementMethods#
Bases:
object
- 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, andfactor
otherwise.See also
EXAMPLES:
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
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
.See also
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#
Bases:
object
- is_unique_factorization_domain(proof=True)#
Return True, since this in an object of the category of unique factorization domains.
EXAMPLES:
sage: UFD = UniqueFactorizationDomains() sage: Parent(QQ, category=UFD).is_unique_factorization_domain() True
- additional_structure()#
Return whether
self
is a structure category.See also
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]