Principal ideal domains#

class sage.categories.principal_ideal_domains.PrincipalIdealDomains[source]#

Bases: Category_singleton

The category of (constructive) principal ideal domains

By constructive, we mean that a single generator can be constructively found for any ideal given by a finite set of generators. Note that this constructive definition only implies that finitely generated ideals are principal. It is not clear what we would mean by an infinitely generated ideal.

EXAMPLES:

sage: PrincipalIdealDomains()
Category of principal ideal domains
sage: PrincipalIdealDomains().super_categories()
[Category of unique factorization domains]

>>> from sage.all import *
>>> PrincipalIdealDomains()
Category of principal ideal domains
>>> PrincipalIdealDomains().super_categories()
[Category of unique factorization domains]

See also Wikipedia article Principal_ideal_domain

class ElementMethods[source]#

Bases: object

class ParentMethods[source]#

Bases: object

class_group()[source]#

Return the trivial group, since the class group of a PID is trivial.

EXAMPLES:

sage: QQ.class_group()                                                      # needs sage.groups
Trivial Abelian group

>>> from sage.all import *
>>> QQ.class_group()                                                      # needs sage.groups
Trivial Abelian group
content(x, y, coerce=True)[source]#

Return the content of \(x\) and \(y\).

This is the unique element \(c\) of self such that \(x/c\) and \(y/c\) are coprime and integral.

EXAMPLES:

sage: QQ.content(ZZ(42), ZZ(48)); type(QQ.content(ZZ(42), ZZ(48)))
6
<class 'sage.rings.rational.Rational'>
sage: QQ.content(1/2, 1/3)
1/6
sage: factor(1/2); factor(1/3); factor(1/6)
2^-1
3^-1
2^-1 * 3^-1
sage: a = (2*3)/(7*11); b = (13*17)/(19*23)
sage: factor(a); factor(b); factor(QQ.content(a,b))
2 * 3 * 7^-1 * 11^-1
13 * 17 * 19^-1 * 23^-1
7^-1 * 11^-1 * 19^-1 * 23^-1

>>> from sage.all import *
>>> QQ.content(ZZ(Integer(42)), ZZ(Integer(48))); type(QQ.content(ZZ(Integer(42)), ZZ(Integer(48))))
6
<class 'sage.rings.rational.Rational'>
>>> QQ.content(Integer(1)/Integer(2), Integer(1)/Integer(3))
1/6
>>> factor(Integer(1)/Integer(2)); factor(Integer(1)/Integer(3)); factor(Integer(1)/Integer(6))
2^-1
3^-1
2^-1 * 3^-1
>>> a = (Integer(2)*Integer(3))/(Integer(7)*Integer(11)); b = (Integer(13)*Integer(17))/(Integer(19)*Integer(23))
>>> factor(a); factor(b); factor(QQ.content(a,b))
2 * 3 * 7^-1 * 11^-1
13 * 17 * 19^-1 * 23^-1
7^-1 * 11^-1 * 19^-1 * 23^-1

Note the changes to the second entry:

sage: c = (2*3)/(7*11); d = (13*17)/(7*19*23)
sage: factor(c); factor(d); factor(QQ.content(c,d))
2 * 3 * 7^-1 * 11^-1
7^-1 * 13 * 17 * 19^-1 * 23^-1
7^-1 * 11^-1 * 19^-1 * 23^-1
sage: e = (2*3)/(7*11); f = (13*17)/(7^3*19*23)
sage: factor(e); factor(f); factor(QQ.content(e,f))
2 * 3 * 7^-1 * 11^-1
7^-3 * 13 * 17 * 19^-1 * 23^-1
7^-3 * 11^-1 * 19^-1 * 23^-1

>>> from sage.all import *
>>> c = (Integer(2)*Integer(3))/(Integer(7)*Integer(11)); d = (Integer(13)*Integer(17))/(Integer(7)*Integer(19)*Integer(23))
>>> factor(c); factor(d); factor(QQ.content(c,d))
2 * 3 * 7^-1 * 11^-1
7^-1 * 13 * 17 * 19^-1 * 23^-1
7^-1 * 11^-1 * 19^-1 * 23^-1
>>> e = (Integer(2)*Integer(3))/(Integer(7)*Integer(11)); f = (Integer(13)*Integer(17))/(Integer(7)**Integer(3)*Integer(19)*Integer(23))
>>> factor(e); factor(f); factor(QQ.content(e,f))
2 * 3 * 7^-1 * 11^-1
7^-3 * 13 * 17 * 19^-1 * 23^-1
7^-3 * 11^-1 * 19^-1 * 23^-1
gcd(x, y, coerce=True)[source]#

Return the greatest common divisor of x and y, as elements of self.

EXAMPLES:

The integers are a principal ideal domain and hence a GCD domain:

sage: ZZ.gcd(42, 48)
6
sage: 42.factor(); 48.factor()
2 * 3 * 7
2^4 * 3
sage: ZZ.gcd(2^4*7^2*11, 2^3*11*13)
88
sage: 88.factor()
2^3 * 11

>>> from sage.all import *
>>> ZZ.gcd(Integer(42), Integer(48))
6
>>> Integer(42).factor(); Integer(48).factor()
2 * 3 * 7
2^4 * 3
>>> ZZ.gcd(Integer(2)**Integer(4)*Integer(7)**Integer(2)*Integer(11), Integer(2)**Integer(3)*Integer(11)*Integer(13))
88
>>> Integer(88).factor()
2^3 * 11

In a field, any nonzero element is a GCD of any nonempty set of nonzero elements. In previous versions, Sage used to return 1 in the case of the rational field. However, since Issue #10771, the rational field is considered as the fraction field of the integer ring. For the fraction field of an integral domain that provides both GCD and LCM, it is possible to pick a GCD that is compatible with the GCD of the base ring:

sage: QQ.gcd(ZZ(42), ZZ(48)); type(QQ.gcd(ZZ(42), ZZ(48)))
6
<class 'sage.rings.rational.Rational'>
sage: QQ.gcd(1/2, 1/3)
1/6

>>> from sage.all import *
>>> QQ.gcd(ZZ(Integer(42)), ZZ(Integer(48))); type(QQ.gcd(ZZ(Integer(42)), ZZ(Integer(48))))
6
<class 'sage.rings.rational.Rational'>
>>> QQ.gcd(Integer(1)/Integer(2), Integer(1)/Integer(3))
1/6

Polynomial rings over fields are GCD domains as well. Here is a simple example over the ring of polynomials over the rationals as well as over an extension ring. Note that gcd requires x and y to be coercible:

sage: # needs sage.rings.number_field
sage: R.<x> = PolynomialRing(QQ)
sage: S.<a> = NumberField(x^2 - 2, 'a')
sage: f = (x - a)*(x + a); g = (x - a)*(x^2 - 2)
sage: print(f); print(g)
x^2 - 2
x^3 - a*x^2 - 2*x + 2*a
sage: f in R
True
sage: g in R
False
sage: R.gcd(f, g)
Traceback (most recent call last):
...
TypeError: Unable to coerce 2*a to a rational
sage: R.base_extend(S).gcd(f,g)
x^2 - 2
sage: R.base_extend(S).gcd(f, (x - a)*(x^2 - 3))
x - a

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> R = PolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1)
>>> S = NumberField(x**Integer(2) - Integer(2), 'a', names=('a',)); (a,) = S._first_ngens(1)
>>> f = (x - a)*(x + a); g = (x - a)*(x**Integer(2) - Integer(2))
>>> print(f); print(g)
x^2 - 2
x^3 - a*x^2 - 2*x + 2*a
>>> f in R
True
>>> g in R
False
>>> R.gcd(f, g)
Traceback (most recent call last):
...
TypeError: Unable to coerce 2*a to a rational
>>> R.base_extend(S).gcd(f,g)
x^2 - 2
>>> R.base_extend(S).gcd(f, (x - a)*(x**Integer(2) - Integer(3)))
x - a
is_noetherian()[source]#

Every principal ideal domain is Noetherian, so we return True.

EXAMPLES:

sage: Zp(5).is_noetherian()                                                 # needs sage.rings.padics
True

>>> from sage.all import *
>>> Zp(Integer(5)).is_noetherian()                                                 # needs sage.rings.padics
True
additional_structure()[source]#

Return None.

Indeed, the category of principal ideal domains defines no additional structure: a ring morphism between two principal ideal domains is a principal ideal domain morphism.

EXAMPLES:

sage: PrincipalIdealDomains().additional_structure()

>>> from sage.all import *
>>> PrincipalIdealDomains().additional_structure()
super_categories()[source]#

EXAMPLES:

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

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