Rings#

This module provides the abstract base class Ring from which all rings in Sage (used to) derive, as well as a selection of more specific base classes.

Warning

Those classes, except maybe for the lowest ones like CommutativeRing and Field, are being progressively deprecated in favor of the corresponding categories. which are more flexible, in particular with respect to multiple inheritance.

The class inheritance hierarchy is:

Ring (to be deprecated)
- Algebra (to be deprecated)
- CommutativeRing
  - NoetherianRing (deprecated)
  - CommutativeAlgebra (deprecated and essentially removed)
  - IntegralDomain (deprecated)
    - DedekindDomain (deprecated and essentially removed)
    - PrincipalIdealDomain (deprecated and essentially removed)

Subclasses of CommutativeRing are

Field
- FiniteField

Some aspects of this structure may seem strange, but this is an unfortunate consequence of the fact that Cython classes do not support multiple inheritance.

(A distinct but equally awkward issue is that sometimes we may not know in advance whether or not a ring belongs in one of these classes; e.g. some orders in number fields are Dedekind domains, but others are not, and we still want to offer a unified interface, so orders are never instances of the deprecated DedekindDomain class.)

AUTHORS:

David Harvey (2006-10-16): changed CommutativeAlgebra to derive from CommutativeRing instead of from Algebra.
David Loeffler (2009-07-09): documentation fixes, added to reference manual.
Simon King (2011-03-29): Proper use of the category framework for rings.
Simon King (2011-05-20): Modify multiplication and _ideal_class_ to support ideals of non-commutative rings.

class sage.rings.ring.Algebra[source]#

Bases: Ring

Generic algebra

class sage.rings.ring.CommutativeAlgebra[source]#: Bases: CommutativeRing

class sage.rings.ring.CommutativeRing[source]#

Bases: Ring

Generic commutative ring.

extension(poly, name=None, names=None, **kwds)[source]#

Algebraically extends self by taking the quotient self[x] / (f(x)).

INPUT:

poly – A polynomial whose coefficients are coercible into self
name – (optional) name for the root of \(f\)

Note

Using this method on an algebraically complete field does not return this field; the construction self[x] / (f(x)) is done anyway.

EXAMPLES:

Sage

sage: R = QQ['x']
sage: y = polygen(R)
sage: R.extension(y^2 - 5, 'a')                                             # needs sage.libs.pari
Univariate Quotient Polynomial Ring in a over
 Univariate Polynomial Ring in x over Rational Field with modulus a^2 - 5

Python

>>> from sage.all import *
>>> R = QQ['x']
>>> y = polygen(R)
>>> R.extension(y**Integer(2) - Integer(5), 'a')                                             # needs sage.libs.pari
Univariate Quotient Polynomial Ring in a over
 Univariate Polynomial Ring in x over Rational Field with modulus a^2 - 5

Sage

sage: # needs sage.rings.finite_rings
sage: P.<x> = PolynomialRing(GF(5))
sage: F.<a> = GF(5).extension(x^2 - 2)
sage: P.<t> = F[]
sage: R.<b> = F.extension(t^2 - a); R
Univariate Quotient Polynomial Ring in b over
 Finite Field in a of size 5^2 with modulus b^2 + 4*a

Python

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> P = PolynomialRing(GF(Integer(5)), names=('x',)); (x,) = P._first_ngens(1)
>>> F = GF(Integer(5)).extension(x**Integer(2) - Integer(2), names=('a',)); (a,) = F._first_ngens(1)
>>> P = F['t']; (t,) = P._first_ngens(1)
>>> R = F.extension(t**Integer(2) - a, names=('b',)); (b,) = R._first_ngens(1); R
Univariate Quotient Polynomial Ring in b over
 Finite Field in a of size 5^2 with modulus b^2 + 4*a

fraction_field()[source]#

Return the fraction field of self.

EXAMPLES:

Sage

sage: R = Integers(389)['x,y']
sage: Frac(R)
Fraction Field of Multivariate Polynomial Ring in x, y over Ring of integers modulo 389
sage: R.fraction_field()
Fraction Field of Multivariate Polynomial Ring in x, y over Ring of integers modulo 389

Python

>>> from sage.all import *
>>> R = Integers(Integer(389))['x,y']
>>> Frac(R)
Fraction Field of Multivariate Polynomial Ring in x, y over Ring of integers modulo 389
>>> R.fraction_field()
Fraction Field of Multivariate Polynomial Ring in x, y over Ring of integers modulo 389

ideal_monoid()[source]#

Return the monoid of ideals of this ring.

EXAMPLES:

Sage

sage: ZZ.ideal_monoid()
Monoid of ideals of Integer Ring
sage: R.<x>=QQ[]; R.ideal_monoid()
Monoid of ideals of Univariate Polynomial Ring in x over Rational Field

Python

>>> from sage.all import *
>>> ZZ.ideal_monoid()
Monoid of ideals of Integer Ring
>>> R = QQ['x']; (x,) = R._first_ngens(1); R.ideal_monoid()
Monoid of ideals of Univariate Polynomial Ring in x over Rational Field

is_commutative()[source]#

Return True, since this ring is commutative.

EXAMPLES:

Sage

sage: QQ.is_commutative()
True
sage: ZpCA(7).is_commutative()                                              # needs sage.rings.padics
True
sage: A = QuaternionAlgebra(QQ, -1, -3, names=('i','j','k')); A             # needs sage.combinat sage.modules
Quaternion Algebra (-1, -3) with base ring Rational Field
sage: A.is_commutative()                                                    # needs sage.combinat sage.modules
False

Python

>>> from sage.all import *
>>> QQ.is_commutative()
True
>>> ZpCA(Integer(7)).is_commutative()                                              # needs sage.rings.padics
True
>>> A = QuaternionAlgebra(QQ, -Integer(1), -Integer(3), names=('i','j','k')); A             # needs sage.combinat sage.modules
Quaternion Algebra (-1, -3) with base ring Rational Field
>>> A.is_commutative()                                                    # needs sage.combinat sage.modules
False

krull_dimension()[source]#

Return the Krull dimension of this commutative ring.

The Krull dimension is the length of the longest ascending chain of prime ideals.

localization(additional_units, names=None, normalize=True, category=None)[source]#

Return the localization of self at the given additional units.

EXAMPLES:

Sage

sage: R.<x, y> = GF(3)[]
sage: R.localization((x*y, x**2 + y**2))                                    # needs sage.rings.finite_rings
Multivariate Polynomial Ring in x, y over Finite Field of size 3
 localized at (y, x, x^2 + y^2)
sage: ~y in _                                                               # needs sage.rings.finite_rings
True

Python

>>> from sage.all import *
>>> R = GF(Integer(3))['x, y']; (x, y,) = R._first_ngens(2)
>>> R.localization((x*y, x**Integer(2) + y**Integer(2)))                                    # needs sage.rings.finite_rings
Multivariate Polynomial Ring in x, y over Finite Field of size 3
 localized at (y, x, x^2 + y^2)
>>> ~y in _                                                               # needs sage.rings.finite_rings
True

class sage.rings.ring.DedekindDomain[source]#: Bases: CommutativeRing

class sage.rings.ring.Field[source]#

Bases: CommutativeRing

Generic field

algebraic_closure()[source]#

Return the algebraic closure of self.

Note

This is only implemented for certain classes of field.

EXAMPLES:

Sage

sage: K = PolynomialRing(QQ,'x').fraction_field(); K
Fraction Field of Univariate Polynomial Ring in x over Rational Field
sage: K.algebraic_closure()
Traceback (most recent call last):
...
NotImplementedError: Algebraic closures of general fields not implemented.

Python

>>> from sage.all import *
>>> K = PolynomialRing(QQ,'x').fraction_field(); K
Fraction Field of Univariate Polynomial Ring in x over Rational Field
>>> K.algebraic_closure()
Traceback (most recent call last):
...
NotImplementedError: Algebraic closures of general fields not implemented.

divides(x, y, coerce=True)[source]#

Return True if x divides y in this field (usually True in a field!). If coerce is True (the default), first coerce x and y into self.

EXAMPLES:

Sage

sage: QQ.divides(2, 3/4)
True
sage: QQ.divides(0, 5)
False

Python

>>> from sage.all import *
>>> QQ.divides(Integer(2), Integer(3)/Integer(4))
True
>>> QQ.divides(Integer(0), Integer(5))
False

fraction_field()[source]#

Return the fraction field of self.

EXAMPLES:

Since fields are their own field of fractions, we simply get the original field in return:

Sage

sage: QQ.fraction_field()
Rational Field
sage: RR.fraction_field()                                                   # needs sage.rings.real_mpfr
Real Field with 53 bits of precision
sage: CC.fraction_field()                                                   # needs sage.rings.real_mpfr
Complex Field with 53 bits of precision

sage: x = polygen(ZZ, 'x')
sage: F = NumberField(x^2 + 1, 'i')                                         # needs sage.rings.number_field
sage: F.fraction_field()                                                    # needs sage.rings.number_field
Number Field in i with defining polynomial x^2 + 1

Python

>>> from sage.all import *
>>> QQ.fraction_field()
Rational Field
>>> RR.fraction_field()                                                   # needs sage.rings.real_mpfr
Real Field with 53 bits of precision
>>> CC.fraction_field()                                                   # needs sage.rings.real_mpfr
Complex Field with 53 bits of precision

>>> x = polygen(ZZ, 'x')
>>> F = NumberField(x**Integer(2) + Integer(1), 'i')                                         # needs sage.rings.number_field
>>> F.fraction_field()                                                    # needs sage.rings.number_field
Number Field in i with defining polynomial x^2 + 1

ideal(*gens, **kwds)[source]#

Return the ideal generated by gens.

EXAMPLES:

Sage

sage: QQ.ideal(2)
Principal ideal (1) of Rational Field
sage: QQ.ideal(0)
Principal ideal (0) of Rational Field

Python

>>> from sage.all import *
>>> QQ.ideal(Integer(2))
Principal ideal (1) of Rational Field
>>> QQ.ideal(Integer(0))
Principal ideal (0) of Rational Field

integral_closure()[source]#

Return this field, since fields are integrally closed in their fraction field.

EXAMPLES:

Sage

sage: QQ.integral_closure()
Rational Field
sage: Frac(ZZ['x,y']).integral_closure()
Fraction Field of Multivariate Polynomial Ring in x, y over Integer Ring

Python

>>> from sage.all import *
>>> QQ.integral_closure()
Rational Field
>>> Frac(ZZ['x,y']).integral_closure()
Fraction Field of Multivariate Polynomial Ring in x, y over Integer Ring

is_field(proof=True)[source]#

Return True since this is a field.

EXAMPLES:

Sage

sage: Frac(ZZ['x,y']).is_field()
True

Python

>>> from sage.all import *
>>> Frac(ZZ['x,y']).is_field()
True

is_integrally_closed()[source]#

Return True since fields are trivially integrally closed in their fraction field (since they are their own fraction field).

EXAMPLES:

Sage

sage: Frac(ZZ['x,y']).is_integrally_closed()
True

Python

>>> from sage.all import *
>>> Frac(ZZ['x,y']).is_integrally_closed()
True

is_noetherian()[source]#

Return True since fields are Noetherian rings.

EXAMPLES:

Sage

sage: QQ.is_noetherian()
True

Python

>>> from sage.all import *
>>> QQ.is_noetherian()
True

krull_dimension()[source]#

Return the Krull dimension of this field, which is 0.

EXAMPLES:

Sage

sage: QQ.krull_dimension()
0
sage: Frac(QQ['x,y']).krull_dimension()
0

Python

>>> from sage.all import *
>>> QQ.krull_dimension()
0
>>> Frac(QQ['x,y']).krull_dimension()
0

prime_subfield()[source]#

Return the prime subfield of self.

EXAMPLES:

Sage

sage: k = GF(9, 'a')                                                        # needs sage.rings.finite_rings
sage: k.prime_subfield()                                                    # needs sage.rings.finite_rings
Finite Field of size 3

Python

>>> from sage.all import *
>>> k = GF(Integer(9), 'a')                                                        # needs sage.rings.finite_rings
>>> k.prime_subfield()                                                    # needs sage.rings.finite_rings
Finite Field of size 3

class sage.rings.ring.IntegralDomain[source]#

Bases: CommutativeRing

Generic integral domain class.

This class is deprecated. Please use the sage.categories.integral_domains.IntegralDomains category instead.

is_field(proof=True)[source]#

Return True if this ring is a field.

EXAMPLES:

Sage

sage: GF(7).is_field()
True

Python

>>> from sage.all import *
>>> GF(Integer(7)).is_field()
True

The following examples have their own is_field implementations:

Sage

sage: ZZ.is_field(); QQ.is_field()
False
True
sage: R.<x> = PolynomialRing(QQ); R.is_field()
False

Python

>>> from sage.all import *
>>> ZZ.is_field(); QQ.is_field()
False
True
>>> R = PolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1); R.is_field()
False

is_integrally_closed()[source]#

Return True if this ring is integrally closed in its field of fractions; otherwise return False.

When no algorithm is implemented for this, then this function raises a NotImplementedError.

Note that is_integrally_closed has a naive implementation in fields. For every field \(F\), \(F\) is its own field of fractions, hence every element of \(F\) is integral over \(F\).

EXAMPLES:

Sage

sage: ZZ.is_integrally_closed()
True
sage: QQ.is_integrally_closed()
True
sage: QQbar.is_integrally_closed()                                          # needs sage.rings.number_field
True
sage: GF(5).is_integrally_closed()
True
sage: Z5 = Integers(5); Z5
Ring of integers modulo 5
sage: Z5.is_integrally_closed()
Traceback (most recent call last):
...
AttributeError: 'IntegerModRing_generic_with_category' object has no attribute 'is_integrally_closed'...

Python

>>> from sage.all import *
>>> ZZ.is_integrally_closed()
True
>>> QQ.is_integrally_closed()
True
>>> QQbar.is_integrally_closed()                                          # needs sage.rings.number_field
True
>>> GF(Integer(5)).is_integrally_closed()
True
>>> Z5 = Integers(Integer(5)); Z5
Ring of integers modulo 5
>>> Z5.is_integrally_closed()
Traceback (most recent call last):
...
AttributeError: 'IntegerModRing_generic_with_category' object has no attribute 'is_integrally_closed'...

class sage.rings.ring.NoetherianRing[source]#: Bases: CommutativeRing

class sage.rings.ring.PrincipalIdealDomain[source]#: Bases: CommutativeRing

class sage.rings.ring.Ring[source]#

Bases: ParentWithGens

Generic ring class.

base_extend(R)[source]#

EXAMPLES:

Sage

sage: QQ.base_extend(GF(7))
Traceback (most recent call last):
...
TypeError: no base extension defined
sage: ZZ.base_extend(GF(7))
Finite Field of size 7

Python

>>> from sage.all import *
>>> QQ.base_extend(GF(Integer(7)))
Traceback (most recent call last):
...
TypeError: no base extension defined
>>> ZZ.base_extend(GF(Integer(7)))
Finite Field of size 7

category()[source]#

Return the category to which this ring belongs.

Note

This method exists because sometimes a ring is its own base ring. During initialisation of a ring \(R\), it may be checked whether the base ring (hence, the ring itself) is a ring. Hence, it is necessary that R.category() tells that R is a ring, even before its category is properly initialised.

EXAMPLES:

Sage

sage: FreeAlgebra(QQ, 3, 'x').category()  # todo: use a ring which is not an algebra!   # needs sage.combinat sage.modules
Category of algebras with basis over Rational Field

Python

>>> from sage.all import *
>>> FreeAlgebra(QQ, Integer(3), 'x').category()  # todo: use a ring which is not an algebra!   # needs sage.combinat sage.modules
Category of algebras with basis over Rational Field

Since a quotient of the integers is its own base ring, and during initialisation of a ring it is tested whether the base ring belongs to the category of rings, the following is an indirect test that the category() method of rings returns the category of rings even before the initialisation was successful:

Sage

sage: I = Integers(15)
sage: I.base_ring() is I
True
sage: I.category()
Join of Category of finite commutative rings
    and Category of subquotients of monoids
    and Category of quotients of semigroups
    and Category of finite enumerated sets

Python

>>> from sage.all import *
>>> I = Integers(Integer(15))
>>> I.base_ring() is I
True
>>> I.category()
Join of Category of finite commutative rings
    and Category of subquotients of monoids
    and Category of quotients of semigroups
    and Category of finite enumerated sets

epsilon()[source]#

Return the precision error of elements in this ring.

EXAMPLES:

Sage

sage: RDF.epsilon()
2.220446049250313e-16
sage: ComplexField(53).epsilon()                                            # needs sage.rings.real_mpfr
2.22044604925031e-16
sage: RealField(10).epsilon()                                               # needs sage.rings.real_mpfr
0.0020

Python

>>> from sage.all import *
>>> RDF.epsilon()
2.220446049250313e-16
>>> ComplexField(Integer(53)).epsilon()                                            # needs sage.rings.real_mpfr
2.22044604925031e-16
>>> RealField(Integer(10)).epsilon()                                               # needs sage.rings.real_mpfr
0.0020

For exact rings, zero is returned:

Sage

sage: ZZ.epsilon()
0

Python

>>> from sage.all import *
>>> ZZ.epsilon()
0

This also works over derived rings:

Sage

sage: RR['x'].epsilon()                                                     # needs sage.rings.real_mpfr
2.22044604925031e-16
sage: QQ['x'].epsilon()
0

Python

>>> from sage.all import *
>>> RR['x'].epsilon()                                                     # needs sage.rings.real_mpfr
2.22044604925031e-16
>>> QQ['x'].epsilon()
0

For the symbolic ring, there is no reasonable answer:

Sage

sage: SR.epsilon()                                                          # needs sage.symbolic
Traceback (most recent call last):
...
NotImplementedError

Python

>>> from sage.all import *
>>> SR.epsilon()                                                          # needs sage.symbolic
Traceback (most recent call last):
...
NotImplementedError

ideal(*args, **kwds)[source]#

Return the ideal defined by x, i.e., generated by x.

INPUT:

*x – list or tuple of generators (or several input arguments)
coerce – bool (default: True); this must be a keyword argument. Only set it to False if you are certain that each generator is already in the ring.
ideal_class – callable (default: self._ideal_class_()); this must be a keyword argument. A constructor for ideals, taking the ring as the first argument and then the generators. Usually a subclass of Ideal_generic or Ideal_nc.
Further named arguments (such as side in the case of non-commutative rings) are forwarded to the ideal class.

EXAMPLES:

Sage

sage: R.<x,y> = QQ[]
sage: R.ideal(x,y)
Ideal (x, y) of Multivariate Polynomial Ring in x, y over Rational Field
sage: R.ideal(x+y^2)
Ideal (y^2 + x) of Multivariate Polynomial Ring in x, y over Rational Field
sage: R.ideal( [x^3,y^3+x^3] )
Ideal (x^3, x^3 + y^3) of Multivariate Polynomial Ring in x, y over Rational Field

Python

>>> from sage.all import *
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> R.ideal(x,y)
Ideal (x, y) of Multivariate Polynomial Ring in x, y over Rational Field
>>> R.ideal(x+y**Integer(2))
Ideal (y^2 + x) of Multivariate Polynomial Ring in x, y over Rational Field
>>> R.ideal( [x**Integer(3),y**Integer(3)+x**Integer(3)] )
Ideal (x^3, x^3 + y^3) of Multivariate Polynomial Ring in x, y over Rational Field

Here is an example over a non-commutative ring:

Sage

sage: A = SteenrodAlgebra(2)                                                # needs sage.combinat sage.modules
sage: A.ideal(A.1, A.2^2)                                                   # needs sage.combinat sage.modules
Twosided Ideal (Sq(2), Sq(2,2)) of mod 2 Steenrod algebra, milnor basis
sage: A.ideal(A.1, A.2^2, side='left')                                      # needs sage.combinat sage.modules
Left Ideal (Sq(2), Sq(2,2)) of mod 2 Steenrod algebra, milnor basis

Python

>>> from sage.all import *
>>> A = SteenrodAlgebra(Integer(2))                                                # needs sage.combinat sage.modules
>>> A.ideal(A.gen(1), A.gen(2)**Integer(2))                                                   # needs sage.combinat sage.modules
Twosided Ideal (Sq(2), Sq(2,2)) of mod 2 Steenrod algebra, milnor basis
>>> A.ideal(A.gen(1), A.gen(2)**Integer(2), side='left')                                      # needs sage.combinat sage.modules
Left Ideal (Sq(2), Sq(2,2)) of mod 2 Steenrod algebra, milnor basis

ideal_monoid()[source]#

Return the monoid of ideals of this ring.

EXAMPLES:

Sage

sage: # needs sage.combinat sage.modules
sage: F.<x,y,z> = FreeAlgebra(ZZ, 3)
sage: I = F * [x*y + y*z, x^2 + x*y - y*x - y^2] * F
sage: Q = F.quotient(I)
sage: Q.ideal_monoid()
Monoid of ideals of Quotient of Free Algebra on 3 generators (x, y, z)
 over Integer Ring by the ideal (x*y + y*z, x^2 + x*y - y*x - y^2)
sage: F.<x,y,z> = FreeAlgebra(ZZ, implementation='letterplace')
sage: I = F * [x*y + y*z, x^2 + x*y - y*x - y^2] * F
sage: Q = F.quo(I)
sage: Q.ideal_monoid()
Monoid of ideals of Quotient of Free Associative Unital Algebra
 on 3 generators (x, y, z) over Integer Ring
 by the ideal (x*y + y*z, x*x + x*y - y*x - y*y)

Python

>>> from sage.all import *
>>> # needs sage.combinat sage.modules
>>> F = FreeAlgebra(ZZ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = F._first_ngens(3)
>>> I = F * [x*y + y*z, x**Integer(2) + x*y - y*x - y**Integer(2)] * F
>>> Q = F.quotient(I)
>>> Q.ideal_monoid()
Monoid of ideals of Quotient of Free Algebra on 3 generators (x, y, z)
 over Integer Ring by the ideal (x*y + y*z, x^2 + x*y - y*x - y^2)
>>> F = FreeAlgebra(ZZ, implementation='letterplace', names=('x', 'y', 'z',)); (x, y, z,) = F._first_ngens(3)
>>> I = F * [x*y + y*z, x**Integer(2) + x*y - y*x - y**Integer(2)] * F
>>> Q = F.quo(I)
>>> Q.ideal_monoid()
Monoid of ideals of Quotient of Free Associative Unital Algebra
 on 3 generators (x, y, z) over Integer Ring
 by the ideal (x*y + y*z, x*x + x*y - y*x - y*y)

is_exact()[source]#

Return True if elements of this ring are represented exactly, i.e., there is no precision loss when doing arithmetic.

Note

This defaults to True, so even if it does return True you have no guarantee (unless the ring has properly overloaded this).

EXAMPLES:

Sage

sage: QQ.is_exact()    # indirect doctest
True
sage: ZZ.is_exact()
True
sage: Qp(7).is_exact()                                                      # needs sage.rings.padics
False
sage: Zp(7, type='capped-abs').is_exact()                                   # needs sage.rings.padics
False

Python

>>> from sage.all import *
>>> QQ.is_exact()    # indirect doctest
True
>>> ZZ.is_exact()
True
>>> Qp(Integer(7)).is_exact()                                                      # needs sage.rings.padics
False
>>> Zp(Integer(7), type='capped-abs').is_exact()                                   # needs sage.rings.padics
False

is_field(proof=True)[source]#

Return True if this ring is a field.

INPUT:

proof – (default: True) Determines what to do in unknown cases

ALGORITHM:

If the parameter proof is set to True, the returned value is correct but the method might throw an error. Otherwise, if it is set to False, the method returns True if it can establish that self is a field and False otherwise.

EXAMPLES:

Sage

sage: QQ.is_field()
True
sage: GF(9, 'a').is_field()                                                 # needs sage.rings.finite_rings
True
sage: ZZ.is_field()
False
sage: QQ['x'].is_field()
False
sage: Frac(QQ['x']).is_field()
True

Python

>>> from sage.all import *
>>> QQ.is_field()
True
>>> GF(Integer(9), 'a').is_field()                                                 # needs sage.rings.finite_rings
True
>>> ZZ.is_field()
False
>>> QQ['x'].is_field()
False
>>> Frac(QQ['x']).is_field()
True

This illustrates the use of the proof parameter:

Sage

sage: R.<a,b> = QQ[]
sage: S.<x,y> = R.quo((b^3))                                                # needs sage.libs.singular
sage: S.is_field(proof=True)                                                # needs sage.libs.singular
Traceback (most recent call last):
...
NotImplementedError
sage: S.is_field(proof=False)                                               # needs sage.libs.singular
False

Python

>>> from sage.all import *
>>> R = QQ['a, b']; (a, b,) = R._first_ngens(2)
>>> S = R.quo((b**Integer(3)), names=('x', 'y',)); (x, y,) = S._first_ngens(2)# needs sage.libs.singular
>>> S.is_field(proof=True)                                                # needs sage.libs.singular
Traceback (most recent call last):
...
NotImplementedError
>>> S.is_field(proof=False)                                               # needs sage.libs.singular
False

is_prime_field()[source]#

Return True if this ring is one of the prime fields \(\QQ\) or \(\GF{p}\).

EXAMPLES:

Sage

sage: QQ.is_prime_field()
True
sage: GF(3).is_prime_field()
True
sage: GF(9, 'a').is_prime_field()                                           # needs sage.rings.finite_rings
False
sage: ZZ.is_prime_field()
False
sage: QQ['x'].is_prime_field()
False
sage: Qp(19).is_prime_field()                                               # needs sage.rings.padics
False

Python

>>> from sage.all import *
>>> QQ.is_prime_field()
True
>>> GF(Integer(3)).is_prime_field()
True
>>> GF(Integer(9), 'a').is_prime_field()                                           # needs sage.rings.finite_rings
False
>>> ZZ.is_prime_field()
False
>>> QQ['x'].is_prime_field()
False
>>> Qp(Integer(19)).is_prime_field()                                               # needs sage.rings.padics
False

is_subring(other)[source]#

Return True if the canonical map from self to other is injective.

Raises a NotImplementedError if not known.

EXAMPLES:

Sage

sage: ZZ.is_subring(QQ)
True
sage: ZZ.is_subring(GF(19))
False

Python

>>> from sage.all import *
>>> ZZ.is_subring(QQ)
True
>>> ZZ.is_subring(GF(Integer(19)))
False

one()[source]#

Return the one element of this ring (cached), if it exists.

EXAMPLES:

Sage

sage: ZZ.one()
1
sage: QQ.one()
1
sage: QQ['x'].one()
1

Python

>>> from sage.all import *
>>> ZZ.one()
1
>>> QQ.one()
1
>>> QQ['x'].one()
1

The result is cached:

Sage

sage: ZZ.one() is ZZ.one()
True

Python

>>> from sage.all import *
>>> ZZ.one() is ZZ.one()
True

order()[source]#

The number of elements of self.

EXAMPLES:

Sage

sage: GF(19).order()
19
sage: QQ.order()
+Infinity

Python

>>> from sage.all import *
>>> GF(Integer(19)).order()
19
>>> QQ.order()
+Infinity

principal_ideal(gen, coerce=True)[source]#

Return the principal ideal generated by gen.

EXAMPLES:

Sage

sage: R.<x,y> = ZZ[]
sage: R.principal_ideal(x+2*y)
Ideal (x + 2*y) of Multivariate Polynomial Ring in x, y over Integer Ring

Python

>>> from sage.all import *
>>> R = ZZ['x, y']; (x, y,) = R._first_ngens(2)
>>> R.principal_ideal(x+Integer(2)*y)
Ideal (x + 2*y) of Multivariate Polynomial Ring in x, y over Integer Ring

random_element(bound=2)[source]#

Return a random integer coerced into this ring, where the integer is chosen uniformly from the interval [-bound,bound].

INPUT:

bound – integer (default: 2)

ALGORITHM:

Uses Python’s randint.

unit_ideal()[source]#

Return the unit ideal of this ring.

EXAMPLES:

Sage

sage: Zp(7).unit_ideal()                                                    # needs sage.rings.padics
Principal ideal (1 + O(7^20)) of 7-adic Ring with capped relative precision 20

Python

>>> from sage.all import *
>>> Zp(Integer(7)).unit_ideal()                                                    # needs sage.rings.padics
Principal ideal (1 + O(7^20)) of 7-adic Ring with capped relative precision 20

zero()[source]#

Return the zero element of this ring (cached).

EXAMPLES:

Sage

sage: ZZ.zero()
0
sage: QQ.zero()
0
sage: QQ['x'].zero()
0

Python

>>> from sage.all import *
>>> ZZ.zero()
0
>>> QQ.zero()
0
>>> QQ['x'].zero()
0

The result is cached:

Sage

sage: ZZ.zero() is ZZ.zero()
True

Python

>>> from sage.all import *
>>> ZZ.zero() is ZZ.zero()
True

zero_ideal()[source]#

Return the zero ideal of this ring (cached).

EXAMPLES:

Sage

sage: ZZ.zero_ideal()
Principal ideal (0) of Integer Ring
sage: QQ.zero_ideal()
Principal ideal (0) of Rational Field
sage: QQ['x'].zero_ideal()
Principal ideal (0) of Univariate Polynomial Ring in x over Rational Field

Python

>>> from sage.all import *
>>> ZZ.zero_ideal()
Principal ideal (0) of Integer Ring
>>> QQ.zero_ideal()
Principal ideal (0) of Rational Field
>>> QQ['x'].zero_ideal()
Principal ideal (0) of Univariate Polynomial Ring in x over Rational Field

The result is cached:

Sage

sage: ZZ.zero_ideal() is ZZ.zero_ideal()
True

Python

>>> from sage.all import *
>>> ZZ.zero_ideal() is ZZ.zero_ideal()
True

zeta(n=2, all=False)[source]#

Return a primitive n-th root of unity in self if there is one, or raise a ValueError otherwise.

INPUT:

n – positive integer
all – bool (default: False); whether to return a list of all primitive \(n\)-th roots of unity. If True, raise a ValueError if self is not an integral domain.

OUTPUT:

Element of self of finite order

EXAMPLES:

Sage

sage: QQ.zeta()
-1
sage: QQ.zeta(1)
1
sage: CyclotomicField(6).zeta(6)                                            # needs sage.rings.number_field
zeta6
sage: CyclotomicField(3).zeta(3)                                            # needs sage.rings.number_field
zeta3
sage: CyclotomicField(3).zeta(3).multiplicative_order()                     # needs sage.rings.number_field
3

sage: # needs sage.rings.finite_rings
sage: a = GF(7).zeta(); a
3
sage: a.multiplicative_order()
6
sage: a = GF(49,'z').zeta(); a
z
sage: a.multiplicative_order()
48
sage: a = GF(49,'z').zeta(2); a
6
sage: a.multiplicative_order()
2

sage: QQ.zeta(3)
Traceback (most recent call last):
...
ValueError: no n-th root of unity in rational field
sage: Zp(7, prec=8).zeta()                                                  # needs sage.rings.padics
3 + 4*7 + 6*7^2 + 3*7^3 + 2*7^5 + 6*7^6 + 2*7^7 + O(7^8)

Python

>>> from sage.all import *
>>> QQ.zeta()
-1
>>> QQ.zeta(Integer(1))
1
>>> CyclotomicField(Integer(6)).zeta(Integer(6))                                            # needs sage.rings.number_field
zeta6
>>> CyclotomicField(Integer(3)).zeta(Integer(3))                                            # needs sage.rings.number_field
zeta3
>>> CyclotomicField(Integer(3)).zeta(Integer(3)).multiplicative_order()                     # needs sage.rings.number_field
3

>>> # needs sage.rings.finite_rings
>>> a = GF(Integer(7)).zeta(); a
3
>>> a.multiplicative_order()
6
>>> a = GF(Integer(49),'z').zeta(); a
z
>>> a.multiplicative_order()
48
>>> a = GF(Integer(49),'z').zeta(Integer(2)); a
6
>>> a.multiplicative_order()
2

>>> QQ.zeta(Integer(3))
Traceback (most recent call last):
...
ValueError: no n-th root of unity in rational field
>>> Zp(Integer(7), prec=Integer(8)).zeta()                                                  # needs sage.rings.padics
3 + 4*7 + 6*7^2 + 3*7^3 + 2*7^5 + 6*7^6 + 2*7^7 + O(7^8)

zeta_order()[source]#

Return the order of the distinguished root of unity in self.

EXAMPLES:

Sage

sage: CyclotomicField(19).zeta_order()                                      # needs sage.rings.number_field
38
sage: GF(19).zeta_order()
18
sage: GF(5^3,'a').zeta_order()                                              # needs sage.rings.finite_rings
124
sage: Zp(7, prec=8).zeta_order()                                            # needs sage.rings.padics
6

Python

>>> from sage.all import *
>>> CyclotomicField(Integer(19)).zeta_order()                                      # needs sage.rings.number_field
38
>>> GF(Integer(19)).zeta_order()
18
>>> GF(Integer(5)**Integer(3),'a').zeta_order()                                              # needs sage.rings.finite_rings
124
>>> Zp(Integer(7), prec=Integer(8)).zeta_order()                                            # needs sage.rings.padics
6

sage.rings.ring.is_Ring(x)[source]#

Return True if x is a ring.

EXAMPLES:

Sage

sage: from sage.rings.ring import is_Ring
sage: is_Ring(ZZ)
True
sage: MS = MatrixSpace(QQ, 2)                                                   # needs sage.modules
sage: is_Ring(MS)                                                               # needs sage.modules
True

Python

>>> from sage.all import *
>>> from sage.rings.ring import is_Ring
>>> is_Ring(ZZ)
True
>>> MS = MatrixSpace(QQ, Integer(2))                                                   # needs sage.modules
>>> is_Ring(MS)                                                               # needs sage.modules
True