Quotients of Univariate Polynomial Rings

EXAMPLES:

sage: R.<x> = QQ[]
sage: S = R.quotient(x**3 - 3*x + 1, 'alpha')
sage: S.gen()**2 in S
True
sage: x in S
True
sage: S.gen() in R
False
sage: 1 in S
True

class sage.rings.polynomial.polynomial_quotient_ring.PolynomialQuotientRingFactory

Bases: UniqueFactory

Create a quotient of a polynomial ring.

INPUT:

• ring - a univariate polynomial ring

• polynomial - an element of ring with a unit leading coefficient

• names - (optional) name for the variable

OUTPUT: Creates the quotient ring \(R/I\), where \(R\) is the ring and \(I\) is the principal ideal generated by polynomial.

EXAMPLES:

We create the quotient ring \(\ZZ[x]/(x^3+7)\), and demonstrate many basic functions with it:

sage: Z = IntegerRing()
sage: R = PolynomialRing(Z, 'x'); x = R.gen()
sage: S = R.quotient(x^3 + 7, 'a'); a = S.gen()
sage: S
Univariate Quotient Polynomial Ring in a
 over Integer Ring with modulus x^3 + 7
sage: a^3
-7
sage: S.is_field()
False
sage: a in S
True
sage: x in S
True
sage: a in R
False
sage: S.polynomial_ring()
Univariate Polynomial Ring in x over Integer Ring
sage: S.modulus()
x^3 + 7
sage: S.degree()
3

We create the “iterated” polynomial ring quotient

\[R = (\GF{2}[y]/(y^{2}+y+1))[x]/(x^3 - 5).\]

sage: # needs sage.libs.ntl
sage: A.<y> = PolynomialRing(GF(2)); A
Univariate Polynomial Ring in y over Finite Field of size 2 (using GF2X)
sage: B = A.quotient(y^2 + y + 1, 'y2'); B
Univariate Quotient Polynomial Ring in y2 over Finite Field of size 2
 with modulus y^2 + y + 1
sage: C = PolynomialRing(B, 'x'); x = C.gen(); C
Univariate Polynomial Ring in x
 over Univariate Quotient Polynomial Ring in y2
  over Finite Field of size 2 with modulus y^2 + y + 1
sage: R = C.quotient(x^3 - 5); R
Univariate Quotient Polynomial Ring in xbar
 over Univariate Quotient Polynomial Ring in y2
  over Finite Field of size 2 with modulus y^2 + y + 1
  with modulus x^3 + 1

Next we create a number field, but viewed as a quotient of a polynomial ring over \(\QQ\):

sage: R = PolynomialRing(RationalField(), 'x'); x = R.gen()
sage: S = R.quotient(x^3 + 2*x - 5, 'a'); S
Univariate Quotient Polynomial Ring in a over Rational Field
 with modulus x^3 + 2*x - 5
sage: S.is_field()
True
sage: S.degree()
3

There are conversion functions for easily going back and forth between quotients of polynomial rings over \(\QQ\) and number fields:

sage: K = S.number_field(); K                                                   # needs sage.rings.number_field
Number Field in a with defining polynomial x^3 + 2*x - 5
sage: K.polynomial_quotient_ring()                                              # needs sage.rings.number_field
Univariate Quotient Polynomial Ring in a
 over Rational Field with modulus x^3 + 2*x - 5

The leading coefficient must be a unit (but need not be 1).

sage: R = PolynomialRing(Integers(9), 'x'); x = R.gen()
sage: S = R.quotient(2*x^4 + 2*x^3 + x + 2, 'a')
sage: S = R.quotient(3*x^4 + 2*x^3 + x + 2, 'a')
Traceback (most recent call last):
...
TypeError: polynomial must have unit leading coefficient

Another example:

sage: R.<x> = PolynomialRing(IntegerRing())
sage: f = x^2 + 1
sage: R.quotient(f)
Univariate Quotient Polynomial Ring in xbar over Integer Ring with modulus x^2 + 1

This shows that the issue at github issue #5482 is solved:

sage: R.<x> = PolynomialRing(QQ)
sage: f = x^2 - 1
sage: R.quotient_by_principal_ideal(f)
Univariate Quotient Polynomial Ring in xbar over Rational Field with modulus x^2 - 1

create_key(ring, polynomial, names=None)

Return a unique description of the quotient ring specified by the arguments.

EXAMPLES:

sage: R.<x> = QQ[]
sage: PolynomialQuotientRing.create_key(R, x + 1)
(Univariate Polynomial Ring in x over Rational Field, x + 1, ('xbar',))

create_object(version, key)

Return the quotient ring specified by key.

EXAMPLES:

sage: R.<x> = QQ[]
sage: PolynomialQuotientRing.create_object((8, 0, 0),
....:                                      (R, x^2 - 1, ('xbar')))
Univariate Quotient Polynomial Ring in xbar over Rational Field with modulus x^2 - 1

class sage.rings.polynomial.polynomial_quotient_ring.PolynomialQuotientRing_coercion

Bases: DefaultConvertMap_unique

A coercion map from a PolynomialQuotientRing to a PolynomialQuotientRing that restricts to the coercion map on the underlying ring of constants.

EXAMPLES:

sage: R.<x> = ZZ[]
sage: S.<x> = QQ[]
sage: f = S.quo(x^2 + 1).coerce_map_from(R.quo(x^2 + 1)); f
Coercion map:
  From: Univariate Quotient Polynomial Ring in xbar over Integer Ring
        with modulus x^2 + 1
  To:   Univariate Quotient Polynomial Ring in xbar over Rational Field
        with modulus x^2 + 1

is_injective()

Return whether this coercion is injective.

EXAMPLES:

If the modulus of the domain and the codomain is the same and the leading coefficient is a unit in the domain, then the map is injective if the underlying map on the constants is:

sage: R.<x> = ZZ[]
sage: S.<x> = QQ[]
sage: f = S.quo(x^2 + 1).coerce_map_from(R.quo(x^2 + 1))
sage: f.is_injective()
True

is_surjective()

Return whether this coercion is surjective.

EXAMPLES:

If the underlying map on constants is surjective, then this coercion is surjective since the modulus of the codomain divides the modulus of the domain:

sage: R.<x> = ZZ[]
sage: f = R.quo(x).coerce_map_from(R.quo(x^2))
sage: f.is_surjective()
True

If the modulus of the domain and the codomain is the same, then the map is surjective iff the underlying map on the constants is:

sage: # needs sage.rings.padics
sage: A.<a> = ZqCA(9)
sage: R.<x> = A[]
sage: S.<x> = A.fraction_field()[]
sage: f = S.quo(x^2 + 2).coerce_map_from(R.quo(x^2 + 2))
sage: f.is_surjective()
False

class sage.rings.polynomial.polynomial_quotient_ring.PolynomialQuotientRing_domain(ring, polynomial, name=None, category=None)

Bases: PolynomialQuotientRing_generic, IntegralDomain

EXAMPLES:

sage: R.<x> = PolynomialRing(ZZ)
sage: S.<xbar> = R.quotient(x^2 + 1)
sage: S
Univariate Quotient Polynomial Ring in xbar
 over Integer Ring with modulus x^2 + 1
sage: loads(S.dumps()) == S
True
sage: loads(xbar.dumps()) == xbar
True

field_extension(names)

Take a polynomial quotient ring, and return a tuple with three elements: the NumberField defined by the same polynomial quotient ring, a homomorphism from its parent to the NumberField sending the generators to one another, and the inverse isomorphism.

OUTPUT:

• field

• homomorphism from self to field

• homomorphism from field to self

EXAMPLES:

sage: # needs sage.rings.number_field
sage: R.<x> = PolynomialRing(Rationals())
sage: S.<alpha> = R.quotient(x^3 - 2)
sage: F.<b>, f, g = S.field_extension()
sage: F
Number Field in b with defining polynomial x^3 - 2
sage: a = F.gen()
sage: f(alpha)
b
sage: g(a)
alpha

Note that the parent ring must be an integral domain:

sage: R.<x> = GF(25, 'f25')['x']                                            # needs sage.rings.finite_rings
sage: S.<a> = R.quo(x^3 - 2)                                                # needs sage.rings.finite_rings
sage: F, g, h = S.field_extension('b')                                      # needs sage.rings.finite_rings
Traceback (most recent call last):
...
AttributeError: 'PolynomialQuotientRing_generic_with_category' object has no attribute 'field_extension'...

Over a finite field, the corresponding field extension is not a number field:

sage: # needs sage.modules sage.rings.finite_rings
sage: R.<x> = GF(25, 'a')['x']
sage: S.<a> = R.quo(x^3 + 2*x + 1)
sage: F, g, h = S.field_extension('b')
sage: h(F.0^2 + 3)
a^2 + 3
sage: g(x^2 + 2)
b^2 + 2

We do an example involving a relative number field:

sage: # needs sage.rings.number_field
sage: R.<x> = QQ['x']
sage: K.<a> = NumberField(x^3 - 2)
sage: S.<X> = K['X']
sage: Q.<b> = S.quo(X^3 + 2*X + 1)
sage: Q.field_extension('b')
(Number Field in b with defining polynomial X^3 + 2*X + 1 over its base field, ...
  Defn: b |--> b, Relative number field morphism:
  From: Number Field in b with defining polynomial X^3 + 2*X + 1 over its base field
  To:   Univariate Quotient Polynomial Ring in b over Number Field in a with defining polynomial x^3 - 2 with modulus X^3 + 2*X + 1
  Defn: b |--> b
        a |--> a)

We slightly change the example above so it works.

sage: # needs sage.rings.number_field
sage: R.<x> = QQ['x']
sage: K.<a> = NumberField(x^3 - 2)
sage: S.<X> = K['X']
sage: f = (X+a)^3 + 2*(X+a) + 1
sage: f
X^3 + 3*a*X^2 + (3*a^2 + 2)*X + 2*a + 3
sage: Q.<z> = S.quo(f)
sage: F.<w>, g, h = Q.field_extension()
sage: c = g(z)
sage: f(c)
0
sage: h(g(z))
z
sage: g(h(w))
w

AUTHORS:

• Craig Citro (2006-08-07)

• William Stein (2006-08-06)

class sage.rings.polynomial.polynomial_quotient_ring.PolynomialQuotientRing_field(ring, polynomial, name=None, category=None)

Bases: PolynomialQuotientRing_domain, Field

EXAMPLES:

sage: # needs sage.rings.number_field
sage: R.<x> = PolynomialRing(QQ)
sage: S.<xbar> = R.quotient(x^2 + 1)
sage: S
Univariate Quotient Polynomial Ring in xbar over Rational Field
 with modulus x^2 + 1
sage: loads(S.dumps()) == S
True
sage: loads(xbar.dumps()) == xbar
True

base_field()

Alias for base_ring(), when we’re defined over a field.

complex_embeddings(prec=53)

Return all homomorphisms of this ring into the approximate complex field with precision prec.

EXAMPLES:

sage: # needs sage.rings.number_field
sage: R.<x> = QQ[]
sage: f = x^5 + x + 17
sage: k = R.quotient(f)
sage: v = k.complex_embeddings(100)
sage: [phi(k.0^2) for phi in v]
[2.9757207403766761469671194565,
 -2.4088994371613850098316292196 + 1.9025410530350528612407363802*I,
 -2.4088994371613850098316292196 - 1.9025410530350528612407363802*I,
 0.92103906697304693634806949137 - 3.0755331188457794473265418086*I,
 0.92103906697304693634806949137 + 3.0755331188457794473265418086*I]

class sage.rings.polynomial.polynomial_quotient_ring.PolynomialQuotientRing_generic(ring, polynomial, name=None, category=None)

Bases: QuotientRing_generic

Quotient of a univariate polynomial ring by an ideal.

EXAMPLES:

sage: R.<x> = PolynomialRing(Integers(8)); R
Univariate Polynomial Ring in x over Ring of integers modulo 8
sage: S.<xbar> = R.quotient(x^2 + 1); S
Univariate Quotient Polynomial Ring in xbar over Ring of integers modulo 8
 with modulus x^2 + 1

We demonstrate object persistence.

sage: loads(S.dumps()) == S
True
sage: loads(xbar.dumps()) == xbar
True

We create some sample homomorphisms;

sage: R.<x> = PolynomialRing(ZZ)
sage: S = R.quo(x^2 - 4)
sage: f = S.hom([2])
sage: f
Ring morphism:
  From: Univariate Quotient Polynomial Ring in xbar over Integer Ring
        with modulus x^2 - 4
  To:   Integer Ring
  Defn: xbar |--> 2
sage: f(x)
2
sage: f(x^2 - 4)
0
sage: f(x^2)
4

Element

alias of PolynomialQuotientRingElement

S_class_group(S, proof=True)

If self is an étale algebra \(D\) over a number field \(K\) (i.e. a quotient of \(K[x]\) by a squarefree polynomial) and \(S\) is a finite set of places of \(K\), return a list of generators of the \(S\)-class group of \(D\).

NOTE:

Since the ideal function behaves differently over number fields than over polynomial quotient rings (the quotient does not even know its ring of integers), we return a set of pairs (gen, order), where gen is a tuple of generators of an ideal \(I\) and order is the order of \(I\) in the \(S\)-class group.

INPUT:

• S - a set of primes of the coefficient ring

• proof - if False, assume the GRH in computing the class group

OUTPUT:

A list of generators of the \(S\)-class group, in the form (gen, order), where gen is a tuple of elements generating a fractional ideal \(I\) and order is the order of \(I\) in the \(S\)-class group.

EXAMPLES:

A trivial algebra over \(\QQ(\sqrt{-5})\) has the same class group as its base:

sage: # needs sage.rings.number_field
sage: K.<a> = QuadraticField(-5)
sage: R.<x> = K[]
sage: S.<xbar> = R.quotient(x)
sage: S.S_class_group([])
[((2, -a + 1), 2)]

When we include the prime \((2, -a+1)\), the \(S\)-class group becomes trivial:

sage: S.S_class_group([K.ideal(2, -a+1)])                                   # needs sage.rings.number_field
[]

Here is an example where the base and the extension both contribute to the class group:

sage: # needs sage.rings.number_field
sage: K.<a> = QuadraticField(-5)
sage: K.class_group()
Class group of order 2 with structure C2 of Number Field in a
 with defining polynomial x^2 + 5 with a = 2.236067977499790?*I
sage: R.<x> = K[]
sage: S.<xbar> = R.quotient(x^2 + 23)
sage: S.S_class_group([])
[((2, -a + 1, 1/2*xbar + 1/2, -1/2*a*xbar + 1/2*a + 1), 6)]
sage: S.S_class_group([K.ideal(3, a-1)])
[]
sage: S.S_class_group([K.ideal(2, a+1)])
[]
sage: S.S_class_group([K.ideal(a)])
[((2, -a + 1, 1/2*xbar + 1/2, -1/2*a*xbar + 1/2*a + 1), 6)]

Now we take an example over a nontrivial base with two factors, each contributing to the class group:

sage: # needs sage.rings.number_field
sage: K.<a> = QuadraticField(-5)
sage: R.<x> = K[]
sage: S.<xbar> = R.quotient((x^2 + 23) * (x^2 + 31))
sage: S.S_class_group([])           # not tested
[((1/4*xbar^2 + 31/4,
   (-1/8*a + 1/8)*xbar^2 - 31/8*a + 31/8,
   1/16*xbar^3 + 1/16*xbar^2 + 31/16*xbar + 31/16,
   -1/16*a*xbar^3 + (1/16*a + 1/8)*xbar^2 - 31/16*a*xbar + 31/16*a + 31/8),
  6),
 ((-1/4*xbar^2 - 23/4,
   (1/8*a - 1/8)*xbar^2 + 23/8*a - 23/8,
   -1/16*xbar^3 - 1/16*xbar^2 - 23/16*xbar - 23/16,
   1/16*a*xbar^3 + (-1/16*a - 1/8)*xbar^2 + 23/16*a*xbar - 23/16*a - 23/8),
  6),
 ((-5/4*xbar^2 - 115/4,
   1/4*a*xbar^2 + 23/4*a,
   -1/16*xbar^3 - 7/16*xbar^2 - 23/16*xbar - 161/16,
   1/16*a*xbar^3 - 1/16*a*xbar^2 + 23/16*a*xbar - 23/16*a),
  2)]

By using the ideal \((a)\), we cut the part of the class group coming from \(x^2 + 31\) from 12 to 2, i.e. we lose a generator of order 6 (this was fixed in github issue #14489):

sage: S.S_class_group([K.ideal(a)])  # representation varies        # not tested, needs sage.rings.number_field
[((1/4*xbar^2 + 31/4, (-1/8*a + 1/8)*xbar^2 - 31/8*a + 31/8,
   1/16*xbar^3 + 1/16*xbar^2 + 31/16*xbar + 31/16,
   -1/16*a*xbar^3 + (1/16*a + 1/8)*xbar^2 - 31/16*a*xbar + 31/16*a + 31/8),
  6),
 ((-1/4*xbar^2 - 23/4, (1/8*a - 1/8)*xbar^2 + 23/8*a - 23/8,
   -1/16*xbar^3 - 1/16*xbar^2 - 23/16*xbar - 23/16,
   1/16*a*xbar^3 + (-1/16*a - 1/8)*xbar^2 + 23/16*a*xbar - 23/16*a - 23/8),
  2)]

Note that all the returned values live where we expect them to:

sage: # needs sage.rings.number_field
sage: CG = S.S_class_group([])
sage: type(CG[0][0][1])
<class 'sage.rings.polynomial.polynomial_quotient_ring.PolynomialQuotientRing_generic_with_category.element_class'>
sage: type(CG[0][1])
<class 'sage.rings.integer.Integer'>

S_units(S, proof=True)

If self is an étale algebra \(D\) over a number field \(K\) (i.e. a quotient of \(K[x]\) by a squarefree polynomial) and \(S\) is a finite set of places of \(K\), return a list of generators of the group of \(S\)-units of \(D\).

INPUT:

• S - a set of primes of the base field

• proof - if False, assume the GRH in computing the class group

OUTPUT:

A list of generators of the \(S\)-unit group, in the form (gen, order), where gen is a unit of order order.

EXAMPLES:

sage: K.<a> = QuadraticField(-3)                                            # needs sage.rings.number_field
sage: K.unit_group()                                                        # needs sage.rings.number_field
Unit group with structure C6 of Number Field in a
 with defining polynomial x^2 + 3 with a = 1.732050807568878?*I

sage: # needs sage.rings.number_field
sage: x = polygen(ZZ, 'x')
sage: K.<a> = QQ['x'].quotient(x^2 + 3)
sage: u, o = K.S_units([])[0]; o
6
sage: 2*u - 1 in {a, -a}
True
sage: u^6
1
sage: u^3
-1
sage: 2*u^2 + 1 in {a, -a}
True

sage: # needs sage.rings.number_field
sage: K.<a> = QuadraticField(-3)
sage: y = polygen(K)
sage: L.<b> = K['y'].quotient(y^3 + 5); L
Univariate Quotient Polynomial Ring in b over Number Field in a
 with defining polynomial x^2 + 3 with a = 1.732050807568878?*I
 with modulus y^3 + 5
sage: [u for u, o in L.S_units([]) if o is Infinity]
[(-1/3*a - 1)*b^2 - 4/3*a*b - 5/6*a + 7/2,
 2/3*a*b^2 + (2/3*a - 2)*b - 5/6*a - 7/2]
sage: [u for u, o in L.S_units([K.ideal(1/2*a - 3/2)])
....:  if o is Infinity]
[(-1/6*a - 1/2)*b^2 + (1/3*a - 1)*b + 4/3*a,
 (-1/3*a - 1)*b^2 - 4/3*a*b - 5/6*a + 7/2,
 2/3*a*b^2 + (2/3*a - 2)*b - 5/6*a - 7/2]
sage: [u for u, o in L.S_units([K.ideal(2)]) if o is Infinity]
[(1/2*a - 1/2)*b^2 + (a + 1)*b + 3,
 (1/6*a + 1/2)*b^2 + (-1/3*a + 1)*b - 5/6*a + 1/2,
 (1/6*a + 1/2)*b^2 + (-1/3*a + 1)*b - 5/6*a - 1/2,
 (-1/3*a - 1)*b^2 - 4/3*a*b - 5/6*a + 7/2,
 2/3*a*b^2 + (2/3*a - 2)*b - 5/6*a - 7/2]

Note that all the returned values live where we expect them to:

sage: # needs sage.rings.number_field
sage: U = L.S_units([])
sage: type(U[0][0])
<class 'sage.rings.polynomial.polynomial_quotient_ring.PolynomialQuotientRing_field_with_category.element_class'>
sage: type(U[0][1])
<class 'sage.rings.integer.Integer'>
sage: type(U[1][1])
<class 'sage.rings.infinity.PlusInfinity'>

ambient()

base_ring()

Return the base ring of the polynomial ring, of which this ring is a quotient.

EXAMPLES:

The base ring of \(\ZZ[z]/(z^3 + z^2 + z + 1)\) is \(\ZZ\).

sage: R.<z> = PolynomialRing(ZZ)
sage: S.<beta> = R.quo(z^3 + z^2 + z + 1)
sage: S.base_ring()
Integer Ring

Next we make a polynomial quotient ring over \(S\) and ask for its base ring.

sage: T.<t> = PolynomialRing(S)
sage: W = T.quotient(t^99 + 99)
sage: W.base_ring()
Univariate Quotient Polynomial Ring in beta
 over Integer Ring with modulus z^3 + z^2 + z + 1

cardinality()

Return the number of elements of this quotient ring.

order is an alias of cardinality.

EXAMPLES:

sage: R.<x> = ZZ[]
sage: R.quo(1).cardinality()
1
sage: R.quo(x^3 - 2).cardinality()
+Infinity

sage: R.quo(1).order()
1
sage: R.quo(x^3 - 2).order()
+Infinity

sage: # needs sage.rings.finite_rings
sage: R.<x> = GF(9, 'a')[]
sage: R.quo(2*x^3 + x + 1).cardinality()
729
sage: GF(9, 'a').extension(2*x^3 + x + 1).cardinality()
729
sage: R.quo(2).cardinality()
1

characteristic()

Return the characteristic of this quotient ring.

This is always the same as the characteristic of the base ring.

EXAMPLES:

sage: R.<z> = PolynomialRing(ZZ)
sage: S.<a> = R.quo(z - 19)
sage: S.characteristic()
0
sage: R.<x> = PolynomialRing(GF(9, 'a'))                                    # needs sage.rings.finite_rings
sage: S = R.quotient(x^3 + 1)                                               # needs sage.rings.finite_rings
sage: S.characteristic()                                                    # needs sage.rings.finite_rings
3

class_group(proof=True)

If self is a quotient ring of a polynomial ring over a number field \(K\), by a polynomial of nonzero discriminant, return a list of generators of the class group.

NOTE:

Since the ideal function behaves differently over number fields than over polynomial quotient rings (the quotient does not even know its ring of integers), we return a set of pairs (gen, order), where gen is a tuple of generators of an ideal \(I\) and order is the order of \(I\) in the class group.

INPUT:

• proof - if False, assume the GRH in computing the class group

OUTPUT:

A list of pairs (gen, order), where gen is a tuple of elements generating a fractional ideal and order is the order of \(I\) in the class group.

EXAMPLES:

sage: # needs sage.rings.number_field
sage: K.<a> = QuadraticField(-3)
sage: K.class_group()
Class group of order 1 of Number Field in a
 with defining polynomial x^2 + 3 with a = 1.732050807568878?*I
sage: x = polygen(QQ, 'x')
sage: K.<a> = QQ['x'].quotient(x^2 + 3)
sage: K.class_group()
[]

A trivial algebra over \(\QQ(\sqrt{-5})\) has the same class group as its base:

sage: # needs sage.rings.number_field
sage: K.<a> = QuadraticField(-5)
sage: R.<x> = K[]
sage: S.<xbar> = R.quotient(x)
sage: S.class_group()
[((2, -a + 1), 2)]

The same algebra constructed in a different way:

sage: x = polygen(ZZ, 'x')
sage: K.<a> = QQ['x'].quotient(x^2 + 5)
sage: K.class_group(())                                                     # needs sage.rings.number_field
[((2, a + 1), 2)]

Here is an example where the base and the extension both contribute to the class group:

sage: # needs sage.rings.number_field
sage: K.<a> = QuadraticField(-5)
sage: K.class_group()
Class group of order 2 with structure C2 of Number Field in a
 with defining polynomial x^2 + 5 with a = 2.236067977499790?*I
sage: R.<x> = K[]
sage: S.<xbar> = R.quotient(x^2 + 23)
sage: S.class_group()
[((2, -a + 1, 1/2*xbar + 1/2, -1/2*a*xbar + 1/2*a + 1), 6)]

Here is an example of a product of number fields, both of which contribute to the class group:

sage: # needs sage.rings.number_field
sage: R.<x> = QQ[]
sage: S.<xbar> = R.quotient((x^2 + 23) * (x^2 + 47))
sage: S.class_group()
[((1/12*xbar^2 + 47/12,
   1/48*xbar^3 - 1/48*xbar^2 + 47/48*xbar - 47/48),
  3),
 ((-1/12*xbar^2 - 23/12,
   -1/48*xbar^3 - 1/48*xbar^2 - 23/48*xbar - 23/48),
  5)]

Now we take an example over a nontrivial base with two factors, each contributing to the class group:

sage: # needs sage.rings.number_field
sage: K.<a> = QuadraticField(-5)
sage: R.<x> = K[]
sage: S.<xbar> = R.quotient((x^2 + 23) * (x^2 + 31))
sage: S.class_group()               # not tested
[((1/4*xbar^2 + 31/4,
   (-1/8*a + 1/8)*xbar^2 - 31/8*a + 31/8,
   1/16*xbar^3 + 1/16*xbar^2 + 31/16*xbar + 31/16,
   -1/16*a*xbar^3 + (1/16*a + 1/8)*xbar^2 - 31/16*a*xbar + 31/16*a + 31/8),
  6),
 ((-1/4*xbar^2 - 23/4,
   (1/8*a - 1/8)*xbar^2 + 23/8*a - 23/8,
   -1/16*xbar^3 - 1/16*xbar^2 - 23/16*xbar - 23/16,
   1/16*a*xbar^3 + (-1/16*a - 1/8)*xbar^2 + 23/16*a*xbar - 23/16*a - 23/8),
  6),
 ((-5/4*xbar^2 - 115/4,
   1/4*a*xbar^2 + 23/4*a,
   -1/16*xbar^3 - 7/16*xbar^2 - 23/16*xbar - 161/16,
   1/16*a*xbar^3 - 1/16*a*xbar^2 + 23/16*a*xbar - 23/16*a),
  2)]

Note that all the returned values live where we expect them to:

sage: # needs sage.rings.number_field
sage: CG = S.class_group()
sage: type(CG[0][0][1])
<class 'sage.rings.polynomial.polynomial_quotient_ring.PolynomialQuotientRing_generic_with_category.element_class'>
sage: type(CG[0][1])
<class 'sage.rings.integer.Integer'>

construction()

Functorial construction of self

EXAMPLES:

sage: P.<t> = ZZ[]
sage: Q = P.quo(5 + t^2)
sage: F, R = Q.construction()
sage: F(R) == Q
True
sage: P.<t> = GF(3)[]
sage: Q = P.quo([2 + t^2])
sage: F, R = Q.construction()
sage: F(R) == Q
True

AUTHOR:

– Simon King (2010-05)

cover_ring()

Return the polynomial ring of which this ring is the quotient.

EXAMPLES:

sage: R.<x> = PolynomialRing(QQ)
sage: S = R.quotient(x^2 - 2)
sage: S.polynomial_ring()
Univariate Polynomial Ring in x over Rational Field

degree()

Return the degree of this quotient ring. The degree is the degree of the polynomial that we quotiented out by.

EXAMPLES:

sage: R.<x> = PolynomialRing(GF(3))
sage: S = R.quotient(x^2005 + 1)
sage: S.degree()
2005

discriminant(v=None)

Return the discriminant of this ring over the base ring. This is by definition the discriminant of the polynomial that we quotiented out by.

EXAMPLES:

sage: R.<x> = PolynomialRing(QQ)
sage: S = R.quotient(x^3 + x^2 + x + 1)
sage: S.discriminant()
-16
sage: S = R.quotient((x + 1) * (x + 1))
sage: S.discriminant()
0

The discriminant of the quotient polynomial ring need not equal the discriminant of the corresponding number field, since the discriminant of a number field is by definition the discriminant of the ring of integers of the number field:

sage: S = R.quotient(x^2 - 8)
sage: S.number_field().discriminant()                                       # needs sage.rings.number_field
8
sage: S.discriminant()
32

gen(n=0)

Return the generator of this quotient ring. This is the equivalence class of the image of the generator of the polynomial ring.

EXAMPLES:

sage: R.<x> = PolynomialRing(QQ)
sage: S = R.quotient(x^2 - 8, 'gamma')
sage: S.gen()
gamma

is_field(proof=True)

Return whether or not this quotient ring is a field.

EXAMPLES:

sage: R.<z> = PolynomialRing(ZZ)
sage: S = R.quo(z^2 - 2)
sage: S.is_field()
False
sage: R.<x> = PolynomialRing(QQ)
sage: S = R.quotient(x^2 - 2)
sage: S.is_field()
True

If proof is True, requires the is_irreducible method of the modulus to be implemented:

sage: # needs sage.rings.padics
sage: R1.<x> = Qp(2)[]
sage: F1 = R1.quotient_ring(x^2 + x + 1)
sage: R2.<x> = F1[]
sage: F2 = R2.quotient_ring(x^2 + x + 1)
sage: F2.is_field()
Traceback (most recent call last):
...
NotImplementedError: cannot rewrite Univariate Quotient Polynomial Ring in
 xbar over 2-adic Field with capped relative precision 20 with modulus
 (1 + O(2^20))*x^2 + (1 + O(2^20))*x + 1 + O(2^20) as an isomorphic ring
sage: F2.is_field(proof = False)
False

is_finite()

Return whether or not this quotient ring is finite.

EXAMPLES:

sage: R.<x> = ZZ[]
sage: R.quo(1).is_finite()
True
sage: R.quo(x^3 - 2).is_finite()
False

sage: R.<x> = GF(9, 'a')[]                                                  # needs sage.rings.finite_rings
sage: R.quo(2*x^3 + x + 1).is_finite()                                      # needs sage.rings.finite_rings
True
sage: R.quo(2).is_finite()                                                  # needs sage.rings.finite_rings
True

sage: P.<v> = GF(2)[]
sage: P.quotient(v^2 - v).is_finite()
True

is_integral_domain(proof=True)

Return whether or not this quotient ring is an integral domain.

EXAMPLES:

sage: R.<z> = PolynomialRing(ZZ)

sage: S = R.quotient(z^2 - z)
sage: S.is_integral_domain()
False
sage: T = R.quotient(z^2 + 1)
sage: T.is_integral_domain()
True
sage: U = R.quotient(-1)
sage: U.is_integral_domain()
False

sage: # needs sage.libs.singular
sage: R2.<y> = PolynomialRing(R)
sage: S2 = R2.quotient(z^2 - y^3)
sage: S2.is_integral_domain()
True
sage: S3 = R2.quotient(z^2 - 2*y*z + y^2)
sage: S3.is_integral_domain()
False

sage: R.<z> = PolynomialRing(ZZ.quotient(4))
sage: S = R.quotient(z - 1)
sage: S.is_integral_domain()
False

krull_dimension()

Return the Krull dimension.

This is the Krull dimension of the base ring, unless the quotient is zero.

EXAMPLES:

sage: x = polygen(ZZ, 'x')
sage: R = PolynomialRing(ZZ, 'x').quotient(x**6 - 1)
sage: R.krull_dimension()
1
sage: R = PolynomialRing(ZZ, 'x').quotient(1)
sage: R.krull_dimension()
-1

lift(x)

Return an element of the ambient ring mapping to the given argument.

EXAMPLES:

sage: P.<x> = QQ[]
sage: Q = P.quotient(x^2 + 2)
sage: Q.lift(Q.0^3)
-2*x
sage: Q(-2*x)
-2*xbar
sage: Q.0^3
-2*xbar

modulus()

Return the polynomial modulus of this quotient ring.

EXAMPLES:

sage: R.<x> = PolynomialRing(GF(3))
sage: S = R.quotient(x^2 - 2)
sage: S.modulus()
x^2 + 1

ngens()

Return the number of generators of this quotient ring over the base ring. This function always returns 1.

EXAMPLES:

sage: R.<x> = PolynomialRing(QQ)
sage: S.<y> = PolynomialRing(R)
sage: T.<z> = S.quotient(y + x)
sage: T
Univariate Quotient Polynomial Ring in z over
 Univariate Polynomial Ring in x over Rational Field with modulus y + x
sage: T.ngens()
1

number_field()

Return the number field isomorphic to this quotient polynomial ring, if possible.

EXAMPLES:

sage: # needs sage.rings.number_field
sage: R.<x> = PolynomialRing(QQ)
sage: S.<alpha> = R.quotient(x^29 - 17*x - 1)
sage: K = S.number_field(); K
Number Field in alpha with defining polynomial x^29 - 17*x - 1
sage: alpha = K.gen()
sage: alpha^29
17*alpha + 1

order()

Return the number of elements of this quotient ring.

order is an alias of cardinality.

EXAMPLES:

sage: R.<x> = ZZ[]
sage: R.quo(1).cardinality()
1
sage: R.quo(x^3 - 2).cardinality()
+Infinity

sage: R.quo(1).order()
1
sage: R.quo(x^3 - 2).order()
+Infinity

sage: # needs sage.rings.finite_rings
sage: R.<x> = GF(9, 'a')[]
sage: R.quo(2*x^3 + x + 1).cardinality()
729
sage: GF(9, 'a').extension(2*x^3 + x + 1).cardinality()
729
sage: R.quo(2).cardinality()
1

polynomial_ring()

Return the polynomial ring of which this ring is the quotient.

EXAMPLES:

sage: R.<x> = PolynomialRing(QQ)
sage: S = R.quotient(x^2 - 2)
sage: S.polynomial_ring()
Univariate Polynomial Ring in x over Rational Field

random_element(degree=None, *args, **kwds)

Return a random element of this quotient ring.

INPUT:

• degree - Optional argument: either an integer for fixing the degree, or a tuple of the minimum and maximum degree. By default the degree is n - 1 with n the degree of the polynomial ring. Note that the degree of the polynomial is fixed before the modulo calculation. So when \(degree\) is bigger than the degree of the polynomial ring, the degree of the returned polynomial would be lower than \(degree\).

• *args, **kwds - Arguments for randomization that are passed on to the random_element method of the polynomial ring, and from there to the base ring

OUTPUT:

Element of this quotient ring

EXAMPLES:

sage: # needs sage.modules sage.rings.finite_rings
sage: F1.<a> = GF(2^7)
sage: P1.<x> = F1[]
sage: F2 = F1.extension(x^2 + x + 1, 'u')
sage: F2.random_element().parent() is F2
True

retract(x)

Return the coercion of x into this polynomial quotient ring.

The rings that coerce into the quotient ring canonically are:

• this ring

• any canonically isomorphic ring

• anything that coerces into the ring of which this is the quotient

selmer_generators(S, m, proof=True)

If self is an étale algebra \(D\) over a number field \(K\) (i.e. a quotient of \(K[x]\) by a squarefree polynomial) and \(S\) is a finite set of places of \(K\), compute the Selmer group \(D(S,m)\). This is the subgroup of \(D^*/(D^*)^m\) consisting of elements \(a\) such that \(D(\sqrt[m]{a})/D\) is unramified at all primes of \(D\) lying above a place outside of \(S\).

INPUT:

• S - A set of primes of the coefficient ring (which is a number field).

• m - a positive integer

• proof - if False, assume the GRH in computing the class group

OUTPUT:

A list of generators of \(D(S,m)\).

EXAMPLES:

sage: # needs sage.rings.number_field
sage: K.<a> = QuadraticField(-5)
sage: R.<x> = K[]
sage: D.<T> = R.quotient(x)
sage: D.selmer_generators((), 2)
[-1, 2]
sage: D.selmer_generators([K.ideal(2, -a + 1)], 2)
[2, -1]
sage: D.selmer_generators([K.ideal(2, -a + 1), K.ideal(3, a + 1)], 2)
[2, a + 1, -1]
sage: D.selmer_generators((K.ideal(2, -a + 1), K.ideal(3, a + 1)), 4)
[2, a + 1, -1]
sage: D.selmer_generators([K.ideal(2, -a + 1)], 3)
[2]
sage: D.selmer_generators([K.ideal(2, -a + 1), K.ideal(3, a + 1)], 3)
[2, a + 1]
sage: D.selmer_generators([K.ideal(2, -a + 1),
....:                      K.ideal(3, a + 1),
....:                      K.ideal(a)], 3)
[2, a + 1, -a]

selmer_group(S, m, proof=True)

If self is an étale algebra \(D\) over a number field \(K\) (i.e. a quotient of \(K[x]\) by a squarefree polynomial) and \(S\) is a finite set of places of \(K\), compute the Selmer group \(D(S,m)\). This is the subgroup of \(D^*/(D^*)^m\) consisting of elements \(a\) such that \(D(\sqrt[m]{a})/D\) is unramified at all primes of \(D\) lying above a place outside of \(S\).

INPUT:

• S - A set of primes of the coefficient ring (which is a number field).

• m - a positive integer

• proof - if False, assume the GRH in computing the class group

OUTPUT:

A list of generators of \(D(S,m)\).

EXAMPLES:

sage: # needs sage.rings.number_field
sage: K.<a> = QuadraticField(-5)
sage: R.<x> = K[]
sage: D.<T> = R.quotient(x)
sage: D.selmer_generators((), 2)
[-1, 2]
sage: D.selmer_generators([K.ideal(2, -a + 1)], 2)
[2, -1]
sage: D.selmer_generators([K.ideal(2, -a + 1), K.ideal(3, a + 1)], 2)
[2, a + 1, -1]
sage: D.selmer_generators((K.ideal(2, -a + 1), K.ideal(3, a + 1)), 4)
[2, a + 1, -1]
sage: D.selmer_generators([K.ideal(2, -a + 1)], 3)
[2]
sage: D.selmer_generators([K.ideal(2, -a + 1), K.ideal(3, a + 1)], 3)
[2, a + 1]
sage: D.selmer_generators([K.ideal(2, -a + 1),
....:                      K.ideal(3, a + 1),
....:                      K.ideal(a)], 3)
[2, a + 1, -a]

units(proof=True)

If this quotient ring is over a number field K, by a polynomial of nonzero discriminant, returns a list of generators of the units.

INPUT:

• proof - if False, assume the GRH in computing the class group

OUTPUT:

A list of generators of the unit group, in the form (gen, order), where gen is a unit of order order.

EXAMPLES:

sage: K.<a> = QuadraticField(-3)                                            # needs sage.rings.number_field
sage: K.unit_group()                                                        # needs sage.rings.number_field
Unit group with structure C6 of
 Number Field in a with defining polynomial x^2 + 3 with a = 1.732050807568878?*I

sage: # needs sage.rings.number_field
sage: x = polygen(ZZ, 'x')
sage: K.<a> = QQ['x'].quotient(x^2 + 3)
sage: u = K.units()[0][0]
sage: 2*u - 1 in {a, -a}
True
sage: u^6
1
sage: u^3
-1
sage: 2*u^2 + 1 in {a, -a}
True
sage: x = polygen(ZZ, 'x')
sage: K.<a> = QQ['x'].quotient(x^2 + 5)
sage: K.units(())
[(-1, 2)]

sage: # needs sage.rings.number_field
sage: K.<a> = QuadraticField(-3)
sage: y = polygen(K)
sage: L.<b> = K['y'].quotient(y^3 + 5); L
Univariate Quotient Polynomial Ring in b over Number Field in a
 with defining polynomial x^2 + 3 with a = 1.732050807568878?*I
 with modulus y^3 + 5
sage: [u for u, o in L.units() if o is Infinity]
[(-1/3*a - 1)*b^2 - 4/3*a*b - 5/6*a + 7/2,
 2/3*a*b^2 + (2/3*a - 2)*b - 5/6*a - 7/2]
sage: L.<b> = K.extension(y^3 + 5)
sage: L.unit_group()
Unit group with structure C6 x Z x Z of
 Number Field in b with defining polynomial x^3 + 5 over its base field
sage: L.unit_group().gens()    # abstract generators
(u0, u1, u2)
sage: L.unit_group().gens_values()[1:]
[(-1/3*a - 1)*b^2 - 4/3*a*b - 5/6*a + 7/2,
 2/3*a*b^2 + (2/3*a - 2)*b - 5/6*a - 7/2]

Note that all the returned values live where we expect them to:

sage: # needs sage.rings.number_field
sage: L.<b> = K['y'].quotient(y^3 + 5)
sage: U = L.units()
sage: type(U[0][0])
<class 'sage.rings.polynomial.polynomial_quotient_ring.PolynomialQuotientRing_field_with_category.element_class'>
sage: type(U[0][1])
<class 'sage.rings.integer.Integer'>
sage: type(U[1][1])
<class 'sage.rings.infinity.PlusInfinity'>

sage.rings.polynomial.polynomial_quotient_ring.is_PolynomialQuotientRing(x)