Ideals of function fields#

Ideals of an order of a function field include all fractional ideals of the order. Sage provides basic arithmetic with fractional ideals.

The fractional ideals of the maximal order of a global function field forms a multiplicative monoid. Sage allows advanced arithmetic with the fractional ideals. For example, an ideal of the maximal order can be factored into a product of prime ideals.

EXAMPLES:

Ideals in the maximal order of a rational function field:

sage: K.<x> = FunctionField(QQ)
sage: O = K.maximal_order()
sage: I = O.ideal(x^3 + 1); I
Ideal (x^3 + 1) of Maximal order of Rational function field in x over Rational Field
sage: I^2
Ideal (x^6 + 2*x^3 + 1) of Maximal order of Rational function field in x over Rational Field
sage: ~I
Ideal (1/(x^3 + 1)) of Maximal order of Rational function field in x over Rational Field
sage: ~I * I
Ideal (1) of Maximal order of Rational function field in x over Rational Field

Ideals in the equation order of an extension of a rational function field:

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x^3 - 1)                                                                            # optional - sage.rings.function_field
sage: O = L.equation_order()                                                                                        # optional - sage.rings.function_field
sage: I = O.ideal(y); I                                                                                             # optional - sage.rings.function_field
Ideal (x^3 + 1, -y) of Order in Function field in y defined by y^2 - x^3 - 1
sage: I^2                                                                                                           # optional - sage.rings.function_field
Ideal (x^3 + 1, (-x^3 - 1)*y) of Order in Function field in y defined by y^2 - x^3 - 1

Ideals in the maximal order of a global function field:

sage: K.<x> = FunctionField(GF(2)); R.<y> = K[]                                                                     # optional - sage.rings.finite_rings
sage: L.<y> = K.extension(y^2 - x^3*y - x)                                                                          # optional - sage.rings.finite_rings sage.rings.function_field
sage: O = L.maximal_order()                                                                                         # optional - sage.rings.finite_rings sage.rings.function_field
sage: I = O.ideal(y)                                                                                                # optional - sage.rings.finite_rings sage.rings.function_field
sage: I^2                                                                                                           # optional - sage.rings.finite_rings sage.rings.function_field
Ideal (x) of Maximal order of Function field in y defined by y^2 + x^3*y + x
sage: ~I                                                                                                            # optional - sage.rings.finite_rings sage.rings.function_field
Ideal (1/x*y) of Maximal order of Function field in y defined by y^2 + x^3*y + x
sage: ~I * I                                                                                                        # optional - sage.rings.finite_rings sage.rings.function_field
Ideal (1) of Maximal order of Function field in y defined by y^2 + x^3*y + x

sage: J = O.ideal(x + y) * I                                                                                        # optional - sage.rings.finite_rings sage.rings.function_field
sage: J.factor()                                                                                                    # optional - sage.rings.finite_rings sage.rings.function_field
(Ideal (y) of Maximal order of Function field in y defined by y^2 + x^3*y + x)^2 *
(Ideal (x^3 + x + 1, y + x) of Maximal order of Function field in y defined by y^2 + x^3*y + x)

Ideals in the maximal infinite order of a global function field:

sage: K.<x> = FunctionField(GF(3^2)); R.<t> = K[]                                                                   # optional - sage.rings.finite_rings
sage: F.<y> = K.extension(t^3 + t^2 - x^4)                                                                          # optional - sage.rings.finite_rings sage.rings.function_field
sage: Oinf = F.maximal_order_infinite()                                                                             # optional - sage.rings.finite_rings sage.rings.function_field
sage: I = Oinf.ideal(1/y)                                                                                           # optional - sage.rings.finite_rings sage.rings.function_field
sage: I + I == I                                                                                                    # optional - sage.rings.finite_rings sage.rings.function_field
True
sage: I^2                                                                                                           # optional - sage.rings.finite_rings sage.rings.function_field
Ideal (1/x^4*y) of Maximal infinite order of Function field in y defined by y^3 + y^2 + 2*x^4
sage: ~I                                                                                                            # optional - sage.rings.finite_rings sage.rings.function_field
Ideal (y) of Maximal infinite order of Function field in y defined by y^3 + y^2 + 2*x^4
sage: ~I * I                                                                                                        # optional - sage.rings.finite_rings sage.rings.function_field
Ideal (1) of Maximal infinite order of Function field in y defined by y^3 + y^2 + 2*x^4
sage: I.factor()                                                                                                    # optional - sage.rings.finite_rings sage.rings.function_field
(Ideal (1/x^3*y^2) of Maximal infinite order of Function field in y defined by y^3 + y^2 + 2*x^4)^4

AUTHORS:

  • William Stein (2010): initial version

  • Maarten Derickx (2011-09-14): fixed ideal_with_gens_over_base()

  • Kwankyu Lee (2017-04-30): added ideals for global function fields

class sage.rings.function_field.ideal.FunctionFieldIdeal(ring)#

Bases: Element

Base class of fractional ideals of function fields.

INPUT:

  • ring – ring of the ideal

EXAMPLES:

sage: K.<x> = FunctionField(GF(7))                                                                              # optional - sage.rings.finite_rings
sage: O = K.equation_order()                                                                                    # optional - sage.rings.finite_rings
sage: O.ideal(x^3 + 1)                                                                                          # optional - sage.rings.finite_rings
Ideal (x^3 + 1) of Maximal order of Rational function field in x over Finite Field of size 7
base_ring()#

Return the base ring of this ideal.

EXAMPLES:

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x^3 - 1)                                                                    # optional - sage.rings.function_field
sage: O = L.equation_order()                                                                                # optional - sage.rings.function_field
sage: I = O.ideal(x^2 + 1)                                                                                  # optional - sage.rings.function_field
sage: I.base_ring()                                                                                         # optional - sage.rings.function_field
Order in Function field in y defined by y^2 - x^3 - 1
divisor()#

Return the divisor corresponding to the ideal.

EXAMPLES:

sage: K.<x> = FunctionField(GF(4))                                                                          # optional - sage.rings.finite_rings
sage: O = K.maximal_order()                                                                                 # optional - sage.rings.finite_rings
sage: I = O.ideal(x*(x + 1)^2/(x^2 + x + 1))                                                                # optional - sage.rings.finite_rings
sage: I.divisor()                                                                                           # optional - sage.rings.finite_rings
Place (x) + 2*Place (x + 1) - Place (x + z2) - Place (x + z2 + 1)

sage: Oinf = K.maximal_order_infinite()                                                                     # optional - sage.rings.finite_rings
sage: I = Oinf.ideal((x + 1)/(x^3 + 1))                                                                     # optional - sage.rings.finite_rings
sage: I.divisor()                                                                                           # optional - sage.rings.finite_rings
2*Place (1/x)

sage: K.<x> = FunctionField(GF(2)); _.<T> = PolynomialRing(K)                                               # optional - sage.rings.finite_rings
sage: F.<y> = K.extension(T^3 - x^2*(x^2 + x + 1)^2)                                                        # optional - sage.rings.finite_rings sage.rings.function_field
sage: O = F.maximal_order()                                                                                 # optional - sage.rings.finite_rings sage.rings.function_field
sage: I = O.ideal(y)                                                                                        # optional - sage.rings.finite_rings sage.rings.function_field
sage: I.divisor()                                                                                           # optional - sage.rings.finite_rings sage.rings.function_field
2*Place (x, (1/(x^3 + x^2 + x))*y^2)
 + 2*Place (x^2 + x + 1, (1/(x^3 + x^2 + x))*y^2)

sage: Oinf = F.maximal_order_infinite()                                                                     # optional - sage.rings.finite_rings sage.rings.function_field
sage: I = Oinf.ideal(y)                                                                                     # optional - sage.rings.finite_rings sage.rings.function_field
sage: I.divisor()                                                                                           # optional - sage.rings.finite_rings sage.rings.function_field
-2*Place (1/x, 1/x^4*y^2 + 1/x^2*y + 1)
 - 2*Place (1/x, 1/x^2*y + 1)

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]                                                             # optional - sage.rings.finite_rings
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)                                                                # optional - sage.rings.finite_rings sage.rings.function_field
sage: O = L.maximal_order()                                                                                 # optional - sage.rings.finite_rings sage.rings.function_field
sage: I = O.ideal(y)                                                                                        # optional - sage.rings.finite_rings sage.rings.function_field
sage: I.divisor()                                                                                           # optional - sage.rings.finite_rings sage.rings.function_field
- Place (x, x*y)
 + 2*Place (x + 1, x*y)

sage: Oinf = L.maximal_order_infinite()                                                                     # optional - sage.rings.finite_rings sage.rings.function_field
sage: I = Oinf.ideal(y)                                                                                     # optional - sage.rings.finite_rings sage.rings.function_field
sage: I.divisor()                                                                                           # optional - sage.rings.finite_rings sage.rings.function_field
- Place (1/x, 1/x*y)
divisor_of_poles()#

Return the divisor of poles corresponding to the ideal.

EXAMPLES:

sage: K.<x> = FunctionField(GF(4))                                                                          # optional - sage.rings.finite_rings
sage: O = K.maximal_order()                                                                                 # optional - sage.rings.finite_rings
sage: I = O.ideal(x*(x + 1)^2/(x^2 + x + 1))                                                                # optional - sage.rings.finite_rings
sage: I.divisor_of_poles()                                                                                  # optional - sage.rings.finite_rings
Place (x + z2) + Place (x + z2 + 1)

sage: K.<x> = FunctionField(GF(2))                                                                          # optional - sage.rings.finite_rings
sage: Oinf = K.maximal_order_infinite()                                                                     # optional - sage.rings.finite_rings
sage: I = Oinf.ideal((x + 1)/(x^3 + 1))                                                                     # optional - sage.rings.finite_rings
sage: I.divisor_of_poles()                                                                                  # optional - sage.rings.finite_rings
0

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]                                                             # optional - sage.rings.finite_rings
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)                                                                # optional - sage.rings.finite_rings sage.rings.function_field
sage: O = L.maximal_order()                                                                                 # optional - sage.rings.finite_rings sage.rings.function_field
sage: I = O.ideal(y)                                                                                        # optional - sage.rings.finite_rings sage.rings.function_field
sage: I.divisor_of_poles()                                                                                  # optional - sage.rings.finite_rings sage.rings.function_field
Place (x, x*y)
divisor_of_zeros()#

Return the divisor of zeros corresponding to the ideal.

EXAMPLES:

sage: K.<x> = FunctionField(GF(4))                                                                          # optional - sage.rings.finite_rings
sage: O = K.maximal_order()                                                                                 # optional - sage.rings.finite_rings
sage: I = O.ideal(x*(x + 1)^2/(x^2 + x + 1))                                                                # optional - sage.rings.finite_rings
sage: I.divisor_of_zeros()                                                                                  # optional - sage.rings.finite_rings
Place (x) + 2*Place (x + 1)

sage: K.<x> = FunctionField(GF(2))                                                                          # optional - sage.rings.finite_rings
sage: Oinf = K.maximal_order_infinite()                                                                     # optional - sage.rings.finite_rings
sage: I = Oinf.ideal((x + 1)/(x^3 + 1))                                                                     # optional - sage.rings.finite_rings
sage: I.divisor_of_zeros()                                                                                  # optional - sage.rings.finite_rings
2*Place (1/x)

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]                                                             # optional - sage.rings.finite_rings
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)                                                                # optional - sage.rings.finite_rings sage.rings.function_field
sage: O = L.maximal_order()                                                                                 # optional - sage.rings.finite_rings sage.rings.function_field
sage: I = O.ideal(y)                                                                                        # optional - sage.rings.finite_rings sage.rings.function_field
sage: I.divisor_of_zeros()                                                                                  # optional - sage.rings.finite_rings sage.rings.function_field
2*Place (x + 1, x*y)
factor()#

Return the factorization of this ideal.

Subclass of this class should define _factor() method that returns a list of prime ideal and multiplicity pairs.

EXAMPLES:

sage: K.<x> = FunctionField(GF(4))                                                                          # optional - sage.rings.finite_rings
sage: O = K.maximal_order()                                                                                 # optional - sage.rings.finite_rings
sage: I = O.ideal(x^3*(x + 1)^2)                                                                            # optional - sage.rings.finite_rings
sage: I.factor()                                                                                            # optional - sage.rings.finite_rings
(Ideal (x) of Maximal order of Rational function field in x
over Finite Field in z2 of size 2^2)^3 *
(Ideal (x + 1) of Maximal order of Rational function field in x
over Finite Field in z2 of size 2^2)^2

sage: Oinf = K.maximal_order_infinite()                                                                     # optional - sage.rings.finite_rings
sage: I = Oinf.ideal((x + 1)/(x^3 + 1))                                                                     # optional - sage.rings.finite_rings
sage: I.factor()                                                                                            # optional - sage.rings.finite_rings
(Ideal (1/x) of Maximal infinite order of Rational function field in x
over Finite Field in z2 of size 2^2)^2

sage: K.<x> = FunctionField(GF(2)); _.<T> = PolynomialRing(K)                                               # optional - sage.rings.finite_rings
sage: F.<y> = K.extension(T^3 - x^2*(x^2 + x + 1)^2)                                                        # optional - sage.rings.finite_rings sage.rings.function_field
sage: O = F.maximal_order()                                                                                 # optional - sage.rings.finite_rings sage.rings.function_field
sage: I = O.ideal(y)                                                                                        # optional - sage.rings.finite_rings sage.rings.function_field
sage: I == I.factor().prod()                                                                                # optional - sage.rings.finite_rings sage.rings.function_field
True

sage: Oinf = F.maximal_order_infinite()                                                                     # optional - sage.rings.finite_rings sage.rings.function_field
sage: f= 1/x                                                                                                # optional - sage.rings.finite_rings sage.rings.function_field
sage: I = Oinf.ideal(f)                                                                                     # optional - sage.rings.finite_rings sage.rings.function_field
sage: I.factor()                                                                                            # optional - sage.rings.finite_rings sage.rings.function_field
(Ideal ((1/(x^4 + x^3 + x^2))*y^2 + 1/x^2*y + 1) of Maximal infinite order
of Function field in y defined by y^3 + x^6 + x^4 + x^2) *
(Ideal ((1/(x^4 + x^3 + x^2))*y^2 + 1) of Maximal infinite order
of Function field in y defined by y^3 + x^6 + x^4 + x^2)

sage: K.<x> = FunctionField(QQ); _.<Y> = K[]
sage: F.<y> = K.extension(Y^3 - x^2*(x^2 + x + 1)^2)                                                        # optional - sage.rings.function_field
sage: O = F.maximal_order()                                                                                 # optional - sage.rings.function_field
sage: I = O.ideal(y)                                                                                        # optional - sage.rings.function_field
sage: I == I.factor().prod()                                                                                # optional - sage.rings.function_field
True

sage: K.<x> = FunctionField(QQ); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)                                                                # optional - sage.rings.function_field
sage: O = L.maximal_order()                                                                                 # optional - sage.rings.function_field
sage: I = O.ideal(y)                                                                                        # optional - sage.rings.function_field
sage: I == I.factor().prod()                                                                                # optional - sage.rings.function_field
True
gens_reduced()#

Return reduced generators.

For now, this method just looks at the generators and sees if any can be removed without changing the ideal. It prefers principal representations (a single generator) over all others, and otherwise picks the generator set with the shortest print representation.

This method is provided so that ideals in function fields have the method gens_reduced(), just like ideals of number fields. Sage linear algebra machinery sometimes requires this.

EXAMPLES:

sage: K.<x> = FunctionField(GF(7))                                                                          # optional - sage.rings.finite_rings
sage: O = K.equation_order()                                                                                # optional - sage.rings.finite_rings
sage: I = O.ideal(x, x^2, x^2 + x)                                                                          # optional - sage.rings.finite_rings
sage: I.gens_reduced()                                                                                      # optional - sage.rings.finite_rings
(x,)
place()#

Return the place associated with this prime ideal.

EXAMPLES:

sage: K.<x> = FunctionField(GF(4))                                                                          # optional - sage.rings.finite_rings
sage: O = K.maximal_order()                                                                                 # optional - sage.rings.finite_rings
sage: I = O.ideal(x^2 + x + 1)                                                                              # optional - sage.rings.finite_rings
sage: I.place()                                                                                             # optional - sage.rings.finite_rings
Traceback (most recent call last):
...
TypeError: not a prime ideal
sage: I = O.ideal(x^3 + x + 1)                                                                              # optional - sage.rings.finite_rings
sage: I.place()                                                                                             # optional - sage.rings.finite_rings
Place (x^3 + x + 1)

sage: K.<x> = FunctionField(GF(2))                                                                          # optional - sage.rings.finite_rings
sage: Oinf = K.maximal_order_infinite()                                                                     # optional - sage.rings.finite_rings
sage: I = Oinf.ideal((x + 1)/(x^3 + 1))                                                                     # optional - sage.rings.finite_rings
sage: p = I.factor()[0][0]                                                                                  # optional - sage.rings.finite_rings
sage: p.place()                                                                                             # optional - sage.rings.finite_rings
Place (1/x)

sage: K.<x> = FunctionField(GF(2)); _.<t> = PolynomialRing(K)                                               # optional - sage.rings.finite_rings
sage: F.<y> = K.extension(t^3 - x^2*(x^2+x+1)^2)                                                            # optional - sage.rings.finite_rings sage.rings.function_field
sage: O = F.maximal_order()                                                                                 # optional - sage.rings.finite_rings sage.rings.function_field
sage: I = O.ideal(y)                                                                                        # optional - sage.rings.finite_rings sage.rings.function_field
sage: [f.place() for f,_ in I.factor()]                                                                     # optional - sage.rings.finite_rings sage.rings.function_field
[Place (x, (1/(x^3 + x^2 + x))*y^2),
 Place (x^2 + x + 1, (1/(x^3 + x^2 + x))*y^2)]

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]                                                             # optional - sage.rings.finite_rings
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)                                                                # optional - sage.rings.finite_rings sage.rings.function_field
sage: O = L.maximal_order()                                                                                 # optional - sage.rings.finite_rings sage.rings.function_field
sage: I = O.ideal(y)                                                                                        # optional - sage.rings.finite_rings sage.rings.function_field
sage: [f.place() for f,_ in I.factor()]                                                                     # optional - sage.rings.finite_rings sage.rings.function_field
[Place (x, x*y), Place (x + 1, x*y)]

sage: K.<x> = FunctionField(GF(3^2)); R.<t> = PolynomialRing(K)                                             # optional - sage.rings.finite_rings
sage: F.<y> = K.extension(t^3 + t^2 - x^4)                                                                  # optional - sage.rings.finite_rings sage.rings.function_field
sage: Oinf = F.maximal_order_infinite()                                                                     # optional - sage.rings.finite_rings sage.rings.function_field
sage: I = Oinf.ideal(1/x)                                                                                   # optional - sage.rings.finite_rings sage.rings.function_field
sage: I.factor()                                                                                            # optional - sage.rings.finite_rings sage.rings.function_field
(Ideal (1/x^3*y^2) of Maximal infinite order of Function field
in y defined by y^3 + y^2 + 2*x^4)^3
sage: J = I.factor()[0][0]                                                                                  # optional - sage.rings.finite_rings sage.rings.function_field
sage: J.is_prime()                                                                                          # optional - sage.rings.finite_rings sage.rings.function_field
True
sage: J.place()                                                                                             # optional - sage.rings.finite_rings sage.rings.function_field
Place (1/x, 1/x^3*y^2)

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]                                                             # optional - sage.rings.finite_rings
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)                                                                # optional - sage.rings.finite_rings sage.rings.function_field
sage: Oinf = L.maximal_order_infinite()                                                                     # optional - sage.rings.finite_rings sage.rings.function_field
sage: I = Oinf.ideal(1/x)                                                                                   # optional - sage.rings.finite_rings sage.rings.function_field
sage: I.factor()                                                                                            # optional - sage.rings.finite_rings sage.rings.function_field
(Ideal (1/x*y) of Maximal infinite order of Function field in y
defined by y^2 + y + (x^2 + 1)/x)^2
sage: J = I.factor()[0][0]                                                                                  # optional - sage.rings.finite_rings sage.rings.function_field
sage: J.is_prime()                                                                                          # optional - sage.rings.finite_rings sage.rings.function_field
True
sage: J.place()                                                                                             # optional - sage.rings.finite_rings sage.rings.function_field
Place (1/x, 1/x*y)
ring()#

Return the ring to which this ideal belongs.

EXAMPLES:

sage: K.<x> = FunctionField(GF(7))                                                                          # optional - sage.rings.finite_rings
sage: O = K.equation_order()                                                                                # optional - sage.rings.finite_rings
sage: I = O.ideal(x, x^2, x^2 + x)                                                                          # optional - sage.rings.finite_rings
sage: I.ring()                                                                                              # optional - sage.rings.finite_rings
Maximal order of Rational function field in x over Finite Field of size 7
class sage.rings.function_field.ideal.FunctionFieldIdealInfinite(ring)#

Bases: FunctionFieldIdeal

Base class of ideals of maximal infinite orders

class sage.rings.function_field.ideal.FunctionFieldIdealInfinite_module(ring, module)#

Bases: FunctionFieldIdealInfinite, Ideal_generic

A fractional ideal specified by a finitely generated module over the integers of the base field.

INPUT:

  • ring – order in a function field

  • module – module

EXAMPLES:

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x^3 - 1)                                                                        # optional - sage.rings.function_field
sage: O = L.equation_order()                                                                                    # optional - sage.rings.function_field
sage: O.ideal(y)                                                                                                # optional - sage.rings.function_field
Ideal (x^3 + 1, -y) of Order in Function field in y defined by y^2 - x^3 - 1
module()#

Return the module over the maximal order of the base field that underlies this ideal.

The formation of the module is compatible with the vector space corresponding to the function field.

EXAMPLES:

sage: K.<x> = FunctionField(GF(7))                                                                          # optional - sage.rings.finite_rings
sage: O = K.maximal_order(); O                                                                              # optional - sage.rings.finite_rings
Maximal order of Rational function field in x over Finite Field of size 7
sage: K.polynomial_ring()                                                                                   # optional - sage.rings.finite_rings
Univariate Polynomial Ring in x over Rational function field in x over Finite Field of size 7
sage: I = O.ideal([x^2 + 1, x*(x^2+1)])                                                                     # optional - sage.rings.finite_rings
sage: I.gens()                                                                                              # optional - sage.rings.finite_rings
(x^2 + 1,)
sage: I.module()                                                                                            # optional - sage.rings.finite_rings
Free module of degree 1 and rank 1 over Maximal order of Rational function field in x over Finite Field of size 7
Echelon basis matrix:
[x^2 + 1]
sage: V, from_V, to_V = K.vector_space(); V                                                                 # optional - sage.rings.finite_rings
Vector space of dimension 1 over Rational function field in x over Finite Field of size 7
sage: I.module().is_submodule(V)                                                                            # optional - sage.rings.finite_rings
True
class sage.rings.function_field.ideal.FunctionFieldIdeal_module(ring, module)#

Bases: FunctionFieldIdeal, Ideal_generic

A fractional ideal specified by a finitely generated module over the integers of the base field.

INPUT:

  • ring – an order in a function field

  • module – a module of the order

EXAMPLES:

An ideal in an extension of a rational function field:

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x^3 - 1)                                                                        # optional - sage.rings.function_field
sage: O = L.equation_order()                                                                                    # optional - sage.rings.function_field
sage: I = O.ideal(y)                                                                                            # optional - sage.rings.function_field
sage: I                                                                                                         # optional - sage.rings.function_field
Ideal (x^3 + 1, -y) of Order in Function field in y defined by y^2 - x^3 - 1
sage: I^2                                                                                                       # optional - sage.rings.function_field
Ideal (x^3 + 1, (-x^3 - 1)*y) of Order in Function field in y defined by y^2 - x^3 - 1
gen(i)#

Return the i-th generator in the current basis of this ideal.

EXAMPLES:

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x^3 - 1)                                                                    # optional - sage.rings.function_field
sage: O = L.equation_order()                                                                                # optional - sage.rings.function_field
sage: I = O.ideal(x^2 + 1)                                                                                  # optional - sage.rings.function_field
sage: I.gen(1)                                                                                              # optional - sage.rings.function_field
(x^2 + 1)*y
gens()#

Return a set of generators of this ideal.

EXAMPLES:

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x^3 - 1)                                                                    # optional - sage.rings.function_field
sage: O = L.equation_order()                                                                                # optional - sage.rings.function_field
sage: I = O.ideal(x^2 + 1)                                                                                  # optional - sage.rings.function_field
sage: I.gens()                                                                                              # optional - sage.rings.function_field
(x^2 + 1, (x^2 + 1)*y)
intersection(other)#

Return the intersection of this ideal and other.

EXAMPLES:

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x^3 - 1)                                                                    # optional - sage.rings.function_field
sage: O = L.equation_order()                                                                                # optional - sage.rings.function_field
sage: I = O.ideal(y^3); J = O.ideal(y^2)                                                                    # optional - sage.rings.function_field
sage: Z = I.intersection(J); Z                                                                              # optional - sage.rings.function_field
Ideal (x^6 + 2*x^3 + 1, (-x^3 - 1)*y) of Order in Function field in y defined by y^2 - x^3 - 1
sage: y^2 in Z                                                                                              # optional - sage.rings.function_field
False
sage: y^3 in Z                                                                                              # optional - sage.rings.function_field
True
module()#

Return the module over the maximal order of the base field that underlies this ideal.

The formation of the module is compatible with the vector space corresponding to the function field.

OUTPUT:

  • a module over the maximal order of the base field of the ideal

EXAMPLES:

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x^3 - 1)                                                                    # optional - sage.rings.function_field
sage: O = L.equation_order();  O                                                                            # optional - sage.rings.function_field
Order in Function field in y defined by y^2 - x^3 - 1
sage: I = O.ideal(x^2 + 1)                                                                                  # optional - sage.rings.function_field
sage: I.gens()                                                                                              # optional - sage.rings.function_field
(x^2 + 1, (x^2 + 1)*y)
sage: I.module()                                                                                            # optional - sage.rings.function_field
Free module of degree 2 and rank 2 over Maximal order of Rational function field in x over Rational Field
Echelon basis matrix:
[x^2 + 1       0]
[      0 x^2 + 1]
sage: V, from_V, to_V = L.vector_space(); V                                                                 # optional - sage.rings.function_field
Vector space of dimension 2 over Rational function field in x over Rational Field
sage: I.module().is_submodule(V)                                                                            # optional - sage.rings.function_field
True
ngens()#

Return the number of generators in the basis.

EXAMPLES:

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x^3 - 1)                                                                    # optional - sage.rings.function_field
sage: O = L.equation_order()                                                                                # optional - sage.rings.function_field
sage: I = O.ideal(x^2 + 1)                                                                                  # optional - sage.rings.function_field
sage: I.ngens()                                                                                             # optional - sage.rings.function_field
2
class sage.rings.function_field.ideal.IdealMonoid(R)#

Bases: UniqueRepresentation, Parent

The monoid of ideals in orders of function fields.

INPUT:

  • R – order

EXAMPLES:

sage: K.<x> = FunctionField(GF(2))                                                                              # optional - sage.rings.finite_rings
sage: O = K.maximal_order()                                                                                     # optional - sage.rings.finite_rings
sage: M = O.ideal_monoid(); M                                                                                   # optional - sage.rings.finite_rings
Monoid of ideals of Maximal order of Rational function field in x over Finite Field of size 2
ring()#

Return the ring of which this is the ideal monoid.

EXAMPLES:

sage: K.<x> = FunctionField(GF(2))                                                                          # optional - sage.rings.finite_rings
sage: O = K.maximal_order()                                                                                 # optional - sage.rings.finite_rings
sage: M = O.ideal_monoid(); M.ring() is O                                                                   # optional - sage.rings.finite_rings
True