Ideals of function fields: extension

class sage.rings.function_field.ideal_polymod.FunctionFieldIdealInfinite_polymod(ring, ideal)

Bases: FunctionFieldIdealInfinite

Ideals of the infinite maximal order of an algebraic function field.

INPUT:

• ring – infinite maximal order of the function field

• ideal – ideal in the inverted function field

EXAMPLES:

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

gens()

Return a set of generators of this ideal.

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(3^2)); R.<t> = PolynomialRing(K)
sage: F.<y> = K.extension(t^3 + t^2 - x^4)
sage: Oinf = F.maximal_order_infinite()
sage: I = Oinf.ideal(x + y)
sage: I.gens()
(x, y, 1/x^2*y^2)

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: Oinf = L.maximal_order_infinite()
sage: I = Oinf.ideal(x + y)
sage: I.gens()
(x, y)

gens_over_base()

Return a set of generators of this ideal.

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(3^2)); _.<t> = K[]
sage: F.<y> = K.extension(t^3 + t^2 - x^4)
sage: Oinf = F.maximal_order_infinite()
sage: I = Oinf.ideal(x + y)
sage: I.gens_over_base()
(x, y, 1/x^2*y^2)

gens_two()

Return a set of at most two generators of this ideal.

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(3^2)); R.<t> = PolynomialRing(K)
sage: F.<y> = K.extension(t^3 + t^2 - x^4)
sage: Oinf = F.maximal_order_infinite()
sage: I = Oinf.ideal(x + y)
sage: I.gens_two()
(x, y)

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: Oinf = L.maximal_order_infinite()
sage: I = Oinf.ideal(x + y)
sage: I.gens_two()
(x,)

ideal_below()

Return a set of generators of this ideal.

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(3^2)); _.<t> = K[]
sage: F.<y> = K.extension(t^3 + t^2 - x^4)
sage: Oinf = F.maximal_order_infinite()
sage: I = Oinf.ideal(1/y^2)
sage: I.ideal_below()
Ideal (x^3) of Maximal order of Rational function field
in x over Finite Field in z2 of size 3^2

is_prime()

Return True if this ideal is a prime ideal.

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(3^2)); _.<t> = PolynomialRing(K)
sage: F.<y> = K.extension(t^3 + t^2 - x^4)
sage: Oinf = F.maximal_order_infinite()
sage: I = Oinf.ideal(1/x)
sage: I.factor()
(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: I.is_prime()
False
sage: J = I.factor()[0][0]
sage: J.is_prime()
True

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

prime_below()

Return the prime of the base order that underlies this prime ideal.

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(3^2)); _.<t> = PolynomialRing(K)
sage: F.<y> = K.extension(t^3 + t^2 - x^4)
sage: Oinf = F.maximal_order_infinite()
sage: I = Oinf.ideal(1/x)
sage: I.factor()
(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]
sage: J.is_prime()
True
sage: J.prime_below()
Ideal (1/x) of Maximal infinite order of Rational function field
in x over Finite Field in z2 of size 3^2

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: Oinf = L.maximal_order_infinite()
sage: I = Oinf.ideal(1/x)
sage: I.factor()
(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]
sage: J.is_prime()
True
sage: J.prime_below()
Ideal (1/x) of Maximal infinite order of Rational function field in x
over Finite Field of size 2

valuation(ideal)

Return the valuation of ideal with respect to this prime ideal.

INPUT:

• ideal – fractional ideal

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: Oinf = L.maximal_order_infinite()
sage: I = Oinf.ideal(y)
sage: [f.valuation(I) for f,_ in I.factor()]
[-1]

class sage.rings.function_field.ideal_polymod.FunctionFieldIdeal_global(ring, hnf, denominator=1)

Bases: FunctionFieldIdeal_polymod

Fractional ideals of canonical function fields

INPUT:

• ring – order in a function field

• hnf – matrix in hermite normal form

• denominator – denominator

The rows of hnf is a basis of the ideal, which itself is denominator times the fractional ideal.

EXAMPLES:

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

gens()

Return a set of generators of this ideal.

This provides whatever set of generators as quickly as possible.

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 - x^3*Y - x)
sage: O = L.maximal_order()
sage: I = O.ideal(x + y)
sage: I.gens()
(x^4 + x^2 + x, y + x)

sage: # needs sage.rings.finite_rings
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: O = L.maximal_order()
sage: I = O.ideal(x + y)
sage: I.gens()
(x^3 + 1, y + x)

gens_two()

Return two generators of this fractional ideal.

If the ideal is principal, one generator may be returned.

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(2)); _.<t> = K[]
sage: F.<y> = K.extension(t^3 - x^2*(x^2 + x + 1)^2)
sage: O = F.maximal_order()
sage: I = O.ideal(y)
sage: I  # indirect doctest
Ideal (y) of Maximal order of Function field
in y defined by y^3 + x^6 + x^4 + x^2
sage: ~I  # indirect doctest
Ideal ((1/(x^6 + x^4 + x^2))*y^2) of Maximal order of Function field
in y defined by y^3 + x^6 + x^4 + x^2

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

class sage.rings.function_field.ideal_polymod.FunctionFieldIdeal_polymod(ring, hnf, denominator=1)

Bases: FunctionFieldIdeal

Fractional ideals of algebraic function fields

INPUT:

• ring – order in a function field

• hnf – matrix in hermite normal form

• denominator – denominator

The rows of hnf is a basis of the ideal, which itself is denominator times the fractional ideal.

EXAMPLES:

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

basis_matrix()

Return the matrix of basis vectors of this ideal as a module.

The basis matrix is by definition the hermite norm form of the ideal divided by the denominator.

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(2)); R.<t> = PolynomialRing(K)
sage: F.<y> = K.extension(t^3 - x^2*(x^2+x+1)^2)
sage: O = F.maximal_order()
sage: I = O.ideal(x, 1/y)
sage: I.denominator() * I.basis_matrix() == I.hnf()
True

denominator()

Return the denominator of this fractional ideal.

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(7)); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x^3 - 1)
sage: O = L.maximal_order()
sage: I = O.ideal(y/(y+1))
sage: d = I.denominator(); d
x^3
sage: d in O
True

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x^3 - 1)
sage: O = L.maximal_order()
sage: I = O.ideal(y/(y+1))
sage: d = I.denominator(); d
x^3
sage: d in O
True

gens()

Return a set of generators of this ideal.

This provides whatever set of generators as quickly as possible.

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 - x^3*Y - x)
sage: O = L.maximal_order()
sage: I = O.ideal(x + y)
sage: I.gens()
(x^4 + x^2 + x, y + x)

sage: # needs sage.rings.finite_rings
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: O = L.maximal_order()
sage: I = O.ideal(x + y)
sage: I.gens()
(x^3 + 1, y + x)

gens_over_base()

Return the generators of this ideal as a module over the maximal order of the base rational function field.

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 - x^3*Y - x)
sage: O = L.maximal_order()
sage: I = O.ideal(x + y)
sage: I.gens_over_base()
(x^4 + x^2 + x, y + x)

sage: # needs sage.rings.finite_rings
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: O = L.maximal_order()
sage: I = O.ideal(x + y)
sage: I.gens_over_base()
(x^3 + 1, y + x)

hnf()

Return the matrix in hermite normal form representing this ideal.

See also denominator()

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(7)); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x^3 - 1)
sage: O = L.maximal_order()
sage: I = O.ideal(y*(y+1)); I.hnf()
[x^6 + x^3         0]
[  x^3 + 1         1]

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x^3 - 1)
sage: O = L.maximal_order()
sage: I = O.ideal(y*(y+1)); I.hnf()
[x^6 + x^3         0]
[  x^3 + 1         1]

ideal_below()

Return the ideal below this ideal.

This is defined only for integral ideals.

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(2)); _.<t> = K[]
sage: F.<y> = K.extension(t^3 - x^2*(x^2+x+1)^2)
sage: O = F.maximal_order()
sage: I = O.ideal(x, 1/y)
sage: I.ideal_below()
Traceback (most recent call last):
...
TypeError: not an integral ideal
sage: J = I.denominator() * I
sage: J.ideal_below()
Ideal (x^3 + x^2 + x) of Maximal order of Rational function field
in x over Finite Field of size 2

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: O = L.maximal_order()
sage: I = O.ideal(x, 1/y)
sage: I.ideal_below()
Traceback (most recent call last):
...
TypeError: not an integral ideal
sage: J = I.denominator() * I
sage: J.ideal_below()
Ideal (x^3 + x) of Maximal order of Rational function field
in x over Finite Field of size 2

sage: K.<x> = FunctionField(QQ); _.<t> = K[]
sage: F.<y> = K.extension(t^3 - x^2*(x^2+x+1)^2)
sage: O = F.maximal_order()
sage: I = O.ideal(x, 1/y)
sage: I.ideal_below()
Traceback (most recent call last):
...
TypeError: not an integral ideal
sage: J = I.denominator() * I
sage: J.ideal_below()
Ideal (x^3 + x^2 + x) of Maximal order of Rational function field
in x over Rational Field

intersect(other)

Intersect this ideal with the other ideal as fractional ideals.

INPUT:

• other – ideal

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 - x^3*Y - x)
sage: O = L.maximal_order()
sage: I = O.ideal(x + y)
sage: J = O.ideal(x)
sage: I.intersect(J) == I * J * (I + J)^-1
True

is_integral()

Return True if this is an integral ideal.

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(2)); _.<t> = PolynomialRing(K)
sage: F.<y> = K.extension(t^3 - x^2*(x^2+x+1)^2)
sage: O = F.maximal_order()
sage: I = O.ideal(x, 1/y)
sage: I.is_integral()
False
sage: J = I.denominator() * I
sage: J.is_integral()
True

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: O = L.maximal_order()
sage: I = O.ideal(x, 1/y)
sage: I.is_integral()
False
sage: J = I.denominator() * I
sage: J.is_integral()
True

sage: K.<x> = FunctionField(QQ); _.<t> = PolynomialRing(K)
sage: F.<y> = K.extension(t^3 - x^2*(x^2+x+1)^2)
sage: O = F.maximal_order()
sage: I = O.ideal(x, 1/y)
sage: I.is_integral()
False
sage: J = I.denominator() * I
sage: J.is_integral()
True

is_prime()

Return True if this ideal is a prime ideal.

EXAMPLES:

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

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

sage: K.<x> = FunctionField(QQ); _.<t> = PolynomialRing(K)
sage: F.<y> = K.extension(t^3 - x^2*(x^2+x+1)^2)
sage: O = F.maximal_order()
sage: I = O.ideal(y)
sage: [f.is_prime() for f,_ in I.factor()]
[True, True]

module()

Return the module, that is the ideal viewed as a module over the base maximal order.

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(7)); R.<y> = K[]
sage: F.<y> = K.extension(y^2 - x^3 - 1)
sage: O = F.maximal_order()
sage: I = O.ideal(x, 1/y)
sage: I.module()
Free module of degree 2 and rank 2 over Maximal order
of Rational function field in x over Finite Field of size 7
Echelon basis matrix:
[          1           0]
[          0 1/(x^3 + 1)]

norm()

Return the norm of this fractional ideal.

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(2)); _.<t> = PolynomialRing(K)
sage: F.<y> = K.extension(t^3 - x^2*(x^2+x+1)^2)
sage: O = F.maximal_order()
sage: i1 = O.ideal(x)
sage: i2 = O.ideal(y)
sage: i3 = i1 * i2
sage: i3.norm() == i1.norm() * i2.norm()
True
sage: i1.norm()
x^3
sage: i1.norm() == x ** F.degree()
True
sage: i2.norm()
x^6 + x^4 + x^2
sage: i2.norm() == y.norm()
True

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: O = L.maximal_order()
sage: i1 = O.ideal(x)
sage: i2 = O.ideal(y)
sage: i3 = i1 * i2
sage: i3.norm() == i1.norm() * i2.norm()
True
sage: i1.norm()
x^2
sage: i1.norm() == x ** L.degree()
True
sage: i2.norm()
(x^2 + 1)/x
sage: i2.norm() == y.norm()
True

prime_below()

Return the prime lying below this prime ideal.

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: F.<y> = K.extension(Y^3 - x^2*(x^2 + x + 1)^2)
sage: O = F.maximal_order()
sage: I = O.ideal(y)
sage: [f.prime_below() for f,_ in I.factor()]
[Ideal (x) of Maximal order of Rational function field in x
over Finite Field of size 2, Ideal (x^2 + x + 1) of Maximal order
of Rational function field in x over Finite Field of size 2]

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^2 + Y + x + 1/x)
sage: O = L.maximal_order()
sage: I = O.ideal(y)
sage: [f.prime_below() for f,_ in I.factor()]
[Ideal (x) of Maximal order of Rational function field in x over Finite Field of size 2,
 Ideal (x + 1) of Maximal order of Rational function field in x over Finite Field of size 2]

sage: K.<x> = FunctionField(QQ); _.<Y> = K[]
sage: F.<y> = K.extension(Y^3 - x^2*(x^2 + x + 1)^2)
sage: O = F.maximal_order()
sage: I = O.ideal(y)
sage: [f.prime_below() for f,_ in I.factor()]
[Ideal (x) of Maximal order of Rational function field in x over Rational Field,
 Ideal (x^2 + x + 1) of Maximal order of Rational function field in x over Rational Field]

valuation(ideal)

Return the valuation of ideal at this prime ideal.

INPUT:

• ideal – fractional ideal

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(2)); _.<t> = K[]
sage: F.<y> = K.extension(t^3 - x^2*(x^2 + x + 1)^2)
sage: O = F.maximal_order()
sage: I = O.ideal(x, (1/(x^3 + x^2 + x))*y^2)
sage: I.is_prime()
True
sage: J = O.ideal(y)
sage: I.valuation(J)
2

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

The method closely follows Algorithm 4.8.17 of [Coh1993].