Orders of function fields: rational

class sage.rings.function_field.order_rational.FunctionFieldMaximalOrderInfinite_rational(field, category=None)

Bases: FunctionFieldMaximalOrderInfinite

Maximal infinite orders of rational function fields.

INPUT:

• field – a rational function field

EXAMPLES:

Sage

sage: K.<t> = FunctionField(GF(19)); K
Rational function field in t over Finite Field of size 19
sage: R = K.maximal_order_infinite(); R
Maximal infinite order of Rational function field in t over Finite Field of size 19

Python

>>> from sage.all import *
>>> K = FunctionField(GF(Integer(19)), names=('t',)); (t,) = K._first_ngens(1); K
Rational function field in t over Finite Field of size 19
>>> R = K.maximal_order_infinite(); R
Maximal infinite order of Rational function field in t over Finite Field of size 19

Sage Live

K.<t> = FunctionField(GF(19)); K
R = K.maximal_order_infinite(); R

basis()

Return the basis (=1) of the order as a module over the polynomial ring.

EXAMPLES:

Sage

sage: K.<t> = FunctionField(GF(19))
sage: O = K.maximal_order()
sage: O.basis()
(1,)

Python

>>> from sage.all import *
>>> K = FunctionField(GF(Integer(19)), names=('t',)); (t,) = K._first_ngens(1)
>>> O = K.maximal_order()
>>> O.basis()
(1,)

Sage Live

K.<t> = FunctionField(GF(19))
O = K.maximal_order()
O.basis()

gen(n=0)

Return the \(n\)-th generator of self. Since there is only one generator \(n\) must be \(0\).

EXAMPLES:

Sage

sage: O = FunctionField(QQ,'y').maximal_order()
sage: O.gen()
y
sage: O.gen(1)
Traceback (most recent call last):
...
IndexError: there is only one generator

Python

>>> from sage.all import *
>>> O = FunctionField(QQ,'y').maximal_order()
>>> O.gen()
y
>>> O.gen(Integer(1))
Traceback (most recent call last):
...
IndexError: there is only one generator

Sage Live

O = FunctionField(QQ,'y').maximal_order()
O.gen()
O.gen(1)

ideal(*gens)

Return the fractional ideal generated by gens.

INPUT:

• gens – elements of the function field

EXAMPLES:

Sage

sage: K.<x> = FunctionField(QQ)
sage: O = K.maximal_order_infinite()
sage: O.ideal(x)
Ideal (x) of Maximal infinite order of Rational function field in x over Rational Field
sage: O.ideal([x, 1/x]) == O.ideal(x ,1/x)  # multiple generators may be given as a list
True
sage: O.ideal(x^3 + 1, x^3 + 6)
Ideal (x^3) of Maximal infinite order of Rational function field in x over Rational Field
sage: I = O.ideal((x^2+1)*(x^3+1), (x^3+6)*(x^2+1)); I
Ideal (x^5) of Maximal infinite order of Rational function field in x over Rational Field
sage: O.ideal(I)
Ideal (x^5) of Maximal infinite order of Rational function field in x over Rational Field

Python

>>> from sage.all import *
>>> K = FunctionField(QQ, names=('x',)); (x,) = K._first_ngens(1)
>>> O = K.maximal_order_infinite()
>>> O.ideal(x)
Ideal (x) of Maximal infinite order of Rational function field in x over Rational Field
>>> O.ideal([x, Integer(1)/x]) == O.ideal(x ,Integer(1)/x)  # multiple generators may be given as a list
True
>>> O.ideal(x**Integer(3) + Integer(1), x**Integer(3) + Integer(6))
Ideal (x^3) of Maximal infinite order of Rational function field in x over Rational Field
>>> I = O.ideal((x**Integer(2)+Integer(1))*(x**Integer(3)+Integer(1)), (x**Integer(3)+Integer(6))*(x**Integer(2)+Integer(1))); I
Ideal (x^5) of Maximal infinite order of Rational function field in x over Rational Field
>>> O.ideal(I)
Ideal (x^5) of Maximal infinite order of Rational function field in x over Rational Field

Sage Live

K.<x> = FunctionField(QQ)
O = K.maximal_order_infinite()
O.ideal(x)
O.ideal([x, 1/x]) == O.ideal(x ,1/x)  # multiple generators may be given as a list
O.ideal(x^3 + 1, x^3 + 6)
I = O.ideal((x^2+1)*(x^3+1), (x^3+6)*(x^2+1)); I
O.ideal(I)

ngens()

Return 1 the number of generators of the order.

EXAMPLES:

Sage

sage: FunctionField(QQ,'y').maximal_order().ngens()
1

Python

>>> from sage.all import *
>>> FunctionField(QQ,'y').maximal_order().ngens()
1

Sage Live

FunctionField(QQ,'y').maximal_order().ngens()

prime_ideal()

Return the unique prime ideal of the order.

EXAMPLES:

Sage

sage: K.<t> = FunctionField(GF(19))
sage: O = K.maximal_order_infinite()
sage: O.prime_ideal()
Ideal (1/t) of Maximal infinite order of Rational function field in t
over Finite Field of size 19

Python

>>> from sage.all import *
>>> K = FunctionField(GF(Integer(19)), names=('t',)); (t,) = K._first_ngens(1)
>>> O = K.maximal_order_infinite()
>>> O.prime_ideal()
Ideal (1/t) of Maximal infinite order of Rational function field in t
over Finite Field of size 19

Sage Live

K.<t> = FunctionField(GF(19))
O = K.maximal_order_infinite()
O.prime_ideal()

class sage.rings.function_field.order_rational.FunctionFieldMaximalOrder_rational(field)

Bases: FunctionFieldMaximalOrder

Maximal orders of rational function fields.

INPUT:

• field – a function field

EXAMPLES:

Sage

sage: K.<t> = FunctionField(GF(19)); K
Rational function field in t over Finite Field of size 19
sage: R = K.maximal_order(); R
Maximal order of Rational function field in t over Finite Field of size 19

Python

>>> from sage.all import *
>>> K = FunctionField(GF(Integer(19)), names=('t',)); (t,) = K._first_ngens(1); K
Rational function field in t over Finite Field of size 19
>>> R = K.maximal_order(); R
Maximal order of Rational function field in t over Finite Field of size 19

Sage Live

K.<t> = FunctionField(GF(19)); K
R = K.maximal_order(); R

basis()

Return the basis (=1) of the order as a module over the polynomial ring.

EXAMPLES:

Sage

sage: K.<t> = FunctionField(GF(19))
sage: O = K.maximal_order()
sage: O.basis()
(1,)

Python

>>> from sage.all import *
>>> K = FunctionField(GF(Integer(19)), names=('t',)); (t,) = K._first_ngens(1)
>>> O = K.maximal_order()
>>> O.basis()
(1,)

Sage Live

K.<t> = FunctionField(GF(19))
O = K.maximal_order()
O.basis()

gen(n=0)

Return the n-th generator of the order. Since there is only one generator n must be 0.

EXAMPLES:

Sage

sage: O = FunctionField(QQ,'y').maximal_order()
sage: O.gen()
y
sage: O.gen(1)
Traceback (most recent call last):
...
IndexError: there is only one generator

Python

>>> from sage.all import *
>>> O = FunctionField(QQ,'y').maximal_order()
>>> O.gen()
y
>>> O.gen(Integer(1))
Traceback (most recent call last):
...
IndexError: there is only one generator

Sage Live

O = FunctionField(QQ,'y').maximal_order()
O.gen()
O.gen(1)

ideal(*gens)

Return the fractional ideal generated by gens.

INPUT:

• gens – elements of the function field

EXAMPLES:

Sage

sage: K.<x> = FunctionField(QQ)
sage: O = K.maximal_order()
sage: O.ideal(x)
Ideal (x) of Maximal order of Rational function field in x over Rational Field
sage: O.ideal([x, 1/x]) == O.ideal(x, 1/x)  # multiple generators may be given as a list
True
sage: O.ideal(x^3 + 1, x^3 + 6)
Ideal (1) of Maximal order of Rational function field in x over Rational Field
sage: I = O.ideal((x^2+1)*(x^3+1), (x^3+6)*(x^2+1)); I
Ideal (x^2 + 1) of Maximal order of Rational function field in x over Rational Field
sage: O.ideal(I)
Ideal (x^2 + 1) of Maximal order of Rational function field in x over Rational Field

Python

>>> from sage.all import *
>>> K = FunctionField(QQ, names=('x',)); (x,) = K._first_ngens(1)
>>> O = K.maximal_order()
>>> O.ideal(x)
Ideal (x) of Maximal order of Rational function field in x over Rational Field
>>> O.ideal([x, Integer(1)/x]) == O.ideal(x, Integer(1)/x)  # multiple generators may be given as a list
True
>>> O.ideal(x**Integer(3) + Integer(1), x**Integer(3) + Integer(6))
Ideal (1) of Maximal order of Rational function field in x over Rational Field
>>> I = O.ideal((x**Integer(2)+Integer(1))*(x**Integer(3)+Integer(1)), (x**Integer(3)+Integer(6))*(x**Integer(2)+Integer(1))); I
Ideal (x^2 + 1) of Maximal order of Rational function field in x over Rational Field
>>> O.ideal(I)
Ideal (x^2 + 1) of Maximal order of Rational function field in x over Rational Field

Sage Live

K.<x> = FunctionField(QQ)
O = K.maximal_order()
O.ideal(x)
O.ideal([x, 1/x]) == O.ideal(x, 1/x)  # multiple generators may be given as a list
O.ideal(x^3 + 1, x^3 + 6)
I = O.ideal((x^2+1)*(x^3+1), (x^3+6)*(x^2+1)); I
O.ideal(I)

ideal_with_gens_over_base(gens)

Return the fractional ideal with generators gens.

INPUT:

• gens – elements of the function field

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> K = FunctionField(QQ, names=('x',)); (x,) = K._first_ngens(1); R = K['y']; (y,) = R._first_ngens(1)
>>> L = K.extension(y**Integer(2) - x**Integer(3) - Integer(1), names=('y',)); (y,) = L._first_ngens(1)# needs sage.rings.function_field
>>> O = L.equation_order()                                                            # needs sage.rings.function_field
>>> O.ideal_with_gens_over_base([x**Integer(3) + Integer(1), -y])                                        # needs sage.rings.function_field
Ideal (x^3 + 1, -y) of Order in Function field in y defined by y^2 - x^3 - 1

Sage Live

K.<x> = FunctionField(QQ); R.<y> = K[]
L.<y> = K.extension(y^2 - x^3 - 1)                                                # needs sage.rings.function_field
O = L.equation_order()                                                            # needs sage.rings.function_field
O.ideal_with_gens_over_base([x^3 + 1, -y])                                        # needs sage.rings.function_field

ngens()

Return 1 the number of generators of the order.

EXAMPLES:

Sage

sage: FunctionField(QQ,'y').maximal_order().ngens()
1

Python

>>> from sage.all import *
>>> FunctionField(QQ,'y').maximal_order().ngens()
1

Sage Live

FunctionField(QQ,'y').maximal_order().ngens()

Make live