Function Fields: rational

class sage.rings.function_field.function_field_rational.RationalFunctionField(constant_field, names, category=None)

Bases: FunctionField

Rational function field in one variable, over an arbitrary base field.

INPUT:

• constant_field – arbitrary field

• names – string or tuple of length 1

EXAMPLES:

sage: K.<t> = FunctionField(GF(3)); K
Rational function field in t over Finite Field of size 3
sage: K.gen()
t
sage: 1/t + t^3 + 5
(t^4 + 2*t + 1)/t

sage: K.<t> = FunctionField(QQ); K
Rational function field in t over Rational Field
sage: K.gen()
t
sage: 1/t + t^3 + 5
(t^4 + 5*t + 1)/t

There are various ways to get at the underlying fields and rings associated to a rational function field:

sage: K.<t> = FunctionField(GF(7))
sage: K.base_field()
Rational function field in t over Finite Field of size 7
sage: K.field()
Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 7
sage: K.constant_field()
Finite Field of size 7
sage: K.maximal_order()
Maximal order of Rational function field in t over Finite Field of size 7

sage: K.<t> = FunctionField(QQ)
sage: K.base_field()
Rational function field in t over Rational Field
sage: K.field()
Fraction Field of Univariate Polynomial Ring in t over Rational Field
sage: K.constant_field()
Rational Field
sage: K.maximal_order()
Maximal order of Rational function field in t over Rational Field

We define a morphism:

sage: K.<t> = FunctionField(QQ)
sage: L = FunctionField(QQ, 'tbar') # give variable name as second input
sage: K.hom(L.gen())
Function Field morphism:
  From: Rational function field in t over Rational Field
  To:   Rational function field in tbar over Rational Field
  Defn: t |--> tbar

Here are some calculations over a number field:

sage: R.<x> = FunctionField(QQ)
sage: L.<y> = R[]
sage: F.<y> = R.extension(y^2 - (x^2+1))                                        # needs sage.rings.function_field
sage: (y/x).divisor()                                                           # needs sage.modules sage.rings.function_field
- Place (x, y - 1)
 - Place (x, y + 1)
 + Place (x^2 + 1, y)

sage: # needs sage.rings.number_field
sage: A.<z> = QQ[]
sage: NF.<i> = NumberField(z^2 + 1)
sage: R.<x> = FunctionField(NF)
sage: L.<y> = R[]
sage: F.<y> = R.extension(y^2 - (x^2+1))                                        # needs sage.modules sage.rings.function_field

sage: (x/y*x.differential()).divisor()                                          # needs sage.modules sage.rings.function_field sage.rings.number_field
-2*Place (1/x, 1/x*y - 1)
 - 2*Place (1/x, 1/x*y + 1)
 + Place (x, y - 1)
 + Place (x, y + 1)

sage: (x/y).divisor()                                                           # needs sage.modules sage.rings.function_field sage.rings.number_field
- Place (x - i, y)
 + Place (x, y - 1)
 + Place (x, y + 1)
 - Place (x + i, y)

Element

alias of FunctionFieldElement_rational

base_field()

Return the base field of the rational function field, which is just the function field itself.

EXAMPLES:

sage: K.<t> = FunctionField(GF(7))
sage: K.base_field()
Rational function field in t over Finite Field of size 7

change_variable_name(name)

Return a field isomorphic to this field with variable name.

INPUT:

• name – a string or a tuple consisting of a single string, the name of the new variable

OUTPUT:

A triple F,f,t where F is a rational function field, f is an isomorphism from F to this field, and t is the inverse of f.

EXAMPLES:

sage: K.<x> = FunctionField(QQ)
sage: L,f,t = K.change_variable_name('y')
sage: L,f,t
(Rational function field in y over Rational Field,
 Function Field morphism:
  From: Rational function field in y over Rational Field
  To:   Rational function field in x over Rational Field
  Defn: y |--> x,
 Function Field morphism:
  From: Rational function field in x over Rational Field
  To:   Rational function field in y over Rational Field
  Defn: x |--> y)
sage: L.change_variable_name('x')[0] is K
True

constant_base_field()

Return the field of which the rational function field is a transcendental extension.

EXAMPLES:

sage: K.<t> = FunctionField(QQ)
sage: K.constant_base_field()
Rational Field

constant_field()

Return the field of which the rational function field is a transcendental extension.

EXAMPLES:

sage: K.<t> = FunctionField(QQ)
sage: K.constant_base_field()
Rational Field

degree(base=None)

Return the degree over the base field of the rational function field. Since the base field is the rational function field itself, the degree is 1.

INPUT:

• base – the base field of the vector space; must be the function field itself (the default)

EXAMPLES:

sage: K.<t> = FunctionField(QQ)
sage: K.degree()
1

different()

Return the different of the rational function field.

For a rational function field, the different is simply the zero divisor.

EXAMPLES:

sage: K.<t> = FunctionField(QQ)
sage: K.different()                                                         # needs sage.modules
0

equation_order()

Return the maximal order of the function field.

Since this is a rational function field it is of the form \(K(t)\), and the maximal order is by definition \(K[t]\), where \(K\) is the constant field.

EXAMPLES:

sage: K.<t> = FunctionField(QQ)
sage: K.maximal_order()
Maximal order of Rational function field in t over Rational Field
sage: K.equation_order()
Maximal order of Rational function field in t over Rational Field

equation_order_infinite()

Return the maximal infinite order of the function field.

By definition, this is the valuation ring of the degree valuation of the rational function field.

EXAMPLES:

sage: K.<t> = FunctionField(QQ)
sage: K.maximal_order_infinite()
Maximal infinite order of Rational function field in t over Rational Field
sage: K.equation_order_infinite()
Maximal infinite order of Rational function field in t over Rational Field

extension(f, names=None)

Create an extension \(L = K[y]/(f(y))\) of the rational function field.

INPUT:

• f – univariate polynomial over self

• names – string or length-1 tuple

OUTPUT:

• a function field

EXAMPLES:

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: K.extension(y^5 - x^3 - 3*x + x*y)                                    # needs sage.rings.function_field
Function field in y defined by y^5 + x*y - x^3 - 3*x

A nonintegral defining polynomial:

sage: K.<t> = FunctionField(QQ); R.<y> = K[]
sage: K.extension(y^3 + (1/t)*y + t^3/(t+1))                                # needs sage.rings.function_field
Function field in y defined by y^3 + 1/t*y + t^3/(t + 1)

The defining polynomial need not be monic or integral:

sage: K.extension(t*y^3 + (1/t)*y + t^3/(t+1))                              # needs sage.rings.function_field
Function field in y defined by t*y^3 + 1/t*y + t^3/(t + 1)

field()

Return the underlying field, forgetting the function field structure.

EXAMPLES:

sage: K.<t> = FunctionField(GF(7))
sage: K.field()
Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 7

See also

sage.rings.fraction_field.FractionField_1poly_field.function_field()

free_module(base=None, basis=None, map=True)

Return a vector space \(V\) and isomorphisms from the field to \(V\) and from \(V\) to the field.

This function allows us to identify the elements of this field with elements of a one-dimensional vector space over the field itself. This method exists so that all function fields (rational or not) have the same interface.

INPUT:

• base – the base field of the vector space; must be the function field itself (the default)

• basis – (ignored) a basis for the vector space

• map – (default True), whether to return maps to and from the vector space

OUTPUT:

• a vector space \(V\) over base field

• an isomorphism from \(V\) to the field

• the inverse isomorphism from the field to \(V\)

EXAMPLES:

sage: K.<x> = FunctionField(QQ)
sage: K.free_module()                                                                                       # needs sage.modules
(Vector space of dimension 1 over Rational function field in x over Rational Field,
 Isomorphism:
  From: Vector space of dimension 1 over Rational function field in x over Rational Field
  To:   Rational function field in x over Rational Field,
 Isomorphism:
  From: Rational function field in x over Rational Field
  To:   Vector space of dimension 1 over Rational function field in x over Rational Field)

gen(n=0)

Return the n-th generator of the function field. If n is not 0, then an :class:` IndexError` is raised.

EXAMPLES:

sage: K.<t> = FunctionField(QQ); K.gen()
t
sage: K.gen().parent()
Rational function field in t over Rational Field
sage: K.gen(1)
Traceback (most recent call last):
...
IndexError: Only one generator.

genus()

Return the genus of the function field, namely 0.

EXAMPLES:

sage: K.<x> = FunctionField(QQ)
sage: K.genus()
0

hom(im_gens, base_morphism=None)

Create a homomorphism from self to another ring.

INPUT:

• im_gens – exactly one element of some ring. It must be invertible and transcendental over the image of base_morphism; this is not checked.

• base_morphism – a homomorphism from the base field into the other ring. If None, try to use a coercion map.

OUTPUT:

• a map between function fields

EXAMPLES:

We make a map from a rational function field to itself:

sage: K.<x> = FunctionField(GF(7))
sage: K.hom((x^4 + 2)/x)
Function Field endomorphism of Rational function field in x over Finite Field of size 7
  Defn: x |--> (x^4 + 2)/x

We construct a map from a rational function field into a non-rational extension field:

sage: # needs sage.rings.function_field
sage: K.<x> = FunctionField(GF(7)); R.<y> = K[]
sage: L.<y> = K.extension(y^3 + 6*x^3 + x)
sage: f = K.hom(y^2 + y  + 2); f
Function Field morphism:
  From: Rational function field in x over Finite Field of size 7
  To:   Function field in y defined by y^3 + 6*x^3 + x
  Defn: x |--> y^2 + y + 2
sage: f(x)
y^2 + y + 2
sage: f(x^2)
5*y^2 + (x^3 + 6*x + 4)*y + 2*x^3 + 5*x + 4

maximal_order()

Return the maximal order of the function field.

Since this is a rational function field it is of the form \(K(t)\), and the maximal order is by definition \(K[t]\), where \(K\) is the constant field.

EXAMPLES:

sage: K.<t> = FunctionField(QQ)
sage: K.maximal_order()
Maximal order of Rational function field in t over Rational Field
sage: K.equation_order()
Maximal order of Rational function field in t over Rational Field

maximal_order_infinite()

Return the maximal infinite order of the function field.

By definition, this is the valuation ring of the degree valuation of the rational function field.

EXAMPLES:

sage: K.<t> = FunctionField(QQ)
sage: K.maximal_order_infinite()
Maximal infinite order of Rational function field in t over Rational Field
sage: K.equation_order_infinite()
Maximal infinite order of Rational function field in t over Rational Field

ngens()

Return the number of generators, which is 1.

EXAMPLES:

sage: K.<t> = FunctionField(QQ)
sage: K.ngens()
1

polynomial_ring(var='x')

Return a polynomial ring in one variable over the rational function field.

INPUT:

• var – string; name of the variable

EXAMPLES:

sage: K.<x> = FunctionField(QQ)
sage: K.polynomial_ring()
Univariate Polynomial Ring in x over Rational function field in x over Rational Field
sage: K.polynomial_ring('T')
Univariate Polynomial Ring in T over Rational function field in x over Rational Field

random_element(*args, **kwds)

Create a random element of the rational function field.

Parameters are passed to the random_element method of the underlying fraction field.

EXAMPLES:

sage: FunctionField(QQ,'alpha').random_element()   # random
(-1/2*alpha^2 - 4)/(-12*alpha^2 + 1/2*alpha - 1/95)

residue_field(place, name=None)

Return the residue field of the place along with the maps from and to it.

INPUT:

• place – place of the function field

• name – string; name of the generator of the residue field

EXAMPLES:

sage: F.<x> = FunctionField(GF(5))
sage: p = F.places_finite(2)[0]                                             # needs sage.libs.pari
sage: R, fr_R, to_R = F.residue_field(p)                                    # needs sage.libs.pari sage.rings.function_field
sage: R                                                                     # needs sage.libs.pari sage.rings.function_field
Finite Field in z2 of size 5^2
sage: to_R(x) in R                                                          # needs sage.libs.pari sage.rings.function_field
True

class sage.rings.function_field.function_field_rational.RationalFunctionField_char_zero(constant_field, names, category=None)

Bases: RationalFunctionField

Rational function fields of characteristic zero.

higher_derivation()

Return the higher derivation for the function field.

This is also called the Hasse-Schmidt derivation.

EXAMPLES:

sage: F.<x> = FunctionField(QQ)
sage: d = F.higher_derivation()                                             # needs sage.libs.singular sage.modules
sage: [d(x^5,i) for i in range(10)]                                         # needs sage.libs.singular sage.modules
[x^5, 5*x^4, 10*x^3, 10*x^2, 5*x, 1, 0, 0, 0, 0]
sage: [d(x^9,i) for i in range(10)]                                         # needs sage.libs.singular sage.modules
[x^9, 9*x^8, 36*x^7, 84*x^6, 126*x^5, 126*x^4, 84*x^3, 36*x^2, 9*x, 1]

class sage.rings.function_field.function_field_rational.RationalFunctionField_global(constant_field, names, category=None)

Bases: RationalFunctionField

Rational function field over finite fields.

get_place(degree)

Return a place of degree.

INPUT:

• degree – a positive integer

EXAMPLES:

sage: F.<a> = GF(2)
sage: K.<x> = FunctionField(F)
sage: K.get_place(1)                                                        # needs sage.libs.pari
Place (x)
sage: K.get_place(2)                                                        # needs sage.libs.pari
Place (x^2 + x + 1)
sage: K.get_place(3)                                                        # needs sage.libs.pari
Place (x^3 + x + 1)
sage: K.get_place(4)                                                        # needs sage.libs.pari
Place (x^4 + x + 1)
sage: K.get_place(5)                                                        # needs sage.libs.pari
Place (x^5 + x^2 + 1)

higher_derivation()

Return the higher derivation for the function field.

This is also called the Hasse-Schmidt derivation.

EXAMPLES:

sage: F.<x> = FunctionField(GF(5))
sage: d = F.higher_derivation()                                             # needs sage.rings.function_field
sage: [d(x^5,i) for i in range(10)]                                         # needs sage.rings.function_field
[x^5, 0, 0, 0, 0, 1, 0, 0, 0, 0]
sage: [d(x^7,i) for i in range(10)]                                         # needs sage.rings.function_field
[x^7, 2*x^6, x^5, 0, 0, x^2, 2*x, 1, 0, 0]

place_infinite()

Return the unique place at infinity.

EXAMPLES:

sage: F.<x> = FunctionField(GF(5))
sage: F.place_infinite()
Place (1/x)

places(degree=1)

Return all places of the degree.

INPUT:

• degree – (default: 1) a positive integer

EXAMPLES:

sage: F.<x> = FunctionField(GF(5))
sage: F.places()                                                            # needs sage.libs.pari
[Place (1/x),
 Place (x),
 Place (x + 1),
 Place (x + 2),
 Place (x + 3),
 Place (x + 4)]

places_finite(degree=1)

Return the finite places of the degree.

INPUT:

• degree – (default: 1) a positive integer

EXAMPLES:

sage: F.<x> = FunctionField(GF(5))
sage: F.places_finite()                                                     # needs sage.libs.pari
[Place (x), Place (x + 1), Place (x + 2), Place (x + 3), Place (x + 4)]