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 fieldnames
– 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
whereF
is a rational function field,f
is an isomorphism fromF
to this field, andt
is the inverse off
.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 selfnames
– 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
- 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 spacemap
– (defaultTrue
), 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. Ifn
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 ofbase_morphism
; this is not checked.base_morphism
– a homomorphism from the base field into the other ring. IfNone
, 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 fieldname
– 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)]