Function Fields: rational¶
- class sage.rings.function_field.function_field_rational.RationalFunctionField(constant_field, names, category=None)[source]¶
- 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 - >>> from sage.all import * >>> K = FunctionField(GF(Integer(3)), names=('t',)); (t,) = K._first_ngens(1); K Rational function field in t over Finite Field of size 3 >>> K.gen() t >>> Integer(1)/t + t**Integer(3) + Integer(5) (t^4 + 2*t + 1)/t >>> K = FunctionField(QQ, names=('t',)); (t,) = K._first_ngens(1); K Rational function field in t over Rational Field >>> K.gen() t >>> Integer(1)/t + t**Integer(3) + Integer(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 - >>> from sage.all import * >>> K = FunctionField(GF(Integer(7)), names=('t',)); (t,) = K._first_ngens(1) >>> K.base_field() Rational function field in t over Finite Field of size 7 >>> K.field() Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 7 >>> K.constant_field() Finite Field of size 7 >>> K.maximal_order() Maximal order of Rational function field in t over Finite Field of size 7 >>> K = FunctionField(QQ, names=('t',)); (t,) = K._first_ngens(1) >>> K.base_field() Rational function field in t over Rational Field >>> K.field() Fraction Field of Univariate Polynomial Ring in t over Rational Field >>> K.constant_field() Rational Field >>> 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 - >>> from sage.all import * >>> K = FunctionField(QQ, names=('t',)); (t,) = K._first_ngens(1) >>> L = FunctionField(QQ, 'tbar') # give variable name as second input >>> 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.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.rings.function_field sage: (x/y*x.differential()).divisor() # needs sage.rings.function_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.rings.function_field - Place (x - i, y) + Place (x, y - 1) + Place (x, y + 1) - Place (x + i, y) - >>> from sage.all import * >>> R = FunctionField(QQ, names=('x',)); (x,) = R._first_ngens(1) >>> L = R['y']; (y,) = L._first_ngens(1) >>> F = R.extension(y**Integer(2) - (x**Integer(2)+Integer(1)), names=('y',)); (y,) = F._first_ngens(1)# needs sage.rings.function_field >>> (y/x).divisor() # needs sage.rings.function_field - Place (x, y - 1) - Place (x, y + 1) + Place (x^2 + 1, y) >>> # needs sage.rings.number_field >>> A = QQ['z']; (z,) = A._first_ngens(1) >>> NF = NumberField(z**Integer(2) + Integer(1), names=('i',)); (i,) = NF._first_ngens(1) >>> R = FunctionField(NF, names=('x',)); (x,) = R._first_ngens(1) >>> L = R['y']; (y,) = L._first_ngens(1) >>> F = R.extension(y**Integer(2) - (x**Integer(2)+Integer(1)), names=('y',)); (y,) = F._first_ngens(1)# needs sage.rings.function_field >>> (x/y*x.differential()).divisor() # needs sage.rings.function_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) >>> (x/y).divisor() # needs sage.rings.function_field - Place (x - i, y) + Place (x, y - 1) + Place (x, y + 1) - Place (x + i, y) - Element[source]¶
- alias of - FunctionFieldElement_rational
 - base_field()[source]¶
- 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 - >>> from sage.all import * >>> K = FunctionField(GF(Integer(7)), names=('t',)); (t,) = K._first_ngens(1) >>> K.base_field() Rational function field in t over Finite Field of size 7 
 - change_variable_name(name)[source]¶
- Return a field isomorphic to this field with variable - name.- INPUT: - name– string or tuple consisting of a single string; the name of the new variable
 - OUTPUT: - A triple - F,f,twhere- Fis a rational function field,- fis an isomorphism from- Fto this field, and- tis 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 - >>> from sage.all import * >>> K = FunctionField(QQ, names=('x',)); (x,) = K._first_ngens(1) >>> L,f,t = K.change_variable_name('y') >>> 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) >>> L.change_variable_name('x')[Integer(0)] is K True 
 - constant_base_field()[source]¶
- 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 - >>> from sage.all import * >>> K = FunctionField(QQ, names=('t',)); (t,) = K._first_ngens(1) >>> K.constant_base_field() Rational Field 
 - constant_field()[source]¶
- 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 - >>> from sage.all import * >>> K = FunctionField(QQ, names=('t',)); (t,) = K._first_ngens(1) >>> K.constant_base_field() Rational Field 
 - degree(base=None)[source]¶
- 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 - >>> from sage.all import * >>> K = FunctionField(QQ, names=('t',)); (t,) = K._first_ngens(1) >>> K.degree() 1 
 - different()[source]¶
- 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 - >>> from sage.all import * >>> K = FunctionField(QQ, names=('t',)); (t,) = K._first_ngens(1) >>> K.different() # needs sage.modules 0 
 - equation_order()[source]¶
- 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 - >>> from sage.all import * >>> K = FunctionField(QQ, names=('t',)); (t,) = K._first_ngens(1) >>> K.maximal_order() Maximal order of Rational function field in t over Rational Field >>> K.equation_order() Maximal order of Rational function field in t over Rational Field 
 - equation_order_infinite()[source]¶
- 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 - >>> from sage.all import * >>> K = FunctionField(QQ, names=('t',)); (t,) = K._first_ngens(1) >>> K.maximal_order_infinite() Maximal infinite order of Rational function field in t over Rational Field >>> K.equation_order_infinite() Maximal infinite order of Rational function field in t over Rational Field 
 - extension(f, names=None)[source]¶
- 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 - >>> from sage.all import * >>> K = FunctionField(QQ, names=('x',)); (x,) = K._first_ngens(1); R = K['y']; (y,) = R._first_ngens(1) >>> K.extension(y**Integer(5) - x**Integer(3) - Integer(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) - >>> from sage.all import * >>> K = FunctionField(QQ, names=('t',)); (t,) = K._first_ngens(1); R = K['y']; (y,) = R._first_ngens(1) >>> K.extension(y**Integer(3) + (Integer(1)/t)*y + t**Integer(3)/(t+Integer(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) - >>> from sage.all import * >>> K.extension(t*y**Integer(3) + (Integer(1)/t)*y + t**Integer(3)/(t+Integer(1))) # needs sage.rings.function_field Function field in y defined by t*y^3 + 1/t*y + t^3/(t + 1) 
 - field()[source]¶
- 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 - >>> from sage.all import * >>> K = FunctionField(GF(Integer(7)), names=('t',)); (t,) = K._first_ngens(1) >>> K.field() Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 7 
 - free_module(base=None, basis=None, map=True)[source]¶
- 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) - >>> from sage.all import * >>> K = FunctionField(QQ, names=('x',)); (x,) = K._first_ngens(1) >>> 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)[source]¶
- Return the - n-th generator of the function field. If- nis 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. - >>> from sage.all import * >>> K = FunctionField(QQ, names=('t',)); (t,) = K._first_ngens(1); K.gen() t >>> K.gen().parent() Rational function field in t over Rational Field >>> K.gen(Integer(1)) Traceback (most recent call last): ... IndexError: Only one generator. 
 - genus()[source]¶
- Return the genus of the function field, namely 0. - EXAMPLES: - sage: K.<x> = FunctionField(QQ) sage: K.genus() 0 - >>> from sage.all import * >>> K = FunctionField(QQ, names=('x',)); (x,) = K._first_ngens(1) >>> K.genus() 0 
 - hom(im_gens, base_morphism=None)[source]¶
- Create a homomorphism from - selfto 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 - >>> from sage.all import * >>> K = FunctionField(GF(Integer(7)), names=('x',)); (x,) = K._first_ngens(1) >>> K.hom((x**Integer(4) + Integer(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 - >>> from sage.all import * >>> # needs sage.rings.function_field >>> K = FunctionField(GF(Integer(7)), names=('x',)); (x,) = K._first_ngens(1); R = K['y']; (y,) = R._first_ngens(1) >>> L = K.extension(y**Integer(3) + Integer(6)*x**Integer(3) + x, names=('y',)); (y,) = L._first_ngens(1) >>> f = K.hom(y**Integer(2) + y + Integer(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 >>> f(x) y^2 + y + 2 >>> f(x**Integer(2)) 5*y^2 + (x^3 + 6*x + 4)*y + 2*x^3 + 5*x + 4 
 - maximal_order()[source]¶
- 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 - >>> from sage.all import * >>> K = FunctionField(QQ, names=('t',)); (t,) = K._first_ngens(1) >>> K.maximal_order() Maximal order of Rational function field in t over Rational Field >>> K.equation_order() Maximal order of Rational function field in t over Rational Field 
 - maximal_order_infinite()[source]¶
- 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 - >>> from sage.all import * >>> K = FunctionField(QQ, names=('t',)); (t,) = K._first_ngens(1) >>> K.maximal_order_infinite() Maximal infinite order of Rational function field in t over Rational Field >>> K.equation_order_infinite() Maximal infinite order of Rational function field in t over Rational Field 
 - ngens()[source]¶
- Return the number of generators, which is 1. - EXAMPLES: - sage: K.<t> = FunctionField(QQ) sage: K.ngens() 1 - >>> from sage.all import * >>> K = FunctionField(QQ, names=('t',)); (t,) = K._first_ngens(1) >>> K.ngens() 1 
 - polynomial_ring(var='x')[source]¶
- 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 - >>> from sage.all import * >>> K = FunctionField(QQ, names=('x',)); (x,) = K._first_ngens(1) >>> K.polynomial_ring() Univariate Polynomial Ring in x over Rational function field in x over Rational Field >>> K.polynomial_ring('T') Univariate Polynomial Ring in T over Rational function field in x over Rational Field 
 - random_element(*args, **kwds)[source]¶
- 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) - >>> from sage.all import * >>> FunctionField(QQ,'alpha').random_element() # random (-1/2*alpha^2 - 4)/(-12*alpha^2 + 1/2*alpha - 1/95) 
 - residue_field(place, name=None)[source]¶
- 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 - >>> from sage.all import * >>> F = FunctionField(GF(Integer(5)), names=('x',)); (x,) = F._first_ngens(1) >>> p = F.places_finite(Integer(2))[Integer(0)] # needs sage.libs.pari >>> R, fr_R, to_R = F.residue_field(p) # needs sage.libs.pari sage.rings.function_field >>> R # needs sage.libs.pari sage.rings.function_field Finite Field in z2 of size 5^2 >>> 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)[source]¶
- Bases: - RationalFunctionField- Rational function fields of characteristic zero. - higher_derivation()[source]¶
- 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] - >>> from sage.all import * >>> F = FunctionField(QQ, names=('x',)); (x,) = F._first_ngens(1) >>> d = F.higher_derivation() # needs sage.libs.singular sage.modules >>> [d(x**Integer(5),i) for i in range(Integer(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] >>> [d(x**Integer(9),i) for i in range(Integer(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)[source]¶
- Bases: - RationalFunctionField- Rational function field over finite fields. - get_place(degree)[source]¶
- Return a place of - degree.- INPUT: - degree– 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) - >>> from sage.all import * >>> F = GF(Integer(2), names=('a',)); (a,) = F._first_ngens(1) >>> K = FunctionField(F, names=('x',)); (x,) = K._first_ngens(1) >>> K.get_place(Integer(1)) # needs sage.libs.pari Place (x) >>> K.get_place(Integer(2)) # needs sage.libs.pari Place (x^2 + x + 1) >>> K.get_place(Integer(3)) # needs sage.libs.pari Place (x^3 + x + 1) >>> K.get_place(Integer(4)) # needs sage.libs.pari Place (x^4 + x + 1) >>> K.get_place(Integer(5)) # needs sage.libs.pari Place (x^5 + x^2 + 1) 
 - higher_derivation()[source]¶
- 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] - >>> from sage.all import * >>> F = FunctionField(GF(Integer(5)), names=('x',)); (x,) = F._first_ngens(1) >>> d = F.higher_derivation() # needs sage.rings.function_field >>> [d(x**Integer(5),i) for i in range(Integer(10))] # needs sage.rings.function_field [x^5, 0, 0, 0, 0, 1, 0, 0, 0, 0] >>> [d(x**Integer(7),i) for i in range(Integer(10))] # needs sage.rings.function_field [x^7, 2*x^6, x^5, 0, 0, x^2, 2*x, 1, 0, 0] 
 - place_infinite()[source]¶
- Return the unique place at infinity. - EXAMPLES: - sage: F.<x> = FunctionField(GF(5)) sage: F.place_infinite() Place (1/x) - >>> from sage.all import * >>> F = FunctionField(GF(Integer(5)), names=('x',)); (x,) = F._first_ngens(1) >>> F.place_infinite() Place (1/x) 
 - places(degree=1)[source]¶
- 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)] - >>> from sage.all import * >>> F = FunctionField(GF(Integer(5)), names=('x',)); (x,) = F._first_ngens(1) >>> 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)[source]¶
- 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)] - >>> from sage.all import * >>> F = FunctionField(GF(Integer(5)), names=('x',)); (x,) = F._first_ngens(1) >>> F.places_finite() # needs sage.libs.pari [Place (x), Place (x + 1), Place (x + 2), Place (x + 3), Place (x + 4)]