Function Fields: extension¶
- class sage.rings.function_field.function_field_polymod.FunctionField_char_zero(polynomial, names, category=None)[source]¶
Bases:
FunctionField_simpleFunction fields of characteristic zero.
EXAMPLES:
sage: K.<x> = FunctionField(QQ); _.<Y> = K[] sage: L.<y> = K.extension(Y^3 - (x^3 - 1)/(x^3 - 2)) sage: L Function field in y defined by y^3 + (-x^3 + 1)/(x^3 - 2) sage: L.characteristic() 0
>>> from sage.all import * >>> K = FunctionField(QQ, names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1) >>> L = K.extension(Y**Integer(3) - (x**Integer(3) - Integer(1))/(x**Integer(3) - Integer(2)), names=('y',)); (y,) = L._first_ngens(1) >>> L Function field in y defined by y^3 + (-x^3 + 1)/(x^3 - 2) >>> L.characteristic() 0
K.<x> = FunctionField(QQ); _.<Y> = K[] L.<y> = K.extension(Y^3 - (x^3 - 1)/(x^3 - 2)) L L.characteristic()
- higher_derivation()[source]¶
Return the higher derivation (also called the Hasse-Schmidt derivation) for the function field.
The higher derivation of the function field is uniquely determined with respect to the separating element \(x\) of the base rational function field \(k(x)\).
EXAMPLES:
sage: K.<x> = FunctionField(QQ); _.<Y> = K[] sage: L.<y> = K.extension(Y^3 - (x^3 - 1)/(x^3 - 2)) sage: L.higher_derivation() # needs sage.modules Higher derivation map: From: Function field in y defined by y^3 + (-x^3 + 1)/(x^3 - 2) To: Function field in y defined by y^3 + (-x^3 + 1)/(x^3 - 2)
>>> from sage.all import * >>> K = FunctionField(QQ, names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1) >>> L = K.extension(Y**Integer(3) - (x**Integer(3) - Integer(1))/(x**Integer(3) - Integer(2)), names=('y',)); (y,) = L._first_ngens(1) >>> L.higher_derivation() # needs sage.modules Higher derivation map: From: Function field in y defined by y^3 + (-x^3 + 1)/(x^3 - 2) To: Function field in y defined by y^3 + (-x^3 + 1)/(x^3 - 2)
K.<x> = FunctionField(QQ); _.<Y> = K[] L.<y> = K.extension(Y^3 - (x^3 - 1)/(x^3 - 2)) L.higher_derivation() # needs sage.modules
- class sage.rings.function_field.function_field_polymod.FunctionField_char_zero_integral(polynomial, names, category=None)[source]¶
Bases:
FunctionField_char_zero,FunctionField_integralFunction fields of characteristic zero, defined by an irreducible and separable polynomial, integral over the maximal order of the base rational function field with a finite constant field.
- class sage.rings.function_field.function_field_polymod.FunctionField_global(polynomial, names)[source]¶
Bases:
FunctionField_simpleGlobal function fields.
INPUT:
polynomial– monic irreducible and separable polynomialnames– name of the generator of the function field
EXAMPLES:
sage: K.<x> = FunctionField(GF(5)); _.<Y> = K[] # needs sage.rings.finite_rings sage: L.<y> = K.extension(Y^3 - (x^3 - 1)/(x^3 - 2)) # needs sage.rings.finite_rings sage: L # needs sage.rings.finite_rings Function field in y defined by y^3 + (4*x^3 + 1)/(x^3 + 3)
>>> from sage.all import * >>> K = FunctionField(GF(Integer(5)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)# needs sage.rings.finite_rings >>> L = K.extension(Y**Integer(3) - (x**Integer(3) - Integer(1))/(x**Integer(3) - Integer(2)), names=('y',)); (y,) = L._first_ngens(1)# needs sage.rings.finite_rings >>> L # needs sage.rings.finite_rings Function field in y defined by y^3 + (4*x^3 + 1)/(x^3 + 3)
K.<x> = FunctionField(GF(5)); _.<Y> = K[] # needs sage.rings.finite_rings L.<y> = K.extension(Y^3 - (x^3 - 1)/(x^3 - 2)) # needs sage.rings.finite_rings L # needs sage.rings.finite_rings
The defining equation needs not be monic:
sage: K.<x> = FunctionField(GF(4)); _.<Y> = K[] # needs sage.rings.finite_rings sage: L.<y> = K.extension((1 - x)*Y^7 - x^3) # needs sage.rings.finite_rings sage: L.gaps() # long time (6s) # needs sage.rings.finite_rings [1, 2, 3]
>>> from sage.all import * >>> K = FunctionField(GF(Integer(4)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)# needs sage.rings.finite_rings >>> L = K.extension((Integer(1) - x)*Y**Integer(7) - x**Integer(3), names=('y',)); (y,) = L._first_ngens(1)# needs sage.rings.finite_rings >>> L.gaps() # long time (6s) # needs sage.rings.finite_rings [1, 2, 3]
K.<x> = FunctionField(GF(4)); _.<Y> = K[] # needs sage.rings.finite_rings L.<y> = K.extension((1 - x)*Y^7 - x^3) # needs sage.rings.finite_rings L.gaps() # long time (6s) # needs sage.rings.finite_rings
or may define a trivial extension:
sage: K.<x> = FunctionField(GF(5)); _.<Y> = K[] # needs sage.rings.finite_rings sage: L.<y> = K.extension(Y-1) # needs sage.rings.finite_rings sage: L.genus() # needs sage.rings.finite_rings 0
>>> from sage.all import * >>> K = FunctionField(GF(Integer(5)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)# needs sage.rings.finite_rings >>> L = K.extension(Y-Integer(1), names=('y',)); (y,) = L._first_ngens(1)# needs sage.rings.finite_rings >>> L.genus() # needs sage.rings.finite_rings 0
K.<x> = FunctionField(GF(5)); _.<Y> = K[] # needs sage.rings.finite_rings L.<y> = K.extension(Y-1) # needs sage.rings.finite_rings L.genus() # needs sage.rings.finite_rings
- L_polynomial(name='t')[source]¶
Return the L-polynomial of the function field.
INPUT:
name– (default:t) name of the variable of the polynomial
EXAMPLES:
sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[] # needs sage.rings.finite_rings sage: F.<y> = K.extension(Y^2 + Y + x + 1/x) # needs sage.rings.finite_rings sage: F.L_polynomial() # needs sage.rings.finite_rings 2*t^2 + t + 1
>>> from sage.all import * >>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)# needs sage.rings.finite_rings >>> F = K.extension(Y**Integer(2) + Y + x + Integer(1)/x, names=('y',)); (y,) = F._first_ngens(1)# needs sage.rings.finite_rings >>> F.L_polynomial() # needs sage.rings.finite_rings 2*t^2 + t + 1
K.<x> = FunctionField(GF(2)); _.<Y> = K[] # needs sage.rings.finite_rings F.<y> = K.extension(Y^2 + Y + x + 1/x) # needs sage.rings.finite_rings F.L_polynomial() # needs sage.rings.finite_rings
- gaps()[source]¶
Return the gaps of the function field.
These are the gaps at the ordinary places, that is, places which are not Weierstrass places.
EXAMPLES:
sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[] # needs sage.rings.finite_rings sage: L.<y> = K.extension(Y^3 + x^3 * Y + x) # needs sage.rings.finite_rings sage: L.gaps() # needs sage.modules sage.rings.finite_rings [1, 2, 3]
>>> from sage.all import * >>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)# needs sage.rings.finite_rings >>> L = K.extension(Y**Integer(3) + x**Integer(3) * Y + x, names=('y',)); (y,) = L._first_ngens(1)# needs sage.rings.finite_rings >>> L.gaps() # needs sage.modules sage.rings.finite_rings [1, 2, 3]
K.<x> = FunctionField(GF(2)); _.<Y> = K[] # needs sage.rings.finite_rings L.<y> = K.extension(Y^3 + x^3 * Y + x) # needs sage.rings.finite_rings L.gaps() # needs sage.modules sage.rings.finite_rings
- get_place(degree)[source]¶
Return a place of
degree.INPUT:
degree– positive integer
OUTPUT: a place of
degreeif any exists; otherwiseNoneEXAMPLES:
sage: # needs sage.rings.finite_rings sage: F.<a> = GF(2) sage: K.<x> = FunctionField(F) sage: R.<Y> = PolynomialRing(K) sage: L.<y> = K.extension(Y^4 + Y - x^5) sage: L.get_place(1) Place (x, y) sage: L.get_place(2) Place (x, y^2 + y + 1) sage: L.get_place(3) Place (x^3 + x^2 + 1, y + x^2 + x) sage: L.get_place(4) Place (x + 1, x^5 + 1) sage: L.get_place(5) Place (x^5 + x^3 + x^2 + x + 1, y + x^4 + 1) sage: L.get_place(6) Place (x^3 + x^2 + 1, y^2 + y + x^2) sage: L.get_place(7) Place (x^7 + x + 1, y + x^6 + x^5 + x^4 + x^3 + x) sage: L.get_place(8)
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> F = GF(Integer(2), names=('a',)); (a,) = F._first_ngens(1) >>> K = FunctionField(F, names=('x',)); (x,) = K._first_ngens(1) >>> R = PolynomialRing(K, names=('Y',)); (Y,) = R._first_ngens(1) >>> L = K.extension(Y**Integer(4) + Y - x**Integer(5), names=('y',)); (y,) = L._first_ngens(1) >>> L.get_place(Integer(1)) Place (x, y) >>> L.get_place(Integer(2)) Place (x, y^2 + y + 1) >>> L.get_place(Integer(3)) Place (x^3 + x^2 + 1, y + x^2 + x) >>> L.get_place(Integer(4)) Place (x + 1, x^5 + 1) >>> L.get_place(Integer(5)) Place (x^5 + x^3 + x^2 + x + 1, y + x^4 + 1) >>> L.get_place(Integer(6)) Place (x^3 + x^2 + 1, y^2 + y + x^2) >>> L.get_place(Integer(7)) Place (x^7 + x + 1, y + x^6 + x^5 + x^4 + x^3 + x) >>> L.get_place(Integer(8))
# needs sage.rings.finite_rings F.<a> = GF(2) K.<x> = FunctionField(F) R.<Y> = PolynomialRing(K) L.<y> = K.extension(Y^4 + Y - x^5) L.get_place(1) L.get_place(2) L.get_place(3) L.get_place(4) L.get_place(5) L.get_place(6) L.get_place(7) L.get_place(8)
- higher_derivation()[source]¶
Return the higher derivation (also called the Hasse-Schmidt derivation) for the function field.
The higher derivation of the function field is uniquely determined with respect to the separating element \(x\) of the base rational function field \(k(x)\).
EXAMPLES:
sage: K.<x> = FunctionField(GF(5)); _.<Y> = K[] # needs sage.rings.finite_rings sage: L.<y> = K.extension(Y^3 - (x^3 - 1)/(x^3 - 2)) # needs sage.rings.finite_rings sage: L.higher_derivation() # needs sage.modules sage.rings.finite_rings Higher derivation map: From: Function field in y defined by y^3 + (4*x^3 + 1)/(x^3 + 3) To: Function field in y defined by y^3 + (4*x^3 + 1)/(x^3 + 3)
>>> from sage.all import * >>> K = FunctionField(GF(Integer(5)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)# needs sage.rings.finite_rings >>> L = K.extension(Y**Integer(3) - (x**Integer(3) - Integer(1))/(x**Integer(3) - Integer(2)), names=('y',)); (y,) = L._first_ngens(1)# needs sage.rings.finite_rings >>> L.higher_derivation() # needs sage.modules sage.rings.finite_rings Higher derivation map: From: Function field in y defined by y^3 + (4*x^3 + 1)/(x^3 + 3) To: Function field in y defined by y^3 + (4*x^3 + 1)/(x^3 + 3)
K.<x> = FunctionField(GF(5)); _.<Y> = K[] # needs sage.rings.finite_rings L.<y> = K.extension(Y^3 - (x^3 - 1)/(x^3 - 2)) # needs sage.rings.finite_rings L.higher_derivation() # needs sage.modules sage.rings.finite_rings
- maximal_order()[source]¶
Return the maximal order of the function field.
EXAMPLES:
sage: # needs sage.rings.finite_rings sage: K.<x> = FunctionField(GF(2)) sage: R.<t> = PolynomialRing(K) sage: F.<y> = K.extension(t^4 + x^12*t^2 + x^18*t + x^21 + x^18) sage: O = F.maximal_order() sage: O.basis() (1, 1/x^4*y, 1/x^11*y^2 + 1/x^2, 1/x^15*y^3 + 1/x^6*y)
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1) >>> R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1) >>> F = K.extension(t**Integer(4) + x**Integer(12)*t**Integer(2) + x**Integer(18)*t + x**Integer(21) + x**Integer(18), names=('y',)); (y,) = F._first_ngens(1) >>> O = F.maximal_order() >>> O.basis() (1, 1/x^4*y, 1/x^11*y^2 + 1/x^2, 1/x^15*y^3 + 1/x^6*y)
# needs sage.rings.finite_rings K.<x> = FunctionField(GF(2)) R.<t> = PolynomialRing(K) F.<y> = K.extension(t^4 + x^12*t^2 + x^18*t + x^21 + x^18) O = F.maximal_order() O.basis()
- number_of_rational_places(r=1)[source]¶
Return the number of rational places of the function field whose constant field extended by degree
r.INPUT:
r– positive integer (default: \(1\))
EXAMPLES:
sage: # needs sage.rings.finite_rings sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[] sage: F.<y> = K.extension(Y^2 + Y + x + 1/x) sage: F.number_of_rational_places() 4 sage: [F.number_of_rational_places(r) for r in [1..10]] [4, 8, 4, 16, 44, 56, 116, 288, 508, 968]
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1) >>> F = K.extension(Y**Integer(2) + Y + x + Integer(1)/x, names=('y',)); (y,) = F._first_ngens(1) >>> F.number_of_rational_places() 4 >>> [F.number_of_rational_places(r) for r in (ellipsis_range(Integer(1),Ellipsis,Integer(10)))] [4, 8, 4, 16, 44, 56, 116, 288, 508, 968]
# needs sage.rings.finite_rings K.<x> = FunctionField(GF(2)); _.<Y> = K[] F.<y> = K.extension(Y^2 + Y + x + 1/x) F.number_of_rational_places() [F.number_of_rational_places(r) for r in [1..10]]
- places(degree=1)[source]¶
Return a list of the places with
degree.INPUT:
degree– positive integer (default: \(1\))
EXAMPLES:
sage: # needs sage.rings.finite_rings sage: F.<a> = GF(2) sage: K.<x> = FunctionField(F) sage: R.<t> = PolynomialRing(K) sage: L.<y> = K.extension(t^4 + t - x^5) sage: L.places(1) [Place (1/x, 1/x^4*y^3), Place (x, y), Place (x, y + 1)]
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> F = GF(Integer(2), names=('a',)); (a,) = F._first_ngens(1) >>> K = FunctionField(F, names=('x',)); (x,) = K._first_ngens(1) >>> R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1) >>> L = K.extension(t**Integer(4) + t - x**Integer(5), names=('y',)); (y,) = L._first_ngens(1) >>> L.places(Integer(1)) [Place (1/x, 1/x^4*y^3), Place (x, y), Place (x, y + 1)]
# needs sage.rings.finite_rings F.<a> = GF(2) K.<x> = FunctionField(F) R.<t> = PolynomialRing(K) L.<y> = K.extension(t^4 + t - x^5) L.places(1)
- places_finite(degree=1)[source]¶
Return a list of the finite places with
degree.INPUT:
degree– positive integer (default: \(1\))
EXAMPLES:
sage: # needs sage.rings.finite_rings sage: F.<a> = GF(2) sage: K.<x> = FunctionField(F) sage: R.<t> = PolynomialRing(K) sage: L.<y> = K.extension(t^4 + t - x^5) sage: L.places_finite(1) [Place (x, y), Place (x, y + 1)]
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> F = GF(Integer(2), names=('a',)); (a,) = F._first_ngens(1) >>> K = FunctionField(F, names=('x',)); (x,) = K._first_ngens(1) >>> R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1) >>> L = K.extension(t**Integer(4) + t - x**Integer(5), names=('y',)); (y,) = L._first_ngens(1) >>> L.places_finite(Integer(1)) [Place (x, y), Place (x, y + 1)]
# needs sage.rings.finite_rings F.<a> = GF(2) K.<x> = FunctionField(F) R.<t> = PolynomialRing(K) L.<y> = K.extension(t^4 + t - x^5) L.places_finite(1)
- weierstrass_places()[source]¶
Return all Weierstrass places of the function field.
EXAMPLES:
sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[] # needs sage.rings.finite_rings sage: L.<y> = K.extension(Y^3 + x^3 * Y + x) # needs sage.rings.finite_rings sage: L.weierstrass_places() # needs sage.modules sage.rings.finite_rings [Place (1/x, 1/x^3*y^2 + 1/x), Place (1/x, 1/x^3*y^2 + 1/x^2*y + 1), Place (x, y), Place (x + 1, (x^3 + 1)*y + x + 1), Place (x^3 + x + 1, y + 1), Place (x^3 + x + 1, y + x^2), Place (x^3 + x + 1, y + x^2 + 1), Place (x^3 + x^2 + 1, y + x), Place (x^3 + x^2 + 1, y + x^2 + 1), Place (x^3 + x^2 + 1, y + x^2 + x + 1)]
>>> from sage.all import * >>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)# needs sage.rings.finite_rings >>> L = K.extension(Y**Integer(3) + x**Integer(3) * Y + x, names=('y',)); (y,) = L._first_ngens(1)# needs sage.rings.finite_rings >>> L.weierstrass_places() # needs sage.modules sage.rings.finite_rings [Place (1/x, 1/x^3*y^2 + 1/x), Place (1/x, 1/x^3*y^2 + 1/x^2*y + 1), Place (x, y), Place (x + 1, (x^3 + 1)*y + x + 1), Place (x^3 + x + 1, y + 1), Place (x^3 + x + 1, y + x^2), Place (x^3 + x + 1, y + x^2 + 1), Place (x^3 + x^2 + 1, y + x), Place (x^3 + x^2 + 1, y + x^2 + 1), Place (x^3 + x^2 + 1, y + x^2 + x + 1)]
K.<x> = FunctionField(GF(2)); _.<Y> = K[] # needs sage.rings.finite_rings L.<y> = K.extension(Y^3 + x^3 * Y + x) # needs sage.rings.finite_rings L.weierstrass_places() # needs sage.modules sage.rings.finite_rings
- class sage.rings.function_field.function_field_polymod.FunctionField_global_integral(polynomial, names)[source]¶
Bases:
FunctionField_global,FunctionField_integralGlobal function fields, defined by an irreducible and separable polynomial, integral over the maximal order of the base rational function field with a finite constant field.
- class sage.rings.function_field.function_field_polymod.FunctionField_integral(polynomial, names, category=None)[source]¶
Bases:
FunctionField_simpleIntegral function fields.
A function field is integral if it is defined by an irreducible separable polynomial, which is integral over the maximal order of the base rational function field.
- equation_order()[source]¶
Return the equation order of the function field.
EXAMPLES:
sage: K.<x> = FunctionField(GF(2)); R.<t> = PolynomialRing(K) # needs sage.rings.finite_rings sage: F.<y> = K.extension(t^3 - x^2*(x^2+x+1)^2) # needs sage.rings.finite_rings sage: F.equation_order() # needs sage.rings.finite_rings Order in Function field in y defined by y^3 + x^6 + x^4 + x^2 sage: K.<x> = FunctionField(QQ); R.<t> = PolynomialRing(K) sage: F.<y> = K.extension(t^3 - x^2*(x^2+x+1)^2) sage: F.equation_order() Order in Function field in y defined by y^3 - x^6 - 2*x^5 - 3*x^4 - 2*x^3 - x^2
>>> from sage.all import * >>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1)# needs sage.rings.finite_rings >>> F = K.extension(t**Integer(3) - x**Integer(2)*(x**Integer(2)+x+Integer(1))**Integer(2), names=('y',)); (y,) = F._first_ngens(1)# needs sage.rings.finite_rings >>> F.equation_order() # needs sage.rings.finite_rings Order in Function field in y defined by y^3 + x^6 + x^4 + x^2 >>> K = FunctionField(QQ, names=('x',)); (x,) = K._first_ngens(1); R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1) >>> F = K.extension(t**Integer(3) - x**Integer(2)*(x**Integer(2)+x+Integer(1))**Integer(2), names=('y',)); (y,) = F._first_ngens(1) >>> F.equation_order() Order in Function field in y defined by y^3 - x^6 - 2*x^5 - 3*x^4 - 2*x^3 - x^2
K.<x> = FunctionField(GF(2)); R.<t> = PolynomialRing(K) # needs sage.rings.finite_rings F.<y> = K.extension(t^3 - x^2*(x^2+x+1)^2) # needs sage.rings.finite_rings F.equation_order() # needs sage.rings.finite_rings K.<x> = FunctionField(QQ); R.<t> = PolynomialRing(K) F.<y> = K.extension(t^3 - x^2*(x^2+x+1)^2) F.equation_order()
- equation_order_infinite()[source]¶
Return the infinite equation order of the function field.
This is by definition \(o[b]\) where \(b\) is the primitive integral element from
primitive_integal_element_infinite()and \(o\) is the maximal infinite order of the base rational function field.EXAMPLES:
sage: K.<x> = FunctionField(GF(2)); R.<t> = PolynomialRing(K) # needs sage.rings.finite_rings sage: F.<y> = K.extension(t^3 - x^2*(x^2+x+1)^2) # needs sage.rings.finite_rings sage: F.equation_order_infinite() # needs sage.rings.finite_rings Infinite order in Function field in y defined by y^3 + x^6 + x^4 + x^2 sage: K.<x> = FunctionField(QQ); R.<t> = PolynomialRing(K) sage: F.<y> = K.extension(t^3 - x^2*(x^2+x+1)^2) sage: F.equation_order_infinite() Infinite order in Function field in y defined by y^3 - x^6 - 2*x^5 - 3*x^4 - 2*x^3 - x^2
>>> from sage.all import * >>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1)# needs sage.rings.finite_rings >>> F = K.extension(t**Integer(3) - x**Integer(2)*(x**Integer(2)+x+Integer(1))**Integer(2), names=('y',)); (y,) = F._first_ngens(1)# needs sage.rings.finite_rings >>> F.equation_order_infinite() # needs sage.rings.finite_rings Infinite order in Function field in y defined by y^3 + x^6 + x^4 + x^2 >>> K = FunctionField(QQ, names=('x',)); (x,) = K._first_ngens(1); R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1) >>> F = K.extension(t**Integer(3) - x**Integer(2)*(x**Integer(2)+x+Integer(1))**Integer(2), names=('y',)); (y,) = F._first_ngens(1) >>> F.equation_order_infinite() Infinite order in Function field in y defined by y^3 - x^6 - 2*x^5 - 3*x^4 - 2*x^3 - x^2
K.<x> = FunctionField(GF(2)); R.<t> = PolynomialRing(K) # needs sage.rings.finite_rings F.<y> = K.extension(t^3 - x^2*(x^2+x+1)^2) # needs sage.rings.finite_rings F.equation_order_infinite() # needs sage.rings.finite_rings K.<x> = FunctionField(QQ); R.<t> = PolynomialRing(K) F.<y> = K.extension(t^3 - x^2*(x^2+x+1)^2) F.equation_order_infinite()
- primitive_integal_element_infinite()[source]¶
Return a primitive integral element over the base maximal infinite order.
This element is integral over the maximal infinite order of the base rational function field and the function field is a simple extension by this element over the base order.
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: b = F.primitive_integal_element_infinite(); b 1/x^2*y sage: b.minimal_polynomial('t') t^3 + (x^4 + x^2 + 1)/x^4
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1) >>> F = K.extension(t**Integer(3) - x**Integer(2)*(x**Integer(2)+x+Integer(1))**Integer(2), names=('y',)); (y,) = F._first_ngens(1) >>> b = F.primitive_integal_element_infinite(); b 1/x^2*y >>> b.minimal_polynomial('t') t^3 + (x^4 + x^2 + 1)/x^4
# needs sage.rings.finite_rings K.<x> = FunctionField(GF(2)); R.<t> = PolynomialRing(K) F.<y> = K.extension(t^3 - x^2*(x^2+x+1)^2) b = F.primitive_integal_element_infinite(); b b.minimal_polynomial('t')
- class sage.rings.function_field.function_field_polymod.FunctionField_polymod(polynomial, names, category=None)[source]¶
Bases:
FunctionFieldFunction fields defined by a univariate polynomial, as an extension of the base field.
INPUT:
polynomial– univariate polynomial over a function fieldnames– tuple of length 1 or string; variable namescategory– category (default: category of function fields)
EXAMPLES:
We make a function field defined by a degree 5 polynomial over the rational function field over the rational numbers:
sage: K.<x> = FunctionField(QQ) sage: R.<y> = K[] sage: L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)); L Function field in y defined by y^5 - 2*x*y + (-x^4 - 1)/x
>>> 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(5) - (x**Integer(3) + Integer(2)*x*y + Integer(1)/x), names=('y',)); (y,) = L._first_ngens(1); L Function field in y defined by y^5 - 2*x*y + (-x^4 - 1)/x
K.<x> = FunctionField(QQ) R.<y> = K[] L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)); L
We next make a function field over the above nontrivial function field L:
sage: S.<z> = L[] sage: M.<z> = L.extension(z^2 + y*z + y); M Function field in z defined by z^2 + y*z + y sage: 1/z ((-x/(x^4 + 1))*y^4 + 2*x^2/(x^4 + 1))*z - 1 sage: z * (1/z) 1
>>> from sage.all import * >>> S = L['z']; (z,) = S._first_ngens(1) >>> M = L.extension(z**Integer(2) + y*z + y, names=('z',)); (z,) = M._first_ngens(1); M Function field in z defined by z^2 + y*z + y >>> Integer(1)/z ((-x/(x^4 + 1))*y^4 + 2*x^2/(x^4 + 1))*z - 1 >>> z * (Integer(1)/z) 1
S.<z> = L[] M.<z> = L.extension(z^2 + y*z + y); M 1/z z * (1/z)
We drill down the tower of function fields:
sage: M.base_field() Function field in y defined by y^5 - 2*x*y + (-x^4 - 1)/x sage: M.base_field().base_field() Rational function field in x over Rational Field sage: M.base_field().base_field().constant_field() Rational Field sage: M.constant_base_field() Rational Field
>>> from sage.all import * >>> M.base_field() Function field in y defined by y^5 - 2*x*y + (-x^4 - 1)/x >>> M.base_field().base_field() Rational function field in x over Rational Field >>> M.base_field().base_field().constant_field() Rational Field >>> M.constant_base_field() Rational Field
M.base_field() M.base_field().base_field() M.base_field().base_field().constant_field() M.constant_base_field()
Warning
It is not checked if the polynomial used to define the function field is irreducible Hence it is not guaranteed that this object really is a field! This is illustrated below.
sage: K.<x> = FunctionField(QQ) sage: R.<y> = K[] sage: L.<y> = K.extension(x^2 - y^2) sage: (y - x)*(y + x) 0 sage: 1/(y - x) 1 sage: y - x == 0; y + x == 0 False False
>>> 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(x**Integer(2) - y**Integer(2), names=('y',)); (y,) = L._first_ngens(1) >>> (y - x)*(y + x) 0 >>> Integer(1)/(y - x) 1 >>> y - x == Integer(0); y + x == Integer(0) False False
K.<x> = FunctionField(QQ) R.<y> = K[] L.<y> = K.extension(x^2 - y^2) (y - x)*(y + x) 1/(y - x) y - x == 0; y + x == 0
- Element[source]¶
alias of
FunctionFieldElement_polymod
- base_field()[source]¶
Return the base field of the function field. This function field is presented as \(L = K[y]/(f(y))\), and the base field is by definition the field \(K\).
EXAMPLES:
sage: K.<x> = FunctionField(QQ); R.<y> = K[] sage: L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)) sage: L.base_field() Rational function field in x over Rational Field
>>> 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(5) - (x**Integer(3) + Integer(2)*x*y + Integer(1)/x), names=('y',)); (y,) = L._first_ngens(1) >>> L.base_field() Rational function field in x over Rational Field
K.<x> = FunctionField(QQ); R.<y> = K[] L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)) L.base_field()
- change_variable_name(name)[source]¶
Return a field isomorphic to this field with variable(s)
name.INPUT:
name– string or tuple consisting of a strings; the names of the new variables starting with a generator of this field and going down to the rational function field
OUTPUT:
A triple
F,f,twhereFis a function field,fis an isomorphism fromFto this field, andtis the inverse off.EXAMPLES:
sage: K.<x> = FunctionField(QQ) sage: R.<y> = K[] sage: L.<y> = K.extension(y^2 - x) sage: R.<z> = L[] sage: M.<z> = L.extension(z^2 - y) sage: M.change_variable_name('zz') (Function field in zz defined by zz^2 - y, Function Field morphism: From: Function field in zz defined by zz^2 - y To: Function field in z defined by z^2 - y Defn: zz |--> z y |--> y x |--> x, Function Field morphism: From: Function field in z defined by z^2 - y To: Function field in zz defined by zz^2 - y Defn: z |--> zz y |--> y x |--> x) sage: M.change_variable_name(('zz','yy')) (Function field in zz defined by zz^2 - yy, Function Field morphism: From: Function field in zz defined by zz^2 - yy To: Function field in z defined by z^2 - y Defn: zz |--> z yy |--> y x |--> x, Function Field morphism: From: Function field in z defined by z^2 - y To: Function field in zz defined by zz^2 - yy Defn: z |--> zz y |--> yy x |--> x) sage: M.change_variable_name(('zz','yy','xx')) (Function field in zz defined by zz^2 - yy, Function Field morphism: From: Function field in zz defined by zz^2 - yy To: Function field in z defined by z^2 - y Defn: zz |--> z yy |--> y xx |--> x, Function Field morphism: From: Function field in z defined by z^2 - y To: Function field in zz defined by zz^2 - yy Defn: z |--> zz y |--> yy x |--> xx)
>>> 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, names=('y',)); (y,) = L._first_ngens(1) >>> R = L['z']; (z,) = R._first_ngens(1) >>> M = L.extension(z**Integer(2) - y, names=('z',)); (z,) = M._first_ngens(1) >>> M.change_variable_name('zz') (Function field in zz defined by zz^2 - y, Function Field morphism: From: Function field in zz defined by zz^2 - y To: Function field in z defined by z^2 - y Defn: zz |--> z y |--> y x |--> x, Function Field morphism: From: Function field in z defined by z^2 - y To: Function field in zz defined by zz^2 - y Defn: z |--> zz y |--> y x |--> x) >>> M.change_variable_name(('zz','yy')) (Function field in zz defined by zz^2 - yy, Function Field morphism: From: Function field in zz defined by zz^2 - yy To: Function field in z defined by z^2 - y Defn: zz |--> z yy |--> y x |--> x, Function Field morphism: From: Function field in z defined by z^2 - y To: Function field in zz defined by zz^2 - yy Defn: z |--> zz y |--> yy x |--> x) >>> M.change_variable_name(('zz','yy','xx')) (Function field in zz defined by zz^2 - yy, Function Field morphism: From: Function field in zz defined by zz^2 - yy To: Function field in z defined by z^2 - y Defn: zz |--> z yy |--> y xx |--> x, Function Field morphism: From: Function field in z defined by z^2 - y To: Function field in zz defined by zz^2 - yy Defn: z |--> zz y |--> yy x |--> xx)
K.<x> = FunctionField(QQ) R.<y> = K[] L.<y> = K.extension(y^2 - x) R.<z> = L[] M.<z> = L.extension(z^2 - y) M.change_variable_name('zz') M.change_variable_name(('zz','yy')) M.change_variable_name(('zz','yy','xx'))
- constant_base_field()[source]¶
Return the base constant field of the function field.
EXAMPLES:
sage: K.<x> = FunctionField(QQ); R.<y> = K[] sage: L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)); L Function field in y defined by y^5 - 2*x*y + (-x^4 - 1)/x sage: L.constant_base_field() Rational Field sage: S.<z> = L[] sage: M.<z> = L.extension(z^2 - y) sage: M.constant_base_field() Rational Field
>>> 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(5) - (x**Integer(3) + Integer(2)*x*y + Integer(1)/x), names=('y',)); (y,) = L._first_ngens(1); L Function field in y defined by y^5 - 2*x*y + (-x^4 - 1)/x >>> L.constant_base_field() Rational Field >>> S = L['z']; (z,) = S._first_ngens(1) >>> M = L.extension(z**Integer(2) - y, names=('z',)); (z,) = M._first_ngens(1) >>> M.constant_base_field() Rational Field
K.<x> = FunctionField(QQ); R.<y> = K[] L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)); L L.constant_base_field() S.<z> = L[] M.<z> = L.extension(z^2 - y) M.constant_base_field()
- constant_field()[source]¶
Return the algebraic closure of the constant field of the function field.
EXAMPLES:
sage: K.<x> = FunctionField(GF(5)); _.<Y> = K[] # needs sage.rings.finite_rings sage: L.<y> = K.extension(Y^5 - x) # needs sage.rings.finite_rings sage: L.constant_field() # needs sage.rings.finite_rings Traceback (most recent call last): ... NotImplementedError
>>> from sage.all import * >>> K = FunctionField(GF(Integer(5)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)# needs sage.rings.finite_rings >>> L = K.extension(Y**Integer(5) - x, names=('y',)); (y,) = L._first_ngens(1)# needs sage.rings.finite_rings >>> L.constant_field() # needs sage.rings.finite_rings Traceback (most recent call last): ... NotImplementedError
K.<x> = FunctionField(GF(5)); _.<Y> = K[] # needs sage.rings.finite_rings L.<y> = K.extension(Y^5 - x) # needs sage.rings.finite_rings L.constant_field() # needs sage.rings.finite_rings
- degree(base=None)[source]¶
Return the degree of the function field over the function field
base.INPUT:
base– a function field (default:None), a function field from which this field has been constructed as a finite extension
EXAMPLES:
sage: K.<x> = FunctionField(QQ) sage: R.<y> = K[] sage: L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)); L Function field in y defined by y^5 - 2*x*y + (-x^4 - 1)/x sage: L.degree() 5 sage: L.degree(L) 1 sage: R.<z> = L[] sage: M.<z> = L.extension(z^2 - y) sage: M.degree(L) 2 sage: M.degree(K) 10
>>> 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(5) - (x**Integer(3) + Integer(2)*x*y + Integer(1)/x), names=('y',)); (y,) = L._first_ngens(1); L Function field in y defined by y^5 - 2*x*y + (-x^4 - 1)/x >>> L.degree() 5 >>> L.degree(L) 1 >>> R = L['z']; (z,) = R._first_ngens(1) >>> M = L.extension(z**Integer(2) - y, names=('z',)); (z,) = M._first_ngens(1) >>> M.degree(L) 2 >>> M.degree(K) 10
K.<x> = FunctionField(QQ) R.<y> = K[] L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)); L L.degree() L.degree(L) R.<z> = L[] M.<z> = L.extension(z^2 - y) M.degree(L) M.degree(K)
- different()[source]¶
Return the different of the function field.
EXAMPLES:
sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[] # needs sage.rings.finite_rings sage: F.<y> = K.extension(Y^3 - x^2*(x^2 + x + 1)^2) # needs sage.rings.finite_rings sage: F.different() # needs sage.rings.finite_rings 2*Place (x, (1/(x^3 + x^2 + x))*y^2) + 2*Place (x^2 + x + 1, (1/(x^3 + x^2 + x))*y^2)
>>> from sage.all import * >>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)# needs sage.rings.finite_rings >>> F = K.extension(Y**Integer(3) - x**Integer(2)*(x**Integer(2) + x + Integer(1))**Integer(2), names=('y',)); (y,) = F._first_ngens(1)# needs sage.rings.finite_rings >>> F.different() # needs sage.rings.finite_rings 2*Place (x, (1/(x^3 + x^2 + x))*y^2) + 2*Place (x^2 + x + 1, (1/(x^3 + x^2 + x))*y^2)
K.<x> = FunctionField(GF(2)); _.<Y> = K[] # needs sage.rings.finite_rings F.<y> = K.extension(Y^3 - x^2*(x^2 + x + 1)^2) # needs sage.rings.finite_rings F.different() # needs sage.rings.finite_rings
- equation_order()[source]¶
Return the equation order of the function field.
If we view the function field as being presented as \(K[y]/(f(y))\), then the order generated by the class of \(y\) is returned. If \(f\) is not monic, then
_make_monic_integral()is called, and instead we get the order generated by some integral multiple of a root of \(f\).EXAMPLES:
sage: K.<x> = FunctionField(QQ); R.<y> = K[] sage: L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)) sage: O = L.equation_order() sage: O.basis() (1, x*y, x^2*y^2, x^3*y^3, x^4*y^4)
>>> 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(5) - (x**Integer(3) + Integer(2)*x*y + Integer(1)/x), names=('y',)); (y,) = L._first_ngens(1) >>> O = L.equation_order() >>> O.basis() (1, x*y, x^2*y^2, x^3*y^3, x^4*y^4)
K.<x> = FunctionField(QQ); R.<y> = K[] L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)) O = L.equation_order() O.basis()
We try an example, in which the defining polynomial is not monic and is not integral:
sage: K.<x> = FunctionField(QQ); R.<y> = K[] sage: L.<y> = K.extension(x^2*y^5 - 1/x); L Function field in y defined by x^2*y^5 - 1/x sage: O = L.equation_order() sage: O.basis() (1, x^3*y, x^6*y^2, x^9*y^3, x^12*y^4)
>>> 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(x**Integer(2)*y**Integer(5) - Integer(1)/x, names=('y',)); (y,) = L._first_ngens(1); L Function field in y defined by x^2*y^5 - 1/x >>> O = L.equation_order() >>> O.basis() (1, x^3*y, x^6*y^2, x^9*y^3, x^12*y^4)
K.<x> = FunctionField(QQ); R.<y> = K[] L.<y> = K.extension(x^2*y^5 - 1/x); L O = L.equation_order() O.basis()
- free_module(base=None, basis=None, map=True)[source]¶
Return a vector space and isomorphisms from the field to and from the vector space.
This function allows us to identify the elements of this field with elements of a vector space over the base field, which is useful for representation and arithmetic with orders, ideals, etc.
INPUT:
base– a function field (default:None), the returned vector space is over this subfield \(R\), which defaults to the base field of this function field.basis– a basis for this field over the basemaps– boolean (default:True); whether to return \(R\)-linear maps to and from \(V\)
OUTPUT:
a vector space over the base function field
an isomorphism from the vector space to the field (if requested)
an isomorphism from the field to the vector space (if requested)
EXAMPLES:
We define a function field:
sage: K.<x> = FunctionField(QQ); R.<y> = K[] sage: L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)); L Function field in y defined by y^5 - 2*x*y + (-x^4 - 1)/x
>>> 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(5) - (x**Integer(3) + Integer(2)*x*y + Integer(1)/x), names=('y',)); (y,) = L._first_ngens(1); L Function field in y defined by y^5 - 2*x*y + (-x^4 - 1)/x
K.<x> = FunctionField(QQ); R.<y> = K[] L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)); L
We get the vector spaces, and maps back and forth:
sage: # needs sage.modules sage: V, from_V, to_V = L.free_module() sage: V Vector space of dimension 5 over Rational function field in x over Rational Field sage: from_V Isomorphism: From: Vector space of dimension 5 over Rational function field in x over Rational Field To: Function field in y defined by y^5 - 2*x*y + (-x^4 - 1)/x sage: to_V Isomorphism: From: Function field in y defined by y^5 - 2*x*y + (-x^4 - 1)/x To: Vector space of dimension 5 over Rational function field in x over Rational Field
>>> from sage.all import * >>> # needs sage.modules >>> V, from_V, to_V = L.free_module() >>> V Vector space of dimension 5 over Rational function field in x over Rational Field >>> from_V Isomorphism: From: Vector space of dimension 5 over Rational function field in x over Rational Field To: Function field in y defined by y^5 - 2*x*y + (-x^4 - 1)/x >>> to_V Isomorphism: From: Function field in y defined by y^5 - 2*x*y + (-x^4 - 1)/x To: Vector space of dimension 5 over Rational function field in x over Rational Field
# needs sage.modules V, from_V, to_V = L.free_module() V from_V to_V
We convert an element of the vector space back to the function field:
sage: from_V(V.1) # needs sage.modules y
>>> from sage.all import * >>> from_V(V.gen(1)) # needs sage.modules y
from_V(V.1) # needs sage.modules
We define an interesting element of the function field:
sage: a = 1/L.0; a # needs sage.modules (x/(x^4 + 1))*y^4 - 2*x^2/(x^4 + 1)
>>> from sage.all import * >>> a = Integer(1)/L.gen(0); a # needs sage.modules (x/(x^4 + 1))*y^4 - 2*x^2/(x^4 + 1)
a = 1/L.0; a # needs sage.modules
We convert it to the vector space, and get a vector over the base field:
sage: to_V(a) # needs sage.modules (-2*x^2/(x^4 + 1), 0, 0, 0, x/(x^4 + 1))
>>> from sage.all import * >>> to_V(a) # needs sage.modules (-2*x^2/(x^4 + 1), 0, 0, 0, x/(x^4 + 1))
to_V(a) # needs sage.modules
We convert to and back, and get the same element:
sage: from_V(to_V(a)) == a # needs sage.modules True
>>> from sage.all import * >>> from_V(to_V(a)) == a # needs sage.modules True
from_V(to_V(a)) == a # needs sage.modules
In the other direction:
sage: v = x*V.0 + (1/x)*V.1 # needs sage.modules sage: to_V(from_V(v)) == v # needs sage.modules True
>>> from sage.all import * >>> v = x*V.gen(0) + (Integer(1)/x)*V.gen(1) # needs sage.modules >>> to_V(from_V(v)) == v # needs sage.modules True
v = x*V.0 + (1/x)*V.1 # needs sage.modules to_V(from_V(v)) == v # needs sage.modules
And we show how it works over an extension of an extension field:
sage: R2.<z> = L[]; M.<z> = L.extension(z^2 - y) sage: M.free_module() # needs sage.modules (Vector space of dimension 2 over Function field in y defined by y^5 - 2*x*y + (-x^4 - 1)/x, Isomorphism: From: Vector space of dimension 2 over Function field in y defined by y^5 - 2*x*y + (-x^4 - 1)/x To: Function field in z defined by z^2 - y, Isomorphism: From: Function field in z defined by z^2 - y To: Vector space of dimension 2 over Function field in y defined by y^5 - 2*x*y + (-x^4 - 1)/x)
>>> from sage.all import * >>> R2 = L['z']; (z,) = R2._first_ngens(1); M = L.extension(z**Integer(2) - y, names=('z',)); (z,) = M._first_ngens(1) >>> M.free_module() # needs sage.modules (Vector space of dimension 2 over Function field in y defined by y^5 - 2*x*y + (-x^4 - 1)/x, Isomorphism: From: Vector space of dimension 2 over Function field in y defined by y^5 - 2*x*y + (-x^4 - 1)/x To: Function field in z defined by z^2 - y, Isomorphism: From: Function field in z defined by z^2 - y To: Vector space of dimension 2 over Function field in y defined by y^5 - 2*x*y + (-x^4 - 1)/x)
R2.<z> = L[]; M.<z> = L.extension(z^2 - y) M.free_module() # needs sage.modules
We can also get the vector space of
MoverK:sage: M.free_module(K) # needs sage.modules (Vector space of dimension 10 over Rational function field in x over Rational Field, Isomorphism: From: Vector space of dimension 10 over Rational function field in x over Rational Field To: Function field in z defined by z^2 - y, Isomorphism: From: Function field in z defined by z^2 - y To: Vector space of dimension 10 over Rational function field in x over Rational Field)
>>> from sage.all import * >>> M.free_module(K) # needs sage.modules (Vector space of dimension 10 over Rational function field in x over Rational Field, Isomorphism: From: Vector space of dimension 10 over Rational function field in x over Rational Field To: Function field in z defined by z^2 - y, Isomorphism: From: Function field in z defined by z^2 - y To: Vector space of dimension 10 over Rational function field in x over Rational Field)
M.free_module(K) # needs sage.modules
- gen(n=0)[source]¶
Return the \(n\)-th generator of the function field. By default, \(n\) is 0; any other value of \(n\) leads to an error. The generator is the class of \(y\), if we view the function field as being presented as \(K[y]/(f(y))\).
EXAMPLES:
sage: K.<x> = FunctionField(QQ); R.<y> = K[] sage: L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)) sage: L.gen() y sage: L.gen(1) Traceback (most recent call last): ... IndexError: there is only one generator
>>> 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(5) - (x**Integer(3) + Integer(2)*x*y + Integer(1)/x), names=('y',)); (y,) = L._first_ngens(1) >>> L.gen() y >>> L.gen(Integer(1)) Traceback (most recent call last): ... IndexError: there is only one generator
K.<x> = FunctionField(QQ); R.<y> = K[] L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)) L.gen() L.gen(1)
- genus()[source]¶
Return the genus of the function field.
For now, the genus is computed using Singular.
EXAMPLES:
sage: K.<x> = FunctionField(QQ); R.<y> = K[] sage: L.<y> = K.extension(y^3 - (x^3 + 2*x*y + 1/x)) sage: L.genus() 3
>>> 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(3) - (x**Integer(3) + Integer(2)*x*y + Integer(1)/x), names=('y',)); (y,) = L._first_ngens(1) >>> L.genus() 3
K.<x> = FunctionField(QQ); R.<y> = K[] L.<y> = K.extension(y^3 - (x^3 + 2*x*y + 1/x)) L.genus()
- hom(im_gens, base_morphism=None)[source]¶
Create a homomorphism from the function field to another function field.
INPUT:
im_gens– list of images of the generators of the function field and of successive base ringsbase_morphism– homomorphism of the base ring, after theim_gensare used. Thus ifim_genshas length 2, thenbase_morphismshould be a morphism from the base ring of the base ring of the function field.
EXAMPLES:
We create a rational function field, and a quadratic extension of it:
sage: K.<x> = FunctionField(QQ); R.<y> = K[] sage: L.<y> = K.extension(y^2 - x^3 - 1)
>>> 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)
K.<x> = FunctionField(QQ); R.<y> = K[] L.<y> = K.extension(y^2 - x^3 - 1)
We make the field automorphism that sends y to -y:
sage: f = L.hom(-y); f Function Field endomorphism of Function field in y defined by y^2 - x^3 - 1 Defn: y |--> -y
>>> from sage.all import * >>> f = L.hom(-y); f Function Field endomorphism of Function field in y defined by y^2 - x^3 - 1 Defn: y |--> -y
f = L.hom(-y); f
Evaluation works:
sage: f(y*x - 1/x) -x*y - 1/x
>>> from sage.all import * >>> f(y*x - Integer(1)/x) -x*y - 1/x
f(y*x - 1/x)
We try to define an invalid morphism:
sage: f = L.hom(y + 1) Traceback (most recent call last): ... ValueError: invalid morphism
>>> from sage.all import * >>> f = L.hom(y + Integer(1)) Traceback (most recent call last): ... ValueError: invalid morphism
f = L.hom(y + 1)
We make a morphism of the base rational function field:
sage: phi = K.hom(x + 1); phi Function Field endomorphism of Rational function field in x over Rational Field Defn: x |--> x + 1 sage: phi(x^3 - 3) x^3 + 3*x^2 + 3*x - 2 sage: (x+1)^3 - 3 x^3 + 3*x^2 + 3*x - 2
>>> from sage.all import * >>> phi = K.hom(x + Integer(1)); phi Function Field endomorphism of Rational function field in x over Rational Field Defn: x |--> x + 1 >>> phi(x**Integer(3) - Integer(3)) x^3 + 3*x^2 + 3*x - 2 >>> (x+Integer(1))**Integer(3) - Integer(3) x^3 + 3*x^2 + 3*x - 2
phi = K.hom(x + 1); phi phi(x^3 - 3) (x+1)^3 - 3
We make a morphism by specifying where the generators and the base generators go:
sage: L.hom([-y, x]) Function Field endomorphism of Function field in y defined by y^2 - x^3 - 1 Defn: y |--> -y x |--> x
>>> from sage.all import * >>> L.hom([-y, x]) Function Field endomorphism of Function field in y defined by y^2 - x^3 - 1 Defn: y |--> -y x |--> x
L.hom([-y, x])
You can also specify a morphism on the base:
sage: R1.<q> = K[] sage: L1.<q> = K.extension(q^2 - (x+1)^3 - 1) sage: L.hom(q, base_morphism=phi) Function Field morphism: From: Function field in y defined by y^2 - x^3 - 1 To: Function field in q defined by q^2 - x^3 - 3*x^2 - 3*x - 2 Defn: y |--> q x |--> x + 1
>>> from sage.all import * >>> R1 = K['q']; (q,) = R1._first_ngens(1) >>> L1 = K.extension(q**Integer(2) - (x+Integer(1))**Integer(3) - Integer(1), names=('q',)); (q,) = L1._first_ngens(1) >>> L.hom(q, base_morphism=phi) Function Field morphism: From: Function field in y defined by y^2 - x^3 - 1 To: Function field in q defined by q^2 - x^3 - 3*x^2 - 3*x - 2 Defn: y |--> q x |--> x + 1
R1.<q> = K[] L1.<q> = K.extension(q^2 - (x+1)^3 - 1) L.hom(q, base_morphism=phi)
We make another extension of a rational function field:
sage: K2.<t> = FunctionField(QQ); R2.<w> = K2[] sage: L2.<w> = K2.extension((4*w)^2 - (t+1)^3 - 1)
>>> from sage.all import * >>> K2 = FunctionField(QQ, names=('t',)); (t,) = K2._first_ngens(1); R2 = K2['w']; (w,) = R2._first_ngens(1) >>> L2 = K2.extension((Integer(4)*w)**Integer(2) - (t+Integer(1))**Integer(3) - Integer(1), names=('w',)); (w,) = L2._first_ngens(1)
K2.<t> = FunctionField(QQ); R2.<w> = K2[] L2.<w> = K2.extension((4*w)^2 - (t+1)^3 - 1)
We define a morphism, by giving the images of generators:
sage: f = L.hom([4*w, t + 1]); f Function Field morphism: From: Function field in y defined by y^2 - x^3 - 1 To: Function field in w defined by 16*w^2 - t^3 - 3*t^2 - 3*t - 2 Defn: y |--> 4*w x |--> t + 1
>>> from sage.all import * >>> f = L.hom([Integer(4)*w, t + Integer(1)]); f Function Field morphism: From: Function field in y defined by y^2 - x^3 - 1 To: Function field in w defined by 16*w^2 - t^3 - 3*t^2 - 3*t - 2 Defn: y |--> 4*w x |--> t + 1
f = L.hom([4*w, t + 1]); f
Evaluation works, as expected:
sage: f(y+x) 4*w + t + 1 sage: f(x*y + x/(x^2+1)) (4*t + 4)*w + (t + 1)/(t^2 + 2*t + 2)
>>> from sage.all import * >>> f(y+x) 4*w + t + 1 >>> f(x*y + x/(x**Integer(2)+Integer(1))) (4*t + 4)*w + (t + 1)/(t^2 + 2*t + 2)
f(y+x) f(x*y + x/(x^2+1))
We make another extension of a rational function field:
sage: K3.<yy> = FunctionField(QQ); R3.<xx> = K3[] sage: L3.<xx> = K3.extension(yy^2 - xx^3 - 1)
>>> from sage.all import * >>> K3 = FunctionField(QQ, names=('yy',)); (yy,) = K3._first_ngens(1); R3 = K3['xx']; (xx,) = R3._first_ngens(1) >>> L3 = K3.extension(yy**Integer(2) - xx**Integer(3) - Integer(1), names=('xx',)); (xx,) = L3._first_ngens(1)
K3.<yy> = FunctionField(QQ); R3.<xx> = K3[] L3.<xx> = K3.extension(yy^2 - xx^3 - 1)
This is the function field L with the generators exchanged. We define a morphism to L:
sage: g = L3.hom([x,y]); g Function Field morphism: From: Function field in xx defined by -xx^3 + yy^2 - 1 To: Function field in y defined by y^2 - x^3 - 1 Defn: xx |--> x yy |--> y
>>> from sage.all import * >>> g = L3.hom([x,y]); g Function Field morphism: From: Function field in xx defined by -xx^3 + yy^2 - 1 To: Function field in y defined by y^2 - x^3 - 1 Defn: xx |--> x yy |--> y
g = L3.hom([x,y]); g
- is_separable(base=None)[source]¶
Return whether this is a separable extension of
base.INPUT:
base– a function field from which this field has been created as an extension orNone(default:None); ifNone, then return whether this is a separable extension over its base field.
EXAMPLES:
sage: # needs sage.rings.finite_rings sage: K.<x> = FunctionField(GF(2)) sage: R.<y> = K[] sage: L.<y> = K.extension(y^2 - x) sage: L.is_separable() False sage: R.<z> = L[] sage: M.<z> = L.extension(z^3 - y) sage: M.is_separable() True sage: M.is_separable(K) False sage: # needs sage.rings.finite_rings sage: K.<x> = FunctionField(GF(5)) sage: R.<y> = K[] sage: L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)) sage: L.is_separable() True sage: # needs sage.rings.finite_rings sage: K.<x> = FunctionField(GF(5)) sage: R.<y> = K[] sage: L.<y> = K.extension(y^5 - 1) sage: L.is_separable() False
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1) >>> R = K['y']; (y,) = R._first_ngens(1) >>> L = K.extension(y**Integer(2) - x, names=('y',)); (y,) = L._first_ngens(1) >>> L.is_separable() False >>> R = L['z']; (z,) = R._first_ngens(1) >>> M = L.extension(z**Integer(3) - y, names=('z',)); (z,) = M._first_ngens(1) >>> M.is_separable() True >>> M.is_separable(K) False >>> # needs sage.rings.finite_rings >>> K = FunctionField(GF(Integer(5)), names=('x',)); (x,) = K._first_ngens(1) >>> R = K['y']; (y,) = R._first_ngens(1) >>> L = K.extension(y**Integer(5) - (x**Integer(3) + Integer(2)*x*y + Integer(1)/x), names=('y',)); (y,) = L._first_ngens(1) >>> L.is_separable() True >>> # needs sage.rings.finite_rings >>> K = FunctionField(GF(Integer(5)), names=('x',)); (x,) = K._first_ngens(1) >>> R = K['y']; (y,) = R._first_ngens(1) >>> L = K.extension(y**Integer(5) - Integer(1), names=('y',)); (y,) = L._first_ngens(1) >>> L.is_separable() False
# needs sage.rings.finite_rings K.<x> = FunctionField(GF(2)) R.<y> = K[] L.<y> = K.extension(y^2 - x) L.is_separable() R.<z> = L[] M.<z> = L.extension(z^3 - y) M.is_separable() M.is_separable(K) # needs sage.rings.finite_rings K.<x> = FunctionField(GF(5)) R.<y> = K[] L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)) L.is_separable() # needs sage.rings.finite_rings K.<x> = FunctionField(GF(5)) R.<y> = K[] L.<y> = K.extension(y^5 - 1) L.is_separable()
- maximal_order()[source]¶
Return the maximal order of the function field.
EXAMPLES:
sage: K.<x> = FunctionField(QQ); R.<y> = K[] sage: L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)) sage: L.maximal_order() Maximal order of Function field in y defined by y^5 - 2*x*y + (-x^4 - 1)/x
>>> 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(5) - (x**Integer(3) + Integer(2)*x*y + Integer(1)/x), names=('y',)); (y,) = L._first_ngens(1) >>> L.maximal_order() Maximal order of Function field in y defined by y^5 - 2*x*y + (-x^4 - 1)/x
K.<x> = FunctionField(QQ); R.<y> = K[] L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)) L.maximal_order()
- maximal_order_infinite()[source]¶
Return the maximal infinite order of the function field.
EXAMPLES:
sage: K.<x> = FunctionField(QQ); R.<y> = K[] sage: L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)) sage: L.maximal_order_infinite() Maximal infinite order of Function field in y defined by y^5 - 2*x*y + (-x^4 - 1)/x sage: K.<x> = FunctionField(GF(2)); _.<t> = K[] # needs sage.rings.finite_rings sage: F.<y> = K.extension(t^3 - x^2*(x^2 + x + 1)^2) # needs sage.rings.finite_rings sage: F.maximal_order_infinite() # needs sage.rings.finite_rings Maximal infinite order of Function field in y defined by y^3 + x^6 + x^4 + x^2 sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[] # needs sage.rings.finite_rings sage: L.<y> = K.extension(Y^2 + Y + x + 1/x) # needs sage.rings.finite_rings sage: L.maximal_order_infinite() # needs sage.rings.finite_rings Maximal infinite order of Function field in y defined by y^2 + y + (x^2 + 1)/x
>>> 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(5) - (x**Integer(3) + Integer(2)*x*y + Integer(1)/x), names=('y',)); (y,) = L._first_ngens(1) >>> L.maximal_order_infinite() Maximal infinite order of Function field in y defined by y^5 - 2*x*y + (-x^4 - 1)/x >>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); _ = K['t']; (t,) = _._first_ngens(1)# needs sage.rings.finite_rings >>> F = K.extension(t**Integer(3) - x**Integer(2)*(x**Integer(2) + x + Integer(1))**Integer(2), names=('y',)); (y,) = F._first_ngens(1)# needs sage.rings.finite_rings >>> F.maximal_order_infinite() # needs sage.rings.finite_rings Maximal infinite order of Function field in y defined by y^3 + x^6 + x^4 + x^2 >>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)# needs sage.rings.finite_rings >>> L = K.extension(Y**Integer(2) + Y + x + Integer(1)/x, names=('y',)); (y,) = L._first_ngens(1)# needs sage.rings.finite_rings >>> L.maximal_order_infinite() # needs sage.rings.finite_rings Maximal infinite order of Function field in y defined by y^2 + y + (x^2 + 1)/x
K.<x> = FunctionField(QQ); R.<y> = K[] L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)) L.maximal_order_infinite() K.<x> = FunctionField(GF(2)); _.<t> = K[] # needs sage.rings.finite_rings F.<y> = K.extension(t^3 - x^2*(x^2 + x + 1)^2) # needs sage.rings.finite_rings F.maximal_order_infinite() # needs sage.rings.finite_rings K.<x> = FunctionField(GF(2)); _.<Y> = K[] # needs sage.rings.finite_rings L.<y> = K.extension(Y^2 + Y + x + 1/x) # needs sage.rings.finite_rings L.maximal_order_infinite() # needs sage.rings.finite_rings
- monic_integral_model(names=None)[source]¶
Return a function field isomorphic to this field but which is an extension of a rational function field with defining polynomial that is monic and integral over the constant base field.
INPUT:
names– string or tuple of up to two strings (default:None); the name of the generator of the field, and the name of the generator of the underlying rational function field (if a tuple). If not given, then the names are chosen automatically.
OUTPUT:
A triple
(F,f,t)whereFis a function field,fis an isomorphism fromFto this field, andtis the inverse off.EXAMPLES:
sage: K.<x> = FunctionField(QQ) sage: R.<y> = K[] sage: L.<y> = K.extension(x^2*y^5 - 1/x); L Function field in y defined by x^2*y^5 - 1/x sage: A, from_A, to_A = L.monic_integral_model('z') sage: A Function field in z defined by z^5 - x^12 sage: from_A Function Field morphism: From: Function field in z defined by z^5 - x^12 To: Function field in y defined by x^2*y^5 - 1/x Defn: z |--> x^3*y x |--> x sage: to_A Function Field morphism: From: Function field in y defined by x^2*y^5 - 1/x To: Function field in z defined by z^5 - x^12 Defn: y |--> 1/x^3*z x |--> x sage: to_A(y) 1/x^3*z sage: from_A(to_A(y)) y sage: from_A(to_A(1/y)) x^3*y^4 sage: from_A(to_A(1/y)) == 1/y True
>>> 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(x**Integer(2)*y**Integer(5) - Integer(1)/x, names=('y',)); (y,) = L._first_ngens(1); L Function field in y defined by x^2*y^5 - 1/x >>> A, from_A, to_A = L.monic_integral_model('z') >>> A Function field in z defined by z^5 - x^12 >>> from_A Function Field morphism: From: Function field in z defined by z^5 - x^12 To: Function field in y defined by x^2*y^5 - 1/x Defn: z |--> x^3*y x |--> x >>> to_A Function Field morphism: From: Function field in y defined by x^2*y^5 - 1/x To: Function field in z defined by z^5 - x^12 Defn: y |--> 1/x^3*z x |--> x >>> to_A(y) 1/x^3*z >>> from_A(to_A(y)) y >>> from_A(to_A(Integer(1)/y)) x^3*y^4 >>> from_A(to_A(Integer(1)/y)) == Integer(1)/y True
K.<x> = FunctionField(QQ) R.<y> = K[] L.<y> = K.extension(x^2*y^5 - 1/x); L A, from_A, to_A = L.monic_integral_model('z') A from_A to_A to_A(y) from_A(to_A(y)) from_A(to_A(1/y)) from_A(to_A(1/y)) == 1/yThis also works for towers of function fields:
sage: R.<z> = L[] sage: M.<z> = L.extension(z^2*y - 1/x) sage: M.monic_integral_model() (Function field in z_ defined by z_^10 - x^18, Function Field morphism: From: Function field in z_ defined by z_^10 - x^18 To: Function field in z defined by y*z^2 - 1/x Defn: z_ |--> x^2*z x |--> x, Function Field morphism: From: Function field in z defined by y*z^2 - 1/x To: Function field in z_ defined by z_^10 - x^18 Defn: z |--> 1/x^2*z_ y |--> 1/x^15*z_^8 x |--> x)
>>> from sage.all import * >>> R = L['z']; (z,) = R._first_ngens(1) >>> M = L.extension(z**Integer(2)*y - Integer(1)/x, names=('z',)); (z,) = M._first_ngens(1) >>> M.monic_integral_model() (Function field in z_ defined by z_^10 - x^18, Function Field morphism: From: Function field in z_ defined by z_^10 - x^18 To: Function field in z defined by y*z^2 - 1/x Defn: z_ |--> x^2*z x |--> x, Function Field morphism: From: Function field in z defined by y*z^2 - 1/x To: Function field in z_ defined by z_^10 - x^18 Defn: z |--> 1/x^2*z_ y |--> 1/x^15*z_^8 x |--> x)
R.<z> = L[] M.<z> = L.extension(z^2*y - 1/x) M.monic_integral_model()
- ngens()[source]¶
Return the number of generators of the function field over its base field. This is by definition 1.
EXAMPLES:
sage: K.<x> = FunctionField(QQ); R.<y> = K[] sage: L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)) sage: L.ngens() 1
>>> 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(5) - (x**Integer(3) + Integer(2)*x*y + Integer(1)/x), names=('y',)); (y,) = L._first_ngens(1) >>> L.ngens() 1
K.<x> = FunctionField(QQ); R.<y> = K[] L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)) L.ngens()
- polynomial()[source]¶
Return the univariate polynomial that defines the function field, that is, the polynomial \(f(y)\) so that the function field is of the form \(K[y]/(f(y))\).
EXAMPLES:
sage: K.<x> = FunctionField(QQ); R.<y> = K[] sage: L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)) sage: L.polynomial() y^5 - 2*x*y + (-x^4 - 1)/x
>>> 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(5) - (x**Integer(3) + Integer(2)*x*y + Integer(1)/x), names=('y',)); (y,) = L._first_ngens(1) >>> L.polynomial() y^5 - 2*x*y + (-x^4 - 1)/x
K.<x> = FunctionField(QQ); R.<y> = K[] L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)) L.polynomial()
- polynomial_ring()[source]¶
Return the polynomial ring used to represent elements of the function field. If we view the function field as being presented as \(K[y]/(f(y))\), then this function returns the ring \(K[y]\).
EXAMPLES:
sage: K.<x> = FunctionField(QQ); R.<y> = K[] sage: L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)) sage: L.polynomial_ring() Univariate Polynomial Ring in y over Rational function field in x over Rational Field
>>> 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(5) - (x**Integer(3) + Integer(2)*x*y + Integer(1)/x), names=('y',)); (y,) = L._first_ngens(1) >>> L.polynomial_ring() Univariate Polynomial Ring in y over Rational function field in x over Rational Field
K.<x> = FunctionField(QQ); R.<y> = K[] L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)) L.polynomial_ring()
- primitive_element()[source]¶
Return a primitive element over the underlying rational function field.
If this is a finite extension of a rational function field \(K(x)\) with \(K\) perfect, then this is a simple extension of \(K(x)\), i.e., there is a primitive element \(y\) which generates this field over \(K(x)\). This method returns such an element \(y\).
EXAMPLES:
sage: K.<x> = FunctionField(QQ) sage: R.<y> = K[] sage: L.<y> = K.extension(y^2 - x) sage: R.<z> = L[] sage: M.<z> = L.extension(z^2 - y) sage: R.<z> = L[] sage: N.<u> = L.extension(z^2 - x - 1) sage: N.primitive_element() u + y sage: M.primitive_element() z sage: L.primitive_element() y
>>> 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, names=('y',)); (y,) = L._first_ngens(1) >>> R = L['z']; (z,) = R._first_ngens(1) >>> M = L.extension(z**Integer(2) - y, names=('z',)); (z,) = M._first_ngens(1) >>> R = L['z']; (z,) = R._first_ngens(1) >>> N = L.extension(z**Integer(2) - x - Integer(1), names=('u',)); (u,) = N._first_ngens(1) >>> N.primitive_element() u + y >>> M.primitive_element() z >>> L.primitive_element() y
K.<x> = FunctionField(QQ) R.<y> = K[] L.<y> = K.extension(y^2 - x) R.<z> = L[] M.<z> = L.extension(z^2 - y) R.<z> = L[] N.<u> = L.extension(z^2 - x - 1) N.primitive_element() M.primitive_element() L.primitive_element()
This also works for inseparable extensions:
sage: # needs sage.rings.finite_rings sage: K.<x> = FunctionField(GF(2)) sage: R.<Y> = K[] sage: L.<y> = K.extension(Y^2 - x) sage: R.<Z> = L[] sage: M.<z> = L.extension(Z^2 - y) sage: M.primitive_element() z
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1) >>> R = K['Y']; (Y,) = R._first_ngens(1) >>> L = K.extension(Y**Integer(2) - x, names=('y',)); (y,) = L._first_ngens(1) >>> R = L['Z']; (Z,) = R._first_ngens(1) >>> M = L.extension(Z**Integer(2) - y, names=('z',)); (z,) = M._first_ngens(1) >>> M.primitive_element() z
# needs sage.rings.finite_rings K.<x> = FunctionField(GF(2)) R.<Y> = K[] L.<y> = K.extension(Y^2 - x) R.<Z> = L[] M.<z> = L.extension(Z^2 - y) M.primitive_element()
- random_element(*args, **kwds)[source]¶
Create a random element of the function field. Parameters are passed onto the random_element method of the base_field.
EXAMPLES:
sage: K.<x> = FunctionField(QQ); R.<y> = K[] sage: L.<y> = K.extension(y^2 - (x^2 + x)) sage: L.random_element() # random ((x^2 - x + 2/3)/(x^2 + 1/3*x - 1))*y^2 + ((-1/4*x^2 + 1/2*x - 1)/(-5/2*x + 2/3))*y + (-1/2*x^2 - 4)/(-12*x^2 + 1/2*x - 1/95)
>>> 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(2) + x), names=('y',)); (y,) = L._first_ngens(1) >>> L.random_element() # random ((x^2 - x + 2/3)/(x^2 + 1/3*x - 1))*y^2 + ((-1/4*x^2 + 1/2*x - 1)/(-5/2*x + 2/3))*y + (-1/2*x^2 - 4)/(-12*x^2 + 1/2*x - 1/95)
K.<x> = FunctionField(QQ); R.<y> = K[] L.<y> = K.extension(y^2 - (x^2 + x)) L.random_element() # random
- separable_model(names=None)[source]¶
Return a function field isomorphic to this field which is a separable extension of a rational function field.
INPUT:
names– tuple of two strings orNone(default:None); the second entry will be used as the variable name of the rational function field, the first entry will be used as the variable name of its separable extension. IfNone, then the variable names will be chosen automatically.
OUTPUT:
A triple
(F,f,t)whereFis a function field,fis an isomorphism fromFto this function field, andtis the inverse off.ALGORITHM:
Suppose that the constant base field is perfect. If this is a monic integral inseparable extension of a rational function field, then the defining polynomial is separable if we swap the variables (Proposition 4.8 in Chapter VIII of [Lan2002].) The algorithm reduces to this case with
monic_integral_model().EXAMPLES:
sage: # needs sage.rings.finite_rings sage: K.<x> = FunctionField(GF(2)) sage: R.<y> = K[] sage: L.<y> = K.extension(y^2 - x^3) sage: L.separable_model(('t','w')) (Function field in t defined by t^3 + w^2, Function Field morphism: From: Function field in t defined by t^3 + w^2 To: Function field in y defined by y^2 + x^3 Defn: t |--> x w |--> y, Function Field morphism: From: Function field in y defined by y^2 + x^3 To: Function field in t defined by t^3 + w^2 Defn: y |--> w x |--> t)
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1) >>> R = K['y']; (y,) = R._first_ngens(1) >>> L = K.extension(y**Integer(2) - x**Integer(3), names=('y',)); (y,) = L._first_ngens(1) >>> L.separable_model(('t','w')) (Function field in t defined by t^3 + w^2, Function Field morphism: From: Function field in t defined by t^3 + w^2 To: Function field in y defined by y^2 + x^3 Defn: t |--> x w |--> y, Function Field morphism: From: Function field in y defined by y^2 + x^3 To: Function field in t defined by t^3 + w^2 Defn: y |--> w x |--> t)
# needs sage.rings.finite_rings K.<x> = FunctionField(GF(2)) R.<y> = K[] L.<y> = K.extension(y^2 - x^3) L.separable_model(('t','w'))This also works for non-integral polynomials:
sage: # needs sage.rings.finite_rings sage: K.<x> = FunctionField(GF(2)) sage: R.<y> = K[] sage: L.<y> = K.extension(y^2/x - x^2) sage: L.separable_model() (Function field in y_ defined by y_^3 + x_^2, Function Field morphism: From: Function field in y_ defined by y_^3 + x_^2 To: Function field in y defined by 1/x*y^2 + x^2 Defn: y_ |--> x x_ |--> y, Function Field morphism: From: Function field in y defined by 1/x*y^2 + x^2 To: Function field in y_ defined by y_^3 + x_^2 Defn: y |--> x_ x |--> y_)
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1) >>> R = K['y']; (y,) = R._first_ngens(1) >>> L = K.extension(y**Integer(2)/x - x**Integer(2), names=('y',)); (y,) = L._first_ngens(1) >>> L.separable_model() (Function field in y_ defined by y_^3 + x_^2, Function Field morphism: From: Function field in y_ defined by y_^3 + x_^2 To: Function field in y defined by 1/x*y^2 + x^2 Defn: y_ |--> x x_ |--> y, Function Field morphism: From: Function field in y defined by 1/x*y^2 + x^2 To: Function field in y_ defined by y_^3 + x_^2 Defn: y |--> x_ x |--> y_)
# needs sage.rings.finite_rings K.<x> = FunctionField(GF(2)) R.<y> = K[] L.<y> = K.extension(y^2/x - x^2) L.separable_model()
If the base field is not perfect this is only implemented in trivial cases:
sage: # needs sage.rings.finite_rings sage: k.<t> = FunctionField(GF(2)) sage: k.is_perfect() False sage: K.<x> = FunctionField(k) sage: R.<y> = K[] sage: L.<y> = K.extension(y^3 - t) sage: L.separable_model() (Function field in y defined by y^3 + t, Function Field endomorphism of Function field in y defined by y^3 + t Defn: y |--> y x |--> x, Function Field endomorphism of Function field in y defined by y^3 + t Defn: y |--> y x |--> x)
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> k = FunctionField(GF(Integer(2)), names=('t',)); (t,) = k._first_ngens(1) >>> k.is_perfect() False >>> K = FunctionField(k, names=('x',)); (x,) = K._first_ngens(1) >>> R = K['y']; (y,) = R._first_ngens(1) >>> L = K.extension(y**Integer(3) - t, names=('y',)); (y,) = L._first_ngens(1) >>> L.separable_model() (Function field in y defined by y^3 + t, Function Field endomorphism of Function field in y defined by y^3 + t Defn: y |--> y x |--> x, Function Field endomorphism of Function field in y defined by y^3 + t Defn: y |--> y x |--> x)
# needs sage.rings.finite_rings k.<t> = FunctionField(GF(2)) k.is_perfect() K.<x> = FunctionField(k) R.<y> = K[] L.<y> = K.extension(y^3 - t) L.separable_model()
Some other cases for which a separable model could be constructed are not supported yet:
sage: R.<y> = K[] # needs sage.rings.finite_rings sage: L.<y> = K.extension(y^2 - t) # needs sage.rings.finite_rings sage: L.separable_model() # needs sage.rings.finite_rings Traceback (most recent call last): ... NotImplementedError: constructing a separable model is only implemented for function fields over a perfect constant base field
>>> from sage.all import * >>> R = K['y']; (y,) = R._first_ngens(1)# needs sage.rings.finite_rings >>> L = K.extension(y**Integer(2) - t, names=('y',)); (y,) = L._first_ngens(1)# needs sage.rings.finite_rings >>> L.separable_model() # needs sage.rings.finite_rings Traceback (most recent call last): ... NotImplementedError: constructing a separable model is only implemented for function fields over a perfect constant base field
R.<y> = K[] # needs sage.rings.finite_rings L.<y> = K.extension(y^2 - t) # needs sage.rings.finite_rings L.separable_model() # needs sage.rings.finite_rings
- simple_model(name=None)[source]¶
Return a function field isomorphic to this field which is a simple extension of a rational function field.
INPUT:
name– string (default:None); the name of generator of the simple extension. IfNone, then the name of the generator will be the same as the name of the generator of this function field.
OUTPUT:
A triple
(F,f,t)whereFis a field isomorphic to this field,fis an isomorphism fromFto this function field andtis the inverse off.EXAMPLES:
A tower of four function fields:
sage: K.<x> = FunctionField(QQ); R.<z> = K[] sage: L.<z> = K.extension(z^2 - x); R.<u> = L[] sage: M.<u> = L.extension(u^2 - z); R.<v> = M[] sage: N.<v> = M.extension(v^2 - u)
>>> from sage.all import * >>> K = FunctionField(QQ, names=('x',)); (x,) = K._first_ngens(1); R = K['z']; (z,) = R._first_ngens(1) >>> L = K.extension(z**Integer(2) - x, names=('z',)); (z,) = L._first_ngens(1); R = L['u']; (u,) = R._first_ngens(1) >>> M = L.extension(u**Integer(2) - z, names=('u',)); (u,) = M._first_ngens(1); R = M['v']; (v,) = R._first_ngens(1) >>> N = M.extension(v**Integer(2) - u, names=('v',)); (v,) = N._first_ngens(1)
K.<x> = FunctionField(QQ); R.<z> = K[] L.<z> = K.extension(z^2 - x); R.<u> = L[] M.<u> = L.extension(u^2 - z); R.<v> = M[] N.<v> = M.extension(v^2 - u)
The fields N and M as simple extensions of K:
sage: N.simple_model() (Function field in v defined by v^8 - x, Function Field morphism: From: Function field in v defined by v^8 - x To: Function field in v defined by v^2 - u Defn: v |--> v, Function Field morphism: From: Function field in v defined by v^2 - u To: Function field in v defined by v^8 - x Defn: v |--> v u |--> v^2 z |--> v^4 x |--> x) sage: M.simple_model() (Function field in u defined by u^4 - x, Function Field morphism: From: Function field in u defined by u^4 - x To: Function field in u defined by u^2 - z Defn: u |--> u, Function Field morphism: From: Function field in u defined by u^2 - z To: Function field in u defined by u^4 - x Defn: u |--> u z |--> u^2 x |--> x)
>>> from sage.all import * >>> N.simple_model() (Function field in v defined by v^8 - x, Function Field morphism: From: Function field in v defined by v^8 - x To: Function field in v defined by v^2 - u Defn: v |--> v, Function Field morphism: From: Function field in v defined by v^2 - u To: Function field in v defined by v^8 - x Defn: v |--> v u |--> v^2 z |--> v^4 x |--> x) >>> M.simple_model() (Function field in u defined by u^4 - x, Function Field morphism: From: Function field in u defined by u^4 - x To: Function field in u defined by u^2 - z Defn: u |--> u, Function Field morphism: From: Function field in u defined by u^2 - z To: Function field in u defined by u^4 - x Defn: u |--> u z |--> u^2 x |--> x)
N.simple_model() M.simple_model()
An optional parameter
namecan be used to set the name of the generator of the simple extension:sage: M.simple_model(name='t') (Function field in t defined by t^4 - x, Function Field morphism: From: Function field in t defined by t^4 - x To: Function field in u defined by u^2 - z Defn: t |--> u, Function Field morphism: From: Function field in u defined by u^2 - z To: Function field in t defined by t^4 - x Defn: u |--> t z |--> t^2 x |--> x)
>>> from sage.all import * >>> M.simple_model(name='t') (Function field in t defined by t^4 - x, Function Field morphism: From: Function field in t defined by t^4 - x To: Function field in u defined by u^2 - z Defn: t |--> u, Function Field morphism: From: Function field in u defined by u^2 - z To: Function field in t defined by t^4 - x Defn: u |--> t z |--> t^2 x |--> x)
M.simple_model(name='t')
An example with higher degrees:
sage: # needs sage.rings.finite_rings sage: K.<x> = FunctionField(GF(3)); R.<y> = K[] sage: L.<y> = K.extension(y^5 - x); R.<z> = L[] sage: M.<z> = L.extension(z^3 - x) sage: M.simple_model() (Function field in z defined by z^15 + x*z^12 + x^2*z^9 + 2*x^3*z^6 + 2*x^4*z^3 + 2*x^5 + 2*x^3, Function Field morphism: From: Function field in z defined by z^15 + x*z^12 + x^2*z^9 + 2*x^3*z^6 + 2*x^4*z^3 + 2*x^5 + 2*x^3 To: Function field in z defined by z^3 + 2*x Defn: z |--> z + y, Function Field morphism: From: Function field in z defined by z^3 + 2*x To: Function field in z defined by z^15 + x*z^12 + x^2*z^9 + 2*x^3*z^6 + 2*x^4*z^3 + 2*x^5 + 2*x^3 Defn: z |--> 2/x*z^6 + 2*z^3 + z + 2*x y |--> 1/x*z^6 + z^3 + x x |--> x)
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> K = FunctionField(GF(Integer(3)), names=('x',)); (x,) = K._first_ngens(1); R = K['y']; (y,) = R._first_ngens(1) >>> L = K.extension(y**Integer(5) - x, names=('y',)); (y,) = L._first_ngens(1); R = L['z']; (z,) = R._first_ngens(1) >>> M = L.extension(z**Integer(3) - x, names=('z',)); (z,) = M._first_ngens(1) >>> M.simple_model() (Function field in z defined by z^15 + x*z^12 + x^2*z^9 + 2*x^3*z^6 + 2*x^4*z^3 + 2*x^5 + 2*x^3, Function Field morphism: From: Function field in z defined by z^15 + x*z^12 + x^2*z^9 + 2*x^3*z^6 + 2*x^4*z^3 + 2*x^5 + 2*x^3 To: Function field in z defined by z^3 + 2*x Defn: z |--> z + y, Function Field morphism: From: Function field in z defined by z^3 + 2*x To: Function field in z defined by z^15 + x*z^12 + x^2*z^9 + 2*x^3*z^6 + 2*x^4*z^3 + 2*x^5 + 2*x^3 Defn: z |--> 2/x*z^6 + 2*z^3 + z + 2*x y |--> 1/x*z^6 + z^3 + x x |--> x)
# needs sage.rings.finite_rings K.<x> = FunctionField(GF(3)); R.<y> = K[] L.<y> = K.extension(y^5 - x); R.<z> = L[] M.<z> = L.extension(z^3 - x) M.simple_model()
This also works for inseparable extensions:
sage: # needs sage.rings.finite_rings sage: K.<x> = FunctionField(GF(2)); R.<y> = K[] sage: L.<y> = K.extension(y^2 - x); R.<z> = L[] sage: M.<z> = L.extension(z^2 - y) sage: M.simple_model() (Function field in z defined by z^4 + x, Function Field morphism: From: Function field in z defined by z^4 + x To: Function field in z defined by z^2 + y Defn: z |--> z, Function Field morphism: From: Function field in z defined by z^2 + y To: Function field in z defined by z^4 + x Defn: z |--> z y |--> z^2 x |--> x)
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); R = K['y']; (y,) = R._first_ngens(1) >>> L = K.extension(y**Integer(2) - x, names=('y',)); (y,) = L._first_ngens(1); R = L['z']; (z,) = R._first_ngens(1) >>> M = L.extension(z**Integer(2) - y, names=('z',)); (z,) = M._first_ngens(1) >>> M.simple_model() (Function field in z defined by z^4 + x, Function Field morphism: From: Function field in z defined by z^4 + x To: Function field in z defined by z^2 + y Defn: z |--> z, Function Field morphism: From: Function field in z defined by z^2 + y To: Function field in z defined by z^4 + x Defn: z |--> z y |--> z^2 x |--> x)
# needs sage.rings.finite_rings K.<x> = FunctionField(GF(2)); R.<y> = K[] L.<y> = K.extension(y^2 - x); R.<z> = L[] M.<z> = L.extension(z^2 - y) M.simple_model()
- class sage.rings.function_field.function_field_polymod.FunctionField_simple(polynomial, names, category=None)[source]¶
Bases:
FunctionField_polymodFunction fields defined by irreducible and separable polynomials over rational function fields.
- constant_field()[source]¶
Return the algebraic closure of the base constant field in the function field.
EXAMPLES:
sage: K.<x> = FunctionField(GF(3)); _.<y> = K[] # needs sage.rings.finite_rings sage: L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)) # needs sage.rings.finite_rings sage: L.constant_field() # needs sage.rings.finite_rings Finite Field of size 3
>>> from sage.all import * >>> K = FunctionField(GF(Integer(3)), names=('x',)); (x,) = K._first_ngens(1); _ = K['y']; (y,) = _._first_ngens(1)# needs sage.rings.finite_rings >>> L = K.extension(y**Integer(5) - (x**Integer(3) + Integer(2)*x*y + Integer(1)/x), names=('y',)); (y,) = L._first_ngens(1)# needs sage.rings.finite_rings >>> L.constant_field() # needs sage.rings.finite_rings Finite Field of size 3
K.<x> = FunctionField(GF(3)); _.<y> = K[] # needs sage.rings.finite_rings L.<y> = K.extension(y^5 - (x^3 + 2*x*y + 1/x)) # needs sage.rings.finite_rings L.constant_field() # needs sage.rings.finite_rings
- exact_constant_field(name='t')[source]¶
Return the exact constant field and its embedding into the function field.
INPUT:
name– name (default: \(t\)) of the generator of the exact constant field
EXAMPLES:
sage: # needs sage.rings.finite_rings sage: K.<x> = FunctionField(GF(3)); _.<Y> = K[] sage: f = Y^2 - x*Y + x^2 + 1 # irreducible but not absolutely irreducible sage: L.<y> = K.extension(f) sage: L.genus() 0 sage: L.exact_constant_field() (Finite Field in t of size 3^2, Ring morphism: From: Finite Field in t of size 3^2 To: Function field in y defined by y^2 + 2*x*y + x^2 + 1 Defn: t |--> y + x) sage: (y+x).divisor() 0
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> K = FunctionField(GF(Integer(3)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1) >>> f = Y**Integer(2) - x*Y + x**Integer(2) + Integer(1) # irreducible but not absolutely irreducible >>> L = K.extension(f, names=('y',)); (y,) = L._first_ngens(1) >>> L.genus() 0 >>> L.exact_constant_field() (Finite Field in t of size 3^2, Ring morphism: From: Finite Field in t of size 3^2 To: Function field in y defined by y^2 + 2*x*y + x^2 + 1 Defn: t |--> y + x) >>> (y+x).divisor() 0
# needs sage.rings.finite_rings K.<x> = FunctionField(GF(3)); _.<Y> = K[] f = Y^2 - x*Y + x^2 + 1 # irreducible but not absolutely irreducible L.<y> = K.extension(f) L.genus() L.exact_constant_field() (y+x).divisor()
- genus()[source]¶
Return the genus of the function field.
EXAMPLES:
sage: # needs sage.rings.finite_rings sage: F.<a> = GF(16) sage: K.<x> = FunctionField(F); K Rational function field in x over Finite Field in a of size 2^4 sage: R.<t> = PolynomialRing(K) sage: L.<y> = K.extension(t^4 + t - x^5) sage: L.genus() 6 sage: # needs sage.rings.number_field sage: R.<T> = QQ[] sage: N.<a> = NumberField(T^2 + 1) sage: K.<x> = FunctionField(N); K Rational function field in x over Number Field in a with defining polynomial T^2 + 1 sage: K.genus() 0 sage: S.<t> = PolynomialRing(K) sage: L.<y> = K.extension(t^2 - x^3 + x) sage: L.genus() 1
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> F = GF(Integer(16), names=('a',)); (a,) = F._first_ngens(1) >>> K = FunctionField(F, names=('x',)); (x,) = K._first_ngens(1); K Rational function field in x over Finite Field in a of size 2^4 >>> R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1) >>> L = K.extension(t**Integer(4) + t - x**Integer(5), names=('y',)); (y,) = L._first_ngens(1) >>> L.genus() 6 >>> # needs sage.rings.number_field >>> R = QQ['T']; (T,) = R._first_ngens(1) >>> N = NumberField(T**Integer(2) + Integer(1), names=('a',)); (a,) = N._first_ngens(1) >>> K = FunctionField(N, names=('x',)); (x,) = K._first_ngens(1); K Rational function field in x over Number Field in a with defining polynomial T^2 + 1 >>> K.genus() 0 >>> S = PolynomialRing(K, names=('t',)); (t,) = S._first_ngens(1) >>> L = K.extension(t**Integer(2) - x**Integer(3) + x, names=('y',)); (y,) = L._first_ngens(1) >>> L.genus() 1
# needs sage.rings.finite_rings F.<a> = GF(16) K.<x> = FunctionField(F); K R.<t> = PolynomialRing(K) L.<y> = K.extension(t^4 + t - x^5) L.genus() # needs sage.rings.number_field R.<T> = QQ[] N.<a> = NumberField(T^2 + 1) K.<x> = FunctionField(N); K K.genus() S.<t> = PolynomialRing(K) L.<y> = K.extension(t^2 - x^3 + x) L.genus()
The genus is computed by the Hurwitz genus formula.
- places_above(p)[source]¶
Return places lying above
p.INPUT:
p– place of the base rational function field
EXAMPLES:
sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[] # needs sage.rings.finite_rings sage: F.<y> = K.extension(Y^3 - x^2*(x^2 + x + 1)^2) # needs sage.rings.finite_rings sage: all(q.place_below() == p # needs sage.rings.finite_rings ....: for p in K.places() for q in F.places_above(p)) True sage: K.<x> = FunctionField(QQ); _.<Y> = K[] sage: F.<y> = K.extension(Y^3 - x^2*(x^2 + x + 1)^2) sage: O = K.maximal_order() sage: pls = [O.ideal(x - c).place() for c in [-2, -1, 0, 1, 2]] sage: all(q.place_below() == p ....: for p in pls for q in F.places_above(p)) True sage: # needs sage.rings.number_field sage: K.<x> = FunctionField(QQbar); _.<Y> = K[] sage: F.<y> = K.extension(Y^3 - x^2*(x^2 + x + 1)^2) sage: O = K.maximal_order() sage: pls = [O.ideal(x - QQbar(sqrt(c))).place() ....: for c in [-2, -1, 0, 1, 2]] sage: all(q.place_below() == p # long time (4s) ....: for p in pls for q in F.places_above(p)) True
>>> from sage.all import * >>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1)# needs sage.rings.finite_rings >>> F = K.extension(Y**Integer(3) - x**Integer(2)*(x**Integer(2) + x + Integer(1))**Integer(2), names=('y',)); (y,) = F._first_ngens(1)# needs sage.rings.finite_rings >>> all(q.place_below() == p # needs sage.rings.finite_rings ... for p in K.places() for q in F.places_above(p)) True >>> K = FunctionField(QQ, names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1) >>> F = K.extension(Y**Integer(3) - x**Integer(2)*(x**Integer(2) + x + Integer(1))**Integer(2), names=('y',)); (y,) = F._first_ngens(1) >>> O = K.maximal_order() >>> pls = [O.ideal(x - c).place() for c in [-Integer(2), -Integer(1), Integer(0), Integer(1), Integer(2)]] >>> all(q.place_below() == p ... for p in pls for q in F.places_above(p)) True >>> # needs sage.rings.number_field >>> K = FunctionField(QQbar, names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1) >>> F = K.extension(Y**Integer(3) - x**Integer(2)*(x**Integer(2) + x + Integer(1))**Integer(2), names=('y',)); (y,) = F._first_ngens(1) >>> O = K.maximal_order() >>> pls = [O.ideal(x - QQbar(sqrt(c))).place() ... for c in [-Integer(2), -Integer(1), Integer(0), Integer(1), Integer(2)]] >>> all(q.place_below() == p # long time (4s) ... for p in pls for q in F.places_above(p)) True
K.<x> = FunctionField(GF(2)); _.<Y> = K[] # needs sage.rings.finite_rings F.<y> = K.extension(Y^3 - x^2*(x^2 + x + 1)^2) # needs sage.rings.finite_rings all(q.place_below() == p # needs sage.rings.finite_rings for p in K.places() for q in F.places_above(p)) K.<x> = FunctionField(QQ); _.<Y> = K[] F.<y> = K.extension(Y^3 - x^2*(x^2 + x + 1)^2) O = K.maximal_order() pls = [O.ideal(x - c).place() for c in [-2, -1, 0, 1, 2]] all(q.place_below() == p for p in pls for q in F.places_above(p)) # needs sage.rings.number_field K.<x> = FunctionField(QQbar); _.<Y> = K[] F.<y> = K.extension(Y^3 - x^2*(x^2 + x + 1)^2) O = K.maximal_order() pls = [O.ideal(x - QQbar(sqrt(c))).place() for c in [-2, -1, 0, 1, 2]] all(q.place_below() == p # long time (4s) for p in pls for q in F.places_above(p))
- places_infinite(degree=1)[source]¶
Return a list of the infinite places with
degree.INPUT:
degree– positive integer (default: \(1\))
EXAMPLES:
sage: # needs sage.rings.finite_rings sage: F.<a> = GF(2) sage: K.<x> = FunctionField(F) sage: R.<t> = PolynomialRing(K) sage: L.<y> = K.extension(t^4 + t - x^5) sage: L.places_infinite(1) [Place (1/x, 1/x^4*y^3)]
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> F = GF(Integer(2), names=('a',)); (a,) = F._first_ngens(1) >>> K = FunctionField(F, names=('x',)); (x,) = K._first_ngens(1) >>> R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1) >>> L = K.extension(t**Integer(4) + t - x**Integer(5), names=('y',)); (y,) = L._first_ngens(1) >>> L.places_infinite(Integer(1)) [Place (1/x, 1/x^4*y^3)]
# needs sage.rings.finite_rings F.<a> = GF(2) K.<x> = FunctionField(F) R.<t> = PolynomialRing(K) L.<y> = K.extension(t^4 + t - x^5) L.places_infinite(1)
- residue_field(place, name=None)[source]¶
Return the residue field associated with the place along with the maps from and to the residue field.
INPUT:
place– place of the function fieldname– string; name of the generator of the residue field
The domain of the map to the residue field is the discrete valuation ring associated with the place.
The discrete valuation ring is defined as the ring of all elements of the function field with nonnegative valuation at the place. The maximal ideal is the set of elements of positive valuation. The residue field is then the quotient of the discrete valuation ring by its maximal ideal.
If an element not in the valuation ring is applied to the map, an exception
TypeErroris raised.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: p = L.places_finite()[0] sage: R, fr_R, to_R = L.residue_field(p) sage: R Finite Field of size 2 sage: f = 1 + y sage: f.valuation(p) -1 sage: to_R(f) Traceback (most recent call last): ... TypeError: ... sage: (1+1/f).valuation(p) 0 sage: to_R(1 + 1/f) 1 sage: [fr_R(e) for e in R] [0, 1]
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); _ = K['Y']; (Y,) = _._first_ngens(1) >>> L = K.extension(Y**Integer(2) + Y + x + Integer(1)/x, names=('y',)); (y,) = L._first_ngens(1) >>> p = L.places_finite()[Integer(0)] >>> R, fr_R, to_R = L.residue_field(p) >>> R Finite Field of size 2 >>> f = Integer(1) + y >>> f.valuation(p) -1 >>> to_R(f) Traceback (most recent call last): ... TypeError: ... >>> (Integer(1)+Integer(1)/f).valuation(p) 0 >>> to_R(Integer(1) + Integer(1)/f) 1 >>> [fr_R(e) for e in R] [0, 1]
# needs sage.rings.finite_rings K.<x> = FunctionField(GF(2)); _.<Y> = K[] L.<y> = K.extension(Y^2 + Y + x + 1/x) p = L.places_finite()[0] R, fr_R, to_R = L.residue_field(p) R f = 1 + y f.valuation(p) to_R(f) (1+1/f).valuation(p) to_R(1 + 1/f) [fr_R(e) for e in R]