# Special extensions of function fields#

This module currently implements only constant field extension.

## Constant field extensions#

EXAMPLES:

Constant field extension of the rational function field over rational numbers:

sage: K.<x> = FunctionField(QQ)
sage: N.<a> = QuadraticField(2)                                                     # needs sage.rings.number_field
sage: L = K.extension_constant_field(N)                                             # needs sage.rings.number_field
sage: L                                                                             # needs sage.rings.number_field
Rational function field in x over Number Field in a with defining
polynomial x^2 - 2 with a = 1.4142... over its base
sage: d = (x^2 - 2).divisor()                                                       # needs sage.libs.pari sage.modules
sage: d                                                                             # needs sage.libs.pari sage.modules
-2*Place (1/x)
+ Place (x^2 - 2)
sage: L.conorm_divisor(d)                                                           # needs sage.libs.pari sage.modules sage.rings.number_field
-2*Place (1/x)
+ Place (x - a)
+ Place (x + a)

>>> from sage.all import *
>>> K = FunctionField(QQ, names=('x',)); (x,) = K._first_ngens(1)
>>> N = QuadraticField(Integer(2), names=('a',)); (a,) = N._first_ngens(1)# needs sage.rings.number_field
>>> L = K.extension_constant_field(N)                                             # needs sage.rings.number_field
>>> L                                                                             # needs sage.rings.number_field
Rational function field in x over Number Field in a with defining
polynomial x^2 - 2 with a = 1.4142... over its base
>>> d = (x**Integer(2) - Integer(2)).divisor()                                                       # needs sage.libs.pari sage.modules
>>> d                                                                             # needs sage.libs.pari sage.modules
-2*Place (1/x)
+ Place (x^2 - 2)
>>> L.conorm_divisor(d)                                                           # needs sage.libs.pari sage.modules sage.rings.number_field
-2*Place (1/x)
+ Place (x - a)
+ Place (x + a)


Constant field extension of a function field over a finite field:

sage: # needs sage.rings.finite_rings sage.rings.function_field
sage: K.<x> = FunctionField(GF(2)); R.<Y> = K[]
sage: F.<y> = K.extension(Y^3 - x^2*(x^2 + x + 1)^2)
sage: E = F.extension_constant_field(GF(2^3))
sage: E
Function field in y defined by y^3 + x^6 + x^4 + x^2 over its base
sage: p = F.get_place(3)
sage: E.conorm_place(p)  # random
Place (x + z3, y + z3^2 + z3)
+ Place (x + z3^2, y + z3)
+ Place (x + z3^2 + z3, y + z3^2)
sage: q = F.get_place(2)
sage: E.conorm_place(q)  # random
Place (x + 1, y^2 + y + 1)
sage: E.conorm_divisor(p + q)  # random
Place (x + 1, y^2 + y + 1)
+ Place (x + z3, y + z3^2 + z3)
+ Place (x + z3^2, y + z3)
+ Place (x + z3^2 + z3, y + z3^2)

>>> from sage.all import *
>>> # needs sage.rings.finite_rings sage.rings.function_field
>>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); R = K['Y']; (Y,) = R._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)
>>> E = F.extension_constant_field(GF(Integer(2)**Integer(3)))
>>> E
Function field in y defined by y^3 + x^6 + x^4 + x^2 over its base
>>> p = F.get_place(Integer(3))
>>> E.conorm_place(p)  # random
Place (x + z3, y + z3^2 + z3)
+ Place (x + z3^2, y + z3)
+ Place (x + z3^2 + z3, y + z3^2)
>>> q = F.get_place(Integer(2))
>>> E.conorm_place(q)  # random
Place (x + 1, y^2 + y + 1)
>>> E.conorm_divisor(p + q)  # random
Place (x + 1, y^2 + y + 1)
+ Place (x + z3, y + z3^2 + z3)
+ Place (x + z3^2, y + z3)
+ Place (x + z3^2 + z3, y + z3^2)


AUTHORS:

• Kwankyu Lee (2021-12-24): added constant field extension

class sage.rings.function_field.extensions.ConstantFieldExtension(F, k_ext)[source]#

Constant field extension.

INPUT:

• F – a function field whose constant field is $$k$$

• k_ext – an extension of $$k$$

conorm_divisor(d)[source]#

Return the conorm of the divisor d in this extension.

INPUT:

• d – divisor of the base function field

OUTPUT: a divisor of the top function field

EXAMPLES:

sage: # needs sage.rings.finite_rings sage.rings.function_field
sage: K.<x> = FunctionField(GF(2)); R.<Y> = K[]
sage: F.<y> = K.extension(Y^3 - x^2*(x^2 + x + 1)^2)
sage: E = F.extension_constant_field(GF(2^3))
sage: p1 = F.get_place(3)
sage: p2 = F.get_place(2)
sage: c = E.conorm_divisor(2*p1 + 3*p2)
sage: c1 = E.conorm_place(p1)
sage: c2 = E.conorm_place(p2)
sage: c == 2*c1 + 3*c2
True

>>> from sage.all import *
>>> # needs sage.rings.finite_rings sage.rings.function_field
>>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); R = K['Y']; (Y,) = R._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)
>>> E = F.extension_constant_field(GF(Integer(2)**Integer(3)))
>>> p1 = F.get_place(Integer(3))
>>> p2 = F.get_place(Integer(2))
>>> c = E.conorm_divisor(Integer(2)*p1 + Integer(3)*p2)
>>> c1 = E.conorm_place(p1)
>>> c2 = E.conorm_place(p2)
>>> c == Integer(2)*c1 + Integer(3)*c2
True

conorm_place(p)[source]#

Return the conorm of the place $$p$$ in this extension.

INPUT:

• p – place of the base function field

OUTPUT: divisor of the top function field

EXAMPLES:

sage: # needs sage.rings.finite_rings sage.rings.function_field
sage: K.<x> = FunctionField(GF(2)); R.<Y> = K[]
sage: F.<y> = K.extension(Y^3 - x^2*(x^2 + x + 1)^2)
sage: E = F.extension_constant_field(GF(2^3))
sage: p = F.get_place(3)
sage: d = E.conorm_place(p)
sage: [pl.degree() for pl in d.support()]
[1, 1, 1]
sage: p = F.get_place(2)
sage: d = E.conorm_place(p)
sage: [pl.degree() for pl in d.support()]
[2]

>>> from sage.all import *
>>> # needs sage.rings.finite_rings sage.rings.function_field
>>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); R = K['Y']; (Y,) = R._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)
>>> E = F.extension_constant_field(GF(Integer(2)**Integer(3)))
>>> p = F.get_place(Integer(3))
>>> d = E.conorm_place(p)
>>> [pl.degree() for pl in d.support()]
[1, 1, 1]
>>> p = F.get_place(Integer(2))
>>> d = E.conorm_place(p)
>>> [pl.degree() for pl in d.support()]
[2]

defining_morphism()[source]#

Return the defining morphism of this extension.

This is the morphism from the base to the top.

EXAMPLES:

sage: # needs sage.rings.finite_rings sage.rings.function_field
sage: K.<x> = FunctionField(GF(2)); R.<Y> = K[]
sage: F.<y> = K.extension(Y^3 - x^2*(x^2 + x + 1)^2)
sage: E = F.extension_constant_field(GF(2^3))
sage: E.defining_morphism()
Function Field morphism:
From: Function field in y defined by y^3 + x^6 + x^4 + x^2
To:   Function field in y defined by y^3 + x^6 + x^4 + x^2
Defn: y |--> y
x |--> x
1 |--> 1

>>> from sage.all import *
>>> # needs sage.rings.finite_rings sage.rings.function_field
>>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); R = K['Y']; (Y,) = R._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)
>>> E = F.extension_constant_field(GF(Integer(2)**Integer(3)))
>>> E.defining_morphism()
Function Field morphism:
From: Function field in y defined by y^3 + x^6 + x^4 + x^2
To:   Function field in y defined by y^3 + x^6 + x^4 + x^2
Defn: y |--> y
x |--> x
1 |--> 1

top()[source]#

Return the top function field of this extension.

EXAMPLES:

sage: # needs sage.rings.finite_rings sage.rings.function_field
sage: K.<x> = FunctionField(GF(2)); R.<Y> = K[]
sage: F.<y> = K.extension(Y^3 - x^2*(x^2 + x + 1)^2)
sage: E = F.extension_constant_field(GF(2^3))
sage: E.top()
Function field in y defined by y^3 + x^6 + x^4 + x^2

>>> from sage.all import *
>>> # needs sage.rings.finite_rings sage.rings.function_field
>>> K = FunctionField(GF(Integer(2)), names=('x',)); (x,) = K._first_ngens(1); R = K['Y']; (Y,) = R._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)
>>> E = F.extension_constant_field(GF(Integer(2)**Integer(3)))
>>> E.top()
Function field in y defined by y^3 + x^6 + x^4 + x^2

class sage.rings.function_field.extensions.FunctionFieldExtension[source]#

Abstract base class of function field extensions.