# 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)


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)


AUTHORS:

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

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

Constant field extension.

INPUT:

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

• k_ext – an extension of $$k$$

conorm_divisor(d)#

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

conorm_place(p)#

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()]


defining_morphism()#

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

top()#

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

class sage.rings.function_field.extensions.FunctionFieldExtension#

Abstract base class of function field extensions.