AUTHORS:
EXAMPLES:
sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: K.hom(1/x)
Function Field endomorphism of Rational function field in x over Rational Field
Defn: x |--> 1/x
sage: L.<y> = K.extension(y^2-x)
sage: K.hom(y)
Function Field morphism:
From: Rational function field in x over Rational Field
To: Function field in y defined by y^2 - x
Defn: x |--> y
sage: L.hom([y,x])
Function Field endomorphism of Function field in y defined by y^2 - x
Defn: y |--> y
x |--> x
sage: L.hom([x,y])
Traceback (most recent call last):
...
ValueError: invalid morphism
Bases: sage.categories.map.Map
A base class for derivations on function fields.
A derivation on is map
with
and
for all
.
EXAMPLES:
sage: K.<x> = FunctionField(QQ)
sage: d = K.derivation()
sage: isinstance(d, sage.rings.function_field.maps.FunctionFieldDerivation)
True
Return whether this derivation is injective.
OUTPUT:
Returns False since derivations are never injective.
EXAMPLES:
sage: K.<x> = FunctionField(QQ)
sage: d = K.derivation()
sage: d.is_injective()
False
Bases: sage.rings.function_field.maps.FunctionFieldDerivation
A derivation on a rational function field.
INPUT:
EXAMPLES:
sage: K.<x> = FunctionField(QQ)
sage: d = K.derivation()
sage: isinstance(d, sage.rings.function_field.maps.FunctionFieldDerivation_rational)
True
Bases: sage.categories.morphism.Morphism
A base class for isomorphisms between function fields and vector spaces.
EXAMPLES:
sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x*y + 4*x^3)
sage: V, f, t = L.vector_space()
sage: isinstance(f, sage.rings.function_field.maps.FunctionFieldIsomorphism)
True
Return True, since this isomorphism is injective.
EXAMPLES:
sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x*y + 4*x^3)
sage: V, f, t = L.vector_space()
sage: f.is_injective()
True
Return True, since this isomorphism is surjective.
EXAMPLES:
sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x*y + 4*x^3)
sage: V, f, t = L.vector_space()
sage: f.is_surjective()
True
Bases: sage.rings.morphism.RingHomomorphism
Base class for morphisms between function fields.
Returns True since homomorphisms of fields are injective.
EXAMPLES:
sage: K.<x> = FunctionField(QQ)
sage: f = K.hom(1/x); f
Function Field endomorphism of Rational function field in x over Rational Field
Defn: x |--> 1/x
sage: f.is_injective()
True
Bases: sage.rings.function_field.maps.FunctionFieldMorphism
Morphism from a finite extension of a function field to a function field.
EXAMPLES:
sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x)
sage: f = L.hom(-y); f
Function Field endomorphism of Function field in y defined by y^2 - x
Defn: y |--> -y
Bases: sage.rings.function_field.maps.FunctionFieldMorphism
Morphism from a rational function field to a function field.
EXAMPLES:
sage: K.<x> = FunctionField(QQ)
sage: f = K.hom(1/x); f
Function Field endomorphism of Rational function field in x over Rational Field
Defn: x |--> 1/x
Bases: sage.rings.function_field.maps.FunctionFieldIsomorphism
An isomorphism from a function field to a vector space.
EXAMPLES:
sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x*y + 4*x^3)
sage: V, f, t = L.vector_space(); t
Isomorphism morphism:
From: Function field in y defined by y^2 - x*y + 4*x^3
To: Vector space of dimension 2 over Rational function field in x over Rational Field
Return the vector space which is the domain of this isomorphism.
EXAMPLES:
sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x*y + 4*x^3)
sage: V, f, t = L.vector_space()
sage: t.codomain()
Vector space of dimension 2 over Rational function field in x over Rational Field
Return the function field which is the domain of this isomorphism.
EXAMPLES:
sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x*y + 4*x^3)
sage: V, f, t = L.vector_space()
sage: t.domain()
Function field in y defined by y^2 - x*y + 4*x^3
Bases: sage.rings.function_field.maps.FunctionFieldIsomorphism
An isomorphism from a vector space to a function field.
EXAMPLES:
sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x*y + 4*x^3)
sage: V, f, t = L.vector_space(); f
Isomorphism morphism:
From: Vector space of dimension 2 over Rational function field in x over Rational Field
To: Function field in y defined by y^2 - x*y + 4*x^3
Return the function field which is the codomain of this isomorphism.
EXAMPLES:
sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x*y + 4*x^3)
sage: V, f, t = L.vector_space()
sage: f.codomain()
Function field in y defined by y^2 - x*y + 4*x^3
Return the vector space which is the domain of this isomorphism.
EXAMPLES:
sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x*y + 4*x^3)
sage: V, f, t = L.vector_space()
sage: f.domain()
Vector space of dimension 2 over Rational function field in x over Rational Field