def vector_space(self):
"""
Return a vector space V and isomorphisms self --> V and V --> self.
OUTPUT:
- ``V`` -- a vector space over the rational numbers
- ``from_V`` -- an isomorphism from V to self
- ``to_V`` -- an isomorphism from self to V
EXAMPLES::
sage: K.<x> = FunctionField(QQ)
sage: K.vector_space()
(Vector space of dimension 1 over Rational function field in x over Rational Field, Isomorphism morphism:
From: Vector space of dimension 1 over Rational function field in x over Rational Field
To: Rational function field in x over Rational Field, Isomorphism morphism:
From: Rational function field in x over Rational Field
To: Vector space of dimension 1 over Rational function field in x over Rational Field)
"""
V = self.base_field()**1
from maps import MapVectorSpaceToFunctionField, MapFunctionFieldToVectorSpace
from_V = MapVectorSpaceToFunctionField(V, self)
to_V = MapFunctionFieldToVectorSpace(self, V)
return (V, from_V, to_V)
def vector_space(self):
"""
Return a vector space V and isomorphisms self --> V and V --> self.
This function allows us to identify the elements of self with
elements of a vector space over the base field, which is
useful for representation and arithmetic with orders, ideals,
etc.
OUTPUT:
- ``V`` -- a vector space over base field
- ``from_V`` -- an isomorphism from V to self
- ``to_V`` -- an isomorphism from self to V
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
We get the vector spaces, and maps back and forth::
sage: V, from_V, to_V = L.vector_space()
sage: V
Vector space of dimension 5 over Rational function field in x over Rational Field
sage: from_V
Isomorphism morphism:
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 morphism:
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
We convert an element of the vector space back to the function field::
sage: from_V(V.1)
y
We define an interesting element of the function field::
sage: a = 1/L.0; a
(-x/(-x^4 - 1))*y^4 + 2*x^2/(-x^4 - 1)
We convert it to the vector space, and get a vector over the base field::
sage: to_V(a)
(2*x^2/(-x^4 - 1), 0, 0, 0, -x/(-x^4 - 1))
We convert to and back, and get the same element::
sage: from_V(to_V(a)) == a
True
In the other direction::
sage: v = x*V.0 + (1/x)*V.1
sage: to_V(from_V(v)) == v
True
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.vector_space()
(Vector space of dimension 2 over Function field in y defined by y^5 - 2*x*y + (-x^4 - 1)/x, Isomorphism morphism:
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 morphism:
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)
"""
V = self.base_field()**self.degree()
from maps import MapVectorSpaceToFunctionField, MapFunctionFieldToVectorSpace
from_V = MapVectorSpaceToFunctionField(V, self)
to_V = MapFunctionFieldToVectorSpace(self, V)
return (V, from_V, to_V)