Factories to construct function fields#
This module provides factories to construct function fields. These factories are only for internal use.
EXAMPLES:
sage: K.<x> = FunctionField(QQ); K
Rational function field in x over Rational Field
sage: L.<x> = FunctionField(QQ); L
Rational function field in x over Rational Field
sage: K is L
True
AUTHORS:
William Stein (2010): initial version
Maarten Derickx (2011-09-11): added
FunctionField_polymod_Constructor
, use@cached_function
Julian Rueth (2011-09-14): replaced
@cached_function
withUniqueFactory
- class sage.rings.function_field.constructor.FunctionFieldExtensionFactory#
Bases:
UniqueFactory
Create a function field defined as an extension of another function field by adjoining a root of a univariate polynomial. The returned function field is unique in the sense that if you call this function twice with an equal
polynomial
andnames
it returns the same python object in both calls.INPUT:
polynomial
– univariate polynomial over a function fieldnames
– variable names (as a tuple of length 1 or string)category
– category (defaults to category of function fields)
EXAMPLES:
sage: K.<x> = FunctionField(QQ) sage: R.<y> = K[] sage: y2 = y*1 sage: y2 is y False sage: L.<w> = K.extension(x - y^2) # needs sage.rings.function_field sage: M.<w> = K.extension(x - y2^2) # needs sage.rings.function_field sage: L is M # needs sage.rings.function_field True
- create_key(polynomial, names)#
Given the arguments and keywords, create a key that uniquely determines this object.
EXAMPLES:
sage: K.<x> = FunctionField(QQ) sage: R.<y> = K[] sage: L.<w> = K.extension(x - y^2) # indirect doctest # needs sage.rings.function_field
- create_object(version, key, **extra_args)#
Create the object from the key and extra arguments. This is only called if the object was not found in the cache.
EXAMPLES:
sage: K.<x> = FunctionField(QQ) sage: R.<y> = K[] sage: L.<w> = K.extension(x - y^2) # indirect doctest # needs sage.rings.function_field sage: y2 = y*1 sage: M.<w> = K.extension(x - y2^2) # indirect doctest # needs sage.rings.function_field sage: L is M # needs sage.rings.function_field True
- class sage.rings.function_field.constructor.FunctionFieldFactory#
Bases:
UniqueFactory
Return the function field in one variable with constant field
F
. The function field returned is unique in the sense that if you call this function twice with the same base field and name then you get the same python object back.INPUT:
F
– fieldnames
– name of variable as a string or a tuple containing a string
EXAMPLES:
sage: K.<x> = FunctionField(QQ); K Rational function field in x over Rational Field sage: L.<y> = FunctionField(GF(7)); L Rational function field in y over Finite Field of size 7 sage: R.<z> = L[] sage: M.<z> = L.extension(z^7 - z - y); M # needs sage.rings.finite_rings sage.rings.function_field Function field in z defined by z^7 + 6*z + 6*y
- create_key(F, names)#
Given the arguments and keywords, create a key that uniquely determines this object.
EXAMPLES:
sage: K.<x> = FunctionField(QQ) # indirect doctest
- create_object(version, key, **extra_args)#
Create the object from the key and extra arguments. This is only called if the object was not found in the cache.
EXAMPLES:
sage: K.<x> = FunctionField(QQ) # indirect doctest sage: L.<x> = FunctionField(QQ) # indirect doctest sage: K is L True