# Factories to construct Function Fields¶

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 with UniqueFactory

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

class sage.rings.function_field.constructor.FunctionFieldFactory

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 – a field
• names – 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
Function field in z defined by z^7 + 6*z + 6*y


TESTS:

sage: K.<x> = FunctionField(QQ)
sage: L.<x> = FunctionField(QQ)
sage: K is L
True
sage: M.<x> = FunctionField(GF(7))
sage: K is M
False
sage: N.<y> = FunctionField(QQ)
sage: K is N
False

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)
sage: L.<x> = FunctionField(QQ)
sage: K is L
True

class sage.rings.function_field.constructor.FunctionFieldPolymodFactory

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 and names it returns the same python object in both calls.

INPUT:

• polynomial – a univariate polynomial over a function field
• names – variable names (as a tuple of length 1 or string)
• category – a 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)
sage: M.<w>=K.extension(x-y2^2)
sage: L is M
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


TESTS:

Verify that trac ticket #16530 has been resolved:

sage: K.<x> = FunctionField(QQ)
sage: R.<y> = K[]
sage: L.<y> = K.extension(y^2-x)
sage: R.<z> = L[]
sage: M.<z> = L.extension(z-1)
sage: R.<z> = K[]
sage: N.<z> = K.extension(z-1)
sage: M is N
False

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
sage: y2 = y*1
sage: M.<w> = K.extension(x-y2^2) # indirect doctest
sage: L is M
True


#### Previous topic

Function Field Morphisms