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

Bases: sage.structure.factory.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 – a field
  • names – name of variable as a string or a tuple containg 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)

x.__init__(...) initializes x; see help(type(x)) for signature

create_object(version, key, **extra_args)

x.__init__(...) initializes x; see help(type(x)) for signature

class sage.rings.function_field.constructor.FunctionFieldPolymodFactory

Bases: sage.structure.factory.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 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) #indirect doctest
sage: M.<w>=K.extension(x-y2^2) #indirect doctest
sage: L is M
True
create_key(polynomial, names)

x.__init__(...) initializes x; see help(type(x)) for signature

create_object(version, key, **extra_args)

x.__init__(...) initializes x; see help(type(x)) for signature

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