[index]
(Class of Localization of Ring)
This class creates the fraction ring of the given ring. To make a concrete class, use the class method ::create or the function Algebra.LocalizedRing().
none.
Algebra.LocalizedRing(ring)
Algebra.RationalFunctionField(ring, obj)
Creates the rational function field over ring with the variable expressed by obj. This class is equipped with the class method ::var which returns the variable.
Example: the quotient field over the polynomial ring over Rational
require "algebra/localized-ring" require "rational" F = Algebra.RationalFunctionField(Rational, "x") x = F.var p ( 1 / (x**2 - 1) - 1 / (x**3 - 1) ) #=> x^2/(x^4 + x^3 - x - 1)
Algebra.MRationalFunctionField(ring, [obj1[, obj2, ...]])
Creates the rational function field over ring with the variables expressed by obj1, obj2, .... This class is equipped with the class method ::vars which returns the array of variables.
Example: the quotient field over the polynomial ring over Rational
require "algebra/localized-ring" require "rational" G = Algebra.MRationalFunctionField(Rational, "x", "y", "z") x, y, z = G.vars f = (x + z) / (x + y) - z / (x + y) p f #=> (x^2 + xy)/(x^2 + 2xy + y^2) p f.simplify #=> x/(x + y)
::create(ring)
Returns the fraction ring of which the numerator and the denominator are the elements of the ring.
This returns the subclass of Algebra::LocalizedRing. The subclass
has the class method ::ground and ::[]
which
return ring and x/1
respectively.
Example: Yet Another Rational
require "localized-ring" F = Algebra.LocalizedRing(Integer) p F.new(1, 2) + F.new(2, 3) #=> 7/6
Example: rational function field over Integer
require "polynomial" require "localized-ring" P = Algebra.Polynomial(Integer, "x") F = Algebra.LocalizedRing(P) x = F[P.var] p ( 1 / (x**2 - 1) - 1 / (x**3 - 1) ) #=> (x^3 - x^2)/(x^5 - x^3 - x^2 + 1)
::zero
::unity
zero?
zero
unity
==(other)
+(other)
-(other)
*(other)
/(other)
**(n)