[index]
(Class of Residue Class Ring)
This class represents a residue class ring. To create concrete class, use the class method ::create or the function Algebra.ResidueClassRing() designating the base ring and the element of it.
none.
Algebra.ResidueClassRing(ring, mod)Same as ::create(ring, mod).
::create(ring, mod)Returns the class of the residue class ring of the ring and the modulus mod.
This class is a subclass of ResidueClassRing and
has the class methods ::ground, ::modulus and [x]
, which return the fundamental ring ring, the modulus
mod and the representing residue class of x, respectively.
Example: divide the polynomial ring by the modulus x**2 + x + 1.
require "rational" require "polynomial" require "residue-class-ring" Px = Algebra.Polynomial(Rational, "x") x = Px.var F = ResidueClassRing(Px, x**2 + x + 1) p F[x + 1]**100 #=> -x - 1
When ring is Integer, all inverse elements are calculated
in advance. And we can obtain the residue classes of
0, 1, ... , mod-1 by to_ary.
Example: the prime field of modulo 7
require "residue-class-ring"
F7 = Algebra::ResidueClassRing.create(Integer, 7)
a, b, c, d, e, f, g = F7
p [e + c, e - c, e * c, e * 2001, 3 + c, 1/c, 1/c * c]
#=> [6, 2, 1, 3, 5, 4, 1]
p( (1...7).collect{|i| F7[i]**6} )
#=> [1, 1, 1, 1, 1, 1]
::[x]Returns the residue class represented bye x.
::zeroReturns zero.
::unityReturns unity.
liftReturns the representative of self.
zero?Returns true if self is zero.
zeroReturns zero.
unityReturns unity.
==(other)Returns true if self equals other.
+(other)Returns the sum of self and other.
-(other)Returns the difference of self from other.
*(other)Returns the product of self and other.
**(n)Returns the n-th power of self.
/(other)Returns the quotient of self by other using inverse.
inverseReturns the inverse element, assuming the fundamental ring is Euclidian. When it does not exist, this returns nil.