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Algebra::ResidueClassRing

(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.

File Name:

• residue-class-ring.rb

SuperClass:

• Object

Included Modules:

none.

Associated Functions:

Algebra.ResidueClassRing(ring, mod)

Same as ::create(ring, mod).

Class Methods:

::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.

::zero

Returns zero.

::unity

Returns unity.

Methods:

lift

Returns the representative of self.

zero?

Returns true if self is zero.

zero

Returns zero.

unity

Returns 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.

inverse

Returns the inverse element, assuming the fundamental ring is Euclidian. When it does not exist, this returns nil.