
[index]

AlgebraicExtensionField

(Algebraic Extension Field)

A class represents the algebraic extension field.

File Name:

 ¡¦ algebraic-Extension-feild.rb

SuperClass:

 ¡¦ ResidueClassRing

Included Module

(none)

Associated Methods:

Algebra.AlgebraicExtensionField(field, obj){|x| ... }
    Same as ::create.
   

Class Method

::create(k, obj){|x| ... }
    Returns the extension field k[x]/(p(x)) of the field k by the polynomial p
    (x) of the variable x, where obj represents the x.
   
    The class methods ::var, ::def_polys and ::env_ring are defined for the
    return value.
   
    Example: Create the field F extended by x**2 + x + 1 == 0 from Rational.
    require "rational"
    require "polynomial"
    require "residue-class-ring"
    F  = Algebra.AlgebraicExtensionField(Rational, "x") {|x| x**2 + x + 1}
    x = F.var
    p( (x-1)** 3 / (x**2 - 1) ) #=> -3x - 3
   
::var
    Returns the residue class of x. This method is defined for the residue
    class ring, k[x]/(p(x)) which is the return value of ::create.
   
::modulus
    Returns the element p(x) of k[x] . This method is defined for the residue
    class ring, k[x]/(p(x)) which is the return value of ::create.
   
::def_polys
    Returns the array of ::modulus's of size n. Where self is defined
    recursively as the AlgebraicExtensionField of height n and base field k0.
    This method is defined for the residue class ring k[x]/(p(x)) which is the
    return value of ::create.
   
    Example: Make the extension field of cubic roots of 2, 3, 5 over Rational.
    require "algebra"
    # K0 == Rational
    K1 = AlgebraicExtensionField(Rational, "x1") { |x|
      x ** 3 - 2
    }
    K2 = AlgebraicExtensionField(K1, "x2") { |y|
      y ** 3 - 3
    }
    K3 = AlgebraicExtensionField(K2, "x3") { |z|
      z ** 3 - 5
    }
    
    p K3.def_polys #=> [x1^3 - 2, x2^3 - 3, x3^3 - 5]
    
    x1, x2, x3 = K1.var, K2.var, K3.var
    f = x1**2 + 2*x2**2 + 3*x3**2
    f0 = f.abs_lift
    
    p f0.type     #=> (Polynomial/(Polynomial/(Polynomial/Rational)))
    p f0.type == K3.env_ring #=> true
    
    p f #=> 3x3^2 + 2x2^2 + x1^2
    p f0.evaluate(x3.abs_lift, x2.abs_lift, x1.abs_lift)
        #=> x3^2 + 2x2^2 + 3x3^2
   
::env_ring
    Returns the multi-variate polynomial ring k0[x1, x2,.., xn]. Where self is
    defined recursively as the AlgebraicExtensionField of height n and base
    field k0. This method is defined for the residue class ring k[x]/(p(x))
    which is the return value of ::create.
   
::ground
    Return the polnomial ring k[x] which is the ground ring of the residue
    class ring.
   

Methods

abs_lift
    Returns the lift of self in ::env_ring. ( = k0[x1, x2,.., xn] ).
   

