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# AlgebraicExtensionField

(Algebraic Extension Field)

A class represents the algebraic extension field.

## File Name:

• algebraic-Extension-feild.rb

## SuperClass:

• ResidueClassRing

(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] ).