=begin
[((<index|URL:index.html>))] 
= Exercise

== Finite Sets

=== Sets

<<< sample-set01.rb.v.rd

=== Maps

<<< sample-map01.rb.v.rd

=== Groups

<<< sample-group01.rb.v.rd


== Calculation of polynomials

<<< sample-polynomial01.rb.v.rd

== Calculation of multi-variate polynomials

<<< sample-polynomial02.rb.v.rd

== Calculation of multi-variate polynomials (2)

<<< sample-m-polynomial01.rb.v.rd

== Remainder of the division of a polynomial by polynomials

<<< sample-divmod01.rb.v.rd

== Groebner basis

<<< sample-groebner01.rb.v.rd

== Prime field

<<< sample-primefield01.rb.v.rd

== Algebraic field

<<< sample-algebraicfield01.rb.v.rd

=== This can be done as following

<<< sample-algebraicfield02.rb.v.rd

== Quotient fields

=== By taking the quotient field of the ring of Integer, rational numbers are obtained.

<<< sample-quotientfield01.rb.v.rd

=== Rational function field

<<< sample-quotientfield02.rb.v.rd

=== Rational function field over the algebraic extension field

<<< sample-quotientfield03.rb.v.rd

=== Algebraic function field

<<< sample-quotientfield04.rb.v.rd

== Linear Algebra

=== Linear equations

<<< sample-gaussian-elimination01.rb.v.rd

=== Diagonalization of Square Matrix

<<< sample-diagonalization01.rb.v.rd

=== Elementary Divisors of Matrix

<<< sample-elementary-divisor01.rb.v.rd

=== Jordan Canonical Form of Matrix

<<< sample-jordan-form01.rb.v.rd

=== The proofs of the theorem of Cayley-Hamilton

<<< sample-cayleyhamilton01.rb.v.rd

== The expression of Groebner basis by original generators

<<< sample-groebner02.rb.v.rd

== The quotients and the remainder by arbitrary basis

<<< sample-groebner03.rb.v.rd

== Factorization

=== Factorization of Integer coefficient polynomial

<<< sample-factorize01.rb.v.rd

=== Factorization of Zp coefficient polynomial

<<< sample-factorize02.rb.v.rd

=== Factorization of the algebraic extension of Rational polynomial

<<< sample-factorize03.rb.v.rd

=== Factorization of the algebraic extension of the algebraic extension of Rational

<<< sample-factorize04.rb.v.rd

=== Factorization of x^4 + 10x^2 + 1

<<< sample-factorize05.rb.v.rd

=== Factorization of Integer or Rational coefficient multi-variate polynomial

<<< sample-m-factorize01.rb.v.rd

=== Factorization of Zp coefficient multi-variate polynomial

<<< sample-m-factorize02.rb.v.rd

== Algebraic Equations

=== Minimal Polynomial

<<< sample-algebraic-equation01.rb.v.rd

=== Minimal Splitting Field

<<< sample-splitting-field01.rb.v.rd

=== Galois Group of polynomials

<<< sample-galois-group01.rb.v.rd

=end

