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# Exercise

## Finite Sets

### Sets

```# sample-set01.rb

require "algebra"
#intersection
p Set[0, 1, 2, 4] & Set[1, 3, 5] == Set[1]
p Set[0, 1, 2, 4] & Set[7, 3, 5] == Set.phi

#union
p Set[0, 1, 2, 4] | Set[1, 3, 5] == Set[0, 1, 2, 3, 4, 5]

#membership
p Set[1, 3, 2].has? 1

#inclusion
p Set[3, 2, 1, 3] < Set[3, 1, 4, 2, 0]```

### Maps

```# sample-map01.rb

require "algebra"
a = Map[0=>2, 1=>2, 2=>0]
b = Map[0=>1, 1=>1, 2=>1]
p a * b #=> {0=>2, 1=>2, 2=>2}```

### Groups

```# sample-group01.rb

require "algebra"
e = Permutation[0, 1, 2, 3, 4]
a = Permutation[1, 0, 3, 4, 2]
b = Permutation[0, 2, 1, 3, 4]
p a * b #=> [2, 0, 3, 4, 1]

g = Group.new(e, a, b)
g.complete!
p g == PermutationGroup.symmetric(5) #=> true```

## Calculation of polynomials

```# sample-polynomial01.rb

require "algebra"
P = Polynomial.create(Integer, "x")
x = P.var
p((x + 1)**100) #=> x^100 + 100x^99 + ... + 100x + 1```

## Calculation of multi-variate polynomials

```# sample-polynomial02.rb

require "algebra"
P = Polynomial(Integer, "x", "y", "z")
x, y, z = P.vars
p((-x + y + z)*(x + y - z)*(x - y + z))
#=> -z^3 + (y + x)z^2 + (y^2 - 2xy + x^2)z - y^3 + xy^2 + x^2y - x^3```

## Calculation of multi-variate polynomials (2)

```# sample-m-polynomial01.rb

require "algebra"
P = MPolynomial(Integer)
x, y, z, w = P.vars("xyz")
p((-x + y + z)*(x + y - z)*(x - y + z))
#=> -x^3 + x^2y + x^2z + xy^2 - 2xyz + xz^2 - y^3 + y^2z + yz^2 - z^3```

## Remainder of the division of a polynomial by polynomials

```# sample-divmod01.rb

require "algebra"
P = MPolynomial(Rational)
x, y, z = P.vars("xyz")
f = x**2*y + x*y**2 + y*2 + z**3
g = x*y-z**3
h = y*2-6*z

P.set_ord(:lex) # lex, grlex, grevlex
puts "(#{f}).divmod([#{g}, #{h}]) =>", "#{f.divmod(g, h).inspect}"
#=> (x^2y + xy^2 + 2y + z^3).divmod([xy - z^3, 2y - 6z]) =>
#   [[x + y, 1/2z^3 + 1], xz^3 + 3z^4 + z^3 + 6z]
#   = [[Quotient1,Quotient2], Remainder]```

## Groebner basis

```# sample-groebner01.rb

require "algebra"
P = MPolynomial(Rational, "xyz")
x, y, z = P.vars("xyz")
f1 = x**2 + y**2 + z**2 -1
f2 = x**2 + z**2 - y
f3 = x - z
p Groebner.basis([f1, f2, f3])
#=> [x - z, y - 2z^2, z^4 + 1/2z^2 - 1/4]```

## Prime field

```# sample-primefield01.rb

require "algebra"
Z13 = ResidueClassRing(Integer, 13)

a, b, c, d, e, f, g = Z13
p [e + c, e - c, e * c, e * 2001, 3 + c, 1/c, 1/c * c, d / d, b * 1 / b]
#=> [6, 2, 8, 9, 5, 7, 1, 1, 1]
p( (1...13).collect{|i|  Z13[i]**12} )
#=> [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]```

## Algebraic field

```# sample-algebraicfield01.rb

require "algebra"

Px = Polynomial(Rational, "x")
x = Px.var
F = ResidueClassRing(Px, x**2 + x + 1)
x = F[x]
p( (x + 1)**100 )
#=> -x - 1
p( (x-1)** 3 / (x**2 - 1) )
#=> -3x - 3

G = Polynomial(F, "y")
y = G.var
p( (x + y + 1)** 7 )
#=> y^7 + (7x + 7)y^6 + 8xy^5 + 4y^4 + (4x + 4)y^3 + 5xy^2 + 7y + x + 1

H = ResidueClassRing(G, y**5 + x*y + 1)
y = H[y]
p( 1/(x + y + 1)**7 )
#=> (1798/3x + 1825/9)y^4 + (-74x + 5176/9)y^3 +
#     (-6886/9x - 5917/9)y^2 + (1826/3x - 3101/9)y + 2146/9x + 4702/9```

### This can be done as following

```# sample-algebraicfield02.rb

require "algebra"

F = AlgebraicExtensionField(Rational, "x") {|x| x**2 + x + 1}
x = F.var
p( (x + 1)**100 )
p( (x-1)** 3 / (x**2 - 1) )

H = AlgebraicExtensionField(F, "y") {|y| y**5 + x*y + 1}
y = H.var
p( 1/(x + y + 1)**7 )```

## Quotient fields

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

```# sample-quotientfield01.rb

require "algebra"
Q = LocalizedRing(Integer)
a = Q.new(3, 5)
b = Q.new(5, 3)
p [a + b, a - b, a * b, a / b, a + 3, 1 + a]
#=> [34/15, -16/15, 15/15, 9/25, 18/5, 8/5]```

### Rational function field

```# sample-quotientfield02.rb

require "algebra"

F13 = ResidueClassRing(Integer, 13)

P = Polynomial(F13, "x")
Q = LocalizedRing(P)
x = Q[P.var]
p ( 1 / (x**2 - 1) - 1 / (x**3 - 1) )

#This is equivalent to the following
F = RationalFunctionField(F13, "x")
x = F.var
p ( 1 / (x**2 - 1) - 1 / (x**3 - 1) )```

### Rational function field over the algebraic extension field

```# sample-quotientfield03.rb

require "algebra"

F13 = ResidueClassRing(Integer, 13)
F = AlgebraicExtensionField(F13, "a") {|a| a**2 - 2}
a = F.var
RF = RationalFunctionField(F, "x")
x = RF.var

p( (a/4*x + RF.unity/2)/(x**2 + a*x + 1) +
(-a/4*x + RF.unity/2)/(x**2 - a*x + 1) )
#=> 1/(x**4 + 1)```

### Algebraic function field

```# sample-quotientfield04.rb

require "algebra"

F13 = ResidueClassRing(Integer, 13)
F = RationalFunctionField(F13, "x")
x = F.var
AF = AlgebraicExtensionField(F, "a") {|a| a**2 - 2*x}
a = AF.var

p( (a/4*x + AF.unity/2)/(x**2 + a*x + 1) +
(-a/4*x + AF.unity/2)/(x**2 - a*x + 1) )
#=> (-x^3 + x^2 + 1)/(x^4 + 11x^3 + 2x^2 + 1)```

## Linear Algebra

### Linear equations

```# sample-gaussian-elimination01.rb

require "algebra"
M = MatrixAlgebra(Rational, 5, 4)
a = M.matrix{|i, j| i + j}
a.display #=>
#[0, 1, 2, 3]
#[1, 2, 3, 4]
#[2, 3, 4, 5]
#[3, 4, 5, 6]
#[4, 5, 6, 7]
a.kernel_basis.each do |v|
puts "a * #{v} = #{a * v}"
#=> a * [1, -2, 1, 0] = [0, 0, 0, 0, 0]
#=> a * [2, -3, 0, 1] = [0, 0, 0, 0, 0]
end```

### Diagonalization of Square Matrix

```# sample-diagonalization01.rb

require "algebra"
class Rational# < Numeric
def inspect; to_s; end
end

M = SquareMatrix(Rational, 3)
a = M[[1,-1,-1], [-1,1,-1], [2,1,-1]]
puts "A = "; a.display; puts
#A =
#  1,  -1,  -1
# -1,   1,  -1
#  2,   1,  -1

extfield, roots, tmatrix, evalues, addelms, evectors, espaces,
chpoly, facts = a.diagonalize

puts "Charactoristic Poly.: #{chpoly} => #{facts}"
#Charactoristic Poly.: t^3 - t^2 + t - 6 => (t - 2)(t^2 + t + 3)

puts "Algebraic Numbers:"
roots.each do |po, rs|
puts "#{rs.join(', ')} : roots of #{po} == 0"
end
puts
#Algebraic Numbers:
#a, -a - 1 : roots of t^2 + t + 3 == 0

puts "EigenSpaces: "
evalues.uniq.each do |ev|
puts "W_{#{ev}} = <#{espaces[ev].join(', ')}>"
end
puts
#EigenSpaces:
#W_{2} = <[4, -5, 1]>
#W_{a} = <[1/3a + 1/3, 1/3a + 1/3, 1]>
#W_{-a - 1} = <[-1/3a, -1/3a, 1]>

puts "P = "; tmatrix.display; puts
puts "P^-1 * A * P = "; (tmatrix.inverse * a * tmatrix).display; puts
#P =
#  4, 1/3a + 1/3, -1/3a
# -5, 1/3a + 1/3, -1/3a
#  1,   1,   1
#
#P^-1 * A * P =
#  2,   0,   0
#  0,   a,   0
#  0,   0, -a - 1```

### Elementary Divisors of Matrix

```# sample-elementary-divisor01.rb

require "algebra"

M = SquareMatrix(Rational, 4)
a = M[
[2, 0, 0, 0],
[0, 2, 0, 0],
[0, 0, 2, 0],
[5, 0, 0, 2]
]
P = Polynomial(Rational, "x")
MP = SquareMatrix(P, 4)

ac = a.char_matrix(MP)
ac.display; puts #=>
#x - 2,   0,   0,   0
#  0, x - 2,   0,   0
#  0,   0, x - 2,   0
# -5,   0,   0, x - 2

p ac.elementary_divisor #=> [1, x - 2, x - 2, x^2 - 4x + 4]

require "matrix-algebra-triplet"
at = ac.to_triplet.e_diagonalize

at.body.display; puts #=>
#  1,   0,   0,   0
#  0, x - 2,   0,   0
#  0,   0, x - 2,   0
#  0,   0,   0, x^2 - 4x + 4

at.left.display; puts #=>
#  0,   0,   0, -1/5
#  0,   1,   0,   0
#  0,   0,   1,   0
#  5,   0,   0, x - 2

at.right.display; puts #=>
#  1,   0,   0, 1/5x - 2/5
#  0,   1,   0,   0
#  0,   0,   1,   0
#  0,   0,   0,   1

p at.left * ac * at.right == at.body #=> true```

### Jordan Canonical Form of Matrix

```# sample-jordan-form01.rb

require "algebra"

M4 = SquareMatrix(Rational, 4)
m = M4[
[-1, 1, 2, -1],
[-5, 3, 4, -2],
[3, -1, 0, 1],
[5, -2, -2, 3]
]
m.jordan_form.display; #=>
#  2,   0,   0,   0
#  0,   1,   1,   0
#  0,   0,   1,   1
#  0,   0,   0,   1
puts

#-----------------------------------
m = M4[
[3, 1, -1, 1],
[-3, -1, 3, -1],
[-2, -2, 0, 0],
[0, 0, -4, 2]
]
jf, pt, qt, field, modulus  = m.jordan_form_info
p modulus #=> [a^2 + 4]
jf.display; puts #=>
#  2,   1,   0,   0
#  0,   2,   0,   0
#  0,   0,   a,   0
#  0,   0,   0,  -a

m = m.convert_to(jf.type)
p jf == pt * m * qt #=> true

#-----------------------------------
m = M4[
[-1, 1, 2, -1],
[-5, 3, 4, -2],
[3, -1, 0, 1],
[5, -2, -2, 0]
]
jf, pt, qt, field, modulus  = m.jordan_form_info
p modulus #=> [a^3 + 3a - 1, b^2 + ab + a^2 + 3]
jf.display; puts #=>
#  2,   0,   0,   0
#  0,   a,   0,   0
#  0,   0,   b,   0
#  0,   0,   0, -b - a

m = m.convert_to(jf.type)
p jf == pt * m * qt #=> true```

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

```# sample-cayleyhamilton01.rb

require "algebra"

n = 4
R = MPolynomial(Integer)
MR = SquareMatrix(R, n)
m = MR.matrix{|i, j| R.var("x#{i}#{j}") }
Rx = Polynomial(R, "x")
ch = m.char_polynomial(Rx)
p ch.evaluate(m) #=> 0```

## The expression of Groebner basis by original generators

```# sample-groebner02.rb

require "algebra"

P = MPolynomial(Rational)
x, y, z = P.vars "xyz"
f1 = x**2 + y**2 + z**2 -1
f2 = x**2 + z**2 - y
f3 = x - z

coeff, basis = Groebner.basis_coeff([f1, f2, f3])
basis.each_with_index do |b, i|
p [coeff[i].inner_product([f1, f2, f3]), b]
p coeff[i].inner_product([f1, f2, f3]) == b #=> true
end```

## The quotients and the remainder by arbitrary basis

```# sample-groebner03.rb

require "algebra"
F5 = ResidueClassRing(Integer, 2)
F = AlgebraicExtensionField(F5, "a") {|a| a**3 + a + 1}
a = F.var
P = MPolynomial(F)

x, y, z = P.vars("xyz")
f1 = x + y**2 + z**2 - 1
f2 = x**2 + z**2 - y * a
f3 = x - z - a

f = x**3 + y**3 + z**3
q, r = f.divmod_s(f1, f2, f3)
p f == q.inner_product([f1, f2, f3]) + r #=> true```

## Factorization

### Factorization of Integer coefficient polynomial

```# sample-factorize01.rb

require "algebra"

P = Polynomial(Integer, "x")
x = P.var
f = 8*x**7 - 20*x**6 + 6*x**5 - 11*x**4 + 44*x**3 - 9*x**2 - 27
p f.factorize #=> (2x - 3)^3(x^2 + x + 1)^2```

### Factorization of Zp coefficient polynomial

```# sample-factorize02.rb

require "algebra"

Z7 = ResidueClassRing(Integer, 7)
P = Polynomial(Z7, "x")
x = P.var
f = 8*x**7 - 20*x**6 + 6*x**5 - 11*x**4 + 44*x**3 - 9*x**2 - 27
p f.factorize #=> (x + 5)^2(x + 3)^2(x + 2)^3```

### Factorization of the algebraic extension of Rational polynomial

```# sample-factorize03.rb

require "algebra"

A = AlgebraicExtensionField(Rational, "a") {|a| a**2 + a + 1}
a = A.var
P = Polynomial(A, "x")
x = P.var
f = x**4 + (2*a + 1)*x**3 + 3*a*x**2 + (-3*a - 5)*x - a + 1
p f.factorize #=> (x + a)^3(x - a + 1)```

### Factorization of the algebraic extension of the algebraic extension of Rational

```# sample-factorize04.rb

require "algebra"

A = AlgebraicExtensionField(Rational, "a") {|a| a**2 - 2}
B = AlgebraicExtensionField(A, "b"){|b| b**2 + 1}
P = Polynomial(B, "x")
x = P.var
f = x**4 + 1
p f.factorize
#=> (x - 1/2ab - 1/2a)(x + 1/2ab - 1/2a)(x + 1/2ab + 1/2a)(x - 1/2ab + 1/2a)```

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

```# sample-factorize05.rb

require "algebra"

def show(f, mod = 0)
if mod > 0
zp = ResidueClassRing(Integer, mod)
pzp = Polynomial(zp, f.variable)
f = f.convert_to(pzp)
end
fact = f.factorize
printf "mod %2d: %-15s  =>  %s\n", mod, f, fact
end

Px = Polynomial(Integer, "x")
x = Px.var
f = x**4 + 10*x**2 + 1
#f = x**4 - 10*x**2 + 1
show(f)
Primes.new.each do |mod|
break if mod > 100
show(f, mod)
end

#mod  0: x^4 + 10x^2 + 1  =>  x^4 + 10x^2 + 1
#mod  2: x^4 + 1          =>  (x + 1)^4
#mod  3: x^4 + x^2 + 1    =>  (x + 2)^2(x + 1)^2
#mod  5: x^4 + 1          =>  (x^2 + 3)(x^2 + 2)
#mod  7: x^4 + 3x^2 + 1   =>  (x^2 + 4x + 6)(x^2 + 3x + 6)
#mod 11: x^4 - x^2 + 1    =>  (x^2 + 5x + 1)(x^2 + 6x + 1)
#mod 13: x^4 + 10x^2 + 1  =>  (x^2 - x + 12)(x^2 + x + 12)
#mod 17: x^4 + 10x^2 + 1  =>  (x^2 + 3x + 1)(x^2 + 14x + 1)
#mod 19: x^4 + 10x^2 + 1  =>  (x + 17)(x + 10)(x + 9)(x + 2)
#mod 23: x^4 + 10x^2 + 1  =>  (x^2 + 6)(x^2 + 4)
#mod 29: x^4 + 10x^2 + 1  =>  (x^2 + 21)(x^2 + 18)
#mod 31: x^4 + 10x^2 + 1  =>  (x^2 + 22x + 30)(x^2 + 9x + 30)
#mod 37: x^4 + 10x^2 + 1  =>  (x^2 + 32x + 36)(x^2 + 5x + 36)
#mod 41: x^4 + 10x^2 + 1  =>  (x^2 + 19x + 1)(x^2 + 22x + 1)
#mod 43: x^4 + 10x^2 + 1  =>  (x + 40)(x + 29)(x + 14)(x + 3)
#mod 47: x^4 + 10x^2 + 1  =>  (x^2 + 32)(x^2 + 25)
#mod 53: x^4 + 10x^2 + 1  =>  (x^2 + 41)(x^2 + 22)
#mod 59: x^4 + 10x^2 + 1  =>  (x^2 + 13x + 1)(x^2 + 46x + 1)
#mod 61: x^4 + 10x^2 + 1  =>  (x^2 + 54x + 60)(x^2 + 7x + 60)
#mod 67: x^4 + 10x^2 + 1  =>  (x + 55)(x + 39)(x + 28)(x + 12)
#mod 71: x^4 + 10x^2 + 1  =>  (x^2 + 43)(x^2 + 38)
#mod 73: x^4 + 10x^2 + 1  =>  (x + 68)(x + 44)(x + 29)(x + 5)
#mod 79: x^4 + 10x^2 + 1  =>  (x^2 + 64x + 78)(x^2 + 15x + 78)
#mod 83: x^4 + 10x^2 + 1  =>  (x^2 + 18x + 1)(x^2 + 65x + 1)
#mod 89: x^4 + 10x^2 + 1  =>  (x^2 + 9x + 1)(x^2 + 80x + 1)
#mod 97: x^4 + 10x^2 + 1  =>  (x + 88)(x + 54)(x + 43)(x + 9)```

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

```# sample-m-factorize01.rb

require "algebra"

P = MPolynomial(Integer)
x, y, z = P.vars("xyz")
f = x**3 + y**3 + z**3 - 3*x*y*z
p f.factorize #=> (x + y + z)(x^2 - xy - xz + y^2 - yz + z^2)

PQ = MPolynomial(Rational)
x, y, z = PQ.vars("xyz")
f = x**3 + y**3/8 + z**3 - 3*x*y*z/2
p f.factorize #=> (1/8)(2x + y + 2z)(4x^2 - 2xy - 4xz + y^2 - 2yz + 4z^2)```

### Factorization of Zp coefficient multi-variate polynomial

```# sample-m-factorize02.rb

require "algebra"

Z7 = ResidueClassRing(Integer, 7)
P = MPolynomial(Z7)
x, y, z = P.vars("xyz")
f = x**3 + y**3 + z**3 - 3*x*y*z
p f.factorize #=> (x + 4y + 2z)(x + 2y + 4z)(x + y + z)```

## Algebraic Equations

### Minimal Polynomial

```# sample-algebraic-equation01.rb

require "algebra"

PQ = MPolynomial(Rational)
a, b, c = PQ.vars("abc")
p AlgebraicEquation.minimal_polynomial(a + b + c, a**2-2, b**2-3, c**2-5)
#=> x^8 - 40x^6 + 352x^4 - 960x^2 + 576```

### Minimal Splitting Field

```# sample-splitting-field01.rb

require "algebra"

PQ = Polynomial(Rational, "x")
x = PQ.var
f = x**4 + 2
p f #=> x^4 + 2
field, modulus, facts = f.decompose
p modulus #=> [a^4 + 2, b^2 + a^2]
p facts #=> (x - a)(x + a)(x - b)(x + b)

fp = Polynomial(field, "x")
x = fp.var
facts = Factors.new(facts.collect{|g, n| [g.evaluate(x), n]})
p facts.pi == f.convert_to(fp) #=> true```

### Galois Group of polynomials

```# sample-galois-group01.rb

require "rational"
require "polynomial"

P = Algebra.Polynomial(Rational, "x")
x = P.var

(x**3 - 3*x + 1).galois_group.each do |g|
p g
end
#=> [0, 1, 2]
#   [1, 2, 0]
#   [2, 0, 1]]

(x**3 - x + 1).galois_group.each do |g|
p g
end
#=> [0, 1, 2]
#   [1, 0, 2]
#   [2, 0, 1]
#   [0, 2, 1]
#   [1, 2, 0]
#   [2, 1, 0]```