)abbrev domain AN AlgebraicNumber
++ Author: Manuel Bronstein
++ Date Created: 22 March 1988
++ Description: Algebraic closure of the rational numbers.
++ Keywords: algebraic, number.
AlgebraicNumber() : Exports == Implementation where
  Z   ==> Integer
  FE  ==> Expression Z
  K   ==> Kernel %
  P   ==> SparseMultivariatePolynomial(Z, K)
  SUP ==> SparseUnivariatePolynomial

  Exports ==> Join(ExpressionSpace, AlgebraicallyClosedField,
                   RetractableTo Z, RetractableTo Fraction Z,
                   LinearlyExplicitOver Z, RealConstant,
                   LinearlyExplicitOver Fraction Z,
                   PolynomialFactorizationExplicit,
                   CharacteristicZero,
                   ConvertibleTo Complex Float,
                   ConvertibleTo InputForm,
                   DifferentialRing) with
    coerce : P -> %
      ++ coerce(p) returns p viewed as an algebraic number.
    numer  : % -> P
      ++ numer(f) returns the numerator of f viewed as a
      ++ polynomial in the kernels over Z.
    denom  : % -> P
      ++ denom(f) returns the denominator of f viewed as a
      ++ polynomial in the kernels over Z.
    reduce : % -> %
      ++ reduce(f) simplifies all the unreduced algebraic numbers
      ++ present in f by applying their defining relations.
    trueEqual : (%, %) -> Boolean
      ++ trueEqual(x, y) tries to determine if the two numbers are equal
    norm : (SUP(%), Kernel %) -> SUP(%)
      ++ norm(p, k) computes the norm of the polynomial p
      ++ with respect to the extension generated by kernel k
    norm : (SUP(%), List Kernel %) -> SUP(%)
      ++ norm(p, l) computes the norm of the polynomial p
      ++ with respect to the extension generated by kernels l
    norm : (%, Kernel %) -> %
      ++ norm(f, k) computes the norm of the algebraic number f
      ++ with respect to the extension generated by kernel k
    norm : (%, List Kernel %) -> %
      ++ norm(f, l) computes the norm of the algebraic number f
      ++ with respect to the extension generated by kernels l
  Implementation ==> FE add

    Rep := FE

    ALGOP := '%alg

    reducedSystem(m : Matrix(%)) : Matrix(Fraction(Z)) ==
        mz : Matrix(Z) := reducedSystem(m)$Rep
        map((c : Z) : Fraction(Z) +-> c::Fraction(Z), mz
           )$MatrixCategoryFunctions2(Z, Vector(Z), Vector(Z), Matrix(Z),
               Fraction(Z), Vector(Fraction(Z)), Vector(Fraction(Z)),
                   Matrix(Fraction(Z)))

    reducedSystem(m : Matrix(%), v : Vector(%)
      ) : Record(mat : Matrix(Fraction(Z)), vec : Vector(Fraction(Z))) ==
        mzr : Record(mat : Matrix(Z), vec : Vector(Z)) :=
          reducedSystem(m, v)$Rep
        mq := map((c : Z) : Fraction(Z) +-> c::Fraction(Z), mzr.mat
                 )$MatrixCategoryFunctions2(Z, Vector(Z), Vector(Z), Matrix(Z),
                     Fraction(Z), Vector(Fraction(Z)), Vector(Fraction(Z)),
                       Matrix(Fraction(Z)))
        vq := map((c : Z) : Fraction(Z) +-> c::Fraction(Z), mzr.vec
                 )$VectorFunctions2(Z, Fraction(Z))
        [mq, vq]

    -- private
    mainRatDenom(f : %) : % ==
       ratDenom(f::Rep::FE)$AlgebraicManipulations(Integer, FE)::Rep::%
--        mv := mainVariable denom f
--        mv case "failed" => f
--        algv := mv::K
--      q := univariate(f, algv, minPoly(algv))$PolynomialCategoryQuotientFunctions(IndexedExponents K, K, Integer, P, %)
--      q(algv::%)

    findDenominator(z : SUP %) : Record(num : SUP %, den : %) ==
       zz := z
       while not(zz = 0) repeat
          dd := (denom leadingCoefficient zz)::%
          not(dd = 1) =>
             rec := findDenominator(dd*z)
             return [rec.num, rec.den*dd]
          zz := reductum zz
       [z, 1]
    makeUnivariate(p : P, k : Kernel %) : SUP % ==
      map(x +-> x::%,
            univariate(p, k))$SparseUnivariatePolynomialFunctions2(P, %)

    import from P
    import from List(K)
    import from Fraction(Z)

    -- public
    a, b : %
    differentiate(x : %) : % == 0
    zero? a == zero? numer a
    one? a == (numer a = 1) and (denom a = 1)
    x : % / y : %        == mainRatDenom(x /$Rep y)
    x : % ^ n : Integer ==
      n < 0 => mainRatDenom (x ^$Rep n)
      x ^$Rep n
    trueEqual(a, b) ==
       -- if two algebraic numbers have the same norm (after deleting repeated
       -- roots, then they are certainly conjugates. Note that we start with a
       -- monic polynomial, so don't have to check for constant factors.
       -- this will be fooled by sqrt(2) and -sqrt(2), but the = in
       -- AlgebraicNumber knows what to do about this.
       ka := reverse tower a
       kb := reverse tower b
       empty? ka and empty? kb => retract(a)@Fraction Z = retract(b)@Fraction Z
       pa, pb : SparseUnivariatePolynomial %
       pa := monomial(1, 1)-monomial(a, 0)
       pb := monomial(1, 1)-monomial(b, 0)
       na := map(retract, norm(pa, ka))$SparseUnivariatePolynomialFunctions2(%, Fraction Z)
       nb := map(retract, norm(pb, kb))$SparseUnivariatePolynomialFunctions2(%, Fraction Z)
       (sa := squareFreePart(na)) = (sb := squareFreePart(nb)) => true
       g := gcd(sa, sb)
       (dg := degree g) = 0 => false
       -- of course, if these have a factor in common, then the
       -- answer is really ambiguous, so we ought to be using Duval-type
       -- technology
       dg = degree sa or dg = degree sb => true
       false
    norm(z : %, k : Kernel %) : % ==
       p := minPoly k
       n := makeUnivariate(numer z, k)
       d := makeUnivariate(denom z, k)
       resultant(n, p)/resultant(d, p)
    norm(z : %, l : List Kernel %) : % ==
       for k in l repeat
           z := norm(z, k)
       z
    norm(z : SUP %, k : Kernel %) : SUP % ==
       p := map(x +-> x::SUP %,
               minPoly k)$SparseUnivariatePolynomialFunctions2(%, SUP %)
       f := findDenominator z
       zz := map(x +-> makeUnivariate(numer x, k),
             f.num)$SparseUnivariatePolynomialFunctions2(%, SUP %)
       zz := swap(zz)$CommuteUnivariatePolynomialCategory(%, SUP %, SUP SUP %)
       resultant(p, zz)/norm(f.den, k)
    norm(z : SUP %, l : List Kernel %) : SUP % ==
       for k in l repeat
           z := norm(z, k)
       z
    belong? op == belong?(op)$ExpressionSpace2_&(%, K) or has?(op, ALGOP)

    convert(x : %) : InputForm ==
      coerceToType(convert(x pretend Rep)$Rep)$InputFormFunctions1(%)

    convert(x : %) : Float ==
      import from Expression(Float)
      retract map(y +-> y::Float, x pretend FE)$ExpressionFunctions2(Z, Float)

    convert(x : %) : DoubleFloat ==
      import from Expression(DoubleFloat)
      retract map(y +-> y::DoubleFloat,
                  x pretend FE)$ExpressionFunctions2(Z, DoubleFloat)

    convert(x : %) : Complex(Float) ==
      import from Expression(Complex(Float))
      retract map(y +-> y::Complex(Float),
                  x pretend FE)$ExpressionFunctions2(Z, Complex Float)

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