AlgebraicNumber

constant.spad line 1 [edit on github]

Algebraic closure of the rational numbers.

* : (%, %) -> %

from Magma

* : (%, Fraction(Integer)) -> %

from RightModule(Fraction(Integer))

* : (%, Integer) -> %

from RightModule(Integer)

* : (Fraction(Integer), %) -> %

from LeftModule(Fraction(Integer))

* : (Integer, %) -> %

from AbelianGroup

* : (NonNegativeInteger, %) -> %

from AbelianMonoid

* : (PositiveInteger, %) -> %

from AbelianSemiGroup

+ : (%, %) -> %

from AbelianSemiGroup

- : % -> %

from AbelianGroup

- : (%, %) -> %

from AbelianGroup

/ : (%, %) -> %

from Field

0 : () -> %

from AbelianMonoid

1 : () -> %

from MagmaWithUnit

= : (%, %) -> Boolean

from BasicType

D : % -> %

from DifferentialRing

D : (%, NonNegativeInteger) -> %

from DifferentialRing

^ : (%, Fraction(Integer)) -> %

from RadicalCategory

^ : (%, Integer) -> %

from DivisionRing

^ : (%, NonNegativeInteger) -> %

from MagmaWithUnit

^ : (%, PositiveInteger) -> %

from Magma

annihilate? : (%, %) -> Boolean

from Rng

antiCommutator : (%, %) -> %

from NonAssociativeSemiRng

associates? : (%, %) -> Boolean

from EntireRing

associator : (%, %, %) -> %

from NonAssociativeRng

belong? : BasicOperator -> Boolean

from ExpressionSpace2(Kernel(%))

box : % -> %

from ExpressionSpace2(Kernel(%))

characteristic : () -> NonNegativeInteger

from NonAssociativeRing

coerce : % -> %

from Algebra(%)

coerce : Fraction(Integer) -> %

from CoercibleFrom(Fraction(Integer))

coerce : Integer -> %

from CoercibleFrom(Integer)

coerce : Kernel(%) -> %

from CoercibleFrom(Kernel(%))

coerce : SparseMultivariatePolynomial(Integer, Kernel(%)) -> %

coerce(p) returns p viewed as an algebraic number.

coerce : % -> OutputForm

from CoercibleTo(OutputForm)

commutator : (%, %) -> %

from NonAssociativeRng

convert : % -> Complex(Float)

from ConvertibleTo(Complex(Float))

convert : % -> DoubleFloat

from ConvertibleTo(DoubleFloat)

convert : % -> Float

from ConvertibleTo(Float)

convert : % -> InputForm

from ConvertibleTo(InputForm)

definingPolynomial : % -> %

from ExpressionSpace2(Kernel(%))

denom : % -> SparseMultivariatePolynomial(Integer, Kernel(%))

denom(f) returns the denominator of f viewed as a polynomial in the kernels over Z.

differentiate : % -> %

from DifferentialRing

differentiate : (%, NonNegativeInteger) -> %

from DifferentialRing

distribute : % -> %

from ExpressionSpace2(Kernel(%))

distribute : (%, %) -> %

from ExpressionSpace2(Kernel(%))

divide : (%, %) -> Record(quotient : %, remainder : %)

from EuclideanDomain

elt : (BasicOperator, %) -> %

from ExpressionSpace2(Kernel(%))

elt : (BasicOperator, %, %) -> %

from ExpressionSpace2(Kernel(%))

elt : (BasicOperator, %, %, %) -> %

from ExpressionSpace2(Kernel(%))

elt : (BasicOperator, %, %, %, %) -> %

from ExpressionSpace2(Kernel(%))

elt : (BasicOperator, %, %, %, %, %) -> %

from ExpressionSpace2(Kernel(%))

elt : (BasicOperator, %, %, %, %, %, %) -> %

from ExpressionSpace2(Kernel(%))

elt : (BasicOperator, %, %, %, %, %, %, %) -> %

from ExpressionSpace2(Kernel(%))

elt : (BasicOperator, %, %, %, %, %, %, %, %) -> %

from ExpressionSpace2(Kernel(%))

elt : (BasicOperator, %, %, %, %, %, %, %, %, %) -> %

from ExpressionSpace2(Kernel(%))

elt : (BasicOperator, List(%)) -> %

from ExpressionSpace2(Kernel(%))

euclideanSize : % -> NonNegativeInteger

from EuclideanDomain

eval : (%, %, %) -> %

from InnerEvalable(%, %)

eval : (%, BasicOperator, Mapping(%, %)) -> %

from ExpressionSpace2(Kernel(%))

eval : (%, BasicOperator, Mapping(%, List(%))) -> %

from ExpressionSpace2(Kernel(%))

eval : (%, Equation(%)) -> %

from Evalable(%)

eval : (%, Kernel(%), %) -> %

from InnerEvalable(Kernel(%), %)

eval : (%, List(%), List(%)) -> %

from InnerEvalable(%, %)

eval : (%, List(BasicOperator), List(Mapping(%, %))) -> %

from ExpressionSpace2(Kernel(%))

eval : (%, List(BasicOperator), List(Mapping(%, List(%)))) -> %

from ExpressionSpace2(Kernel(%))

eval : (%, List(Equation(%))) -> %

from Evalable(%)

eval : (%, List(Kernel(%)), List(%)) -> %

from InnerEvalable(Kernel(%), %)

eval : (%, List(Symbol), List(Mapping(%, %))) -> %

from ExpressionSpace2(Kernel(%))

eval : (%, List(Symbol), List(Mapping(%, List(%)))) -> %

from ExpressionSpace2(Kernel(%))

eval : (%, Symbol, Mapping(%, %)) -> %

from ExpressionSpace2(Kernel(%))

eval : (%, Symbol, Mapping(%, List(%))) -> %

from ExpressionSpace2(Kernel(%))

even? : % -> Boolean

from ExpressionSpace2(Kernel(%))

expressIdealMember : (List(%), %) -> Union(List(%), "failed")

from PrincipalIdealDomain

exquo : (%, %) -> Union(%, "failed")

from EntireRing

extendedEuclidean : (%, %) -> Record(coef1 : %, coef2 : %, generator : %)

from EuclideanDomain

extendedEuclidean : (%, %, %) -> Union(Record(coef1 : %, coef2 : %), "failed")

from EuclideanDomain

factor : % -> Factored(%)

from UniqueFactorizationDomain

factorPolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%))

from PolynomialFactorizationExplicit

factorSquareFreePolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%))

from PolynomialFactorizationExplicit

freeOf? : (%, %) -> Boolean

from ExpressionSpace2(Kernel(%))

freeOf? : (%, Symbol) -> Boolean

from ExpressionSpace2(Kernel(%))

gcd : (%, %) -> %

from GcdDomain

gcd : List(%) -> %

from GcdDomain

gcdPolynomial : (SparseUnivariatePolynomial(%), SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%)

from PolynomialFactorizationExplicit

height : % -> NonNegativeInteger

from ExpressionSpace2(Kernel(%))

inv : % -> %

from DivisionRing

is? : (%, BasicOperator) -> Boolean

from ExpressionSpace2(Kernel(%))

is? : (%, Symbol) -> Boolean

from ExpressionSpace2(Kernel(%))

kernel : (BasicOperator, %) -> %

from ExpressionSpace2(Kernel(%))

kernel : (BasicOperator, List(%)) -> %

from ExpressionSpace2(Kernel(%))

kernels : % -> List(Kernel(%))

from ExpressionSpace2(Kernel(%))

kernels : List(%) -> List(Kernel(%))

from ExpressionSpace2(Kernel(%))

latex : % -> String

from SetCategory

lcm : (%, %) -> %

from GcdDomain

lcm : List(%) -> %

from GcdDomain

lcmCoef : (%, %) -> Record(llcm_res : %, coeff1 : %, coeff2 : %)

from LeftOreRing

leftPower : (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower : (%, PositiveInteger) -> %

from Magma

leftRecip : % -> Union(%, "failed")

from MagmaWithUnit

mainKernel : % -> Union(Kernel(%), "failed")

from ExpressionSpace2(Kernel(%))

map : (Mapping(%, %), Kernel(%)) -> %

from ExpressionSpace2(Kernel(%))

minPoly : Kernel(%) -> SparseUnivariatePolynomial(%)

from ExpressionSpace2(Kernel(%))

multiEuclidean : (List(%), %) -> Union(List(%), "failed")

from EuclideanDomain

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

norm : (SparseUnivariatePolynomial(%), Kernel(%)) -> SparseUnivariatePolynomial(%)

norm(p, k) computes the norm of the polynomial p with respect to the extension generated by kernel k

norm : (SparseUnivariatePolynomial(%), List(Kernel(%))) -> SparseUnivariatePolynomial(%)

norm(p, l) computes the norm of the polynomial p with respect to the extension generated by kernels l

nthRoot : (%, Integer) -> %

from RadicalCategory

numer : % -> SparseMultivariatePolynomial(Integer, Kernel(%))

numer(f) returns the numerator of f viewed as a polynomial in the kernels over Z.

odd? : % -> Boolean

from ExpressionSpace2(Kernel(%))

one? : % -> Boolean

from MagmaWithUnit

operator : BasicOperator -> BasicOperator

from ExpressionSpace2(Kernel(%))

operators : % -> List(BasicOperator)

from ExpressionSpace2(Kernel(%))

opposite? : (%, %) -> Boolean

from AbelianMonoid

paren : % -> %

from ExpressionSpace2(Kernel(%))

plenaryPower : (%, PositiveInteger) -> %

from NonAssociativeAlgebra(%)

prime? : % -> Boolean

from UniqueFactorizationDomain

principalIdeal : List(%) -> Record(coef : List(%), generator : %)

from PrincipalIdealDomain

quo : (%, %) -> %

from EuclideanDomain

recip : % -> Union(%, "failed")

from MagmaWithUnit

reduce : % -> %

reduce(f) simplifies all the unreduced algebraic numbers present in f by applying their defining relations.

reducedSystem : Matrix(%) -> Matrix(Fraction(Integer))

from LinearlyExplicitOver(Fraction(Integer))

reducedSystem : Matrix(%) -> Matrix(Integer)

from LinearlyExplicitOver(Integer)

reducedSystem : (Matrix(%), Vector(%)) -> Record(mat : Matrix(Fraction(Integer)), vec : Vector(Fraction(Integer)))

from LinearlyExplicitOver(Fraction(Integer))

reducedSystem : (Matrix(%), Vector(%)) -> Record(mat : Matrix(Integer), vec : Vector(Integer))

from LinearlyExplicitOver(Integer)

rem : (%, %) -> %

from EuclideanDomain

retract : % -> Fraction(Integer)

from RetractableTo(Fraction(Integer))

retract : % -> Integer

from RetractableTo(Integer)

retract : % -> Kernel(%)

from RetractableTo(Kernel(%))

retractIfCan : % -> Union(Fraction(Integer), "failed")

from RetractableTo(Fraction(Integer))

retractIfCan : % -> Union(Integer, "failed")

from RetractableTo(Integer)

retractIfCan : % -> Union(Kernel(%), "failed")

from RetractableTo(Kernel(%))

rightPower : (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower : (%, PositiveInteger) -> %

from Magma

rightRecip : % -> Union(%, "failed")

from MagmaWithUnit

rootOf : Polynomial(%) -> %

from AlgebraicallyClosedField

rootOf : SparseUnivariatePolynomial(%) -> %

from AlgebraicallyClosedField

rootOf : (SparseUnivariatePolynomial(%), Symbol) -> %

from AlgebraicallyClosedField

rootsOf : Polynomial(%) -> List(%)

from AlgebraicallyClosedField

rootsOf : SparseUnivariatePolynomial(%) -> List(%)

from AlgebraicallyClosedField

rootsOf : (SparseUnivariatePolynomial(%), Symbol) -> List(%)

from AlgebraicallyClosedField

sample : () -> %

from AbelianMonoid

sizeLess? : (%, %) -> Boolean

from EuclideanDomain

smaller? : (%, %) -> Boolean

from Comparable

solveLinearPolynomialEquation : (List(SparseUnivariatePolynomial(%)), SparseUnivariatePolynomial(%)) -> Union(List(SparseUnivariatePolynomial(%)), "failed")

from PolynomialFactorizationExplicit

sqrt : % -> %

from RadicalCategory

squareFree : % -> Factored(%)

from UniqueFactorizationDomain

squareFreePart : % -> %

from UniqueFactorizationDomain

squareFreePolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%))

from PolynomialFactorizationExplicit

subst : (%, Equation(%)) -> %

from ExpressionSpace2(Kernel(%))

subst : (%, List(Equation(%))) -> %

from ExpressionSpace2(Kernel(%))

subst : (%, List(Kernel(%)), List(%)) -> %

from ExpressionSpace2(Kernel(%))

subtractIfCan : (%, %) -> Union(%, "failed")

from CancellationAbelianMonoid

tower : % -> List(Kernel(%))

from ExpressionSpace2(Kernel(%))

tower : List(%) -> List(Kernel(%))

from ExpressionSpace2(Kernel(%))

trueEqual : (%, %) -> Boolean

trueEqual(x, y) tries to determine if the two numbers are equal

unit? : % -> Boolean

from EntireRing

unitCanonical : % -> %

from EntireRing

unitNormal : % -> Record(unit : %, canonical : %, associate : %)

from EntireRing

zero? : % -> Boolean

from AbelianMonoid

zeroOf : Polynomial(%) -> %

from AlgebraicallyClosedField

zeroOf : SparseUnivariatePolynomial(%) -> %

from AlgebraicallyClosedField

zeroOf : (SparseUnivariatePolynomial(%), Symbol) -> %

from AlgebraicallyClosedField

zerosOf : Polynomial(%) -> List(%)

from AlgebraicallyClosedField

zerosOf : SparseUnivariatePolynomial(%) -> List(%)

from AlgebraicallyClosedField

zerosOf : (SparseUnivariatePolynomial(%), Symbol) -> List(%)

from AlgebraicallyClosedField

~= : (%, %) -> Boolean

from BasicType

Algebra(Fraction(Integer))

Module(Fraction(Integer))

ConvertibleTo(Float)

PrincipalIdealDomain

NonAssociativeSemiRing

BiModule(%, %)

ConvertibleTo(InputForm)

Field

canonicalUnitNormal

Rng

CoercibleFrom(Integer)

TwoSidedRecip

SemiRing

EntireRing

NonAssociativeAlgebra(Fraction(Integer))

unitsKnown

ExpressionSpace

RightModule(Integer)

noZeroDivisors

RetractableTo(Fraction(Integer))

Magma

InnerEvalable(%, %)

RightModule(Fraction(Integer))

GcdDomain

IntegralDomain

LeftModule(%)

NonAssociativeRing

UniqueFactorizationDomain

ExpressionSpace2(Kernel(%))

CharacteristicZero

RetractableTo(Kernel(%))

LinearlyExplicitOver(Fraction(Integer))

Algebra(%)

CommutativeRing

DifferentialRing

RadicalCategory

DivisionRing

canonicalsClosed

CancellationAbelianMonoid

EuclideanDomain

Comparable

RetractableTo(Integer)

AlgebraicallyClosedField

CommutativeStar

AbelianMonoid

MagmaWithUnit

NonAssociativeSemiRng

RightModule(%)

SemiGroup

RealConstant

CoercibleFrom(Kernel(%))

ConvertibleTo(DoubleFloat)

Module(%)

CoercibleTo(OutputForm)

LinearlyExplicitOver(Integer)

SemiRng

Monoid

PolynomialFactorizationExplicit

LeftOreRing

NonAssociativeAlgebra(%)

ConvertibleTo(Complex(Float))

BasicType

Ring

InnerEvalable(Kernel(%), %)

LeftModule(Fraction(Integer))

AbelianSemiGroup

SetCategory

CoercibleFrom(Fraction(Integer))

NonAssociativeRng

Evalable(%)

BiModule(Fraction(Integer), Fraction(Integer))

AbelianGroup