AlgebraicNumberΒΆ

constant.spad line 1 [edit on github]

Algebraic closure of the rational numbers.

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from LeftModule %

*: (%, Fraction Integer) -> %

from RightModule Fraction Integer

*: (Fraction Integer, %) -> %

from LeftModule Fraction Integer

*: (Integer, %) -> %

from AbelianGroup

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

+: (%, %) -> %

from AbelianSemiGroup

-: % -> %

from AbelianGroup

-: (%, %) -> %

from AbelianGroup

/: (%, %) -> %

from Field

=: (%, %) -> Boolean

from BasicType

^: (%, Fraction Integer) -> %

from RadicalCategory

^: (%, Integer) -> %

from DivisionRing

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %

from NonAssociativeSemiRng

associates?: (%, %) -> Boolean

from EntireRing

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

from NonAssociativeRng

belong?: BasicOperator -> Boolean

from ExpressionSpace

box: % -> %

from ExpressionSpace

characteristic: () -> NonNegativeInteger

from NonAssociativeRing

coerce: % -> %

from Algebra %

coerce: % -> OutputForm

from CoercibleTo OutputForm

coerce: Fraction Integer -> %

from CoercibleFrom Fraction Integer

coerce: Integer -> %

from NonAssociativeRing

coerce: Kernel % -> %

from CoercibleFrom Kernel %

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

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

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

D: % -> %

from DifferentialRing

D: (%, NonNegativeInteger) -> %

from DifferentialRing

definingPolynomial: % -> %

from ExpressionSpace

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 ExpressionSpace

distribute: (%, %) -> %

from ExpressionSpace

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

from EuclideanDomain

elt: (BasicOperator, %) -> %

from ExpressionSpace

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

from ExpressionSpace

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

from ExpressionSpace

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

from ExpressionSpace

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

from ExpressionSpace

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

from ExpressionSpace

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

from ExpressionSpace

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

from ExpressionSpace

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

from ExpressionSpace

elt: (BasicOperator, List %) -> %

from ExpressionSpace

euclideanSize: % -> NonNegativeInteger

from EuclideanDomain

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

from InnerEvalable(%, %)

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

from ExpressionSpace

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

from ExpressionSpace

eval: (%, Equation %) -> %

from Evalable %

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

from InnerEvalable(Kernel %, %)

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

from InnerEvalable(%, %)

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

from ExpressionSpace

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

from ExpressionSpace

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

from Evalable %

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

from InnerEvalable(Kernel %, %)

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

from ExpressionSpace

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

from ExpressionSpace

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

from ExpressionSpace

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

from ExpressionSpace

even?: % -> Boolean

from ExpressionSpace

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 ExpressionSpace

freeOf?: (%, Symbol) -> Boolean

from ExpressionSpace

gcd: (%, %) -> %

from GcdDomain

gcd: List % -> %

from GcdDomain

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

from PolynomialFactorizationExplicit

hash: % -> SingleInteger

from SetCategory

hashUpdate!: (HashState, %) -> HashState

from SetCategory

height: % -> NonNegativeInteger

from ExpressionSpace

inv: % -> %

from DivisionRing

is?: (%, BasicOperator) -> Boolean

from ExpressionSpace

is?: (%, Symbol) -> Boolean

from ExpressionSpace

kernel: (BasicOperator, %) -> %

from ExpressionSpace

kernel: (BasicOperator, List %) -> %

from ExpressionSpace

kernels: % -> List Kernel %

from ExpressionSpace

kernels: List % -> List Kernel %

from ExpressionSpace

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 ExpressionSpace

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

from ExpressionSpace

minPoly: Kernel % -> SparseUnivariatePolynomial %

from ExpressionSpace

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 ExpressionSpace

one?: % -> Boolean

from MagmaWithUnit

operator: BasicOperator -> BasicOperator

from ExpressionSpace

operators: % -> List BasicOperator

from ExpressionSpace

opposite?: (%, %) -> Boolean

from AbelianMonoid

paren: % -> %

from ExpressionSpace

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 %, 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

reducedSystem: Matrix % -> Matrix Fraction Integer

from LinearlyExplicitOver Fraction Integer

reducedSystem: Matrix % -> Matrix 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: (SparseUnivariatePolynomial %, Symbol) -> %

from AlgebraicallyClosedField

rootOf: Polynomial % -> %

from AlgebraicallyClosedField

rootOf: SparseUnivariatePolynomial % -> %

from AlgebraicallyClosedField

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

from AlgebraicallyClosedField

rootsOf: Polynomial % -> List %

from AlgebraicallyClosedField

rootsOf: SparseUnivariatePolynomial % -> 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 ExpressionSpace

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

from ExpressionSpace

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

from ExpressionSpace

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

from CancellationAbelianMonoid

tower: % -> List Kernel %

from ExpressionSpace

tower: List % -> List Kernel %

from ExpressionSpace

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: (SparseUnivariatePolynomial %, Symbol) -> %

from AlgebraicallyClosedField

zeroOf: Polynomial % -> %

from AlgebraicallyClosedField

zeroOf: SparseUnivariatePolynomial % -> %

from AlgebraicallyClosedField

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

from AlgebraicallyClosedField

zerosOf: Polynomial % -> List %

from AlgebraicallyClosedField

zerosOf: SparseUnivariatePolynomial % -> List %

from AlgebraicallyClosedField

AbelianGroup

AbelianMonoid

AbelianSemiGroup

Algebra %

Algebra Fraction Integer

AlgebraicallyClosedField

BasicType

BiModule(%, %)

BiModule(Fraction Integer, Fraction Integer)

CancellationAbelianMonoid

canonicalsClosed

canonicalUnitNormal

CharacteristicZero

CoercibleFrom Fraction Integer

CoercibleFrom Integer

CoercibleFrom Kernel %

CoercibleTo OutputForm

CommutativeRing

CommutativeStar

Comparable

ConvertibleTo Complex Float

ConvertibleTo DoubleFloat

ConvertibleTo Float

ConvertibleTo InputForm

DifferentialRing

DivisionRing

EntireRing

EuclideanDomain

Evalable %

ExpressionSpace

Field

GcdDomain

InnerEvalable(%, %)

InnerEvalable(Kernel %, %)

IntegralDomain

LeftModule %

LeftModule Fraction Integer

LeftOreRing

LinearlyExplicitOver Fraction Integer

LinearlyExplicitOver Integer

Magma

MagmaWithUnit

Module %

Module Fraction Integer

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors

PolynomialFactorizationExplicit

PrincipalIdealDomain

RadicalCategory

RealConstant

RetractableTo Fraction Integer

RetractableTo Integer

RetractableTo Kernel %

RightModule %

RightModule Fraction Integer

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

TwoSidedRecip

UniqueFactorizationDomain

unitsKnown