AlgebraicNumberΒΆ

constant.spad line 1

Algebraic closure of the rational numbers.

0: %
from AbelianMonoid
1: %
from MagmaWithUnit
*: (%, %) -> %
from Magma
*: (%, 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 Algebra Fraction Integer
coerce: Integer -> %
from NonAssociativeRing
coerce: Kernel % -> %
from RetractableTo 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 GcdDomain
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

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

UniqueFactorizationDomain

unitsKnown