# AlgebraicNumberΒΆ

Algebraic closure of the rational numbers.

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from LeftModule %

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

from AbelianGroup

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

+: (%, %) -> %

from AbelianSemiGroup

-: % -> %

from AbelianGroup

-: (%, %) -> %

from AbelianGroup

/: (%, %) -> %

from Field

=: (%, %) -> Boolean

from BasicType

^: (%, Fraction Integer) -> %

^: (%, Integer) -> %

from DivisionRing

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %
associates?: (%, %) -> Boolean

from EntireRing

associator: (%, %, %) -> %
belong?: BasicOperator -> Boolean

from ExpressionSpace

box: % -> %

from ExpressionSpace

characteristic: () -> NonNegativeInteger
coerce: % -> %

from Algebra %

coerce: % -> OutputForm
coerce: Fraction Integer -> %
coerce: Integer -> %
coerce: Kernel % -> %

from CoercibleFrom Kernel %

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

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

commutator: (%, %) -> %
convert: % -> Complex Float
convert: % -> DoubleFloat
convert: % -> Float
convert: % -> 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)
exquo: (%, %) -> Union(%, failed)

from EntireRing

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

from EuclideanDomain

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

from EuclideanDomain

factor: % -> Factored %
factorPolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial %
factorSquareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial %
freeOf?: (%, %) -> Boolean

from ExpressionSpace

freeOf?: (%, Symbol) -> Boolean

from ExpressionSpace

gcd: (%, %) -> %

from GcdDomain

gcd: List % -> %

from GcdDomain

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

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

plenaryPower: (%, PositiveInteger) -> %
prime?: % -> Boolean
principalIdeal: List % -> Record(coef: List %, generator: %)
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)
reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix Integer, vec: Vector Integer)
reducedSystem: Matrix % -> Matrix Fraction Integer
reducedSystem: Matrix % -> Matrix Integer
rem: (%, %) -> %

from EuclideanDomain

retract: % -> Fraction Integer
retract: % -> Integer
retract: % -> Kernel %

from RetractableTo Kernel %

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

from RetractableTo Kernel %

rightPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed)

from MagmaWithUnit

rootOf: (SparseUnivariatePolynomial %, Symbol) -> %
rootOf: Polynomial % -> %
rootOf: SparseUnivariatePolynomial % -> %
rootsOf: (SparseUnivariatePolynomial %, Symbol) -> List %
rootsOf: Polynomial % -> List %
rootsOf: SparseUnivariatePolynomial % -> List %
sample: %

from AbelianMonoid

sizeLess?: (%, %) -> Boolean

from EuclideanDomain

smaller?: (%, %) -> Boolean

from Comparable

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

squareFree: % -> Factored %
squareFreePart: % -> %
squareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial %
subst: (%, Equation %) -> %

from ExpressionSpace

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

from ExpressionSpace

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

from ExpressionSpace

subtractIfCan: (%, %) -> Union(%, failed)
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) -> %
zeroOf: Polynomial % -> %
zeroOf: SparseUnivariatePolynomial % -> %
zerosOf: (SparseUnivariatePolynomial %, Symbol) -> List %
zerosOf: Polynomial % -> List %
zerosOf: SparseUnivariatePolynomial % -> List %

AbelianGroup

AbelianMonoid

AbelianSemiGroup

AlgebraicallyClosedField

BasicType

BiModule(%, %)

CancellationAbelianMonoid

canonicalsClosed

canonicalUnitNormal

CharacteristicZero

CommutativeRing

CommutativeStar

Comparable

DifferentialRing

DivisionRing

EntireRing

EuclideanDomain

ExpressionSpace

Field

GcdDomain

InnerEvalable(%, %)

InnerEvalable(Kernel %, %)

IntegralDomain

LeftOreRing

Magma

MagmaWithUnit

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors

PolynomialFactorizationExplicit

PrincipalIdealDomain

RealConstant

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

TwoSidedRecip

UniqueFactorizationDomain

unitsKnown