AlgebraicallyClosedField

algfunc.spad line 1 [edit on github]

Model for algebraically closed fields.

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

characteristic: () -> NonNegativeInteger

from NonAssociativeRing

coerce: % -> %

from Algebra %

coerce: % -> OutputForm

from CoercibleTo OutputForm

coerce: Fraction Integer -> %

from Algebra Fraction Integer

coerce: Integer -> %

from NonAssociativeRing

commutator: (%, %) -> %

from NonAssociativeRng

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

from EuclideanDomain

euclideanSize: % -> NonNegativeInteger

from EuclideanDomain

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

gcd: (%, %) -> %

from GcdDomain

gcd: List % -> %

from GcdDomain

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

from GcdDomain

hash: % -> SingleInteger

from SetCategory

hashUpdate!: (HashState, %) -> HashState

from SetCategory

inv: % -> %

from DivisionRing

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

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

from EuclideanDomain

nthRoot: (%, Integer) -> %

from RadicalCategory

one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

prime?: % -> Boolean

from UniqueFactorizationDomain

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

from PrincipalIdealDomain

quo: (%, %) -> %

from EuclideanDomain

recip: % -> Union(%, failed)

from MagmaWithUnit

rem: (%, %) -> %

from EuclideanDomain

rightPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed)

from MagmaWithUnit

rootOf: (SparseUnivariatePolynomial %, Symbol) -> %

rootOf(p, y) returns y such that p(y) = 0. The object returned displays as 'y.

rootOf: Polynomial % -> %

rootOf(p) returns y such that p(y) = 0. Error: if p has more than one variable y.

rootOf: SparseUnivariatePolynomial % -> %

rootOf(p) returns y such that p(y) = 0.

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

rootsOf(p, z) returns [y1, ..., yn] such that p(yi) = 0; The returned roots contain new symbols '\%z0, '\%z1 …; Note: the new symbols are bound in the interpreter to the respective values.

rootsOf: Polynomial % -> List %

rootsOf(p) returns [y1, ..., yn] such that p(yi) = 0. Note: the returned values y1, …, yn contain new symbols which are bound in the interpreter to the respective values. Error: if p has more than one variable y.

rootsOf: SparseUnivariatePolynomial % -> List %

rootsOf(p) returns [y1, ..., yn] such that p(yi) = 0. Note: the returned values y1, …, yn contain new symbols which are bound in the interpreter to the respective values.

sample: %

from AbelianMonoid

sizeLess?: (%, %) -> Boolean

from EuclideanDomain

sqrt: % -> %

from RadicalCategory

squareFree: % -> Factored %

from UniqueFactorizationDomain

squareFreePart: % -> %

from UniqueFactorizationDomain

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

from CancellationAbelianMonoid

unit?: % -> Boolean

from EntireRing

unitCanonical: % -> %

from EntireRing

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

from EntireRing

zero?: % -> Boolean

from AbelianMonoid

zeroOf: (SparseUnivariatePolynomial %, Symbol) -> %

zeroOf(p, y) returns y such that p(y) = 0; if possible, y is expressed in terms of radicals. Otherwise it is an implicit algebraic quantity which displays as 'y.

zeroOf: Polynomial % -> %

zeroOf(p) returns y such that p(y) = 0. If possible, y is expressed in terms of radicals. Otherwise it is an implicit algebraic quantity. Error: if p has more than one variable y.

zeroOf: SparseUnivariatePolynomial % -> %

zeroOf(p) returns y such that p(y) = 0; if possible, y is expressed in terms of radicals. Otherwise it is an implicit algebraic quantity.

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

zerosOf(p, y) returns [y1, ..., yn] such that p(yi) = 0. The yi's are expressed in radicals if possible, and otherwise as implicit algebraic quantities containing new symbols which display as '\%z0, '\%z1, …; The new symbols are bound in the interpreter to respective values.

zerosOf: Polynomial % -> List %

zerosOf(p) returns [y1, ..., yn] such that p(yi) = 0. The yi's are expressed in radicals if possible. Otherwise they are implicit algebraic quantities containing new symbols. The new symbols are bound in the interpreter to the respective values. Error: if p has more than one variable y.

zerosOf: SparseUnivariatePolynomial % -> List %

zerosOf(p) returns [y1, ..., yn] such that p(yi) = 0. The yi's are expressed in radicals if possible. Otherwise they are implicit algebraic quantities containing new symbols. The new symbols are bound in the interpreter to the respective values.

AbelianGroup

AbelianMonoid

AbelianSemiGroup

Algebra %

Algebra Fraction Integer

BasicType

BiModule(%, %)

BiModule(Fraction Integer, Fraction Integer)

CancellationAbelianMonoid

canonicalsClosed

canonicalUnitNormal

CoercibleTo OutputForm

CommutativeRing

CommutativeStar

DivisionRing

EntireRing

EuclideanDomain

Field

GcdDomain

IntegralDomain

LeftModule %

LeftModule Fraction Integer

LeftOreRing

Magma

MagmaWithUnit

Module %

Module Fraction Integer

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors

PrincipalIdealDomain

RadicalCategory

RightModule %

RightModule Fraction Integer

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

TwoSidedRecip

UniqueFactorizationDomain

unitsKnown