AlgebraicallyClosedFieldΒΆ

algfunc.spad line 1

Model for algebraically closed fields.

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
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

UniqueFactorizationDomain

unitsKnown