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
- ^: (%, Fraction Integer) -> %
from RadicalCategory
- ^: (%, Integer) -> %
from DivisionRing
- ^: (%, NonNegativeInteger) -> %
from MagmaWithUnit
- ^: (%, PositiveInteger) -> %
from Magma
- annihilate?: (%, %) -> Boolean
from Rng
- antiCommutator: (%, %) -> %
- associates?: (%, %) -> Boolean
from EntireRing
- associator: (%, %, %) -> %
from NonAssociativeRng
- characteristic: () -> NonNegativeInteger
from NonAssociativeRing
- coerce: % -> %
from Algebra %
- coerce: % -> OutputForm
from CoercibleTo OutputForm
- coerce: 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
- gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial %
from GcdDomain
- inv: % -> %
from DivisionRing
- latex: % -> String
from SetCategory
- 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
- plenaryPower: (%, PositiveInteger) -> %
from NonAssociativeAlgebra %
- 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)returnsysuch thatp(y) = 0. The object returned displays as'y.
- rootOf: Polynomial % -> %
rootOf(p)returnsysuch thatp(y) = 0. Error: ifphas more than one variabley.
- rootOf: SparseUnivariatePolynomial % -> %
rootOf(p)returnsysuch thatp(y) = 0.
- rootsOf: (SparseUnivariatePolynomial %, Symbol) -> List %
rootsOf(p, z)returns[y1, ..., yn]such thatp(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 thatp(yi) = 0. Note: the returned valuesy1, …,yncontain new symbols which are bound in the interpreter to the respective values. Error: ifphas more than one variabley.
- rootsOf: SparseUnivariatePolynomial % -> List %
rootsOf(p)returns[y1, ..., yn]such thatp(yi) = 0. Note: the returned valuesy1, …,yncontain new symbols which are bound in the interpreter to the respective values.
- sample: %
from AbelianMonoid
- sizeLess?: (%, %) -> Boolean
from EuclideanDomain
- sqrt: % -> %
from RadicalCategory
- squareFree: % -> Factored %
- squareFreePart: % -> %
- subtractIfCan: (%, %) -> Union(%, failed)
- unit?: % -> Boolean
from EntireRing
- unitCanonical: % -> %
from EntireRing
- unitNormal: % -> Record(unit: %, canonical: %, associate: %)
from EntireRing
- zero?: % -> Boolean
from AbelianMonoid
- zeroOf: (SparseUnivariatePolynomial %, Symbol) -> %
zeroOf(p, y)returnsysuch thatp(y) = 0; if possible,yis expressed in terms of radicals. Otherwise it is an implicit algebraic quantity which displays as'y.
- zeroOf: Polynomial % -> %
zeroOf(p)returnsysuch thatp(y) = 0. If possible,yis expressed in terms of radicals. Otherwise it is an implicit algebraic quantity. Error: ifphas more than one variabley.
- zeroOf: SparseUnivariatePolynomial % -> %
zeroOf(p)returnsysuch thatp(y) = 0; if possible,yis expressed in terms of radicals. Otherwise it is an implicit algebraic quantity.
- zerosOf: (SparseUnivariatePolynomial %, Symbol) -> List %
zerosOf(p, y)returns[y1, ..., yn]such thatp(yi) = 0. Theyi'sare 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 thatp(yi) = 0. Theyi'sare 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: ifphas more than one variabley.
- zerosOf: SparseUnivariatePolynomial % -> List %
zerosOf(p)returns[y1, ..., yn]such thatp(yi) = 0. Theyi'sare 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.
Algebra %
BiModule(%, %)
BiModule(Fraction Integer, Fraction Integer)
Module %