AlgebraicallyClosedField

algfunc.spad line 1 [edit on github]

Model for algebraically closed fields.

* : (%, %) -> %

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

0 : () -> %

from AbelianMonoid

1 : () -> %

from MagmaWithUnit

= : (%, %) -> Boolean

from BasicType

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

from RadicalCategory

^ : (%, Integer) -> %

from DivisionRing

^ : (%, NonNegativeInteger) -> %

from MagmaWithUnit

^ : (%, PositiveInteger) -> %

from Magma

annihilate? : (%, %) -> Boolean

from Rng

antiCommutator : (%, %) -> %

from NonAssociativeSemiRng

associates? : (%, %) -> Boolean

from EntireRing

associator : (%, %, %) -> %

from NonAssociativeRng

characteristic : () -> NonNegativeInteger

from NonAssociativeRing

coerce : % -> %

from Algebra(%)

coerce : Fraction(Integer) -> %

from Algebra(Fraction(Integer))

coerce : Integer -> %

from NonAssociativeRing

coerce : % -> OutputForm

from CoercibleTo(OutputForm)

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

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

plenaryPower : (%, PositiveInteger) -> %

from NonAssociativeAlgebra(%)

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 : 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.

rootOf : (SparseUnivariatePolynomial(%), Symbol) -> %

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

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.

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.

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 : 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.

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.

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.

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.

~= : (%, %) -> Boolean

from BasicType

IntegralDomain

Module(Fraction(Integer))

noZeroDivisors

LeftModule(Fraction(Integer))

canonicalsClosed

Algebra(%)

Monoid

GcdDomain

AbelianMonoid

EuclideanDomain

EntireRing

NonAssociativeSemiRng

NonAssociativeAlgebra(Fraction(Integer))

CancellationAbelianMonoid

MagmaWithUnit

NonAssociativeRing

RightModule(Fraction(Integer))

unitsKnown

TwoSidedRecip

LeftModule(%)

canonicalUnitNormal

RadicalCategory

Module(%)

Magma

SetCategory

LeftOreRing

CoercibleTo(OutputForm)

Algebra(Fraction(Integer))

Rng

Field

CommutativeRing

UniqueFactorizationDomain

SemiGroup

DivisionRing

BiModule(%, %)

AbelianGroup

AbelianSemiGroup

CommutativeStar

NonAssociativeSemiRing

RightModule(%)

NonAssociativeAlgebra(%)

PrincipalIdealDomain

BiModule(Fraction(Integer), Fraction(Integer))

NonAssociativeRng

Ring

SemiRng

BasicType

SemiRing