# AlgebraicallyClosedField¶

Model for algebraically closed fields.

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from LeftModule %

*: (%, Fraction 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: (%, %, %) -> %
characteristic: () -> NonNegativeInteger
coerce: % -> %

from Algebra %

coerce: % -> OutputForm
coerce: Fraction Integer -> %
coerce: Integer -> %
commutator: (%, %) -> %
divide: (%, %) -> Record(quotient: %, remainder: %)

from EuclideanDomain

euclideanSize: % -> NonNegativeInteger

from EuclideanDomain

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 %
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) -> %

one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

plenaryPower: (%, PositiveInteger) -> %
prime?: % -> Boolean
principalIdeal: List % -> Record(coef: List %, generator: %)
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: % -> %

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)` 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

BasicType

BiModule(%, %)

CancellationAbelianMonoid

canonicalsClosed

canonicalUnitNormal

CommutativeRing

CommutativeStar

DivisionRing

EntireRing

EuclideanDomain

Field

GcdDomain

IntegralDomain

LeftOreRing

Magma

MagmaWithUnit

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors

PrincipalIdealDomain

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

TwoSidedRecip

UniqueFactorizationDomain

unitsKnown