# PolynomialFactorizationExplicit¶

This is the category of domains that know “enough” about themselves in order to factor univariate polynomials over their fraction field.

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from LeftModule %

*: (Integer, %) -> %

from AbelianGroup

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

+: (%, %) -> %

from AbelianSemiGroup

-: % -> %

from AbelianGroup

-: (%, %) -> %

from AbelianGroup

=: (%, %) -> Boolean

from BasicType

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %
associates?: (%, %) -> Boolean

from EntireRing

associator: (%, %, %) -> %
characteristic: () -> NonNegativeInteger
charthRoot: % -> Union(%, failed) if % has CharacteristicNonZero

`charthRoot(r)` returns the `p``-`th root of `r`, or “failed” if none exists in the domain.

coerce: % -> %

from Algebra %

coerce: % -> OutputForm
coerce: Integer -> %
commutator: (%, %) -> %
conditionP: Matrix % -> Union(Vector %, failed) if % has CharacteristicNonZero

`conditionP(m)` returns a vector of elements, not all zero, whose `p``-`th powers (`p` is the characteristic of the domain) are a solution of the homogenous linear system represented by `m`, or “failed” is there is no such vector.

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

from EntireRing

factor: % -> Factored %
factorPolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial %

`factorPolynomial(p)` returns the factorization into irreducibles of the univariate polynomial `p`.

factorSquareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial %

`factorSquareFreePolynomial(p)` factors the univariate polynomial `p` into irreducibles where `p` is known to be square free and primitive with respect to its main variable.

gcd: (%, %) -> %

from GcdDomain

gcd: List % -> %

from GcdDomain

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

`gcdPolynomial(p, q)` returns the `gcd` of the univariate polynomials `p` `qnd` `q`.

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

one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

plenaryPower: (%, PositiveInteger) -> %
prime?: % -> Boolean
recip: % -> Union(%, failed)

from MagmaWithUnit

rightPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed)

from MagmaWithUnit

sample: %

from AbelianMonoid

solveLinearPolynomialEquation: (List SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> Union(List SparseUnivariatePolynomial %, failed)

`solveLinearPolynomialEquation([f1, ..., fn], g)` (where the `fi` are relatively prime to each other) returns a list of `ai` such that `g/prod fi = sum ai/fi` or returns “failed” if no such list of `ai``'s` exists.

squareFree: % -> Factored %
squareFreePart: % -> %
squareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial %

`squareFreePolynomial(p)` returns the square-free factorization of the univariate polynomial `p`.

subtractIfCan: (%, %) -> Union(%, failed)
unit?: % -> Boolean

from EntireRing

unitCanonical: % -> %

from EntireRing

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

from EntireRing

zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

BasicType

BiModule(%, %)

CancellationAbelianMonoid

CommutativeRing

CommutativeStar

EntireRing

GcdDomain

IntegralDomain

LeftOreRing

Magma

MagmaWithUnit

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

TwoSidedRecip

UniqueFactorizationDomain

unitsKnown