UniqueFactorizationDomain

catdef.spad line 1504 [edit on github]

A constructive unique factorization domain, i.e. where we can constructively factor members into a product of a finite number of irreducible elements.

* : (%, %) -> %

from Magma

* : (Integer, %) -> %

from AbelianGroup

* : (NonNegativeInteger, %) -> %

from AbelianMonoid

* : (PositiveInteger, %) -> %

from AbelianSemiGroup

+ : (%, %) -> %

from AbelianSemiGroup

- : % -> %

from AbelianGroup

- : (%, %) -> %

from AbelianGroup

0 : () -> %

from AbelianMonoid

1 : () -> %

from MagmaWithUnit

= : (%, %) -> Boolean

from BasicType

^ : (%, 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 : Integer -> %

from NonAssociativeRing

coerce : % -> OutputForm

from CoercibleTo(OutputForm)

commutator : (%, %) -> %

from NonAssociativeRng

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

from EntireRing

factor : % -> Factored(%)

factor(x) returns the factorization of x into irreducibles.

gcd : (%, %) -> %

from GcdDomain

gcd : List(%) -> %

from GcdDomain

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

from GcdDomain

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) -> %

from NonAssociativeAlgebra(%)

prime? : % -> Boolean

prime?(x) tests if x can never be written as the product of two non-units of the ring, i.e. x is an irreducible element.

recip : % -> Union(%, "failed")

from MagmaWithUnit

rightPower : (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower : (%, PositiveInteger) -> %

from Magma

rightRecip : % -> Union(%, "failed")

from MagmaWithUnit

sample : () -> %

from AbelianMonoid

squareFree : % -> Factored(%)

squareFree(x) returns the square-free factorization of x i.e. such that the factors are pairwise relatively prime and each has multiple prime factors.

squareFreePart : % -> %

squareFreePart(x) returns a product of prime factors of x each taken with multiplicity one.

subtractIfCan : (%, %) -> Union(%, "failed")

from CancellationAbelianMonoid

unit? : % -> Boolean

from EntireRing

unitCanonical : % -> %

from EntireRing

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

from EntireRing

zero? : % -> Boolean

from AbelianMonoid

~= : (%, %) -> Boolean

from BasicType

IntegralDomain

noZeroDivisors

NonAssociativeSemiRng

Algebra(%)

RightModule(%)

Monoid

GcdDomain

AbelianMonoid

CancellationAbelianMonoid

MagmaWithUnit

NonAssociativeRing

LeftModule(%)

CommutativeStar

Module(%)

SetCategory

LeftOreRing

CoercibleTo(OutputForm)

Rng

CommutativeRing

TwoSidedRecip

Magma

SemiGroup

BiModule(%, %)

AbelianGroup

AbelianSemiGroup

NonAssociativeSemiRing

NonAssociativeAlgebra(%)

NonAssociativeRng

unitsKnown

Ring

SemiRng

EntireRing

BasicType

SemiRing