GcdDomain

catdef.spad line 689 [edit on github]

This category describes domains where gcd can be computed but where there is no guarantee of the existence of factor operation for factorization into irreducibles. However, if such a factor operation exist, factorization will be unique up to order and units.

* : (%, %) -> %

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

gcd : (%, %) -> %

gcd(x, y) returns the greatest common divisor of x and y.

gcd : List(%) -> %

gcd(l) returns the common gcd of the elements in the list l.

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

gcdPolynomial(p, q) returns the greatest common divisor (gcd) of univariate polynomials over the domain

latex : % -> String

from SetCategory

lcm : (%, %) -> %

lcm(x, y) returns the least common multiple of x and y.

lcm : List(%) -> %

lcm(l) returns the least common multiple of the elements of the list l.

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(%)

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

from MagmaWithUnit

rightPower : (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower : (%, PositiveInteger) -> %

from Magma

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

from MagmaWithUnit

sample : () -> %

from AbelianMonoid

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

RightModule(%)

Monoid

AbelianMonoid

Algebra(%)

CancellationAbelianMonoid

MagmaWithUnit

NonAssociativeRing

AbelianGroup

LeftModule(%)

CommutativeStar

Module(%)

SetCategory

LeftOreRing

Rng

CommutativeRing

TwoSidedRecip

Magma

SemiGroup

BiModule(%, %)

unitsKnown

CoercibleTo(OutputForm)

AbelianSemiGroup

NonAssociativeSemiRing

NonAssociativeAlgebra(%)

NonAssociativeRng

Ring

SemiRng

EntireRing

NonAssociativeSemiRng

BasicType

SemiRing