EuclideanModularRing(S, R, Mod, reduction, merge, exactQuo)

S : CommutativeRing
R : UnivariatePolynomialCategory(S)
Mod : AbelianMonoid
reduction : Mapping(R, R, Mod)
merge : Mapping(Union(Mod, "failed"), Mod, Mod)
exactQuo : Mapping(Union(R, "failed"), R, R, Mod)

These domains are used for the factorization and gcds of univariate polynomials over the integers in order to work modulo different primes. See ModularRing, ModularField

* : (%, %) -> %: 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 : % -> R: coerce(x) is undocumented

coerce : % -> OutputForm: from CoercibleTo(OutputForm)

commutator : (%, %) -> %: from NonAssociativeRng

divide : (%, %) -> Record(quotient : %, remainder : %): from EuclideanDomain

elt : (%, R) -> R: elt(x, r) or x.r is undocumented

euclideanSize : % -> NonNegativeInteger: from EuclideanDomain

exQuo : (%, %) -> Union(%, "failed"): exQuo(x, y) is undocumented

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

gcd : (%, %) -> %: from GcdDomain

gcd : List(%) -> %: from GcdDomain

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

inv : % -> %: inv(x) is undocumented

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

modulus : % -> Mod: modulus(x) is undocumented

multiEuclidean : (List(%), %) -> Union(List(%), "failed"): from EuclideanDomain

one? : % -> Boolean: from MagmaWithUnit

opposite? : (%, %) -> Boolean: from AbelianMonoid

plenaryPower : (%, PositiveInteger) -> %: from NonAssociativeAlgebra(%)

principalIdeal : List(%) -> Record(coef : List(%), generator : %): from PrincipalIdealDomain

quo : (%, %) -> %: from EuclideanDomain

recip : % -> Union(%, "failed"): recip(x) is undocumented

reduce : (R, Mod) -> %: reduce(r, m) is undocumented

rem : (%, %) -> %: from EuclideanDomain

rightPower : (%, NonNegativeInteger) -> %: from MagmaWithUnit

rightPower : (%, PositiveInteger) -> %: from Magma

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

sample : () -> %: from AbelianMonoid

sizeLess? : (%, %) -> Boolean: from EuclideanDomain

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

RightModule(%)

Algebra(%)

CancellationAbelianMonoid

LeftModule(%)

Module(%)

BiModule(%, %)

CoercibleTo(OutputForm)

AbelianSemiGroup

NonAssociativeSemiRing

NonAssociativeAlgebra(%)