ModularField(R, Mod, reduction, merge, exactQuo)

modring.spad line 187 [edit on github]

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

* : (%, %) -> %
from Magma
* : (%, Fraction(Integer)) -> %
from RightModule(Fraction(Integer))
* : (Fraction(Integer), %) -> %
from LeftModule(Fraction(Integer))
* : (Integer, %) -> %
from AbelianGroup
* : (NonNegativeInteger, %) -> %
from AbelianMonoid
* : (PositiveInteger, %) -> %
from AbelianSemiGroup
+ : (%, %) -> %
from AbelianSemiGroup
- : % -> %
from AbelianGroup
- : (%, %) -> %
from AbelianGroup
/ : (%, %) -> %
from Field
0 : () -> %
from AbelianMonoid
1 : () -> %
from MagmaWithUnit
= : (%, %) -> Boolean
from BasicType
^ : (%, Integer) -> %
from DivisionRing
^ : (%, 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 : Fraction(Integer) -> %
from Algebra(Fraction(Integer))
coerce : Integer -> %
from NonAssociativeRing
coerce : % -> R

coerce(x) is undocumented

coerce : % -> OutputForm
from CoercibleTo(OutputForm)
commutator : (%, %) -> %
from NonAssociativeRng
divide : (%, %) -> Record(quotient : %, remainder : %)
from EuclideanDomain
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
factor : % -> Factored(%)
from UniqueFactorizationDomain
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
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(%)
prime? : % -> Boolean
from UniqueFactorizationDomain
principalIdeal : List(%) -> Record(coef : List(%), generator : %)
from PrincipalIdealDomain
quo : (%, %) -> %
from EuclideanDomain
recip : % -> Union(%, "failed")
from MagmaWithUnit
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
squareFree : % -> Factored(%)
from UniqueFactorizationDomain
squareFreePart : % -> %
from UniqueFactorizationDomain
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

Module(Fraction(Integer))

LeftModule(Fraction(Integer))

noZeroDivisors

RightModule(%)

Monoid

GcdDomain

AbelianMonoid

Algebra(%)

EuclideanDomain

EntireRing

NonAssociativeSemiRng

NonAssociativeAlgebra(Fraction(Integer))

CancellationAbelianMonoid

MagmaWithUnit

NonAssociativeRing

RightModule(Fraction(Integer))

unitsKnown

LeftModule(%)

canonicalUnitNormal

CommutativeStar

Module(%)

SetCategory

LeftOreRing

CoercibleTo(OutputForm)

Algebra(Fraction(Integer))

Rng

Field

CommutativeRing

TwoSidedRecip

Magma

UniqueFactorizationDomain

SemiGroup

DivisionRing

BiModule(%, %)

AbelianGroup

AbelianSemiGroup

NonAssociativeSemiRing

canonicalsClosed

NonAssociativeAlgebra(%)

PrincipalIdealDomain

BiModule(Fraction(Integer), Fraction(Integer))

NonAssociativeRng

Ring

SemiRng

BasicType

SemiRing