NMResidueField(R, P, poly)

jnemo.spad line 647 [edit on github]

This domain allows the manipulation of Nemo residue field elements using the AbstractAlgebra and Nemo Julia packages. The constructor takes a univariate polynomial from which the ideal generated by this polynomial defines the quotient field. The ideal generated must be maximal, no check are done at jlFriCAS level.

0: %

from SetWithZero

1: %

from MagmaWithUnit

*: (%, %) -> %

from Magma

*: (%, Integer) -> %

x*i is the multiplication by an integer.

*: (%, NMInteger) -> %

x*i is the multiplication by a Nemo integer.

*: (Integer, %) -> %

from AbelianGroup

*: (NMInteger, %) -> JLObject

from JLObjectRing

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

+: (%, %) -> %

from AbelianSemiGroup

-: % -> %

from AbelianGroup

-: (%, %) -> %

from AbelianGroup

/: (%, %) -> %

from Field

=: (%, %) -> Boolean

from BasicType

^: (%, Integer) -> %

from DivisionRing

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %

from NonAssociativeSemiRng

associates?: (%, %) -> Boolean

from EntireRing

associator: (%, %, %) -> %

from NonAssociativeRng

characteristic: () -> NonNegativeInteger

from NonAssociativeRing

coerce: % -> %

from Algebra %

coerce: % -> JLObject

from JLObjectType

coerce: % -> OutputForm

from CoercibleTo OutputForm

coerce: Integer -> %

coerce(i) coerces i.

coerce: NMInteger -> %

coerce(i) coerces i.

coerce: P -> %

coerce(p) coerces p.

commutator: (%, %) -> %

from NonAssociativeRng

convert: % -> String

from ConvertibleTo String

data: % -> P

data(x) returns the data of x. See https://nemocas.github.io/AbstractAlgebra.jl/stable/residue_interface/ for more information for the difference with lift.

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

from EuclideanDomain

equal?: (%, %) -> Boolean

from NMRing

euclideanSize: % -> NonNegativeInteger

from EuclideanDomain

exact?: % -> Boolean

from NMRing

exactDivide: (%, %) -> %

from NMRing

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

extendedGcd: (%, %) -> Record(g: %, s: %, t: %)

extendedGcd(x, y) returns the extended gcd of x and y.

factor: % -> Factored %

from UniqueFactorizationDomain

gcd: (%, %) -> %

gcd(x, y) returns a greatest common divisor of x and y using the data and modulus associated.

gcd: List % -> %

from GcdDomain

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

from GcdDomain

inv: % -> %

from DivisionRing

irreducible?: % -> Boolean

from NMRing

jlAbout: % -> Void

from JLObjectType

jlApply: (String, %) -> JLObject

from JLObjectType

jlApply: (String, %, %) -> JLObject

from JLObjectType

jlApply: (String, %, %, %) -> JLObject

from JLObjectType

jlApply: (String, %, %, %, %) -> JLObject

from JLObjectType

jlApply: (String, %, %, %, %, %) -> JLObject

from JLObjectType

jlDisplay: % -> Void

from JLObjectType

jlDump: JLObject -> Void

from JLObjectType

jlFieldNames: % -> JLObject

from JLObjectType

jlGetField: (%, JLSymbol) -> JLObject

from JLObjectType

jlGetJuliaIndex: % -> String

from JLObjectType

jlGetProperty: (%, JLSymbol) -> JLObject

from JLObjectType

jlId: % -> JLInt64

from JLObjectType

jlimref: String -> %

from JLObjectType

jlNMRing: () -> String

from NMRing

jlObject: () -> String

from JLObjectType

jlPropertyNames: % -> JLObject

from JLObjectType

jlRef: % -> SExpression

from JLObjectType

jlref: String -> %

from JLObjectType

jlText: (%, String) -> List String

from JLObjectType

jlType: % -> Symbol

from JLObjectType

jnresidue: Integer -> %

jnresidue(i) returns i as an element of the residue ring NMResidueRing.

jnresidue: NMInteger -> %

jnresidue(i) returns i as an element of the residue ring NMResidueRing.

jnresidue: P -> %

jnresidue(p) returns p as an element of the residue ring NMResidueRing.

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

lift: % -> P

lift(x) returns the lift of x. See https://nemocas.github.io/AbstractAlgebra.jl/stable/residue_interface/ for more information for the difference with data.

missing?: % -> Boolean

from JLObjectType

modulus: () -> P

modulus returns the modulus of the residue ring.

multiEuclidean: (List %, %) -> Union(List %, failed)

from EuclideanDomain

mutable?: % -> Boolean

from JLObjectType

nothing?: % -> Boolean

from JLObjectType

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

rem: (%, %) -> %

from EuclideanDomain

rightPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed)

from MagmaWithUnit

sample: %

from SetWithZero

sizeLess?: (%, %) -> Boolean

from EuclideanDomain

sqrt: % -> Union(%, failed)

sqrt(x) returns the square root of x if possible. Only for ResFieldElem.

square?: % -> Boolean

square?(x) returns true if x is a square in the field. Only for ResFieldElem.

squareFree: % -> Factored %

from UniqueFactorizationDomain

squareFreePart: % -> %

from UniqueFactorizationDomain

string: % -> String

from JLType

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

from CancellationAbelianMonoid

unit?: % -> Boolean

from NMRing

unitCanonical: % -> %

from EntireRing

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

from EntireRing

zero?: % -> Boolean

from SetWithZero

AbelianGroup

AbelianMonoid

AbelianSemiGroup

Algebra %

BasicType

BiModule(%, %)

CancellationAbelianMonoid

canonicalsClosed

canonicalUnitNormal

CoercibleTo OutputForm

CommutativeRing

CommutativeStar

ConvertibleTo String

DivisionRing

EntireRing

EuclideanDomain

Field

GcdDomain

IntegralDomain

JLObjectRing

JLObjectType

JLRing

JLType

LeftModule %

LeftOreRing

Magma

MagmaWithUnit

Module %

Monoid

NMCommutativeRing

NMField

NMRing

NMType

NonAssociativeAlgebra %

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors

PrincipalIdealDomain

RightModule %

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

SetWithZero

TwoSidedRecip

UniqueFactorizationDomain

unitsKnown