NemoFiniteField(p, n)

jnemo.spad line 724 [edit on github]

• p: PositiveInteger

• n: PositiveInteger

This domain allows the manipulation of Nemo Galois field of p^n elements using the Nemo Julia package (FLINT based). https://flintlib.org/

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from Magma

*: (%, Fraction Integer) -> %

from RightModule Fraction Integer

*: (%, Integer) -> %

*: (%, NemoInteger) -> %

*: (%, NemoPrimeField p) -> %

from RightModule NemoPrimeField p

*: (Fraction Integer, %) -> %

from LeftModule Fraction Integer

*: (Integer, %) -> %

from AbelianGroup

*: (NemoInteger, %) -> %

*: (NemoPrimeField p, %) -> %

from LeftModule NemoPrimeField p

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

+: (%, %) -> %

from AbelianSemiGroup

-: % -> %

from AbelianGroup

-: (%, %) -> %

from AbelianGroup

/: (%, %) -> %

from Field

/: (%, NemoPrimeField p) -> %

from ExtensionField NemoPrimeField p

=: (%, %) -> Boolean

from BasicType

^: (%, Integer) -> %

from DivisionRing

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

absolute?: () -> Boolean

absoluteNorm: % -> NemoPrimeField p

absoluteTrace: % -> NemoPrimeField p

algebraic?: % -> Boolean

from ExtensionField NemoPrimeField p

annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %

from NonAssociativeSemiRng

associates?: (%, %) -> Boolean

from EntireRing

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

from NonAssociativeRng

basis: () -> Vector %

from FramedModule NemoPrimeField p

basis: PositiveInteger -> Vector %

from FiniteAlgebraicExtensionField NemoPrimeField p

characteristic: () -> NonNegativeInteger

from NonAssociativeRing

characteristicPolynomial: % -> SparseUnivariatePolynomial NemoPrimeField p

from FiniteRankAlgebra(NemoPrimeField p, SparseUnivariatePolynomial NemoPrimeField p)

charthRoot: % -> %

from FiniteFieldCategory

charthRoot: % -> Union(%, failed)

from CharacteristicNonZero

coerce: % -> %

from Algebra %

coerce: % -> OutputForm

from CoercibleTo OutputForm

coerce: Fraction Integer -> %

from Algebra Fraction Integer

coerce: Integer -> %

coerce: NemoInteger -> %

coerce: NemoPrimeField p -> %

from Algebra NemoPrimeField p

commutator: (%, %) -> %

from NonAssociativeRng

conditionP: Matrix % -> Union(Vector %, failed)

from PolynomialFactorizationExplicit

convert: % -> InputForm

from ConvertibleTo InputForm

convert: % -> Integer

from ConvertibleTo Integer

convert: % -> String

from ConvertibleTo String

convert: % -> Vector NemoPrimeField p

from FramedModule NemoPrimeField p

convert: Vector NemoPrimeField p -> %

from FramedModule NemoPrimeField p

coordinates: % -> Vector NemoPrimeField p

from FramedModule NemoPrimeField p

coordinates: (%, Vector %) -> Vector NemoPrimeField p

from FiniteRankAlgebra(NemoPrimeField p, SparseUnivariatePolynomial NemoPrimeField p)

coordinates: (Vector %, Vector %) -> Matrix NemoPrimeField p

from FiniteRankAlgebra(NemoPrimeField p, SparseUnivariatePolynomial NemoPrimeField p)

coordinates: Vector % -> Matrix NemoPrimeField p

from FramedModule NemoPrimeField p

createNormalElement: () -> %

from FiniteAlgebraicExtensionField NemoPrimeField p

createPrimitiveElement: () -> %

from FiniteFieldCategory

D: % -> %

from DifferentialRing

D: (%, NonNegativeInteger) -> %

from DifferentialRing

definingPolynomial: () -> SparseUnivariatePolynomial NemoPrimeField p

from FiniteAlgebraicExtensionField NemoPrimeField p

degree: % -> OnePointCompletion PositiveInteger

from ExtensionField NemoPrimeField p

degree: % -> PositiveInteger

from FiniteAlgebraicExtensionField NemoPrimeField p

differentiate: % -> %

from DifferentialRing

differentiate: (%, NonNegativeInteger) -> %

from DifferentialRing

discreteLog: % -> NonNegativeInteger

from FiniteFieldCategory

discreteLog: (%, %) -> Union(NonNegativeInteger, failed)

from FieldOfPrimeCharacteristic

discriminant: () -> NemoPrimeField p

from FramedAlgebra(NemoPrimeField p, SparseUnivariatePolynomial NemoPrimeField p)

discriminant: Vector % -> NemoPrimeField p

from FiniteRankAlgebra(NemoPrimeField p, SparseUnivariatePolynomial NemoPrimeField p)

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

from EuclideanDomain

enumerate: () -> List %

from Finite

euclideanSize: % -> NonNegativeInteger

from EuclideanDomain

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

extensionDegree: () -> OnePointCompletion PositiveInteger

from ExtensionField NemoPrimeField p

extensionDegree: () -> PositiveInteger

from FiniteAlgebraicExtensionField NemoPrimeField p

factor: % -> Factored %

from UniqueFactorizationDomain

factorPolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial %

from PolynomialFactorizationExplicit

factorsOfCyclicGroupSize: () -> List Record(factor: Integer, exponent: NonNegativeInteger)

from FiniteFieldCategory

factorSquareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial %

from PolynomialFactorizationExplicit

Frobenius: % -> %

from ExtensionField NemoPrimeField p

Frobenius: (%, NonNegativeInteger) -> %

from ExtensionField NemoPrimeField p

gcd: (%, %) -> %

from GcdDomain

gcd: List % -> %

from GcdDomain

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

from PolynomialFactorizationExplicit

generator?: % -> Boolean

generator: () -> %

from FiniteAlgebraicExtensionField NemoPrimeField p

hash: % -> SingleInteger

from Hashable

hashUpdate!: (HashState, %) -> HashState

from Hashable

index: PositiveInteger -> %

from Finite

inGroundField?: % -> Boolean

from ExtensionField NemoPrimeField p

init: %

from StepThrough

inv: % -> %

from DivisionRing

jlAbout: % -> Void

from JuliaObjectType

jlApply: (String, %) -> %

from JuliaObjectType

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

from JuliaObjectType

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

from JuliaObjectType

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

from JuliaObjectType

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

from JuliaObjectType

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

from JuliaObjectType

jlId: % -> String

from JuliaObjectType

jlRef: % -> SExpression

from JuliaObjectType

jlref: String -> %

from JuliaObjectType

jlType: % -> String

from JuliaObjectType

jnff: Integer -> %

jnff: NemoInteger -> %

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

linearAssociatedExp: (%, SparseUnivariatePolynomial NemoPrimeField p) -> %

from FiniteAlgebraicExtensionField NemoPrimeField p

linearAssociatedLog: % -> SparseUnivariatePolynomial NemoPrimeField p

from FiniteAlgebraicExtensionField NemoPrimeField p

linearAssociatedLog: (%, %) -> Union(SparseUnivariatePolynomial NemoPrimeField p, failed)

from FiniteAlgebraicExtensionField NemoPrimeField p

linearAssociatedOrder: % -> SparseUnivariatePolynomial NemoPrimeField p

from FiniteAlgebraicExtensionField NemoPrimeField p

lookup: % -> PositiveInteger

from Finite

minimalPolynomial: % -> SparseUnivariatePolynomial NemoPrimeField p

from FiniteRankAlgebra(NemoPrimeField p, SparseUnivariatePolynomial NemoPrimeField p)

minimalPolynomial: (%, PositiveInteger) -> SparseUnivariatePolynomial %

from FiniteAlgebraicExtensionField NemoPrimeField p

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

from EuclideanDomain

mutable?: % -> Boolean

from JuliaObjectType

nextItem: % -> Union(%, failed)

from StepThrough

norm: % -> NemoPrimeField p

norm: (%, PositiveInteger) -> %

from FiniteAlgebraicExtensionField NemoPrimeField p

normal?: % -> Boolean

from FiniteAlgebraicExtensionField NemoPrimeField p

normalElement: () -> %

from FiniteAlgebraicExtensionField NemoPrimeField p

nothing?: % -> Boolean

from JuliaObjectType

one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

order: % -> OnePointCompletion PositiveInteger

from FieldOfPrimeCharacteristic

order: % -> PositiveInteger

from FiniteFieldCategory

order: () -> NonNegativeInteger

plenaryPower: (%, PositiveInteger) -> %

from NonAssociativeAlgebra NemoPrimeField p

prime?: % -> Boolean

from UniqueFactorizationDomain

primeFrobenius: % -> %

from FieldOfPrimeCharacteristic

primeFrobenius: (%, NonNegativeInteger) -> %

from FieldOfPrimeCharacteristic

primitive?: % -> Boolean

from FiniteFieldCategory

primitiveElement: () -> %

from FiniteFieldCategory

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

from PrincipalIdealDomain

quo: (%, %) -> %

from EuclideanDomain

random: () -> %

from Finite

rank: () -> PositiveInteger

from FramedModule NemoPrimeField p

recip: % -> Union(%, failed)

from MagmaWithUnit

regularRepresentation: % -> Matrix NemoPrimeField p

from FramedAlgebra(NemoPrimeField p, SparseUnivariatePolynomial NemoPrimeField p)

regularRepresentation: (%, Vector %) -> Matrix NemoPrimeField p

from FiniteRankAlgebra(NemoPrimeField p, SparseUnivariatePolynomial NemoPrimeField p)

rem: (%, %) -> %

from EuclideanDomain

representationType: () -> Union(prime, polynomial, normal, cyclic)

from FiniteFieldCategory

represents: (Vector NemoPrimeField p, Vector %) -> %

from FiniteRankAlgebra(NemoPrimeField p, SparseUnivariatePolynomial NemoPrimeField p)

represents: Vector NemoPrimeField p -> %

from FramedModule NemoPrimeField p

retract: % -> NemoPrimeField p

from RetractableTo NemoPrimeField p

retractIfCan: % -> Union(NemoPrimeField p, failed)

from RetractableTo NemoPrimeField p

rightPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed)

from MagmaWithUnit

sample: %

from AbelianMonoid

size: () -> NonNegativeInteger

from Finite

sizeLess?: (%, %) -> Boolean

from EuclideanDomain

smaller?: (%, %) -> Boolean

from Comparable

solveLinearPolynomialEquation: (List SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> Union(List SparseUnivariatePolynomial %, failed)

from PolynomialFactorizationExplicit

sqrt: % -> %

sqrt(x) returns a square root of x. Throw a Julia error if there is no square root and returns 0.

squareFree: % -> Factored %

from UniqueFactorizationDomain

squareFreePart: % -> %

from UniqueFactorizationDomain

squareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial %

from PolynomialFactorizationExplicit

string: % -> String

from JuliaObjectType

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

from CancellationAbelianMonoid

tableForDiscreteLogarithm: Integer -> Table(PositiveInteger, NonNegativeInteger)

from FiniteFieldCategory

trace: % -> NemoPrimeField p

trace: (%, PositiveInteger) -> %

from FiniteAlgebraicExtensionField NemoPrimeField p

traceMatrix: () -> Matrix NemoPrimeField p

from FramedAlgebra(NemoPrimeField p, SparseUnivariatePolynomial NemoPrimeField p)

traceMatrix: Vector % -> Matrix NemoPrimeField p

from FiniteRankAlgebra(NemoPrimeField p, SparseUnivariatePolynomial NemoPrimeField p)

transcendenceDegree: () -> NonNegativeInteger

from ExtensionField NemoPrimeField p

transcendent?: % -> Boolean

from ExtensionField NemoPrimeField p

unit?: % -> Boolean

from EntireRing

unitCanonical: % -> %

from EntireRing

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

from EntireRing

zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

Algebra %

Algebra Fraction Integer

Algebra NemoPrimeField p

BasicType

BiModule(%, %)

BiModule(Fraction Integer, Fraction Integer)

BiModule(NemoPrimeField p, NemoPrimeField p)

CancellationAbelianMonoid

canonicalsClosed

canonicalUnitNormal

CharacteristicNonZero

CharacteristicZero if NemoPrimeField p has CharacteristicZero

CoercibleFrom NemoPrimeField p

CoercibleTo OutputForm

CommutativeRing

CommutativeStar

Comparable

ConvertibleTo InputForm

ConvertibleTo Integer

ConvertibleTo String

DifferentialRing

DivisionRing

EntireRing

EuclideanDomain

ExtensionField NemoPrimeField p

Field

FieldOfPrimeCharacteristic

Finite

FiniteAlgebraicExtensionField NemoPrimeField p

FiniteFieldCategory

FiniteRankAlgebra(NemoPrimeField p, SparseUnivariatePolynomial NemoPrimeField p)

FramedAlgebra(NemoPrimeField p, SparseUnivariatePolynomial NemoPrimeField p)

FramedModule NemoPrimeField p

GcdDomain

Hashable

IntegralDomain

JuliaObjectRing

JuliaObjectType

JuliaRing

JuliaType

LeftModule %

LeftModule Fraction Integer

LeftModule NemoPrimeField p

LeftOreRing

Magma

MagmaWithUnit

Module %

Module Fraction Integer

Module NemoPrimeField p

Monoid

NemoRing

NemoType

NonAssociativeAlgebra %

NonAssociativeAlgebra Fraction Integer

NonAssociativeAlgebra NemoPrimeField p

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors

PolynomialFactorizationExplicit

PrincipalIdealDomain

RetractableTo NemoPrimeField p

RightModule %

RightModule Fraction Integer

RightModule NemoPrimeField p

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

StepThrough

TwoSidedRecip

UniqueFactorizationDomain

unitsKnown