FiniteFieldExtension(GF, n)ΒΆ

ffp.spad line 230

FiniteFieldExtensionByPolynomial(GF, n) implements an extension of the finite field GF of degree n generated by the extension polynomial constructed by createIrreduciblePoly from FiniteFieldPolynomialPackage.

0: %
from AbelianMonoid
1: %
from MagmaWithUnit
*: (%, %) -> %
from Magma
*: (%, Fraction Integer) -> %
from RightModule Fraction Integer
*: (%, GF) -> %
from RightModule GF
*: (Fraction Integer, %) -> %
from LeftModule Fraction Integer
*: (GF, %) -> %
from LeftModule GF
*: (Integer, %) -> %
from AbelianGroup
*: (NonNegativeInteger, %) -> %
from AbelianMonoid
*: (PositiveInteger, %) -> %
from AbelianSemiGroup
+: (%, %) -> %
from AbelianSemiGroup
-: % -> %
from AbelianGroup
-: (%, %) -> %
from AbelianGroup
/: (%, %) -> %
from Field
/: (%, GF) -> %
from VectorSpace GF
=: (%, %) -> Boolean
from BasicType
^: (%, Integer) -> %
from DivisionRing
^: (%, NonNegativeInteger) -> %
from MagmaWithUnit
^: (%, PositiveInteger) -> %
from Magma
~=: (%, %) -> Boolean
from BasicType
algebraic?: % -> Boolean
from ExtensionField GF
annihilate?: (%, %) -> Boolean
from Rng
antiCommutator: (%, %) -> %
from NonAssociativeSemiRng
associates?: (%, %) -> Boolean
from EntireRing
associator: (%, %, %) -> %
from NonAssociativeRng
basis: () -> Vector %
from FiniteAlgebraicExtensionField GF
basis: PositiveInteger -> Vector %
from FiniteAlgebraicExtensionField GF
characteristic: () -> NonNegativeInteger
from NonAssociativeRing
charthRoot: % -> %
from FiniteFieldCategory
charthRoot: % -> Union(%, failed)
from CharacteristicNonZero
coerce: % -> %
from Algebra %
coerce: % -> OutputForm
from CoercibleTo OutputForm
coerce: Fraction Integer -> %
from Algebra Fraction Integer
coerce: GF -> %
from RetractableTo GF
coerce: Integer -> %
from NonAssociativeRing
commutator: (%, %) -> %
from NonAssociativeRng
conditionP: Matrix % -> Union(Vector %, failed)
from FiniteFieldCategory
convert: % -> InputForm
from ConvertibleTo InputForm
coordinates: % -> Vector GF
from FiniteAlgebraicExtensionField GF
coordinates: Vector % -> Matrix GF
from FiniteAlgebraicExtensionField GF
createNormalElement: () -> %
from FiniteAlgebraicExtensionField GF
createPrimitiveElement: () -> %
from FiniteFieldCategory
D: % -> %
from DifferentialRing
D: (%, NonNegativeInteger) -> %
from DifferentialRing
definingPolynomial: () -> SparseUnivariatePolynomial GF
from FiniteAlgebraicExtensionField GF
degree: % -> OnePointCompletion PositiveInteger
from ExtensionField GF
degree: % -> PositiveInteger
from FiniteAlgebraicExtensionField GF
differentiate: % -> %
from DifferentialRing
differentiate: (%, NonNegativeInteger) -> %
from DifferentialRing
dimension: () -> CardinalNumber
from VectorSpace GF
discreteLog: % -> NonNegativeInteger
from FiniteFieldCategory
discreteLog: (%, %) -> Union(NonNegativeInteger, failed)
from FieldOfPrimeCharacteristic
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 GF
extensionDegree: () -> PositiveInteger
from FiniteAlgebraicExtensionField GF
factor: % -> Factored %
from UniqueFactorizationDomain
factorsOfCyclicGroupSize: () -> List Record(factor: Integer, exponent: Integer)
from FiniteFieldCategory
Frobenius: % -> %
from ExtensionField GF
Frobenius: (%, NonNegativeInteger) -> %
from ExtensionField GF
gcd: (%, %) -> %
from GcdDomain
gcd: List % -> %
from GcdDomain
gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial %
from GcdDomain
generator: () -> %
from FiniteAlgebraicExtensionField GF
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
index: PositiveInteger -> %
from Finite
inGroundField?: % -> Boolean
from ExtensionField GF
init: %
from StepThrough
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
linearAssociatedExp: (%, SparseUnivariatePolynomial GF) -> %
from FiniteAlgebraicExtensionField GF
linearAssociatedLog: % -> SparseUnivariatePolynomial GF
from FiniteAlgebraicExtensionField GF
linearAssociatedLog: (%, %) -> Union(SparseUnivariatePolynomial GF, failed)
from FiniteAlgebraicExtensionField GF
linearAssociatedOrder: % -> SparseUnivariatePolynomial GF
from FiniteAlgebraicExtensionField GF
lookup: % -> PositiveInteger
from Finite
minimalPolynomial: % -> SparseUnivariatePolynomial GF
from FiniteAlgebraicExtensionField GF
minimalPolynomial: (%, PositiveInteger) -> SparseUnivariatePolynomial %
from FiniteAlgebraicExtensionField GF
multiEuclidean: (List %, %) -> Union(List %, failed)
from EuclideanDomain
nextItem: % -> Union(%, failed)
from StepThrough
norm: % -> GF
from FiniteAlgebraicExtensionField GF
norm: (%, PositiveInteger) -> %
from FiniteAlgebraicExtensionField GF
normal?: % -> Boolean
from FiniteAlgebraicExtensionField GF
normalElement: () -> %
from FiniteAlgebraicExtensionField GF
one?: % -> Boolean
from MagmaWithUnit
opposite?: (%, %) -> Boolean
from AbelianMonoid
order: % -> OnePointCompletion PositiveInteger
from FieldOfPrimeCharacteristic
order: % -> PositiveInteger
from FiniteFieldCategory
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
recip: % -> Union(%, failed)
from MagmaWithUnit
rem: (%, %) -> %
from EuclideanDomain
representationType: () -> Union(prime, polynomial, normal, cyclic)
from FiniteFieldCategory
represents: Vector GF -> %
from FiniteAlgebraicExtensionField GF
retract: % -> GF
from RetractableTo GF
retractIfCan: % -> Union(GF, failed)
from RetractableTo GF
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
squareFree: % -> Factored %
from UniqueFactorizationDomain
squareFreePart: % -> %
from UniqueFactorizationDomain
subtractIfCan: (%, %) -> Union(%, failed)
from CancellationAbelianMonoid
tableForDiscreteLogarithm: Integer -> Table(PositiveInteger, NonNegativeInteger)
from FiniteFieldCategory
trace: % -> GF
from FiniteAlgebraicExtensionField GF
trace: (%, PositiveInteger) -> %
from FiniteAlgebraicExtensionField GF
transcendenceDegree: () -> NonNegativeInteger
from ExtensionField GF
transcendent?: % -> Boolean
from ExtensionField GF
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

BasicType

BiModule(%, %)

BiModule(Fraction Integer, Fraction Integer)

BiModule(GF, GF)

CancellationAbelianMonoid

canonicalsClosed

canonicalUnitNormal

CharacteristicNonZero

CharacteristicZero if GF has CharacteristicZero

CoercibleTo OutputForm

CommutativeRing

CommutativeStar

Comparable

ConvertibleTo InputForm

DifferentialRing

DivisionRing

EntireRing

EuclideanDomain

ExtensionField GF

Field

FieldOfPrimeCharacteristic

Finite

FiniteAlgebraicExtensionField GF

FiniteFieldCategory

GcdDomain

IntegralDomain

LeftModule %

LeftModule Fraction Integer

LeftModule GF

LeftOreRing

Magma

MagmaWithUnit

Module %

Module Fraction Integer

Module GF

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors

PrincipalIdealDomain

RetractableTo GF

RightModule %

RightModule Fraction Integer

RightModule GF

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

StepThrough

UniqueFactorizationDomain

unitsKnown

VectorSpace GF