FiniteFieldExtensionByPolynomial(GF, defpol)

ffdoms.spad line 1316 [edit on github]

• GF : FiniteFieldCategory
• defpol : SparseUnivariatePolynomial(GF)

FiniteFieldExtensionByPolynomial(GF, defpol) implements the extension of the finite field GF generated by the extension polynomial defpol which MUST be irreducible. Note: the user has the responsibility to ensure that defpol is irreducible.

* : (%, %) -> %

from Magma

* : (%, GF) -> %

from RightModule(GF)

* : (%, Fraction(Integer)) -> %

from RightModule(Fraction(Integer))

* : (GF, %) -> %

from LeftModule(GF)

* : (Fraction(Integer), %) -> %

from LeftModule(Fraction(Integer))

* : (Integer, %) -> %

from AbelianGroup

* : (NonNegativeInteger, %) -> %

from AbelianMonoid

* : (PositiveInteger, %) -> %

from AbelianSemiGroup

+ : (%, %) -> %

from AbelianSemiGroup

- : % -> %

from AbelianGroup

- : (%, %) -> %

from AbelianGroup

/ : (%, %) -> %

from Field

/ : (%, GF) -> %

from ExtensionField(GF)

0 : () -> %

from AbelianMonoid

1 : () -> %

from MagmaWithUnit

= : (%, %) -> Boolean

from BasicType

D : % -> %

from DifferentialRing

D : (%, NonNegativeInteger) -> %

from DifferentialRing

Frobenius : % -> %

from ExtensionField(GF)

Frobenius : (%, NonNegativeInteger) -> %

from ExtensionField(GF)

^ : (%, Integer) -> %

from DivisionRing

^ : (%, NonNegativeInteger) -> %

from MagmaWithUnit

^ : (%, PositiveInteger) -> %

from Magma

algebraic? : % -> Boolean

from ExtensionField(GF)

annihilate? : (%, %) -> Boolean

from Rng

antiCommutator : (%, %) -> %

from NonAssociativeSemiRng

associates? : (%, %) -> Boolean

from EntireRing

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

from NonAssociativeRng

basis : () -> Vector(%)

from FramedModule(GF)

basis : PositiveInteger -> Vector(%)

from FiniteAlgebraicExtensionField(GF)

characteristic : () -> NonNegativeInteger

from NonAssociativeRing

characteristicPolynomial : % -> SparseUnivariatePolynomial(GF)

from FiniteRankAlgebra(GF, SparseUnivariatePolynomial(GF))

charthRoot : % -> %

from FiniteFieldCategory

charthRoot : % -> Union(%, "failed")

from CharacteristicNonZero

coerce : % -> %

from Algebra(%)

coerce : GF -> %

from CoercibleFrom(GF)

coerce : Fraction(Integer) -> %

from Algebra(Fraction(Integer))

coerce : Integer -> %

from NonAssociativeRing

coerce : % -> OutputForm

from CoercibleTo(OutputForm)

commutator : (%, %) -> %

from NonAssociativeRng

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

from PolynomialFactorizationExplicit

convert : Vector(GF) -> %

from FramedModule(GF)

convert : % -> InputForm

from ConvertibleTo(InputForm)

convert : % -> Vector(GF)

from FramedModule(GF)

coordinates : Vector(%) -> Matrix(GF)

from FramedModule(GF)

coordinates : (Vector(%), Vector(%)) -> Matrix(GF)

from FiniteRankAlgebra(GF, SparseUnivariatePolynomial(GF))

coordinates : % -> Vector(GF)

from FramedModule(GF)

coordinates : (%, Vector(%)) -> Vector(GF)

from FiniteRankAlgebra(GF, SparseUnivariatePolynomial(GF))

createNormalElement : () -> %

from FiniteAlgebraicExtensionField(GF)

createPrimitiveElement : () -> %

from FiniteFieldCategory

definingPolynomial : () -> SparseUnivariatePolynomial(GF)

from FiniteAlgebraicExtensionField(GF)

degree : % -> OnePointCompletion(PositiveInteger)

from ExtensionField(GF)

degree : % -> PositiveInteger

from FiniteAlgebraicExtensionField(GF)

differentiate : % -> %

from DifferentialRing

differentiate : (%, NonNegativeInteger) -> %

from DifferentialRing

discreteLog : % -> NonNegativeInteger

from FiniteFieldCategory

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

from FieldOfPrimeCharacteristic

discriminant : () -> GF

from FramedAlgebra(GF, SparseUnivariatePolynomial(GF))

discriminant : Vector(%) -> GF

from FiniteRankAlgebra(GF, SparseUnivariatePolynomial(GF))

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

factorPolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%))

from PolynomialFactorizationExplicit

factorSquareFreePolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%))

from PolynomialFactorizationExplicit

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

from FiniteFieldCategory

gcd : (%, %) -> %

from GcdDomain

gcd : List(%) -> %

from GcdDomain

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

from GcdDomain

generator : () -> %

from FiniteAlgebraicExtensionField(GF)

hash : % -> SingleInteger

from Hashable

hashUpdate! : (HashState, %) -> HashState

from Hashable

inGroundField? : % -> Boolean

from ExtensionField(GF)

index : PositiveInteger -> %

from Finite

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

from FiniteAlgebraicExtensionField(GF)

minimalPolynomial : % -> SparseUnivariatePolynomial(GF)

from FiniteRankAlgebra(GF, SparseUnivariatePolynomial(GF))

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

from EuclideanDomain

nextItem : % -> Union(%, "failed")

from StepThrough

norm : (%, PositiveInteger) -> %

from FiniteAlgebraicExtensionField(GF)

norm : % -> GF

from FiniteRankAlgebra(GF, SparseUnivariatePolynomial(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

plenaryPower : (%, PositiveInteger) -> %

from NonAssociativeAlgebra(%)

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 FiniteRankAlgebra(GF, SparseUnivariatePolynomial(GF))

recip : % -> Union(%, "failed")

from MagmaWithUnit

regularRepresentation : % -> Matrix(GF)

from FramedAlgebra(GF, SparseUnivariatePolynomial(GF))

regularRepresentation : (%, Vector(%)) -> Matrix(GF)

from FiniteRankAlgebra(GF, SparseUnivariatePolynomial(GF))

rem : (%, %) -> %

from EuclideanDomain

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

from FiniteFieldCategory

represents : Vector(GF) -> %

from FramedModule(GF)

represents : (Vector(GF), Vector(%)) -> %

from FiniteRankAlgebra(GF, SparseUnivariatePolynomial(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

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

from PolynomialFactorizationExplicit

squareFree : % -> Factored(%)

from UniqueFactorizationDomain

squareFreePart : % -> %

from UniqueFactorizationDomain

squareFreePolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%))

from PolynomialFactorizationExplicit

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

from CancellationAbelianMonoid

tableForDiscreteLogarithm : Integer -> Table(PositiveInteger, NonNegativeInteger)

from FiniteFieldCategory

trace : (%, PositiveInteger) -> %

from FiniteAlgebraicExtensionField(GF)

trace : % -> GF

from FiniteRankAlgebra(GF, SparseUnivariatePolynomial(GF))

traceMatrix : () -> Matrix(GF)

from FramedAlgebra(GF, SparseUnivariatePolynomial(GF))

traceMatrix : Vector(%) -> Matrix(GF)

from FiniteRankAlgebra(GF, SparseUnivariatePolynomial(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

~= : (%, %) -> Boolean

from BasicType

Module(Fraction(Integer))

PrincipalIdealDomain

Algebra(GF)

NonAssociativeSemiRing

BiModule(%, %)

ConvertibleTo(InputForm)

Field

canonicalUnitNormal

Rng

AbelianMonoid

LeftModule(GF)

TwoSidedRecip

SemiRing

EntireRing

NonAssociativeAlgebra(Fraction(Integer))

CharacteristicNonZero

unitsKnown

FiniteRankAlgebra(GF, SparseUnivariatePolynomial(GF))

RightModule(GF)

noZeroDivisors

RightModule(%)

UniqueFactorizationDomain

SemiGroup

RightModule(Fraction(Integer))

Magma

LeftModule(%)

NonAssociativeRing

GcdDomain

CharacteristicZero

Algebra(%)

CommutativeRing

Ring

DifferentialRing

DivisionRing

Module(GF)

IntegralDomain

NonAssociativeSemiRng

CancellationAbelianMonoid

EuclideanDomain

canonicalsClosed

CommutativeStar

MagmaWithUnit

Comparable

Hashable

BiModule(GF, GF)

Module(%)

CoercibleTo(OutputForm)

NonAssociativeAlgebra(GF)

Finite

SemiRng

FramedAlgebra(GF, SparseUnivariatePolynomial(GF))

RetractableTo(GF)

Monoid

PolynomialFactorizationExplicit

FiniteFieldCategory

CoercibleFrom(GF)

LeftOreRing

NonAssociativeAlgebra(%)

Algebra(Fraction(Integer))

BasicType

LeftModule(Fraction(Integer))

FramedModule(GF)

SetCategory

ExtensionField(GF)

FiniteAlgebraicExtensionField(GF)

NonAssociativeRng

FieldOfPrimeCharacteristic

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

StepThrough

AbelianGroup

AbelianSemiGroup