FiniteFieldCategory

ffcat.spad line 426 [edit on github]

FiniteFieldCategory is the category of finite fields

* : (%, %) -> %

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

D : % -> %

from DifferentialRing

D : (%, NonNegativeInteger) -> %

from DifferentialRing

^ : (%, 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

charthRoot : % -> %

charthRoot(a) takes the characteristic'th root of a. Note: such a root is always defined in finite fields.

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

from PolynomialFactorizationExplicit

coerce : % -> %

from Algebra(%)

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 : % -> InputForm

from ConvertibleTo(InputForm)

createPrimitiveElement : () -> %

createPrimitiveElement() computes a generator of the (cyclic) multiplicative group of the field.

differentiate : % -> %

from DifferentialRing

differentiate : (%, NonNegativeInteger) -> %

from DifferentialRing

discreteLog : % -> NonNegativeInteger

discreteLog(a) computes the discrete logarithm of a with respect to primitiveElement() of the field.

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

factor : % -> Factored(%)

from UniqueFactorizationDomain

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

from PolynomialFactorizationExplicit

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

from PolynomialFactorizationExplicit

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

factorsOfCyclicGroupSize() returns the factorization of size()-1

gcd : (%, %) -> %

from GcdDomain

gcd : List(%) -> %

from GcdDomain

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

from PolynomialFactorizationExplicit

hash : % -> SingleInteger

from Hashable

hashUpdate! : (HashState, %) -> HashState

from Hashable

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

lookup : % -> PositiveInteger

from Finite

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

from EuclideanDomain

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

from StepThrough

one? : % -> Boolean

from MagmaWithUnit

opposite? : (%, %) -> Boolean

from AbelianMonoid

order : % -> OnePointCompletion(PositiveInteger)

from FieldOfPrimeCharacteristic

order : % -> PositiveInteger

order(b) computes the order of an element b in the multiplicative group of the field. Error: if b equals 0.

plenaryPower : (%, PositiveInteger) -> %

from NonAssociativeAlgebra(%)

prime? : % -> Boolean

from UniqueFactorizationDomain

primeFrobenius : % -> %

from FieldOfPrimeCharacteristic

primeFrobenius : (%, NonNegativeInteger) -> %

from FieldOfPrimeCharacteristic

primitive? : % -> Boolean

primitive?(b) tests whether the element b is a generator of the (cyclic) multiplicative group of the field, i.e. is a primitive element. Implementation Note: see ch.IX.1.3, th.2 in D. Lipson.

primitiveElement : () -> %

primitiveElement() returns a primitive element stored in a global variable in the domain. At first call, the primitive element is computed by calling createPrimitiveElement.

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")

representationType() returns the type of the representation, one of: prime, polynomial, normal, or cyclic.

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)

tableForDiscreteLogarithm(a, n) returns a table of the discrete logarithms of a^0 up to a^(n-1) which, called with key lookup(a^i) returns i for i in 0..n-1. Error: if not called for prime divisors of order of multiplicative group.

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))

Comparable

ConvertibleTo(InputForm)

NonAssociativeSemiRng

Algebra(%)

RightModule(%)

Monoid

LeftModule(Fraction(Integer))

GcdDomain

UniqueFactorizationDomain

EuclideanDomain

CancellationAbelianMonoid

CharacteristicNonZero

NonAssociativeRing

MagmaWithUnit

RightModule(Fraction(Integer))

StepThrough

AbelianSemiGroup

LeftModule(%)

canonicalUnitNormal

CommutativeStar

Module(%)

Algebra(Fraction(Integer))

SetCategory

LeftOreRing

CoercibleTo(OutputForm)

SemiRing

Rng

Field

CommutativeRing

PolynomialFactorizationExplicit

TwoSidedRecip

Magma

PrincipalIdealDomain

SemiGroup

DivisionRing

BiModule(%, %)

AbelianGroup

AbelianMonoid

noZeroDivisors

NonAssociativeSemiRing

canonicalsClosed

NonAssociativeAlgebra(%)

NonAssociativeAlgebra(Fraction(Integer))

FieldOfPrimeCharacteristic

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

DifferentialRing

NonAssociativeRng

unitsKnown

Ring

SemiRng

EntireRing

Hashable

Finite

BasicType