FiniteFieldCategoryΒΆ

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FiniteFieldCategory is the category of finite fields

0: %
from AbelianMonoid
1: %
from MagmaWithUnit
*: (%, %) -> %
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
=: (%, %) -> 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
charthRoot: % -> %
charthRoot(a) takes the characteristic’th root of a. Note: such a root is alway defined in finite fields.
charthRoot: % -> Union(%, failed)
from CharacteristicNonZero
coerce: % -> %
from Algebra %
coerce: % -> OutputForm
from CoercibleTo OutputForm
coerce: Fraction Integer -> %
from Algebra Fraction Integer
coerce: Integer -> %
from NonAssociativeRing
commutator: (%, %) -> %
from NonAssociativeRng
conditionP: Matrix % -> Union(Vector %, failed)
conditionP(mat), given a matrix representing a homogeneous system of equations, returns a vector whose characteristic’th powers is a non-trivial solution, or “failed” if no such vector exists.
convert: % -> InputForm
from ConvertibleTo InputForm
createPrimitiveElement: () -> %
createPrimitiveElement() computes a generator of the (cyclic) multiplicative group of the field.
D: % -> %
from DifferentialRing
D: (%, NonNegativeInteger) -> %
from DifferentialRing
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
factorsOfCyclicGroupSize: () -> List Record(factor: Integer, exponent: Integer)
factorsOfCyclicGroupSize() returns the factorization of size()-1
gcd: (%, %) -> %
from GcdDomain
gcd: List % -> %
from GcdDomain
gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial %
from GcdDomain
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
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.
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
squareFree: % -> Factored %
from UniqueFactorizationDomain
squareFreePart: % -> %
from UniqueFactorizationDomain
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

AbelianGroup

AbelianMonoid

AbelianSemiGroup

Algebra %

Algebra Fraction Integer

BasicType

BiModule(%, %)

BiModule(Fraction Integer, Fraction Integer)

CancellationAbelianMonoid

canonicalsClosed

canonicalUnitNormal

CharacteristicNonZero

CoercibleTo OutputForm

CommutativeRing

CommutativeStar

Comparable

ConvertibleTo InputForm

DifferentialRing

DivisionRing

EntireRing

EuclideanDomain

Field

FieldOfPrimeCharacteristic

Finite

GcdDomain

IntegralDomain

LeftModule %

LeftModule Fraction Integer

LeftOreRing

Magma

MagmaWithUnit

Module %

Module Fraction Integer

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors

PrincipalIdealDomain

RightModule %

RightModule Fraction Integer

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

StepThrough

UniqueFactorizationDomain

unitsKnown