InnerFiniteFieldExtensionByTriangularSet KΒΆ
iffts.spad line 104 [edit on github]
Implement an ACF according to cite{Steel_AlgebraicallyClosedFields_2010} but here for a finite field of characteristic p.
- 0: %
from AbelianMonoid
- 1: %
from MagmaWithUnit
- *: (%, %) -> %
from LeftModule %
- *: (%, 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
- ^: (%, Integer) -> %
from DivisionRing
- ^: (%, NonNegativeInteger) -> %
from MagmaWithUnit
- ^: (%, PositiveInteger) -> %
from Magma
- annihilate?: (%, %) -> Boolean
from Rng
- antiCommutator: (%, %) -> %
- associates?: (%, %) -> Boolean
from EntireRing
- associator: (%, %, %) -> %
from NonAssociativeRng
- characteristic: () -> NonNegativeInteger
from NonAssociativeRing
- charthRoot: % -> %
from FiniteFieldCategory
- charthRoot: % -> Union(%, failed)
- coerce: % -> %
from Algebra %
- coerce: % -> OutputForm
from CoercibleTo OutputForm
- coerce: Fraction Integer -> %
- coerce: Integer -> %
from NonAssociativeRing
- commutator: (%, %) -> %
from NonAssociativeRng
- conditionP: Matrix % -> Union(Vector %, failed)
- convert: % -> InputForm
from ConvertibleTo InputForm
- createPrimitiveElement: () -> %
from FiniteFieldCategory
- D: % -> %
from DifferentialRing
- D: (%, NonNegativeInteger) -> %
from DifferentialRing
- differentiate: % -> %
from DifferentialRing
- differentiate: (%, NonNegativeInteger) -> %
from DifferentialRing
- discreteLog: % -> NonNegativeInteger
from FiniteFieldCategory
- discreteLog: (%, %) -> Union(NonNegativeInteger, failed)
- divide: (%, %) -> Record(quotient: %, remainder: %)
from EuclideanDomain
- euclideanSize: % -> NonNegativeInteger
from EuclideanDomain
- expressIdealMember: (List %, %) -> Union(List %, failed)
from PrincipalIdealDomain
- exquo: (%, %) -> Union(%, failed)
from EntireRing
- extendBy!: SparseUnivariatePolynomial % -> %
extendBy!(p)adds a new polynomialpto the triangular set that is used to internally describe the extension. It is assumed thatpis irreducible with respect to the current extension. The return value is a root of the given polynomial (then considered to be an element of this domain.
- extendedEuclidean: (%, %) -> Record(coef1: %, coef2: %, generator: %)
from EuclideanDomain
- extendedEuclidean: (%, %, %) -> Union(Record(coef1: %, coef2: %), failed)
from EuclideanDomain
- factorsOfCyclicGroupSize: () -> List Record(factor: Integer, exponent: NonNegativeInteger)
from FiniteFieldCategory
- gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial %
- getGenerator: PositiveInteger -> %
getGenerator(n)forn< rank() returns a root of then-th extension polynomial.
- getRelation: PositiveInteger -> NewSparseMultivariatePolynomial(K, IndexedVariable f)
getRelation(n)forn<=rank() returns then-th extension polynomial.
- getRelations: () -> List NewSparseMultivariatePolynomial(K, IndexedVariable f)
getRelations()returns the triangular set that is used for the extension. The list is sorted descendingly by the main variable of the respective polynomial.
- hash: % -> SingleInteger
from SetCategory
- hashUpdate!: (HashState, %) -> HashState
from SetCategory
- index: PositiveInteger -> %
from Finite
- init: %
from StepThrough
- inv: % -> %
from DivisionRing
- latex: % -> String
from SetCategory
- 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
- minimalPolynomial: PositiveInteger -> SparseUnivariatePolynomial %
minimalPolynomial(n)forn< rank() returns then-th extension polynomial.
- multiEuclidean: (List %, %) -> Union(List %, failed)
from EuclideanDomain
- nextItem: % -> Union(%, failed)
from StepThrough
- one?: % -> Boolean
from MagmaWithUnit
- opposite?: (%, %) -> Boolean
from AbelianMonoid
- order: % -> OnePointCompletion PositiveInteger
- order: % -> PositiveInteger
from FiniteFieldCategory
- primeFrobenius: % -> %
- primeFrobenius: (%, NonNegativeInteger) -> %
- primitive?: % -> Boolean
from FiniteFieldCategory
- primitiveElement: () -> %
from FiniteFieldCategory
- principalIdeal: List % -> Record(coef: List %, generator: %)
from PrincipalIdealDomain
- quo: (%, %) -> %
from EuclideanDomain
- rank: () -> Integer
rank()returns the size of the current triangular set used for the extension.
- recip: % -> Union(%, failed)
from MagmaWithUnit
- rem: (%, %) -> %
from EuclideanDomain
- representationType: () -> Union(prime, polynomial, normal, cyclic)
from FiniteFieldCategory
- rightPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
- rightPower: (%, PositiveInteger) -> %
from Magma
- rightRecip: % -> Union(%, failed)
from MagmaWithUnit
- sample: %
from AbelianMonoid
show: () -> Void
- size: () -> NonNegativeInteger
size()returns the number of elements in the current extension. size() changes with every extension, i.e. with every call to extendBy!(p).
- sizeLess?: (%, %) -> Boolean
from EuclideanDomain
- smaller?: (%, %) -> Boolean
from Comparable
- solveLinearPolynomialEquation: (List SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> Union(List SparseUnivariatePolynomial %, failed)
- squareFree: % -> Factored %
- squareFreePart: % -> %
- subtractIfCan: (%, %) -> Union(%, failed)
- tableForDiscreteLogarithm: Integer -> Table(PositiveInteger, NonNegativeInteger)
from FiniteFieldCategory
- unit?: % -> Boolean
from EntireRing
- unitCanonical: % -> %
from EntireRing
- unitNormal: % -> Record(unit: %, canonical: %, associate: %)
from EntireRing
- zero?: % -> Boolean
from AbelianMonoid
Algebra %
BiModule(%, %)
BiModule(Fraction Integer, Fraction Integer)
Module %