MonogenicAlgebra(R, UP)ΒΆ

algcat.spad line 211

A MonogenicAlgebra is an algebra of finite rank which can be generated by a single element.

0: %
from AbelianMonoid
1: %
from MagmaWithUnit
*: (%, %) -> %
from Magma
*: (%, Fraction Integer) -> % if R has Field
from RightModule Fraction Integer
*: (%, R) -> %
from RightModule R
*: (Fraction Integer, %) -> % if R has Field
from LeftModule Fraction Integer
*: (Integer, %) -> %
from AbelianGroup
*: (NonNegativeInteger, %) -> %
from AbelianMonoid
*: (PositiveInteger, %) -> %
from AbelianSemiGroup
*: (R, %) -> %
from LeftModule R
+: (%, %) -> %
from AbelianSemiGroup
-: % -> %
from AbelianGroup
-: (%, %) -> %
from AbelianGroup
/: (%, %) -> % if R has Field
from Field
=: (%, %) -> Boolean
from BasicType
^: (%, Integer) -> % if R has Field
from DivisionRing
^: (%, NonNegativeInteger) -> %
from MagmaWithUnit
^: (%, PositiveInteger) -> %
from Magma
~=: (%, %) -> Boolean
from BasicType
annihilate?: (%, %) -> Boolean
from Rng
antiCommutator: (%, %) -> %
from NonAssociativeSemiRng
associates?: (%, %) -> Boolean if R has Field
from EntireRing
associator: (%, %, %) -> %
from NonAssociativeRng
basis: () -> Vector %
from FramedModule R
characteristic: () -> NonNegativeInteger
from NonAssociativeRing
characteristicPolynomial: % -> UP
from FiniteRankAlgebra(R, UP)
charthRoot: % -> % if R has FiniteFieldCategory
from FiniteFieldCategory
charthRoot: % -> Union(%, failed) if R has CharacteristicNonZero
from CharacteristicNonZero
coerce: % -> %
from Algebra %
coerce: % -> OutputForm
from CoercibleTo OutputForm
coerce: Fraction Integer -> % if R has Field or R has RetractableTo Fraction Integer
from Algebra Fraction Integer
coerce: Integer -> %
from NonAssociativeRing
coerce: R -> %
from Algebra R
commutator: (%, %) -> %
from NonAssociativeRng
conditionP: Matrix % -> Union(Vector %, failed) if R has FiniteFieldCategory
from PolynomialFactorizationExplicit
convert: % -> InputForm if R has Finite
from ConvertibleTo InputForm
convert: % -> UP
from ConvertibleTo UP
convert: % -> Vector R
from FramedModule R
convert: UP -> %
convert(up) converts the univariate polynomial up to an algebra element, reducing by the definingPolynomial() if necessary.
convert: Vector R -> %
from FramedModule R
coordinates: % -> Vector R
from FramedModule R
coordinates: (%, Vector %) -> Vector R
from FiniteRankAlgebra(R, UP)
coordinates: (Vector %, Vector %) -> Matrix R
from FiniteRankAlgebra(R, UP)
coordinates: Vector % -> Matrix R
from FramedModule R
createPrimitiveElement: () -> % if R has FiniteFieldCategory
from FiniteFieldCategory
D: % -> % if R has FiniteFieldCategory or R has DifferentialRing and R has Field
from DifferentialRing
D: (%, List Symbol) -> % if R has PartialDifferentialRing Symbol and R has Field
from PartialDifferentialRing Symbol
D: (%, List Symbol, List NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol and R has Field
from PartialDifferentialRing Symbol
D: (%, NonNegativeInteger) -> % if R has FiniteFieldCategory or R has DifferentialRing and R has Field
from DifferentialRing
D: (%, R -> R) -> % if R has Field
from DifferentialExtension R
D: (%, R -> R, NonNegativeInteger) -> % if R has Field
from DifferentialExtension R
D: (%, Symbol) -> % if R has PartialDifferentialRing Symbol and R has Field
from PartialDifferentialRing Symbol
D: (%, Symbol, NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol and R has Field
from PartialDifferentialRing Symbol
definingPolynomial: () -> UP
definingPolynomial() returns the minimal polynomial which generator() satisfies.
derivationCoordinates: (Vector %, R -> R) -> Matrix R if R has Field
derivationCoordinates(b, ') returns M such that b' = M b.
differentiate: % -> % if R has FiniteFieldCategory or R has DifferentialRing and R has Field
from DifferentialRing
differentiate: (%, List Symbol) -> % if R has PartialDifferentialRing Symbol and R has Field
from PartialDifferentialRing Symbol
differentiate: (%, List Symbol, List NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol and R has Field
from PartialDifferentialRing Symbol
differentiate: (%, NonNegativeInteger) -> % if R has FiniteFieldCategory or R has DifferentialRing and R has Field
from DifferentialRing
differentiate: (%, R -> R) -> % if R has Field
from DifferentialExtension R
differentiate: (%, R -> R, NonNegativeInteger) -> % if R has Field
from DifferentialExtension R
differentiate: (%, Symbol) -> % if R has PartialDifferentialRing Symbol and R has Field
from PartialDifferentialRing Symbol
differentiate: (%, Symbol, NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol and R has Field
from PartialDifferentialRing Symbol
discreteLog: % -> NonNegativeInteger if R has FiniteFieldCategory
from FiniteFieldCategory
discreteLog: (%, %) -> Union(NonNegativeInteger, failed) if R has FiniteFieldCategory
from FieldOfPrimeCharacteristic
discriminant: () -> R
from FramedAlgebra(R, UP)
discriminant: Vector % -> R
from FiniteRankAlgebra(R, UP)
divide: (%, %) -> Record(quotient: %, remainder: %) if R has Field
from EuclideanDomain
enumerate: () -> List % if R has Finite
from Finite
euclideanSize: % -> NonNegativeInteger if R has Field
from EuclideanDomain
expressIdealMember: (List %, %) -> Union(List %, failed) if R has Field
from PrincipalIdealDomain
exquo: (%, %) -> Union(%, failed) if R has Field
from EntireRing
extendedEuclidean: (%, %) -> Record(coef1: %, coef2: %, generator: %) if R has Field
from EuclideanDomain
extendedEuclidean: (%, %, %) -> Union(Record(coef1: %, coef2: %), failed) if R has Field
from EuclideanDomain
factor: % -> Factored % if R has Field
from UniqueFactorizationDomain
factorPolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if R has FiniteFieldCategory
from PolynomialFactorizationExplicit
factorsOfCyclicGroupSize: () -> List Record(factor: Integer, exponent: Integer) if R has FiniteFieldCategory
from FiniteFieldCategory
factorSquareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if R has FiniteFieldCategory
from PolynomialFactorizationExplicit
gcd: (%, %) -> % if R has Field
from GcdDomain
gcd: List % -> % if R has Field
from GcdDomain
gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial % if R has Field
from GcdDomain
generator: () -> %
generator() returns the generator for this domain.
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
index: PositiveInteger -> % if R has Finite
from Finite
init: % if R has FiniteFieldCategory
from StepThrough
inv: % -> % if R has Field
from DivisionRing
latex: % -> String
from SetCategory
lcm: (%, %) -> % if R has Field
from GcdDomain
lcm: List % -> % if R has Field
from GcdDomain
lcmCoef: (%, %) -> Record(llcm_res: %, coeff1: %, coeff2: %) if R has Field
from LeftOreRing
leftPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
leftPower: (%, PositiveInteger) -> %
from Magma
leftRecip: % -> Union(%, failed)
from MagmaWithUnit
lift: % -> UP
lift(z) returns a minimal degree univariate polynomial up such that z=reduce up.
lookup: % -> PositiveInteger if R has Finite
from Finite
minimalPolynomial: % -> UP if R has Field
from FiniteRankAlgebra(R, UP)
multiEuclidean: (List %, %) -> Union(List %, failed) if R has Field
from EuclideanDomain
nextItem: % -> Union(%, failed) if R has FiniteFieldCategory
from StepThrough
norm: % -> R
from FiniteRankAlgebra(R, UP)
one?: % -> Boolean
from MagmaWithUnit
opposite?: (%, %) -> Boolean
from AbelianMonoid
order: % -> OnePointCompletion PositiveInteger if R has FiniteFieldCategory
from FieldOfPrimeCharacteristic
order: % -> PositiveInteger if R has FiniteFieldCategory
from FiniteFieldCategory
prime?: % -> Boolean if R has Field
from UniqueFactorizationDomain
primeFrobenius: % -> % if R has FiniteFieldCategory
from FieldOfPrimeCharacteristic
primeFrobenius: (%, NonNegativeInteger) -> % if R has FiniteFieldCategory
from FieldOfPrimeCharacteristic
primitive?: % -> Boolean if R has FiniteFieldCategory
from FiniteFieldCategory
primitiveElement: () -> % if R has FiniteFieldCategory
from FiniteFieldCategory
principalIdeal: List % -> Record(coef: List %, generator: %) if R has Field
from PrincipalIdealDomain
quo: (%, %) -> % if R has Field
from EuclideanDomain
random: () -> % if R has Finite
from Finite
rank: () -> PositiveInteger
from FramedModule R
recip: % -> Union(%, failed)
from MagmaWithUnit
reduce: Fraction UP -> Union(%, failed) if R has Field
reduce(frac) converts the fraction frac to an algebra element.
reduce: UP -> %
reduce(up) converts the univariate polynomial up to an algebra element, reducing by the definingPolynomial() if necessary.
reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix Integer, vec: Vector Integer) if R has LinearlyExplicitOver Integer
from LinearlyExplicitOver Integer
reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix R, vec: Vector R)
from LinearlyExplicitOver R
reducedSystem: Matrix % -> Matrix Integer if R has LinearlyExplicitOver Integer
from LinearlyExplicitOver Integer
reducedSystem: Matrix % -> Matrix R
from LinearlyExplicitOver R
regularRepresentation: % -> Matrix R
from FramedAlgebra(R, UP)
regularRepresentation: (%, Vector %) -> Matrix R
from FiniteRankAlgebra(R, UP)
rem: (%, %) -> % if R has Field
from EuclideanDomain
representationType: () -> Union(prime, polynomial, normal, cyclic) if R has FiniteFieldCategory
from FiniteFieldCategory
represents: (Vector R, Vector %) -> %
from FiniteRankAlgebra(R, UP)
represents: Vector R -> %
from FramedModule R
retract: % -> Fraction Integer if R has RetractableTo Fraction Integer
from RetractableTo Fraction Integer
retract: % -> Integer if R has RetractableTo Integer
from RetractableTo Integer
retract: % -> R
from RetractableTo R
retractIfCan: % -> Union(Fraction Integer, failed) if R has RetractableTo Fraction Integer
from RetractableTo Fraction Integer
retractIfCan: % -> Union(Integer, failed) if R has RetractableTo Integer
from RetractableTo Integer
retractIfCan: % -> Union(R, failed)
from RetractableTo R
rightPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
rightPower: (%, PositiveInteger) -> %
from Magma
rightRecip: % -> Union(%, failed)
from MagmaWithUnit
sample: %
from AbelianMonoid
size: () -> NonNegativeInteger if R has Finite
from Finite
sizeLess?: (%, %) -> Boolean if R has Field
from EuclideanDomain
smaller?: (%, %) -> Boolean if R has Finite
from Comparable
solveLinearPolynomialEquation: (List SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> Union(List SparseUnivariatePolynomial %, failed) if R has FiniteFieldCategory
from PolynomialFactorizationExplicit
squareFree: % -> Factored % if R has Field
from UniqueFactorizationDomain
squareFreePart: % -> % if R has Field
from UniqueFactorizationDomain
squareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if R has FiniteFieldCategory
from PolynomialFactorizationExplicit
subtractIfCan: (%, %) -> Union(%, failed)
from CancellationAbelianMonoid
tableForDiscreteLogarithm: Integer -> Table(PositiveInteger, NonNegativeInteger) if R has FiniteFieldCategory
from FiniteFieldCategory
trace: % -> R
from FiniteRankAlgebra(R, UP)
traceMatrix: () -> Matrix R
from FramedAlgebra(R, UP)
traceMatrix: Vector % -> Matrix R
from FiniteRankAlgebra(R, UP)
unit?: % -> Boolean if R has Field
from EntireRing
unitCanonical: % -> % if R has Field
from EntireRing
unitNormal: % -> Record(unit: %, canonical: %, associate: %) if R has Field
from EntireRing
zero?: % -> Boolean
from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

Algebra %

Algebra Fraction Integer if R has Field

Algebra R

BasicType

BiModule(%, %)

BiModule(Fraction Integer, Fraction Integer) if R has Field

BiModule(R, R)

CancellationAbelianMonoid

canonicalsClosed if R has Field

canonicalUnitNormal if R has Field

CharacteristicNonZero if R has CharacteristicNonZero

CharacteristicZero if R has CharacteristicZero

CoercibleTo OutputForm

CommutativeRing

CommutativeStar

Comparable if R has Finite

ConvertibleTo InputForm if R has Finite

ConvertibleTo UP

DifferentialExtension R if R has Field

DifferentialRing if R has FiniteFieldCategory or R has DifferentialRing and R has Field

DivisionRing if R has Field

EntireRing if R has Field

EuclideanDomain if R has Field

Field if R has Field

FieldOfPrimeCharacteristic if R has FiniteFieldCategory

Finite if R has Finite

FiniteFieldCategory if R has FiniteFieldCategory

FiniteRankAlgebra(R, UP)

FramedAlgebra(R, UP)

FramedModule R

FullyLinearlyExplicitOver R

FullyRetractableTo R

GcdDomain if R has Field

IntegralDomain if R has Field

LeftModule %

LeftModule Fraction Integer if R has Field

LeftModule R

LeftOreRing if R has Field

LinearlyExplicitOver Integer if R has LinearlyExplicitOver Integer

LinearlyExplicitOver R

Magma

MagmaWithUnit

Module %

Module Fraction Integer if R has Field

Module R

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors if R has Field

PartialDifferentialRing Symbol if R has PartialDifferentialRing Symbol and R has Field

PolynomialFactorizationExplicit if R has FiniteFieldCategory

PrincipalIdealDomain if R has Field

RetractableTo Fraction Integer if R has RetractableTo Fraction Integer

RetractableTo Integer if R has RetractableTo Integer

RetractableTo R

RightModule %

RightModule Fraction Integer if R has Field

RightModule R

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

StepThrough if R has FiniteFieldCategory

UniqueFactorizationDomain if R has Field

unitsKnown