# MonogenicAlgebra(R, UP)ΒΆ

algcat.spad line 217 [edit on github]

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

- *: (%, Integer) -> % if R has LinearlyExplicitOver Integer
from RightModule 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

- ^: (%, Integer) -> % if R has Field
from DivisionRing

- ^: (%, NonNegativeInteger) -> %
from MagmaWithUnit

- ^: (%, PositiveInteger) -> %
from Magma

- annihilate?: (%, %) -> Boolean
from Rng

- antiCommutator: (%, %) -> %

- 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

- coerce: % -> %
from Algebra %

- coerce: % -> OutputForm
from CoercibleTo OutputForm

- coerce: Fraction Integer -> % if R has Field or R has RetractableTo Fraction Integer
- coerce: Integer -> %
from NonAssociativeRing

- coerce: R -> %
from Algebra R

- commutator: (%, %) -> %
from NonAssociativeRng

- conditionP: Matrix % -> Union(Vector %, failed) if R has FiniteFieldCategory

- 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 DifferentialRing and R has Field or R has FiniteFieldCategory
from DifferentialRing

- D: (%, List Symbol) -> % if R has PartialDifferentialRing Symbol and R has Field
- D: (%, List Symbol, List NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol and R has Field
- D: (%, NonNegativeInteger) -> % if R has DifferentialRing and R has Field or R has FiniteFieldCategory
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
- D: (%, Symbol, NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol and R has Field

- 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 DifferentialRing and R has Field or R has FiniteFieldCategory
from DifferentialRing

- differentiate: (%, List Symbol) -> % if R has PartialDifferentialRing Symbol and R has Field
- differentiate: (%, List Symbol, List NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol and R has Field
- differentiate: (%, NonNegativeInteger) -> % if R has DifferentialRing and R has Field or R has FiniteFieldCategory
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
- differentiate: (%, Symbol, NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol and R has Field

- discreteLog: % -> NonNegativeInteger if R has FiniteFieldCategory
from FiniteFieldCategory

- discreteLog: (%, %) -> Union(NonNegativeInteger, failed) if R has FiniteFieldCategory

- discriminant: () -> R
from FramedAlgebra(R, UP)

- discriminant: Vector % -> R
from FiniteRankAlgebra(R, UP)

- divide: (%, %) -> Record(quotient: %, remainder: %) if R has Field
from EuclideanDomain

- 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

- factorPolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if R has FiniteFieldCategory

- factorsOfCyclicGroupSize: () -> List Record(factor: Integer, exponent: NonNegativeInteger) if R has FiniteFieldCategory
from FiniteFieldCategory

- factorSquareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if R has FiniteFieldCategory

- gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial % if R has Field
from GcdDomain

- generator: () -> %
`generator()`

returns the generator for this domain.

- hash: % -> SingleInteger if R has Hashable
from Hashable

- hashUpdate!: (HashState, %) -> HashState if R has Hashable
from Hashable

- 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

- 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
- order: % -> PositiveInteger if R has FiniteFieldCategory
from FiniteFieldCategory

- plenaryPower: (%, PositiveInteger) -> %
from NonAssociativeAlgebra %

- primeFrobenius: % -> % if R has FiniteFieldCategory
- primeFrobenius: (%, NonNegativeInteger) -> % if R has FiniteFieldCategory

- 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

- rank: () -> PositiveInteger
from FiniteRankAlgebra(R, UP)

- 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
- reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix R, vec: Vector R)
from LinearlyExplicitOver R

- reducedSystem: Matrix % -> Matrix Integer if R has 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

- squareFree: % -> Factored % if R has Field

- squareFreePart: % -> % if R has Field

- squareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if R has FiniteFieldCategory

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

- 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

Algebra %

Algebra Fraction Integer if R has Field

Algebra R

BiModule(%, %)

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

BiModule(R, R)

canonicalsClosed if R has Field

canonicalUnitNormal if R has Field

CharacteristicNonZero if R has CharacteristicNonZero

CharacteristicZero if R has CharacteristicZero

CoercibleFrom Fraction Integer if R has RetractableTo Fraction Integer

CoercibleFrom Integer if R has RetractableTo Integer

Comparable if R has Finite

ConvertibleTo InputForm if R has Finite

DifferentialExtension R if R has Field

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

DivisionRing if R has Field

EntireRing if R has Field

EuclideanDomain if R has Field

FieldOfPrimeCharacteristic if R has FiniteFieldCategory

FiniteFieldCategory if R has FiniteFieldCategory

FiniteRankAlgebra(R, UP)

FramedAlgebra(R, UP)

IntegralDomain if R has Field

LeftModule Fraction Integer if R has Field

LeftOreRing if R has Field

LinearlyExplicitOver Integer if R has LinearlyExplicitOver Integer

Module %

Module Fraction Integer if R has Field

Module R

NonAssociativeAlgebra Fraction Integer if R has Field

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

RightModule Fraction Integer if R has Field

RightModule Integer if R has LinearlyExplicitOver Integer

StepThrough if R has FiniteFieldCategory

UniqueFactorizationDomain if R has Field