# ComplexCategory RΒΆ

This category represents the extension of a ring by a square root of `-1`

.

- 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
- ^: (%, %) -> % if R has TranscendentalFunctionCategory
- from ElementaryFunctionCategory
- ^: (%, Fraction Integer) -> % if R has RadicalCategory and R has TranscendentalFunctionCategory
- from RadicalCategory
- ^: (%, Integer) -> % if R has Field
- from DivisionRing
- ^: (%, NonNegativeInteger) -> %
- from MagmaWithUnit
- ^: (%, PositiveInteger) -> %
- from Magma
- ~=: (%, %) -> Boolean
- from BasicType

- abs: % -> % if R has RealNumberSystem
`abs(x)`

returns the absolute value of`x`

= sqrt(norm(`x`

)).- acos: % -> % if R has TranscendentalFunctionCategory
- from ArcTrigonometricFunctionCategory
- acosh: % -> % if R has TranscendentalFunctionCategory
- from ArcHyperbolicFunctionCategory
- acot: % -> % if R has TranscendentalFunctionCategory
- from ArcTrigonometricFunctionCategory
- acoth: % -> % if R has TranscendentalFunctionCategory
- from ArcHyperbolicFunctionCategory
- acsc: % -> % if R has TranscendentalFunctionCategory
- from ArcTrigonometricFunctionCategory
- acsch: % -> % if R has TranscendentalFunctionCategory
- from ArcHyperbolicFunctionCategory
- annihilate?: (%, %) -> Boolean
- from Rng
- antiCommutator: (%, %) -> %
- from NonAssociativeSemiRng

- argument: % -> R if R has TranscendentalFunctionCategory
`argument(x)`

returns the angle made by (0, 1) and (0,`x`

).- asec: % -> % if R has TranscendentalFunctionCategory
- from ArcTrigonometricFunctionCategory
- asech: % -> % if R has TranscendentalFunctionCategory
- from ArcHyperbolicFunctionCategory
- asin: % -> % if R has TranscendentalFunctionCategory
- from ArcTrigonometricFunctionCategory
- asinh: % -> % if R has TranscendentalFunctionCategory
- from ArcHyperbolicFunctionCategory
- associates?: (%, %) -> Boolean if R has IntegralDomain
- from EntireRing
- associator: (%, %, %) -> %
- from NonAssociativeRng
- atan: % -> % if R has TranscendentalFunctionCategory
- from ArcTrigonometricFunctionCategory
- atanh: % -> % if R has TranscendentalFunctionCategory
- from ArcHyperbolicFunctionCategory
- basis: () -> Vector %
- from FramedModule R
- characteristic: () -> NonNegativeInteger
- from NonAssociativeRing
- characteristicPolynomial: % -> SparseUnivariatePolynomial R
- from FiniteRankAlgebra(R, SparseUnivariatePolynomial R)
- charthRoot: % -> % if R has FiniteFieldCategory
- from FiniteFieldCategory
- charthRoot: % -> Union(%, failed) if % has CharacteristicNonZero and R has PolynomialFactorizationExplicit and R has EuclideanDomain or R has CharacteristicNonZero
- from PolynomialFactorizationExplicit
- coerce: % -> %
- from Algebra %
- coerce: % -> OutputForm
- from CoercibleTo OutputForm
- coerce: Fraction Integer -> % if R has RetractableTo Fraction Integer or R has Field
- from Algebra Fraction Integer
- coerce: Integer -> %
- from NonAssociativeRing
- coerce: R -> %
- from RetractableTo R
- commutator: (%, %) -> %
- from NonAssociativeRng

- complex: (R, R) -> %
`complex(x, y)`

constructs`x`

+ %i*y.- conditionP: Matrix % -> Union(Vector %, failed) if R has FiniteFieldCategory or % has CharacteristicNonZero and R has PolynomialFactorizationExplicit and R has EuclideanDomain
- from PolynomialFactorizationExplicit

- conjugate: % -> %
`conjugate(x + \%i y)`

returns`x`

- %`i`

`y`

.- convert: % -> Complex DoubleFloat if R has RealConstant
- from ConvertibleTo Complex DoubleFloat
- convert: % -> Complex Float if R has RealConstant
- from ConvertibleTo Complex Float
- convert: % -> InputForm if R has ConvertibleTo InputForm
- from ConvertibleTo InputForm
- convert: % -> Pattern Float if R has ConvertibleTo Pattern Float
- from ConvertibleTo Pattern Float
- convert: % -> Pattern Integer if R has ConvertibleTo Pattern Integer
- from ConvertibleTo Pattern Integer
- convert: % -> SparseUnivariatePolynomial R
- from ConvertibleTo SparseUnivariatePolynomial R
- convert: % -> Vector R
- from FramedModule R
- convert: SparseUnivariatePolynomial R -> %
- from MonogenicAlgebra(R, SparseUnivariatePolynomial R)
- convert: Vector R -> %
- from FramedModule R
- coordinates: % -> Vector R
- from FramedModule R
- coordinates: (%, Vector %) -> Vector R
- from FiniteRankAlgebra(R, SparseUnivariatePolynomial R)
- coordinates: (Vector %, Vector %) -> Matrix R
- from FiniteRankAlgebra(R, SparseUnivariatePolynomial R)
- coordinates: Vector % -> Matrix R
- from FramedModule R
- cos: % -> % if R has TranscendentalFunctionCategory
- from TrigonometricFunctionCategory
- cosh: % -> % if R has TranscendentalFunctionCategory
- from HyperbolicFunctionCategory
- cot: % -> % if R has TranscendentalFunctionCategory
- from TrigonometricFunctionCategory
- coth: % -> % if R has TranscendentalFunctionCategory
- from HyperbolicFunctionCategory
- createPrimitiveElement: () -> % if R has FiniteFieldCategory
- from FiniteFieldCategory
- csc: % -> % if R has TranscendentalFunctionCategory
- from TrigonometricFunctionCategory
- csch: % -> % if R has TranscendentalFunctionCategory
- from HyperbolicFunctionCategory
- D: % -> % if R has DifferentialRing
- from DifferentialRing
- D: (%, List Symbol) -> % if R has PartialDifferentialRing Symbol
- from PartialDifferentialRing Symbol
- D: (%, List Symbol, List NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol
- from PartialDifferentialRing Symbol
- D: (%, NonNegativeInteger) -> % if R has DifferentialRing
- from DifferentialRing
- D: (%, R -> R) -> %
- from DifferentialExtension R
- D: (%, R -> R, NonNegativeInteger) -> %
- from DifferentialExtension R
- D: (%, Symbol) -> % if R has PartialDifferentialRing Symbol
- from PartialDifferentialRing Symbol
- D: (%, Symbol, NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol
- from PartialDifferentialRing Symbol
- definingPolynomial: () -> SparseUnivariatePolynomial R
- from MonogenicAlgebra(R, SparseUnivariatePolynomial R)
- derivationCoordinates: (Vector %, R -> R) -> Matrix R if R has Field
- from MonogenicAlgebra(R, SparseUnivariatePolynomial R)
- differentiate: % -> % if R has DifferentialRing
- from DifferentialRing
- differentiate: (%, List Symbol) -> % if R has PartialDifferentialRing Symbol
- from PartialDifferentialRing Symbol
- differentiate: (%, List Symbol, List NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol
- from PartialDifferentialRing Symbol
- differentiate: (%, NonNegativeInteger) -> % if R has DifferentialRing
- from DifferentialRing
- differentiate: (%, R -> R) -> %
- from DifferentialExtension R
- differentiate: (%, R -> R, NonNegativeInteger) -> %
- from DifferentialExtension R
- differentiate: (%, Symbol) -> % if R has PartialDifferentialRing Symbol
- from PartialDifferentialRing Symbol
- differentiate: (%, Symbol, NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol
- 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, SparseUnivariatePolynomial R)
- discriminant: Vector % -> R
- from FiniteRankAlgebra(R, SparseUnivariatePolynomial R)
- divide: (%, %) -> Record(quotient: %, remainder: %) if R has EuclideanDomain
- from EuclideanDomain
- elt: (%, R) -> % if R has Eltable(R, R)
- from Eltable(R, %)
- enumerate: () -> List % if R has Finite
- from Finite
- euclideanSize: % -> NonNegativeInteger if R has EuclideanDomain
- from EuclideanDomain
- eval: (%, Equation R) -> % if R has Evalable R
- from Evalable R
- eval: (%, List Equation R) -> % if R has Evalable R
- from Evalable R
- eval: (%, List R, List R) -> % if R has Evalable R
- from InnerEvalable(R, R)
- eval: (%, List Symbol, List R) -> % if R has InnerEvalable(Symbol, R)
- from InnerEvalable(Symbol, R)
- eval: (%, R, R) -> % if R has Evalable R
- from InnerEvalable(R, R)
- eval: (%, Symbol, R) -> % if R has InnerEvalable(Symbol, R)
- from InnerEvalable(Symbol, R)
- exp: % -> % if R has TranscendentalFunctionCategory
- from ElementaryFunctionCategory
- expressIdealMember: (List %, %) -> Union(List %, failed) if R has EuclideanDomain
- from PrincipalIdealDomain
- exquo: (%, %) -> Union(%, failed) if R has IntegralDomain
- from EntireRing

- exquo: (%, R) -> Union(%, failed) if R has IntegralDomain
`exquo(x, r)`

returns the exact quotient of`x`

by`r`

, or “failed” if`r`

does not divide`x`

exactly.- extendedEuclidean: (%, %) -> Record(coef1: %, coef2: %, generator: %) if R has EuclideanDomain
- from EuclideanDomain
- extendedEuclidean: (%, %, %) -> Union(Record(coef1: %, coef2: %), failed) if R has EuclideanDomain
- from EuclideanDomain
- factor: % -> Factored % if R has Field or R has IntegerNumberSystem or R has PolynomialFactorizationExplicit and R has EuclideanDomain
- from UniqueFactorizationDomain
- factorPolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if R has FiniteFieldCategory or R has PolynomialFactorizationExplicit and R has EuclideanDomain
- from PolynomialFactorizationExplicit
- factorsOfCyclicGroupSize: () -> List Record(factor: Integer, exponent: Integer) if R has FiniteFieldCategory
- from FiniteFieldCategory
- factorSquareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if R has FiniteFieldCategory or R has PolynomialFactorizationExplicit and R has EuclideanDomain
- from PolynomialFactorizationExplicit
- gcd: (%, %) -> % if R has EuclideanDomain
- from GcdDomain
- gcd: List % -> % if R has EuclideanDomain
- from GcdDomain
- gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial % if R has EuclideanDomain
- from PolynomialFactorizationExplicit
- generator: () -> %
- from MonogenicAlgebra(R, SparseUnivariatePolynomial R)
- hash: % -> SingleInteger
- from SetCategory
- hashUpdate!: (HashState, %) -> HashState
- from SetCategory

- imag: % -> R
`imag(x)`

returns imaginary part of`x`

.

- imaginary: () -> %
`imaginary()`

= sqrt(`-1`

) = %`i`

.- 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 EuclideanDomain
- from GcdDomain
- lcm: List % -> % if R has EuclideanDomain
- from GcdDomain
- lcmCoef: (%, %) -> Record(llcm_res: %, coeff1: %, coeff2: %) if R has EuclideanDomain
- from LeftOreRing
- leftPower: (%, NonNegativeInteger) -> %
- from MagmaWithUnit
- leftPower: (%, PositiveInteger) -> %
- from Magma
- leftRecip: % -> Union(%, failed)
- from MagmaWithUnit
- lift: % -> SparseUnivariatePolynomial R
- from MonogenicAlgebra(R, SparseUnivariatePolynomial R)
- log: % -> % if R has TranscendentalFunctionCategory
- from ElementaryFunctionCategory
- lookup: % -> PositiveInteger if R has Finite
- from Finite
- map: (R -> R, %) -> %
- from FullyEvalableOver R
- minimalPolynomial: % -> SparseUnivariatePolynomial R if R has Field
- from FiniteRankAlgebra(R, SparseUnivariatePolynomial R)
- multiEuclidean: (List %, %) -> Union(List %, failed) if R has EuclideanDomain
- from EuclideanDomain
- nextItem: % -> Union(%, failed) if R has FiniteFieldCategory
- from StepThrough

- norm: % -> R
`norm(x)`

returns`x`

* conjugate(`x`

)- nthRoot: (%, Integer) -> % if R has RadicalCategory and R has TranscendentalFunctionCategory
- from RadicalCategory
- 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
- patternMatch: (%, Pattern Float, PatternMatchResult(Float, %)) -> PatternMatchResult(Float, %) if R has PatternMatchable Float
- from PatternMatchable Float
- patternMatch: (%, Pattern Integer, PatternMatchResult(Integer, %)) -> PatternMatchResult(Integer, %) if R has PatternMatchable Integer
- from PatternMatchable Integer
- pi: () -> % if R has TranscendentalFunctionCategory
- from TranscendentalFunctionCategory

- polarCoordinates: % -> Record(r: R, phi: R) if R has RealNumberSystem and R has TranscendentalFunctionCategory
`polarCoordinates(x)`

returns (`r`

, phi) such that`x`

=`r`

* exp(%`i`

* phi).- prime?: % -> Boolean if R has Field or R has IntegerNumberSystem or R has PolynomialFactorizationExplicit and R has EuclideanDomain
- 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 EuclideanDomain
- from PrincipalIdealDomain
- quo: (%, %) -> % if R has EuclideanDomain
- from EuclideanDomain
- random: () -> % if R has Finite
- from Finite
- rank: () -> PositiveInteger
- from FiniteRankAlgebra(R, SparseUnivariatePolynomial R)

- rational: % -> Fraction Integer if R has IntegerNumberSystem
`rational(x)`

returns`x`

as a rational number. Error: if`x`

is not a rational number.

- rational?: % -> Boolean if R has IntegerNumberSystem
`rational?(x)`

tests if`x`

is a rational number.

- rationalIfCan: % -> Union(Fraction Integer, failed) if R has IntegerNumberSystem
`rationalIfCan(x)`

returns`x`

as a rational number, or “failed” if`x`

is not a rational number.

- real: % -> R
`real(x)`

returns real part of`x`

.- recip: % -> Union(%, failed)
- from MagmaWithUnit
- reduce: Fraction SparseUnivariatePolynomial R -> Union(%, failed) if R has Field
- from MonogenicAlgebra(R, SparseUnivariatePolynomial R)
- reduce: SparseUnivariatePolynomial R -> %
- from MonogenicAlgebra(R, SparseUnivariatePolynomial R)
- 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, SparseUnivariatePolynomial R)
- regularRepresentation: (%, Vector %) -> Matrix R
- from FiniteRankAlgebra(R, SparseUnivariatePolynomial R)
- rem: (%, %) -> % if R has EuclideanDomain
- from EuclideanDomain
- representationType: () -> Union(prime, polynomial, normal, cyclic) if R has FiniteFieldCategory
- from FiniteFieldCategory
- represents: (Vector R, Vector %) -> %
- from FiniteRankAlgebra(R, SparseUnivariatePolynomial R)
- 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
- sec: % -> % if R has TranscendentalFunctionCategory
- from TrigonometricFunctionCategory
- sech: % -> % if R has TranscendentalFunctionCategory
- from HyperbolicFunctionCategory
- sin: % -> % if R has TranscendentalFunctionCategory
- from TrigonometricFunctionCategory
- sinh: % -> % if R has TranscendentalFunctionCategory
- from HyperbolicFunctionCategory
- size: () -> NonNegativeInteger if R has Finite
- from Finite
- sizeLess?: (%, %) -> Boolean if R has EuclideanDomain
- from EuclideanDomain
- smaller?: (%, %) -> Boolean if R has Comparable
- from Comparable
- solveLinearPolynomialEquation: (List SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> Union(List SparseUnivariatePolynomial %, failed) if R has FiniteFieldCategory or R has PolynomialFactorizationExplicit and R has EuclideanDomain
- from PolynomialFactorizationExplicit
- sqrt: % -> % if R has RadicalCategory and R has TranscendentalFunctionCategory
- from RadicalCategory
- squareFree: % -> Factored % if R has Field or R has IntegerNumberSystem or R has PolynomialFactorizationExplicit and R has EuclideanDomain
- from UniqueFactorizationDomain
- squareFreePart: % -> % if R has Field or R has IntegerNumberSystem or R has PolynomialFactorizationExplicit and R has EuclideanDomain
- from UniqueFactorizationDomain
- squareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if R has FiniteFieldCategory or R has PolynomialFactorizationExplicit and R has EuclideanDomain
- from PolynomialFactorizationExplicit
- subtractIfCan: (%, %) -> Union(%, failed)
- from CancellationAbelianMonoid
- tableForDiscreteLogarithm: Integer -> Table(PositiveInteger, NonNegativeInteger) if R has FiniteFieldCategory
- from FiniteFieldCategory
- tan: % -> % if R has TranscendentalFunctionCategory
- from TrigonometricFunctionCategory
- tanh: % -> % if R has TranscendentalFunctionCategory
- from HyperbolicFunctionCategory
- trace: % -> R
- from FiniteRankAlgebra(R, SparseUnivariatePolynomial R)
- traceMatrix: () -> Matrix R
- from FramedAlgebra(R, SparseUnivariatePolynomial R)
- traceMatrix: Vector % -> Matrix R
- from FiniteRankAlgebra(R, SparseUnivariatePolynomial R)
- unit?: % -> Boolean if R has IntegralDomain
- from EntireRing
- unitCanonical: % -> % if R has IntegralDomain
- from EntireRing
- unitNormal: % -> Record(unit: %, canonical: %, associate: %) if R has IntegralDomain
- from EntireRing
- zero?: % -> Boolean
- from AbelianMonoid

additiveValuation if R has additiveValuation

Algebra %

Algebra Fraction Integer if R has Field

Algebra R

arbitraryPrecision if R has arbitraryPrecision

ArcHyperbolicFunctionCategory if R has TranscendentalFunctionCategory

ArcTrigonometricFunctionCategory if R has TranscendentalFunctionCategory

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

Comparable if R has Comparable

ConvertibleTo Complex DoubleFloat if R has RealConstant

ConvertibleTo Complex Float if R has RealConstant

ConvertibleTo InputForm if R has ConvertibleTo InputForm

ConvertibleTo Pattern Float if R has ConvertibleTo Pattern Float

ConvertibleTo Pattern Integer if R has ConvertibleTo Pattern Integer

ConvertibleTo SparseUnivariatePolynomial R

DifferentialRing if R has DifferentialRing

DivisionRing if R has Field

ElementaryFunctionCategory if R has TranscendentalFunctionCategory

Eltable(R, %) if R has Eltable(R, R)

EntireRing if R has IntegralDomain

EuclideanDomain if R has EuclideanDomain

Evalable R if R has Evalable R

FieldOfPrimeCharacteristic if R has FiniteFieldCategory

FiniteFieldCategory if R has FiniteFieldCategory

FiniteRankAlgebra(R, SparseUnivariatePolynomial R)

FramedAlgebra(R, SparseUnivariatePolynomial R)

GcdDomain if R has EuclideanDomain

HyperbolicFunctionCategory if R has TranscendentalFunctionCategory

InnerEvalable(R, R) if R has Evalable R

InnerEvalable(Symbol, R) if R has InnerEvalable(Symbol, R)

IntegralDomain if R has IntegralDomain

LeftModule Fraction Integer if R has Field

LeftOreRing if R has EuclideanDomain

LinearlyExplicitOver Integer if R has LinearlyExplicitOver Integer

Module %

Module Fraction Integer if R has Field

Module R

MonogenicAlgebra(R, SparseUnivariatePolynomial R)

multiplicativeValuation if R has multiplicativeValuation

noZeroDivisors if R has IntegralDomain

PartialDifferentialRing Symbol if R has PartialDifferentialRing Symbol

PatternMatchable Float if R has PatternMatchable Float

PatternMatchable Integer if R has PatternMatchable Integer

PolynomialFactorizationExplicit if R has FiniteFieldCategory or R has PolynomialFactorizationExplicit and R has EuclideanDomain

PrincipalIdealDomain if R has EuclideanDomain

RadicalCategory if R has RadicalCategory and R has TranscendentalFunctionCategory

RetractableTo Fraction Integer if R has RetractableTo Fraction Integer

RetractableTo Integer if R has RetractableTo Integer

RightModule Fraction Integer if R has Field

StepThrough if R has FiniteFieldCategory

TranscendentalFunctionCategory if R has TranscendentalFunctionCategory

TrigonometricFunctionCategory if R has TranscendentalFunctionCategory

UniqueFactorizationDomain if R has Field or R has PolynomialFactorizationExplicit and R has EuclideanDomain or R has IntegerNumberSystem