QuotientFieldCategory SΒΆ

fraction.spad line 86

QuotientField(S) is the category of fractions of an Integral Domain S.

0: %
from AbelianMonoid
1: %
from MagmaWithUnit
*: (%, %) -> %
from Magma
*: (%, Fraction Integer) -> %
from RightModule Fraction Integer
*: (%, S) -> %
from RightModule S
*: (Fraction Integer, %) -> %
from LeftModule Fraction Integer
*: (Integer, %) -> %
from AbelianGroup
*: (NonNegativeInteger, %) -> %
from AbelianMonoid
*: (PositiveInteger, %) -> %
from AbelianSemiGroup
*: (S, %) -> %
from LeftModule S
+: (%, %) -> %
from AbelianSemiGroup
-: % -> %
from AbelianGroup
-: (%, %) -> %
from AbelianGroup
/: (%, %) -> %
from Field
/: (S, S) -> %
d1 / d2 returns the fraction d1 divided by d2.
<: (%, %) -> Boolean if S has OrderedSet
from PartialOrder
<=: (%, %) -> Boolean if S has OrderedSet
from PartialOrder
=: (%, %) -> Boolean
from BasicType
>: (%, %) -> Boolean if S has OrderedSet
from PartialOrder
>=: (%, %) -> Boolean if S has OrderedSet
from PartialOrder
^: (%, Integer) -> %
from DivisionRing
^: (%, NonNegativeInteger) -> %
from MagmaWithUnit
^: (%, PositiveInteger) -> %
from Magma
~=: (%, %) -> Boolean
from BasicType
abs: % -> % if S has OrderedIntegralDomain
from OrderedRing
annihilate?: (%, %) -> Boolean
from Rng
antiCommutator: (%, %) -> %
from NonAssociativeSemiRng
associates?: (%, %) -> Boolean
from EntireRing
associator: (%, %, %) -> %
from NonAssociativeRng
ceiling: % -> S if S has IntegerNumberSystem
ceiling(x) returns the smallest integral element above x.
characteristic: () -> NonNegativeInteger
from NonAssociativeRing
charthRoot: % -> Union(%, failed) if S has CharacteristicNonZero or % has CharacteristicNonZero and S has PolynomialFactorizationExplicit
from PolynomialFactorizationExplicit
coerce: % -> %
from Algebra %
coerce: % -> OutputForm
from CoercibleTo OutputForm
coerce: Fraction Integer -> %
from RetractableTo Fraction Integer
coerce: Integer -> %
from NonAssociativeRing
coerce: S -> %
from RetractableTo S
coerce: Symbol -> % if S has RetractableTo Symbol
from RetractableTo Symbol
commutator: (%, %) -> %
from NonAssociativeRng
conditionP: Matrix % -> Union(Vector %, failed) if % has CharacteristicNonZero and S has PolynomialFactorizationExplicit
from PolynomialFactorizationExplicit
convert: % -> DoubleFloat if S has RealConstant
from ConvertibleTo DoubleFloat
convert: % -> Float if S has RealConstant
from ConvertibleTo Float
convert: % -> InputForm if S has ConvertibleTo InputForm
from ConvertibleTo InputForm
convert: % -> Pattern Float if S has ConvertibleTo Pattern Float
from ConvertibleTo Pattern Float
convert: % -> Pattern Integer if S has ConvertibleTo Pattern Integer
from ConvertibleTo Pattern Integer
D: % -> % if S has DifferentialRing
from DifferentialRing
D: (%, List Symbol) -> % if S has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
D: (%, List Symbol, List NonNegativeInteger) -> % if S has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
D: (%, NonNegativeInteger) -> % if S has DifferentialRing
from DifferentialRing
D: (%, S -> S) -> %
from DifferentialExtension S
D: (%, S -> S, NonNegativeInteger) -> %
from DifferentialExtension S
D: (%, Symbol) -> % if S has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
D: (%, Symbol, NonNegativeInteger) -> % if S has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
denom: % -> S
denom(x) returns the denominator of the fraction x.
denominator: % -> %
denominator(x) is the denominator of the fraction x converted to %.
differentiate: % -> % if S has DifferentialRing
from DifferentialRing
differentiate: (%, List Symbol) -> % if S has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
differentiate: (%, List Symbol, List NonNegativeInteger) -> % if S has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
differentiate: (%, NonNegativeInteger) -> % if S has DifferentialRing
from DifferentialRing
differentiate: (%, S -> S) -> %
from DifferentialExtension S
differentiate: (%, S -> S, NonNegativeInteger) -> %
from DifferentialExtension S
differentiate: (%, Symbol) -> % if S has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
differentiate: (%, Symbol, NonNegativeInteger) -> % if S has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
divide: (%, %) -> Record(quotient: %, remainder: %)
from EuclideanDomain
elt: (%, S) -> % if S has Eltable(S, S)
from Eltable(S, %)
euclideanSize: % -> NonNegativeInteger
from EuclideanDomain
eval: (%, Equation S) -> % if S has Evalable S
from Evalable S
eval: (%, List Equation S) -> % if S has Evalable S
from Evalable S
eval: (%, List S, List S) -> % if S has Evalable S
from InnerEvalable(S, S)
eval: (%, List Symbol, List S) -> % if S has InnerEvalable(Symbol, S)
from InnerEvalable(Symbol, S)
eval: (%, S, S) -> % if S has Evalable S
from InnerEvalable(S, S)
eval: (%, Symbol, S) -> % if S has InnerEvalable(Symbol, S)
from InnerEvalable(Symbol, S)
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
factorPolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if S has PolynomialFactorizationExplicit
from PolynomialFactorizationExplicit
factorSquareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if S has PolynomialFactorizationExplicit
from PolynomialFactorizationExplicit
floor: % -> S if S has IntegerNumberSystem
floor(x) returns the largest integral element below x.
fractionPart: % -> % if S has EuclideanDomain
fractionPart(x) returns the fractional part of x. x = wholePart(x) + fractionPart(x)
gcd: (%, %) -> %
from GcdDomain
gcd: List % -> %
from GcdDomain
gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial %
from PolynomialFactorizationExplicit
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
init: % if S has StepThrough
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
map: (S -> S, %) -> %
from FullyEvalableOver S
max: (%, %) -> % if S has OrderedSet
from OrderedSet
min: (%, %) -> % if S has OrderedSet
from OrderedSet
multiEuclidean: (List %, %) -> Union(List %, failed)
from EuclideanDomain
negative?: % -> Boolean if S has OrderedIntegralDomain
from OrderedRing
nextItem: % -> Union(%, failed) if S has StepThrough
from StepThrough
numer: % -> S
numer(x) returns the numerator of the fraction x.
numerator: % -> %
numerator(x) is the numerator of the fraction x converted to %.
one?: % -> Boolean
from MagmaWithUnit
opposite?: (%, %) -> Boolean
from AbelianMonoid
patternMatch: (%, Pattern Float, PatternMatchResult(Float, %)) -> PatternMatchResult(Float, %) if S has PatternMatchable Float
from PatternMatchable Float
patternMatch: (%, Pattern Integer, PatternMatchResult(Integer, %)) -> PatternMatchResult(Integer, %) if S has PatternMatchable Integer
from PatternMatchable Integer
positive?: % -> Boolean if S has OrderedIntegralDomain
from OrderedRing
prime?: % -> Boolean
from UniqueFactorizationDomain
principalIdeal: List % -> Record(coef: List %, generator: %)
from PrincipalIdealDomain
quo: (%, %) -> %
from EuclideanDomain
recip: % -> Union(%, failed)
from MagmaWithUnit
reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix Integer, vec: Vector Integer) if S has LinearlyExplicitOver Integer
from LinearlyExplicitOver Integer
reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix S, vec: Vector S)
from LinearlyExplicitOver S
reducedSystem: Matrix % -> Matrix Integer if S has LinearlyExplicitOver Integer
from LinearlyExplicitOver Integer
reducedSystem: Matrix % -> Matrix S
from LinearlyExplicitOver S
rem: (%, %) -> %
from EuclideanDomain
retract: % -> Fraction Integer if S has RetractableTo Integer
from RetractableTo Fraction Integer
retract: % -> Integer if S has RetractableTo Integer
from RetractableTo Integer
retract: % -> S
from RetractableTo S
retract: % -> Symbol if S has RetractableTo Symbol
from RetractableTo Symbol
retractIfCan: % -> Union(Fraction Integer, failed) if S has RetractableTo Integer
from RetractableTo Fraction Integer
retractIfCan: % -> Union(Integer, failed) if S has RetractableTo Integer
from RetractableTo Integer
retractIfCan: % -> Union(S, failed)
from RetractableTo S
retractIfCan: % -> Union(Symbol, failed) if S has RetractableTo Symbol
from RetractableTo Symbol
rightPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
rightPower: (%, PositiveInteger) -> %
from Magma
rightRecip: % -> Union(%, failed)
from MagmaWithUnit
sample: %
from AbelianMonoid
sign: % -> Integer if S has OrderedIntegralDomain
from OrderedRing
sizeLess?: (%, %) -> Boolean
from EuclideanDomain
smaller?: (%, %) -> Boolean if S has Comparable
from Comparable
solveLinearPolynomialEquation: (List SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> Union(List SparseUnivariatePolynomial %, failed) if S has PolynomialFactorizationExplicit
from PolynomialFactorizationExplicit
squareFree: % -> Factored %
from UniqueFactorizationDomain
squareFreePart: % -> %
from UniqueFactorizationDomain
squareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if S has PolynomialFactorizationExplicit
from PolynomialFactorizationExplicit
subtractIfCan: (%, %) -> Union(%, failed)
from CancellationAbelianMonoid
unit?: % -> Boolean
from EntireRing
unitCanonical: % -> %
from EntireRing
unitNormal: % -> Record(unit: %, canonical: %, associate: %)
from EntireRing
wholePart: % -> S if S has EuclideanDomain
wholePart(x) returns the whole part of the fraction x i.e. the truncated quotient of the numerator by the denominator.
zero?: % -> Boolean
from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

Algebra %

Algebra Fraction Integer

Algebra S

BasicType

BiModule(%, %)

BiModule(Fraction Integer, Fraction Integer)

BiModule(S, S)

CancellationAbelianMonoid

canonicalsClosed

canonicalUnitNormal

CharacteristicNonZero if S has CharacteristicNonZero

CharacteristicZero if S has CharacteristicZero

CoercibleTo OutputForm

CommutativeRing

CommutativeStar

Comparable if S has Comparable

ConvertibleTo DoubleFloat if S has RealConstant

ConvertibleTo Float if S has RealConstant

ConvertibleTo InputForm if S has ConvertibleTo InputForm

ConvertibleTo Pattern Float if S has ConvertibleTo Pattern Float

ConvertibleTo Pattern Integer if S has ConvertibleTo Pattern Integer

DifferentialExtension S

DifferentialRing if S has DifferentialRing

DivisionRing

Eltable(S, %) if S has Eltable(S, S)

EntireRing

EuclideanDomain

Evalable S if S has Evalable S

Field

FullyEvalableOver S

FullyLinearlyExplicitOver S

FullyPatternMatchable S

GcdDomain

InnerEvalable(S, S) if S has Evalable S

InnerEvalable(Symbol, S) if S has InnerEvalable(Symbol, S)

IntegralDomain

LeftModule %

LeftModule Fraction Integer

LeftModule S

LeftOreRing

LinearlyExplicitOver Integer if S has LinearlyExplicitOver Integer

LinearlyExplicitOver S

Magma

MagmaWithUnit

Module %

Module Fraction Integer

Module S

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors

OrderedAbelianGroup if S has OrderedIntegralDomain

OrderedAbelianMonoid if S has OrderedIntegralDomain

OrderedAbelianSemiGroup if S has OrderedIntegralDomain

OrderedCancellationAbelianMonoid if S has OrderedIntegralDomain

OrderedIntegralDomain if S has OrderedIntegralDomain

OrderedRing if S has OrderedIntegralDomain

OrderedSet if S has OrderedSet

PartialDifferentialRing Symbol if S has PartialDifferentialRing Symbol

PartialOrder if S has OrderedSet

Patternable S

PatternMatchable Float if S has PatternMatchable Float

PatternMatchable Integer if S has PatternMatchable Integer

PolynomialFactorizationExplicit if S has PolynomialFactorizationExplicit

PrincipalIdealDomain

RealConstant if S has RealConstant

RetractableTo Fraction Integer if S has RetractableTo Integer

RetractableTo Integer if S has RetractableTo Integer

RetractableTo S

RetractableTo Symbol if S has RetractableTo Symbol

RightModule %

RightModule Fraction Integer

RightModule S

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

StepThrough if S has StepThrough

UniqueFactorizationDomain

unitsKnown