PAdicRational pΒΆ

padic.spad line 508

Stream-based implementation of Qp: numbers are represented as sum(i = k.., a[i] * p^i) where the a[i] lie in 0, 1, ..., (p - 1).

0: %
from AbelianMonoid
1: %
from MagmaWithUnit
*: (%, %) -> %
from Magma
*: (%, Fraction Integer) -> %
from RightModule Fraction Integer
*: (%, PAdicInteger p) -> %
from RightModule PAdicInteger p
*: (Fraction Integer, %) -> %
from LeftModule Fraction Integer
*: (Integer, %) -> %
from AbelianGroup
*: (NonNegativeInteger, %) -> %
from AbelianMonoid
*: (PAdicInteger p, %) -> %
from LeftModule PAdicInteger p
*: (PositiveInteger, %) -> %
from AbelianSemiGroup
+: (%, %) -> %
from AbelianSemiGroup
-: % -> %
from AbelianGroup
-: (%, %) -> %
from AbelianGroup
/: (%, %) -> %
from Field
/: (PAdicInteger p, PAdicInteger p) -> %
from QuotientFieldCategory PAdicInteger p
<: (%, %) -> Boolean if PAdicInteger p has OrderedSet
from PartialOrder
<=: (%, %) -> Boolean if PAdicInteger p has OrderedSet
from PartialOrder
=: (%, %) -> Boolean
from BasicType
>: (%, %) -> Boolean if PAdicInteger p has OrderedSet
from PartialOrder
>=: (%, %) -> Boolean if PAdicInteger p has OrderedSet
from PartialOrder
^: (%, Integer) -> %
from DivisionRing
^: (%, NonNegativeInteger) -> %
from MagmaWithUnit
^: (%, PositiveInteger) -> %
from Magma
~=: (%, %) -> Boolean
from BasicType
abs: % -> % if PAdicInteger p has OrderedIntegralDomain
from OrderedRing
annihilate?: (%, %) -> Boolean
from Rng
antiCommutator: (%, %) -> %
from NonAssociativeSemiRng

approximate: (%, Integer) -> Fraction Integer

associates?: (%, %) -> Boolean
from EntireRing
associator: (%, %, %) -> %
from NonAssociativeRng
ceiling: % -> PAdicInteger p if PAdicInteger p has IntegerNumberSystem
from QuotientFieldCategory PAdicInteger p
characteristic: () -> NonNegativeInteger
from NonAssociativeRing
charthRoot: % -> Union(%, failed) if PAdicInteger p has CharacteristicNonZero
from CharacteristicNonZero
coerce: % -> %
from Algebra %
coerce: % -> OutputForm
from CoercibleTo OutputForm
coerce: Fraction Integer -> %
from RetractableTo Fraction Integer
coerce: Integer -> %
from NonAssociativeRing
coerce: PAdicInteger p -> %
from Algebra PAdicInteger p
coerce: Symbol -> % if PAdicInteger p has RetractableTo Symbol
from RetractableTo Symbol
commutator: (%, %) -> %
from NonAssociativeRng

continuedFraction: % -> ContinuedFraction Fraction Integer

convert: % -> DoubleFloat if PAdicInteger p has RealConstant
from ConvertibleTo DoubleFloat
convert: % -> Float if PAdicInteger p has RealConstant
from ConvertibleTo Float
convert: % -> InputForm if PAdicInteger p has ConvertibleTo InputForm
from ConvertibleTo InputForm
convert: % -> Pattern Float if PAdicInteger p has ConvertibleTo Pattern Float
from ConvertibleTo Pattern Float
convert: % -> Pattern Integer if PAdicInteger p has ConvertibleTo Pattern Integer
from ConvertibleTo Pattern Integer
D: % -> % if PAdicInteger p has DifferentialRing
from DifferentialRing
D: (%, List Symbol) -> % if PAdicInteger p has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
D: (%, List Symbol, List NonNegativeInteger) -> % if PAdicInteger p has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
D: (%, NonNegativeInteger) -> % if PAdicInteger p has DifferentialRing
from DifferentialRing
D: (%, PAdicInteger p -> PAdicInteger p) -> %
from DifferentialExtension PAdicInteger p
D: (%, PAdicInteger p -> PAdicInteger p, NonNegativeInteger) -> %
from DifferentialExtension PAdicInteger p
D: (%, Symbol) -> % if PAdicInteger p has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
D: (%, Symbol, NonNegativeInteger) -> % if PAdicInteger p has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
denom: % -> PAdicInteger p
from QuotientFieldCategory PAdicInteger p
denominator: % -> %
from QuotientFieldCategory PAdicInteger p
differentiate: % -> % if PAdicInteger p has DifferentialRing
from DifferentialRing
differentiate: (%, List Symbol) -> % if PAdicInteger p has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
differentiate: (%, List Symbol, List NonNegativeInteger) -> % if PAdicInteger p has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
differentiate: (%, NonNegativeInteger) -> % if PAdicInteger p has DifferentialRing
from DifferentialRing
differentiate: (%, PAdicInteger p -> PAdicInteger p) -> %
from DifferentialExtension PAdicInteger p
differentiate: (%, PAdicInteger p -> PAdicInteger p, NonNegativeInteger) -> %
from DifferentialExtension PAdicInteger p
differentiate: (%, Symbol) -> % if PAdicInteger p has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
differentiate: (%, Symbol, NonNegativeInteger) -> % if PAdicInteger p has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
divide: (%, %) -> Record(quotient: %, remainder: %)
from EuclideanDomain
elt: (%, PAdicInteger p) -> % if PAdicInteger p has Eltable(PAdicInteger p, PAdicInteger p)
from Eltable(PAdicInteger p, %)
euclideanSize: % -> NonNegativeInteger
from EuclideanDomain
eval: (%, Equation PAdicInteger p) -> % if PAdicInteger p has Evalable PAdicInteger p
from Evalable PAdicInteger p
eval: (%, List Equation PAdicInteger p) -> % if PAdicInteger p has Evalable PAdicInteger p
from Evalable PAdicInteger p
eval: (%, List PAdicInteger p, List PAdicInteger p) -> % if PAdicInteger p has Evalable PAdicInteger p
from InnerEvalable(PAdicInteger p, PAdicInteger p)
eval: (%, List Symbol, List PAdicInteger p) -> % if PAdicInteger p has InnerEvalable(Symbol, PAdicInteger p)
from InnerEvalable(Symbol, PAdicInteger p)
eval: (%, PAdicInteger p, PAdicInteger p) -> % if PAdicInteger p has Evalable PAdicInteger p
from InnerEvalable(PAdicInteger p, PAdicInteger p)
eval: (%, Symbol, PAdicInteger p) -> % if PAdicInteger p has InnerEvalable(Symbol, PAdicInteger p)
from InnerEvalable(Symbol, PAdicInteger p)
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 PAdicInteger p has PolynomialFactorizationExplicit
from PolynomialFactorizationExplicit
factorSquareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if PAdicInteger p has PolynomialFactorizationExplicit
from PolynomialFactorizationExplicit
floor: % -> PAdicInteger p if PAdicInteger p has IntegerNumberSystem
from QuotientFieldCategory PAdicInteger p
fractionPart: % -> %
from QuotientFieldCategory PAdicInteger p
gcd: (%, %) -> %
from GcdDomain
gcd: List % -> %
from GcdDomain
gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial %
from PolynomialFactorizationExplicit
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
init: % if PAdicInteger p 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: (PAdicInteger p -> PAdicInteger p, %) -> %
from FullyEvalableOver PAdicInteger p
max: (%, %) -> % if PAdicInteger p has OrderedSet
from OrderedSet
min: (%, %) -> % if PAdicInteger p has OrderedSet
from OrderedSet
multiEuclidean: (List %, %) -> Union(List %, failed)
from EuclideanDomain
negative?: % -> Boolean if PAdicInteger p has OrderedIntegralDomain
from OrderedRing
nextItem: % -> Union(%, failed) if PAdicInteger p has StepThrough
from StepThrough
numer: % -> PAdicInteger p
from QuotientFieldCategory PAdicInteger p
numerator: % -> %
from QuotientFieldCategory PAdicInteger p
one?: % -> Boolean
from MagmaWithUnit
opposite?: (%, %) -> Boolean
from AbelianMonoid
patternMatch: (%, Pattern Float, PatternMatchResult(Float, %)) -> PatternMatchResult(Float, %) if PAdicInteger p has PatternMatchable Float
from PatternMatchable Float
patternMatch: (%, Pattern Integer, PatternMatchResult(Integer, %)) -> PatternMatchResult(Integer, %) if PAdicInteger p has PatternMatchable Integer
from PatternMatchable Integer
positive?: % -> Boolean if PAdicInteger p 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 PAdicInteger p has LinearlyExplicitOver Integer
from LinearlyExplicitOver Integer
reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix PAdicInteger p, vec: Vector PAdicInteger p)
from LinearlyExplicitOver PAdicInteger p
reducedSystem: Matrix % -> Matrix Integer if PAdicInteger p has LinearlyExplicitOver Integer
from LinearlyExplicitOver Integer
reducedSystem: Matrix % -> Matrix PAdicInteger p
from LinearlyExplicitOver PAdicInteger p
rem: (%, %) -> %
from EuclideanDomain

removeZeroes: % -> %

removeZeroes: (Integer, %) -> %

retract: % -> Fraction Integer if PAdicInteger p has RetractableTo Integer
from RetractableTo Fraction Integer
retract: % -> Integer if PAdicInteger p has RetractableTo Integer
from RetractableTo Integer
retract: % -> PAdicInteger p
from RetractableTo PAdicInteger p
retract: % -> Symbol if PAdicInteger p has RetractableTo Symbol
from RetractableTo Symbol
retractIfCan: % -> Union(Fraction Integer, failed) if PAdicInteger p has RetractableTo Integer
from RetractableTo Fraction Integer
retractIfCan: % -> Union(Integer, failed) if PAdicInteger p has RetractableTo Integer
from RetractableTo Integer
retractIfCan: % -> Union(PAdicInteger p, failed)
from RetractableTo PAdicInteger p
retractIfCan: % -> Union(Symbol, failed) if PAdicInteger p 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 PAdicInteger p has OrderedIntegralDomain
from OrderedRing
sizeLess?: (%, %) -> Boolean
from EuclideanDomain
smaller?: (%, %) -> Boolean if PAdicInteger p has Comparable
from Comparable
solveLinearPolynomialEquation: (List SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> Union(List SparseUnivariatePolynomial %, failed) if PAdicInteger p has PolynomialFactorizationExplicit
from PolynomialFactorizationExplicit
squareFree: % -> Factored %
from UniqueFactorizationDomain
squareFreePart: % -> %
from UniqueFactorizationDomain
squareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if PAdicInteger p has PolynomialFactorizationExplicit
from PolynomialFactorizationExplicit
subtractIfCan: (%, %) -> Union(%, failed)
from CancellationAbelianMonoid
unit?: % -> Boolean
from EntireRing
unitCanonical: % -> %
from EntireRing
unitNormal: % -> Record(unit: %, canonical: %, associate: %)
from EntireRing
wholePart: % -> PAdicInteger p
from QuotientFieldCategory PAdicInteger p
zero?: % -> Boolean
from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

Algebra %

Algebra Fraction Integer

Algebra PAdicInteger p

BasicType

BiModule(%, %)

BiModule(Fraction Integer, Fraction Integer)

BiModule(PAdicInteger p, PAdicInteger p)

CancellationAbelianMonoid

canonicalsClosed

canonicalUnitNormal

CharacteristicNonZero if PAdicInteger p has CharacteristicNonZero

CharacteristicZero

CoercibleTo OutputForm

CommutativeRing

CommutativeStar

Comparable if PAdicInteger p has Comparable

ConvertibleTo DoubleFloat if PAdicInteger p has RealConstant

ConvertibleTo Float if PAdicInteger p has RealConstant

ConvertibleTo InputForm if PAdicInteger p has ConvertibleTo InputForm

ConvertibleTo Pattern Float if PAdicInteger p has ConvertibleTo Pattern Float

ConvertibleTo Pattern Integer if PAdicInteger p has ConvertibleTo Pattern Integer

DifferentialExtension PAdicInteger p

DifferentialRing if PAdicInteger p has DifferentialRing

DivisionRing

Eltable(PAdicInteger p, %) if PAdicInteger p has Eltable(PAdicInteger p, PAdicInteger p)

EntireRing

EuclideanDomain

Evalable PAdicInteger p if PAdicInteger p has Evalable PAdicInteger p

Field

FullyEvalableOver PAdicInteger p

FullyLinearlyExplicitOver PAdicInteger p

FullyPatternMatchable PAdicInteger p

GcdDomain

InnerEvalable(PAdicInteger p, PAdicInteger p) if PAdicInteger p has Evalable PAdicInteger p

InnerEvalable(Symbol, PAdicInteger p) if PAdicInteger p has InnerEvalable(Symbol, PAdicInteger p)

IntegralDomain

LeftModule %

LeftModule Fraction Integer

LeftModule PAdicInteger p

LeftOreRing

LinearlyExplicitOver Integer if PAdicInteger p has LinearlyExplicitOver Integer

LinearlyExplicitOver PAdicInteger p

Magma

MagmaWithUnit

Module %

Module Fraction Integer

Module PAdicInteger p

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors

OrderedAbelianGroup if PAdicInteger p has OrderedIntegralDomain

OrderedAbelianMonoid if PAdicInteger p has OrderedIntegralDomain

OrderedAbelianSemiGroup if PAdicInteger p has OrderedIntegralDomain

OrderedCancellationAbelianMonoid if PAdicInteger p has OrderedIntegralDomain

OrderedIntegralDomain if PAdicInteger p has OrderedIntegralDomain

OrderedRing if PAdicInteger p has OrderedIntegralDomain

OrderedSet if PAdicInteger p has OrderedSet

PartialDifferentialRing Symbol if PAdicInteger p has PartialDifferentialRing Symbol

PartialOrder if PAdicInteger p has OrderedSet

Patternable PAdicInteger p

PatternMatchable Float if PAdicInteger p has PatternMatchable Float

PatternMatchable Integer if PAdicInteger p has PatternMatchable Integer

PolynomialFactorizationExplicit if PAdicInteger p has PolynomialFactorizationExplicit

PrincipalIdealDomain

QuotientFieldCategory PAdicInteger p

RealConstant if PAdicInteger p has RealConstant

RetractableTo Fraction Integer if PAdicInteger p has RetractableTo Integer

RetractableTo Integer if PAdicInteger p has RetractableTo Integer

RetractableTo PAdicInteger p

RetractableTo Symbol if PAdicInteger p has RetractableTo Symbol

RightModule %

RightModule Fraction Integer

RightModule PAdicInteger p

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

StepThrough if PAdicInteger p has StepThrough

UniqueFactorizationDomain

unitsKnown