PAdicRational p

padic.spad line 543 [edit on github]

• p: Integer

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

*: (%, Integer) -> % if PAdicInteger p has LinearlyExplicitOver Integer

from RightModule 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 or PAdicInteger p has PolynomialFactorizationExplicit and % has CharacteristicNonZero

from CharacteristicNonZero

coerce: % -> %

from Algebra %

coerce: % -> OutputForm

from CoercibleTo OutputForm

coerce: Fraction Integer -> %

from CoercibleFrom Fraction Integer

coerce: Integer -> %

from NonAssociativeRing

coerce: PAdicInteger p -> %

from Algebra PAdicInteger p

coerce: Symbol -> % if PAdicInteger p has RetractableTo Symbol

from CoercibleFrom Symbol

commutator: (%, %) -> %

from NonAssociativeRng

conditionP: Matrix % -> Union(Vector %, failed) if PAdicInteger p has PolynomialFactorizationExplicit and % has CharacteristicNonZero

from PolynomialFactorizationExplicit

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

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

plenaryPower: (%, PositiveInteger) -> %

from NonAssociativeAlgebra %

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

CoercibleFrom Fraction Integer if PAdicInteger p has RetractableTo Integer

CoercibleFrom Integer if PAdicInteger p has RetractableTo Integer

CoercibleFrom PAdicInteger p

CoercibleFrom Symbol if PAdicInteger p has RetractableTo Symbol

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

NonAssociativeAlgebra %

NonAssociativeAlgebra Fraction Integer

NonAssociativeAlgebra PAdicInteger p

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 Integer if PAdicInteger p has LinearlyExplicitOver Integer

RightModule PAdicInteger p

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

StepThrough if PAdicInteger p has StepThrough

TwoSidedRecip

UniqueFactorizationDomain

unitsKnown