BalancedPAdicRational pΒΆ

padic.spad line 524

Stream-based implementation of Qp: numbers are represented as sum(i = k.., a[i] * p^i), where the a[i] lie in -(p - 1)/2, ..., (p - 1)/2.

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

approximate: (%, Integer) -> Fraction Integer

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

continuedFraction: % -> ContinuedFraction Fraction Integer

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

removeZeroes: % -> %

removeZeroes: (Integer, %) -> %

retract: % -> BalancedPAdicInteger p
from RetractableTo BalancedPAdicInteger p
retract: % -> Fraction Integer if BalancedPAdicInteger p has RetractableTo Integer
from RetractableTo Fraction Integer
retract: % -> Integer if BalancedPAdicInteger p has RetractableTo Integer
from RetractableTo Integer
retract: % -> Symbol if BalancedPAdicInteger p has RetractableTo Symbol
from RetractableTo Symbol
retractIfCan: % -> Union(BalancedPAdicInteger p, failed)
from RetractableTo BalancedPAdicInteger p
retractIfCan: % -> Union(Fraction Integer, failed) if BalancedPAdicInteger p has RetractableTo Integer
from RetractableTo Fraction Integer
retractIfCan: % -> Union(Integer, failed) if BalancedPAdicInteger p has RetractableTo Integer
from RetractableTo Integer
retractIfCan: % -> Union(Symbol, failed) if BalancedPAdicInteger 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 BalancedPAdicInteger p has OrderedIntegralDomain
from OrderedRing
sizeLess?: (%, %) -> Boolean
from EuclideanDomain
smaller?: (%, %) -> Boolean if BalancedPAdicInteger p has Comparable
from Comparable
solveLinearPolynomialEquation: (List SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> Union(List SparseUnivariatePolynomial %, failed) if BalancedPAdicInteger p has PolynomialFactorizationExplicit
from PolynomialFactorizationExplicit
squareFree: % -> Factored %
from UniqueFactorizationDomain
squareFreePart: % -> %
from UniqueFactorizationDomain
squareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if BalancedPAdicInteger 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: % -> BalancedPAdicInteger p
from QuotientFieldCategory BalancedPAdicInteger p
zero?: % -> Boolean
from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

Algebra %

Algebra BalancedPAdicInteger p

Algebra Fraction Integer

BasicType

BiModule(%, %)

BiModule(BalancedPAdicInteger p, BalancedPAdicInteger p)

BiModule(Fraction Integer, Fraction Integer)

CancellationAbelianMonoid

canonicalsClosed

canonicalUnitNormal

CharacteristicNonZero if BalancedPAdicInteger p has CharacteristicNonZero

CharacteristicZero

CoercibleTo OutputForm

CommutativeRing

CommutativeStar

Comparable if BalancedPAdicInteger p has Comparable

ConvertibleTo DoubleFloat if BalancedPAdicInteger p has RealConstant

ConvertibleTo Float if BalancedPAdicInteger p has RealConstant

ConvertibleTo InputForm if BalancedPAdicInteger p has ConvertibleTo InputForm

ConvertibleTo Pattern Float if BalancedPAdicInteger p has ConvertibleTo Pattern Float

ConvertibleTo Pattern Integer if BalancedPAdicInteger p has ConvertibleTo Pattern Integer

DifferentialExtension BalancedPAdicInteger p

DifferentialRing if BalancedPAdicInteger p has DifferentialRing

DivisionRing

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

EntireRing

EuclideanDomain

Evalable BalancedPAdicInteger p if BalancedPAdicInteger p has Evalable BalancedPAdicInteger p

Field

FullyEvalableOver BalancedPAdicInteger p

FullyLinearlyExplicitOver BalancedPAdicInteger p

FullyPatternMatchable BalancedPAdicInteger p

GcdDomain

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

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

IntegralDomain

LeftModule %

LeftModule BalancedPAdicInteger p

LeftModule Fraction Integer

LeftOreRing

LinearlyExplicitOver BalancedPAdicInteger p

LinearlyExplicitOver Integer if BalancedPAdicInteger p has LinearlyExplicitOver Integer

Magma

MagmaWithUnit

Module %

Module BalancedPAdicInteger p

Module Fraction Integer

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors

OrderedAbelianGroup if BalancedPAdicInteger p has OrderedIntegralDomain

OrderedAbelianMonoid if BalancedPAdicInteger p has OrderedIntegralDomain

OrderedAbelianSemiGroup if BalancedPAdicInteger p has OrderedIntegralDomain

OrderedCancellationAbelianMonoid if BalancedPAdicInteger p has OrderedIntegralDomain

OrderedIntegralDomain if BalancedPAdicInteger p has OrderedIntegralDomain

OrderedRing if BalancedPAdicInteger p has OrderedIntegralDomain

OrderedSet if BalancedPAdicInteger p has OrderedSet

PartialDifferentialRing Symbol if BalancedPAdicInteger p has PartialDifferentialRing Symbol

PartialOrder if BalancedPAdicInteger p has OrderedSet

Patternable BalancedPAdicInteger p

PatternMatchable Float if BalancedPAdicInteger p has PatternMatchable Float

PatternMatchable Integer if BalancedPAdicInteger p has PatternMatchable Integer

PolynomialFactorizationExplicit if BalancedPAdicInteger p has PolynomialFactorizationExplicit

PrincipalIdealDomain

QuotientFieldCategory BalancedPAdicInteger p

RealConstant if BalancedPAdicInteger p has RealConstant

RetractableTo BalancedPAdicInteger p

RetractableTo Fraction Integer if BalancedPAdicInteger p has RetractableTo Integer

RetractableTo Integer if BalancedPAdicInteger p has RetractableTo Integer

RetractableTo Symbol if BalancedPAdicInteger p has RetractableTo Symbol

RightModule %

RightModule BalancedPAdicInteger p

RightModule Fraction Integer

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

StepThrough if BalancedPAdicInteger p has StepThrough

UniqueFactorizationDomain

unitsKnown