PAdicRationalConstructor(p, PADIC)

padic.spad line 327 [edit on github]

This is the category of stream-based representations of Qp.

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from LeftModule %

*: (%, Fraction Integer) -> %

from RightModule Fraction Integer

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

from RightModule Integer

*: (%, PADIC) -> %

from RightModule PADIC

*: (Fraction Integer, %) -> %

from LeftModule Fraction Integer

*: (Integer, %) -> %

from AbelianGroup

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PADIC, %) -> %

from LeftModule PADIC

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

+: (%, %) -> %

from AbelianSemiGroup

-: % -> %

from AbelianGroup

-: (%, %) -> %

from AbelianGroup

/: (%, %) -> %

from Field

/: (PADIC, PADIC) -> %

from QuotientFieldCategory PADIC

<=: (%, %) -> Boolean if PADIC has OrderedSet

from PartialOrder

<: (%, %) -> Boolean if PADIC has OrderedSet

from PartialOrder

=: (%, %) -> Boolean

from BasicType

>=: (%, %) -> Boolean if PADIC has OrderedSet

from PartialOrder

>: (%, %) -> Boolean if PADIC has OrderedSet

from PartialOrder

^: (%, Integer) -> %

from DivisionRing

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

abs: % -> % if PADIC has OrderedIntegralDomain

from OrderedRing

annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %

from NonAssociativeSemiRng

approximate: (%, Integer) -> Fraction Integer

approximate(x, n) returns a rational number y such that y = x (mod p^n).

associates?: (%, %) -> Boolean

from EntireRing

associator: (%, %, %) -> %

from NonAssociativeRng

ceiling: % -> PADIC if PADIC has IntegerNumberSystem

from QuotientFieldCategory PADIC

characteristic: () -> NonNegativeInteger

from NonAssociativeRing

charthRoot: % -> Union(%, failed) if PADIC has PolynomialFactorizationExplicit and % has CharacteristicNonZero or PADIC has CharacteristicNonZero

from PolynomialFactorizationExplicit

coerce: % -> %

from Algebra %

coerce: % -> OutputForm

from CoercibleTo OutputForm

coerce: Fraction Integer -> %

from CoercibleFrom Fraction Integer

coerce: Integer -> %

from NonAssociativeRing

coerce: PADIC -> %

from Algebra PADIC

coerce: Symbol -> % if PADIC has RetractableTo Symbol

from CoercibleFrom Symbol

commutator: (%, %) -> %

from NonAssociativeRng

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

from PolynomialFactorizationExplicit

continuedFraction: % -> ContinuedFraction Fraction Integer

continuedFraction(x) converts the p-adic rational number x to a continued fraction.

convert: % -> DoubleFloat if PADIC has RealConstant

from ConvertibleTo DoubleFloat

convert: % -> Float if PADIC has RealConstant

from ConvertibleTo Float

convert: % -> InputForm if PADIC has ConvertibleTo InputForm

from ConvertibleTo InputForm

convert: % -> Pattern Float if PADIC has ConvertibleTo Pattern Float

from ConvertibleTo Pattern Float

convert: % -> Pattern Integer if PADIC has ConvertibleTo Pattern Integer

from ConvertibleTo Pattern Integer

D: % -> % if PADIC has DifferentialRing

from DifferentialRing

D: (%, List Symbol) -> % if PADIC has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

D: (%, List Symbol, List NonNegativeInteger) -> % if PADIC has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

D: (%, NonNegativeInteger) -> % if PADIC has DifferentialRing

from DifferentialRing

D: (%, PADIC -> PADIC) -> %

from DifferentialExtension PADIC

D: (%, PADIC -> PADIC, NonNegativeInteger) -> %

from DifferentialExtension PADIC

D: (%, Symbol) -> % if PADIC has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

D: (%, Symbol, NonNegativeInteger) -> % if PADIC has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

denom: % -> PADIC

from QuotientFieldCategory PADIC

denominator: % -> %

from QuotientFieldCategory PADIC

differentiate: % -> % if PADIC has DifferentialRing

from DifferentialRing

differentiate: (%, List Symbol) -> % if PADIC has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

differentiate: (%, List Symbol, List NonNegativeInteger) -> % if PADIC has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

differentiate: (%, NonNegativeInteger) -> % if PADIC has DifferentialRing

from DifferentialRing

differentiate: (%, PADIC -> PADIC) -> %

from DifferentialExtension PADIC

differentiate: (%, PADIC -> PADIC, NonNegativeInteger) -> %

from DifferentialExtension PADIC

differentiate: (%, Symbol) -> % if PADIC has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

differentiate: (%, Symbol, NonNegativeInteger) -> % if PADIC has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

divide: (%, %) -> Record(quotient: %, remainder: %)

from EuclideanDomain

elt: (%, PADIC) -> % if PADIC has Eltable(PADIC, PADIC)

from Eltable(PADIC, %)

euclideanSize: % -> NonNegativeInteger

from EuclideanDomain

eval: (%, Equation PADIC) -> % if PADIC has Evalable PADIC

from Evalable PADIC

eval: (%, List Equation PADIC) -> % if PADIC has Evalable PADIC

from Evalable PADIC

eval: (%, List PADIC, List PADIC) -> % if PADIC has Evalable PADIC

from InnerEvalable(PADIC, PADIC)

eval: (%, List Symbol, List PADIC) -> % if PADIC has InnerEvalable(Symbol, PADIC)

from InnerEvalable(Symbol, PADIC)

eval: (%, PADIC, PADIC) -> % if PADIC has Evalable PADIC

from InnerEvalable(PADIC, PADIC)

eval: (%, Symbol, PADIC) -> % if PADIC has InnerEvalable(Symbol, PADIC)

from InnerEvalable(Symbol, PADIC)

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 PADIC has PolynomialFactorizationExplicit

from PolynomialFactorizationExplicit

factorSquareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if PADIC has PolynomialFactorizationExplicit

from PolynomialFactorizationExplicit

floor: % -> PADIC if PADIC has IntegerNumberSystem

from QuotientFieldCategory PADIC

fractionPart: % -> %

from QuotientFieldCategory PADIC

gcd: (%, %) -> %

from GcdDomain

gcd: List % -> %

from GcdDomain

gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial %

from PolynomialFactorizationExplicit

init: % if PADIC 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: (PADIC -> PADIC, %) -> %

from FullyEvalableOver PADIC

max: (%, %) -> % if PADIC has OrderedSet

from OrderedSet

min: (%, %) -> % if PADIC has OrderedSet

from OrderedSet

multiEuclidean: (List %, %) -> Union(List %, failed)

from EuclideanDomain

negative?: % -> Boolean if PADIC has OrderedIntegralDomain

from OrderedRing

nextItem: % -> Union(%, failed) if PADIC has StepThrough

from StepThrough

numer: % -> PADIC

from QuotientFieldCategory PADIC

numerator: % -> %

from QuotientFieldCategory PADIC

one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

patternMatch: (%, Pattern Float, PatternMatchResult(Float, %)) -> PatternMatchResult(Float, %) if PADIC has PatternMatchable Float

from PatternMatchable Float

patternMatch: (%, Pattern Integer, PatternMatchResult(Integer, %)) -> PatternMatchResult(Integer, %) if PADIC has PatternMatchable Integer

from PatternMatchable Integer

plenaryPower: (%, PositiveInteger) -> %

from NonAssociativeAlgebra PADIC

positive?: % -> Boolean if PADIC 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 PADIC has LinearlyExplicitOver Integer

from LinearlyExplicitOver Integer

reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix PADIC, vec: Vector PADIC)

from LinearlyExplicitOver PADIC

reducedSystem: Matrix % -> Matrix Integer if PADIC has LinearlyExplicitOver Integer

from LinearlyExplicitOver Integer

reducedSystem: Matrix % -> Matrix PADIC

from LinearlyExplicitOver PADIC

rem: (%, %) -> %

from EuclideanDomain

removeZeroes: % -> %

removeZeroes(x) removes leading zeroes from the representation of the p-adic rational x. A p-adic rational is represented by (1) an exponent and (2) a p-adic integer which may have leading zero digits. When the p-adic integer has a leading zero digit, a ‘leading zero’ is removed from the p-adic rational as follows: the number is rewritten by increasing the exponent by 1 and dividing the p-adic integer by p. Note: removeZeroes(f) removes all leading zeroes from f.

removeZeroes: (Integer, %) -> %

removeZeroes(n, x) removes up to n leading zeroes from the p-adic rational x.

retract: % -> Fraction Integer if PADIC has RetractableTo Integer

from RetractableTo Fraction Integer

retract: % -> Integer if PADIC has RetractableTo Integer

from RetractableTo Integer

retract: % -> PADIC

from RetractableTo PADIC

retract: % -> Symbol if PADIC has RetractableTo Symbol

from RetractableTo Symbol

retractIfCan: % -> Union(Fraction Integer, failed) if PADIC has RetractableTo Integer

from RetractableTo Fraction Integer

retractIfCan: % -> Union(Integer, failed) if PADIC has RetractableTo Integer

from RetractableTo Integer

retractIfCan: % -> Union(PADIC, failed)

from RetractableTo PADIC

retractIfCan: % -> Union(Symbol, failed) if PADIC 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 PADIC has OrderedIntegralDomain

from OrderedRing

sizeLess?: (%, %) -> Boolean

from EuclideanDomain

smaller?: (%, %) -> Boolean if PADIC has Comparable

from Comparable

solveLinearPolynomialEquation: (List SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> Union(List SparseUnivariatePolynomial %, failed) if PADIC has PolynomialFactorizationExplicit

from PolynomialFactorizationExplicit

squareFree: % -> Factored %

from UniqueFactorizationDomain

squareFreePart: % -> %

from UniqueFactorizationDomain

squareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if PADIC has PolynomialFactorizationExplicit

from PolynomialFactorizationExplicit

subtractIfCan: (%, %) -> Union(%, failed)

from CancellationAbelianMonoid

unit?: % -> Boolean

from EntireRing

unitCanonical: % -> %

from EntireRing

unitNormal: % -> Record(unit: %, canonical: %, associate: %)

from EntireRing

wholePart: % -> PADIC

from QuotientFieldCategory PADIC

zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

Algebra %

Algebra Fraction Integer

Algebra PADIC

BasicType

BiModule(%, %)

BiModule(Fraction Integer, Fraction Integer)

BiModule(PADIC, PADIC)

CancellationAbelianMonoid

canonicalsClosed

canonicalUnitNormal

CharacteristicNonZero if PADIC has CharacteristicNonZero

CharacteristicZero

CoercibleFrom Fraction Integer if PADIC has RetractableTo Integer

CoercibleFrom Integer if PADIC has RetractableTo Integer

CoercibleFrom PADIC

CoercibleFrom Symbol if PADIC has RetractableTo Symbol

CoercibleTo OutputForm

CommutativeRing

CommutativeStar

Comparable if PADIC has Comparable

ConvertibleTo DoubleFloat if PADIC has RealConstant

ConvertibleTo Float if PADIC has RealConstant

ConvertibleTo InputForm if PADIC has ConvertibleTo InputForm

ConvertibleTo Pattern Float if PADIC has ConvertibleTo Pattern Float

ConvertibleTo Pattern Integer if PADIC has ConvertibleTo Pattern Integer

DifferentialExtension PADIC

DifferentialRing if PADIC has DifferentialRing

DivisionRing

Eltable(PADIC, %) if PADIC has Eltable(PADIC, PADIC)

EntireRing

EuclideanDomain

Evalable PADIC if PADIC has Evalable PADIC

Field

FullyEvalableOver PADIC

FullyLinearlyExplicitOver PADIC

FullyPatternMatchable PADIC

GcdDomain

InnerEvalable(PADIC, PADIC) if PADIC has Evalable PADIC

InnerEvalable(Symbol, PADIC) if PADIC has InnerEvalable(Symbol, PADIC)

IntegralDomain

LeftModule %

LeftModule Fraction Integer

LeftModule PADIC

LeftOreRing

LinearlyExplicitOver Integer if PADIC has LinearlyExplicitOver Integer

LinearlyExplicitOver PADIC

Magma

MagmaWithUnit

Module %

Module Fraction Integer

Module PADIC

Monoid

NonAssociativeAlgebra %

NonAssociativeAlgebra Fraction Integer

NonAssociativeAlgebra PADIC

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors

OrderedAbelianGroup if PADIC has OrderedIntegralDomain

OrderedAbelianMonoid if PADIC has OrderedIntegralDomain

OrderedAbelianSemiGroup if PADIC has OrderedIntegralDomain

OrderedCancellationAbelianMonoid if PADIC has OrderedIntegralDomain

OrderedIntegralDomain if PADIC has OrderedIntegralDomain

OrderedRing if PADIC has OrderedIntegralDomain

OrderedSet if PADIC has OrderedSet

PartialDifferentialRing Symbol if PADIC has PartialDifferentialRing Symbol

PartialOrder if PADIC has OrderedSet

Patternable PADIC

PatternMatchable Float if PADIC has PatternMatchable Float

PatternMatchable Integer if PADIC has PatternMatchable Integer

PolynomialFactorizationExplicit if PADIC has PolynomialFactorizationExplicit

PrincipalIdealDomain

QuotientFieldCategory PADIC

RealConstant if PADIC has RealConstant

RetractableTo Fraction Integer if PADIC has RetractableTo Integer

RetractableTo Integer if PADIC has RetractableTo Integer

RetractableTo PADIC

RetractableTo Symbol if PADIC has RetractableTo Symbol

RightModule %

RightModule Fraction Integer

RightModule Integer if PADIC has LinearlyExplicitOver Integer

RightModule PADIC

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

StepThrough if PADIC has StepThrough

TwoSidedRecip

UniqueFactorizationDomain

unitsKnown