PAdicRationalConstructor(p, PADIC)ΒΆ

padic.spad line 327

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

0: %
from AbelianMonoid
1: %
from MagmaWithUnit
*: (%, %) -> %
from Magma
*: (%, Fraction Integer) -> %
from RightModule Fraction 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 % has CharacteristicNonZero and PADIC has PolynomialFactorizationExplicit or PADIC has CharacteristicNonZero
from PolynomialFactorizationExplicit
coerce: % -> %
from Algebra %
coerce: % -> OutputForm
from CoercibleTo OutputForm
coerce: Fraction Integer -> %
from RetractableTo Fraction Integer
coerce: Integer -> %
from NonAssociativeRing
coerce: PADIC -> %
from RetractableTo PADIC
coerce: Symbol -> % if PADIC has RetractableTo Symbol
from RetractableTo Symbol
commutator: (%, %) -> %
from NonAssociativeRng
conditionP: Matrix % -> Union(Vector %, failed) if % has CharacteristicNonZero and PADIC has PolynomialFactorizationExplicit
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
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
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
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

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

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 PADIC

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

StepThrough if PADIC has StepThrough

UniqueFactorizationDomain

unitsKnown