IntegerLocalizedAtPrime(p)

intlocp.spad line 1 [edit on github]

• p : PositiveInteger

IntegerLocalizedAtPrime(p) represents the Euclidean domain of integers localized at a prime p, i.e. the set of rational numbers whose denominator is not divisible by p.

* : (%, %) -> %

from Magma

* : (Integer, %) -> %

from AbelianGroup

* : (NonNegativeInteger, %) -> %

from AbelianMonoid

* : (PositiveInteger, %) -> %

from AbelianSemiGroup

+ : (%, %) -> %

from AbelianSemiGroup

- : % -> %

from AbelianGroup

- : (%, %) -> %

from AbelianGroup

0 : () -> %

from AbelianMonoid

1 : () -> %

from MagmaWithUnit

< : (%, %) -> Boolean

from PartialOrder

<= : (%, %) -> Boolean

from PartialOrder

= : (%, %) -> Boolean

from BasicType

> : (%, %) -> Boolean

from PartialOrder

>= : (%, %) -> Boolean

from PartialOrder

^ : (%, NonNegativeInteger) -> %

from MagmaWithUnit

^ : (%, PositiveInteger) -> %

from Magma

abs : % -> %

from OrderedRing

annihilate? : (%, %) -> Boolean

from Rng

antiCommutator : (%, %) -> %

from NonAssociativeSemiRng

associates? : (%, %) -> Boolean

from EntireRing

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

from NonAssociativeRng

characteristic : () -> NonNegativeInteger

from NonAssociativeRing

coerce : % -> %

from Algebra(%)

coerce : Integer -> %

from NonAssociativeRing

coerce : % -> Fraction(Integer)

from CoercibleTo(Fraction(Integer))

coerce : % -> OutputForm

from CoercibleTo(OutputForm)

commutator : (%, %) -> %

from NonAssociativeRng

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

from EuclideanDomain

euclideanSize : % -> NonNegativeInteger

from EuclideanDomain

exponent : % -> NonNegativeInteger

Each element x can be written as x=p^n*a/b with gcd(p,a)=gcd(p,b)=1. exponent(x) returns n.

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

gcd : (%, %) -> %

from GcdDomain

gcd : List(%) -> %

from GcdDomain

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

from GcdDomain

hash : % -> SingleInteger

from Hashable

hashUpdate! : (HashState, %) -> HashState

from Hashable

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

max : (%, %) -> %

from OrderedSet

min : (%, %) -> %

from OrderedSet

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

from EuclideanDomain

negative? : % -> Boolean

from OrderedRing

one? : % -> Boolean

from MagmaWithUnit

opposite? : (%, %) -> Boolean

from AbelianMonoid

plenaryPower : (%, PositiveInteger) -> %

from NonAssociativeAlgebra(%)

positive? : % -> Boolean

from OrderedRing

principalIdeal : List(%) -> Record(coef : List(%), generator : %)

from PrincipalIdealDomain

quo : (%, %) -> %

from EuclideanDomain

recip : % -> Union(%, "failed")

from MagmaWithUnit

rem : (%, %) -> %

from EuclideanDomain

retract : Fraction(Integer) -> %

from RetractableFrom(Fraction(Integer))

retract : % -> Integer

from RetractableTo(Integer)

retractIfCan : Fraction(Integer) -> Union(%, "failed")

from RetractableFrom(Fraction(Integer))

retractIfCan : % -> Union(Integer, "failed")

from RetractableTo(Integer)

rightPower : (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower : (%, PositiveInteger) -> %

from Magma

rightRecip : % -> Union(%, "failed")

from MagmaWithUnit

sample : () -> %

from AbelianMonoid

sign : % -> Integer

from OrderedRing

sizeLess? : (%, %) -> Boolean

from EuclideanDomain

smaller? : (%, %) -> Boolean

from Comparable

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

from CancellationAbelianMonoid

unit? : % -> Boolean

from EntireRing

unitCanonical : % -> %

from EntireRing

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

from EntireRing

unitPart : % -> Fraction(Integer)

Each element x can be written as x=p^n*a/b with gcd(p,a)=gcd(p,b)=1. unitPart(x) returns a/b.

zero? : % -> Boolean

from AbelianMonoid

~= : (%, %) -> Boolean

from BasicType

Comparable

noZeroDivisors

RetractableFrom(Fraction(Integer))

OrderedAbelianSemiGroup

RightModule(%)

Monoid

Algebra(%)

AbelianMonoid

EuclideanDomain

EntireRing

CancellationAbelianMonoid

MagmaWithUnit

NonAssociativeRing

OrderedCancellationAbelianMonoid

RetractableTo(Integer)

CommutativeStar

LeftModule(%)

canonicalUnitNormal

OrderedSet

Module(%)

SetCategory

LeftOreRing

CoercibleTo(OutputForm)

CommutativeRing

IntegralDomain

TwoSidedRecip

Magma

Rng

SemiGroup

OrderedAbelianMonoid

PartialOrder

BiModule(%, %)

CoercibleFrom(Integer)

unitsKnown

AbelianGroup

AbelianSemiGroup

GcdDomain

NonAssociativeSemiRing

NonAssociativeAlgebra(%)

OrderedAbelianGroup

OrderedRing

PrincipalIdealDomain

NonAssociativeRng

CoercibleTo(Fraction(Integer))

Ring

SemiRng

NonAssociativeSemiRng

Hashable

CharacteristicZero

BasicType

SemiRing