IntegerLocalizedAtPrime pΒΆ
intlocp.spad line 1 [edit on github]
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.
- 0: %
from AbelianMonoid
- 1: %
from MagmaWithUnit
- *: (%, %) -> %
from LeftModule %
- *: (Integer, %) -> %
from AbelianGroup
- *: (NonNegativeInteger, %) -> %
from AbelianMonoid
- *: (PositiveInteger, %) -> %
from AbelianSemiGroup
- +: (%, %) -> %
from AbelianSemiGroup
- -: % -> %
from AbelianGroup
- -: (%, %) -> %
from AbelianGroup
- <=: (%, %) -> Boolean
from PartialOrder
- <: (%, %) -> Boolean
from PartialOrder
- >=: (%, %) -> Boolean
from PartialOrder
- >: (%, %) -> Boolean
from PartialOrder
- ^: (%, NonNegativeInteger) -> %
from MagmaWithUnit
- ^: (%, PositiveInteger) -> %
from Magma
- abs: % -> %
from OrderedRing
- annihilate?: (%, %) -> Boolean
from Rng
- antiCommutator: (%, %) -> %
- associates?: (%, %) -> Boolean
from EntireRing
- associator: (%, %, %) -> %
from NonAssociativeRng
- characteristic: () -> NonNegativeInteger
from NonAssociativeRing
- coerce: % -> %
from Algebra %
- coerce: % -> Fraction Integer
from CoercibleTo Fraction Integer
- coerce: % -> OutputForm
from CoercibleTo OutputForm
- coerce: Integer -> %
from CoercibleFrom Integer
- commutator: (%, %) -> %
from NonAssociativeRng
- divide: (%, %) -> Record(quotient: %, remainder: %)
from EuclideanDomain
- euclideanSize: % -> NonNegativeInteger
from EuclideanDomain
- exponent: % -> NonNegativeInteger
Each element
xcan be written as x=p^n*a/b withgcd(p,a)=gcd(p,b)=1. exponent(x) returnsn.
- 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
- gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial %
from GcdDomain
- hash: % -> SingleInteger
from Hashable
- hashUpdate!: (HashState, %) -> HashState
from Hashable
- latex: % -> String
from SetCategory
- 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
- retractIfCan: % -> Union(Integer, failed)
from RetractableTo Integer
- retractIfCan: Fraction Integer -> Union(%, failed)
- 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)
- unit?: % -> Boolean
from EntireRing
- unitCanonical: % -> %
from EntireRing
- unitNormal: % -> Record(unit: %, canonical: %, associate: %)
from EntireRing
- unitPart: % -> Fraction Integer
Each element
xcan be written as x=p^n*a/b withgcd(p,a)=gcd(p,b)=1. unitPart(x) returns a/b.
- zero?: % -> Boolean
from AbelianMonoid
Algebra %
BiModule(%, %)
Module %
OrderedCancellationAbelianMonoid