Integer Residue Class Rings
- Introduction
- Ideals of \({\mathbb{Z}}\)
- \({\mathbb{Z}}\) as a Number Field Order
Decomposition(R, p): RngInt, RngIntElt → SeqEnumGenerator(I): RngInt → RngIntEltRamificationIndex(I, p): RngInt, RngIntElt → RngIntEltRamificationIndex(I): RngInt → RngIntEltDegree(I): RngInt → RngIntEltTwoElementNormal(I): RngInt → RngIntElt, RngIntEltChineseRemainderTheorem(I, J, a, b): RngInt, RngInt, RngIntElt, RngIntElt → RngIntEltValuation(x, I): RngIntElt, RngInt → RngIntEltClassRepresentative(I): RngInt → RngInt
- Residue Class Rings
- Creation
quo<Z | I>: RngInt, RngInt → RngIntResquo<Z | m>: RngInt, RngIntElt → RngIntResResidueClassRing(m): RngIntElt → RngIntRes, MapIntegerRing(m): RngIntElt → RngIntResIntegers(m): RngIntElt → RngIntResRingOfIntegers(m): RngIntElt → RngIntResResidueClassField(p): RngIntElt → FldFin, MapResidueClassRing(Q): RngIntEltFact → RngIntResIntegerRing(Q): RngIntEltFact → RngIntResIntegers(Q): RngIntEltFact → RngIntRes- Example: Cube Root in a Prime Residue Ring
- Coercion
- Elementary Invariants
- Structure Operations
AdditiveGroup(R): RngIntRes → GrpAb, MapMultiplicativeGroup(R): RngIntRes → GrpAb, MapUnitGroup(R): RngIntRes → GrpAb, Mapsub< R | n >: RngIntRes, RngIntResElt → RngIntResSet(R): RngIntRes → SetEnumCategory(R): RngIntRes → CatParent(R): RngIntRes → PowerStructurePrimeRing(R): RngIntRes → RngIntResCenter(R): RngIntRes → RngIntRes
- Ring Predicates and Booleans
IsCommutative(R): RngIntRes → BoolEltIsUnitary(R): RngIntRes → BoolEltIsFinite(R): RngIntRes → BoolEltIsOrdered(R): RngIntRes → BoolEltIsField(R): RngIntRes → BoolEltIsEuclideanDomain(R): RngIntRes → BoolEltIsPID(R): RngIntRes → BoolEltIsUFD(R): RngIntRes → BoolEltIsDivisionRing(R): RngIntRes → BoolEltIsEuclideanRing(R): RngIntRes → BoolEltIsPrincipalIdealRing(R): RngIntRes → BoolEltIsDomain(R): RngIntRes → BoolEltR eq R: RngIntRes, Rng → BoolEltR ne R: RngIntRes, Rng → BoolElt
- Homomorphisms
- Creation
- Elements of Residue Class Rings
- Creation
- Arithmetic Operators
+ n: RngIntResElt → RngIntResElt- n: RngIntResElt → RngIntResEltm + n: RngIntResElt, RngIntResElt → RngIntResEltm - n: RngIntResElt, RngIntResElt → RngIntResEltm * n: RngIntResElt, RngIntResElt → RngIntResEltn ^ k: RngIntResElt, RngIntResElt → RngIntResEltm / n: RngIntResElt, RngIntResElt → RngIntResEltm div n: RngIntResElt, RngIntResElt → RngIntResEltm +:= n: RngIntResElt, RngIntResElt → RngIntResEltm -:= n: RngIntResElt, RngIntResElt → RngIntResEltm *:= n: RngIntResElt, RngIntResElt → RngIntResEltm /:= n: RngIntResElt, RngIntResElt → RngIntResEltm ^:= k: RngIntResElt, RngIntResElt → RngIntResElt
- Equality and Membership
- Parent and Category
- Predicates on Ring Elements
IsZero(n): RngIntResElt → BoolEltIsOne(n): RngIntResElt → BoolEltIsMinusOne(n): RngIntResElt → BoolEltIsNilpotent(n): RngIntResElt → BoolEltIsIdempotent(n): RngIntResElt → BoolEltIsUnit(n): RngIntResElt → BoolEltIsZeroDivisor(n): RngIntResElt → BoolEltIsRegular(n): RngIntRes → BoolEltIsIrreducible(n): RngIntResElt → BoolEltIsPrime(n): RngIntResElt → BoolElt
- Solving Equations over \({\mathbb{Z}}/m{\mathbb{Z}}\)
Solution(a, b): RngIntResElt, RngIntResElt → RngIntResEltIsSquare(n): RngIntResElt → BoolElt, RngIntResEltSqrt(a): RngIntResElt → RngIntResEltSquareRoot(a): RngIntResElt → RngIntResEltAllSquareRoots(a): RngIntResElt → [ RngIntResElt ]AllSqrts(a): RngIntResElt → [ RngIntResElt ]- Example: Square Roots Modulo 2340
- Ideal Operations
ideal< R | a₁, ..., aᵣ >: RngIntRes, RngIntResElt, ..., RngIntResElt → RngIntResGreatestCommonDivisor(a, b): RngIntResElt, RngIntResElt → RngIntResEltGcd(a, b): RngIntResElt, RngIntResElt → RngIntResEltGCD(a, b): RngIntResElt, RngIntResElt → RngIntResEltGreatestCommonDivisor(Q): [RngIntResElt] → RngIntResEltGcd(Q): [RngIntResElt] → RngIntResEltGCD(Q): [RngIntResElt] → RngIntResEltLeastCommonMultiple(a, b): RngIntResElt, RngIntResElt → RngIntResEltLcm(a, b): RngIntResElt, RngIntResElt → RngIntResEltLCM(a, b): RngIntResElt, RngIntResElt → RngIntResEltLeastCommonMultiple(Q): [RngIntResElt] → RngIntResEltLcm(Q): [RngIntResElt] → RngIntResEltLCM(Q): [RngIntResElt] → RngIntResEltI + J: RngIntRes, RngIntRes → RngIntResI * J: RngIntRes, RngIntRes → RngIntResI meet J: RngIntRes, RngIntRes → RngIntResa in I: RngIntResElt, RngIntRes → BoolElta notin I: RngIntResElt, RngIntRes → BoolEltI eq J: RngIntRes, RngIntRes → BoolEltI ne J: RngIntRes, RngIntRes → BoolEltI subset J: RngIntRes, RngIntRes → BoolEltI notsubset J: RngIntRes, RngIntRes → BoolElt
- The Unit Group
UnitGroup(R): RngIntRes → GrpAb, MapIsPrimitive(n): RngIntResElt → BoolEltPrimitiveElement(R): RngIntRes → RngIntResEltPrimitiveRoot(R): RngIntRes → RngIntResEltOrder(a): RngIntResElt → RngIntEltNormalize(x): RngIntRes → RngIntResElt, RngIntResEltNormalise(x): RngIntRes → RngIntResElt, RngIntResElt- Example: Unit Group of a Residue Class Ring
- Example: Cyclic Unit Group
- Dirichlet Characters
- Creation
DirichletGroup(N): RngIntElt → GrpDrchFullDirichletGroup(N): RngIntElt → GrpDrchDirichletGroup(N,R): RngIntElt, Rng → GrpDrchDirichletGroup(N,R,z,r): RngIntElt, Rng, RngElt, RngIntElt → GrpDrchBaseExtend(G, R): GrpDrch, Rng → GrpDrchBaseExtend(G, R, z): GrpDrch, Rng, RngElt → GrpDrchAssignNames(~G, S): GrpDrch, [MonStgElt]
- Element Creation
- Attributes of Dirichlet Groups
BaseRing(G): GrpDrch → RngModulus(G): GrpDrch → RngIntEltOrder(G): GrpDrch → RngIntEltExponent(G): GrpDrch → RngIntEltNumberOfGenerators(G): GrpDrch → RngIntEltGenerators(G): GrpDrch → [GrpDrchElt]UnitGenerators(G): GrpDrch → [RngIntElt]GaloisConjugacyRepresentatives(G): GrpDrch → [GrpDrchElt]GaloisConjugacyRepresentatives(seq): [GrpDrchElt] → [GrpDrchElt]AbelianGroup(G): GrpDrch → GrpAb, Map
- Attributes of Elements
BaseRing(chi): GrpDrchElt → RngModulus(chi): GrpDrchElt → RngIntEltConductor(chi): GrpDrchElt → RngIntEltElementToSequence(chi): GrpDrchElt → SeqEnumx eq y: GrpDrchElt, GrpDrchElt → BoolEltOrder(chi): GrpDrchElt → RngIntEltIsTrivial(chi): GrpDrchElt → BoolEltIsPrimitive(chi): GrpDrchElt → BoolEltAssociatedPrimitiveCharacter(chi): GrpDrchElt → GrpDrchEltIsEven(chi): GrpDrchElt → BoolEltIsOdd(chi): GrpDrchElt → BoolEltIsTotallyEven(chi): GrpDrchElt → BoolEltDecomposition(chi): GrpDrchElt → ListMinimalBaseRingCharacter(chi): GrpDrchElt → GrpDrchElt
- Evaluation
- Arithmetic
- Example
- Creation