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INTEGER RESIDUE CLASS RINGS

 
Acknowledgements
 
Introduction
 
Ideals of Z
 
Z as a Number Field Order
 
Residue Class Rings
      Creation
      Coercion
      Elementary Invariants
      Structure Operations
      Ring Predicates and Booleans
      Homomorphisms
 
Elements of Residue Class Rings
      Creation
      Arithmetic Operators
      Equality and Membership
      Parent and Category
      Predicates on Ring Elements
      Solving Equations over Z/mZ
 
Ideal Operations
 
The Unit Group
 
Dirichlet Characters
      Creation
      Element Creation
      Properties of Dirichlet Groups
      Properties of Elements
      Evaluation
      Arithmetic
      Example







DETAILS

 
Introduction

 
Ideals of Z
      ideal< R | a > : RngInt, RngIntElt -> RngIntRes
      Example RngIntRes_residue-ring (H19E1)

 
Z as a Number Field Order
      Decomposition(R, p) : RngInt, RngIntElt -> SeqEnum
      Generator(I) : RngInt -> RngIntElt
      RamificationIndex(I, p) : RngInt, RngIntElt -> RngIntElt
      Degree(I) : RngInt -> RngIntElt
      TwoElementNormal(I) : RngInt -> RngIntElt, RngIntElt
      ChineseRemainderTheorem(I, J, a, b) : RngInt, RngInt, RngIntElt, RngIntElt -> RngIntElt
      Valuation(x, I) : RngIntElt, RngInt -> RngIntElt
      ClassRepresentative(I) : RngInt -> RngInt

 
Residue Class Rings

      Creation
            quo< Z | I > : RngInt, RngInt -> RngIntRes
            quo< Z | m > : RngInt, RngIntElt -> RngIntRes
            ResidueClassRing(m) : RngIntElt -> RngIntRes
            ResidueClassRing(Q) : RngIntEltFact -> RngIntRes
            Example RngIntRes_residue-ring (H19E2)

      Coercion
            Example RngIntRes_Coercion (H19E3)

      Elementary Invariants
            Modulus(R) : RngIntRes -> RngInt
            FactoredModulus(R) : RngIntRes -> RngIntEltFact

      Structure Operations
            AdditiveGroup(R) : RngIntRes -> GrpAb, Map
            MultiplicativeGroup(R) : RngIntRes -> GrpAb, Map
            sub< R | n > : RngIntRes, RngIntResElt -> RngIntRes
            Set(R) : RngIntRes -> SetEnum

      Ring Predicates and Booleans

      Homomorphisms
            hom< R -> S | > : RngIntRes, Rng -> Map

 
Elements of Residue Class Rings

      Creation
            elt< R | k > : RngIntRes, RngIntElt -> RngIntResElt
            R ! k : RngIntRes, RngIntElt -> RngIntResElt
            Random(R) : RngIntRes -> RngIntResElt

      Arithmetic Operators

      Equality and Membership

      Parent and Category

      Predicates on Ring Elements

      Solving Equations over Z/mZ
            Solution(a, b) : RngIntResElt, RngIntResElt -> RngIntResElt
            IsSquare(n) : RngIntResElt -> BoolElt, RngIntResElt
            Sqrt(a) : RngIntResElt -> RngIntResElt
            AllSquareRoots(a) : RngIntResElt -> [ RngIntResElt ]
            Example RngIntRes_element-ops (H19E4)

 
Ideal Operations
      ideal< R | a1, ..., ar > : RngIntRes, RngIntResElt, ..., RngIntResElt -> RngIntRes
      GreatestCommonDivisor(a, b) : RngIntResElt, RngIntResElt -> RngIntResElt
      LeastCommonMultiple(a, b) : RngIntResElt, RngIntResElt -> RngIntResElt
      LeastCommonMultiple(Q) : [RngIntResElt] -> RngIntResElt

 
The Unit Group
      UnitGroup(R) : RngIntRes -> GrpAb, Map
      IsPrimitive(n) : RngIntResElt -> BoolElt
      PrimitiveElement(R) : RngIntRes -> RngIntResElt
      Order(a) : RngIntResElt -> RngIntElt
      Normalize(x) : RngIntRes -> RngIntResElt, RngIntResElt
      Example RngIntRes_unit-group (H19E5)
      Example RngIntRes_cyclic-unit-group (H19E6)

 
Dirichlet Characters

      Creation
            DirichletGroup(N) : RngIntElt -> GrpDrch
            DirichletGroup(N,R) : RngIntElt, Rng -> GrpDrch
            DirichletGroup(N,R,z,r) : RngIntElt, Rng, RngElt, RngIntElt -> GrpDrch
            FullDirichletGroup(N) : RngIntElt -> GrpDrch
            BaseExtend(G, R) : GrpDrch, Rng -> GrpDrch
            AssignNames(~G, S) : GrpDrch, [MonStgElt] ->

      Element Creation
            Elements(G) : GrpDrch -> [GrpDrchElt]
            Random(G) : GrpDrch -> GrpDrchElt
            G . i : GrpDrch, RngIntElt -> GrpDrchElt
            G ! x : GrpDrch, Any -> GrpDrchElt
            KroneckerCharacter(D) : RngIntElt -> GrpDrchElt

      Properties of Dirichlet Groups
            BaseRing(G) : GrpDrch -> Rng
            Modulus(G) : GrpDrch -> RngIntElt
            Order(G) : GrpDrch -> RngIntElt
            Exponent(G) : GrpDrch -> RngIntElt
            AbelianGroup(G) : GrpDrch -> GrpAb, Map
            NumberOfGenerators(G) : GrpDrch -> RngIntElt
            Generators(G) : GrpDrch -> [GrpDrchElt]
            G . i : GrpDrch, RngIntElt -> GrpDrchElt
            UnitGenerators(G) : GrpDrch -> [RngIntElt]

      Properties of Elements
            BaseRing(chi) : GrpDrchElt -> Rng
            Modulus(chi) : GrpDrchElt -> RngIntElt
            Conductor(chi) : GrpDrchElt -> RngIntElt
            ElementToSequence(chi) : GrpDrchElt -> SeqEnum
            x eq y : GrpDrchElt, GrpDrchElt -> BoolElt
            Order(chi) : GrpDrchElt -> RngIntElt
            IsTrivial(chi) : GrpDrchElt -> BoolElt
            IsPrimitive(chi) : GrpDrchElt -> BoolElt
            AssociatedPrimitiveCharacter(chi) : GrpDrchElt -> GrpDrchElt
            IsEven(chi) : GrpDrchElt -> BoolElt
            IsOdd(chi) : GrpDrchElt -> BoolElt
            IsTotallyEven(chi) : GrpDrchElt -> BoolElt
            Decomposition(chi) : GrpDrchElt -> List
            GaloisConjugacyRepresentatives(G) : GrpDrch -> [GrpDrchElt]
            MinimalBaseRingCharacter(chi) : GrpDrchElt -> GrpDrchElt

      Evaluation
            Evaluate(chi,n) : GrpDrchElt, RngIntElt -> RngElt
            ValueList(chi) : GrpDrchElt -> [RngElt]
            ValuesOnUnitGenerators(chi) : GrpDrchElt -> [RngElt]
            OrderOfRootOfUnity(r, n) : RngElt, RngIntElt -> RngIntElt

      Arithmetic
            x * y : GrpDrchElt, GrpDrchElt -> GrpDrchElt
            x ^ n : GrpDrchElt, RngIntElt -> GrpDrchElt
            x ^ phi : GrpDrchElt, Map -> GrpDrchElt
            Sqrt(x) : GrpDrchElt -> GrpDrchElt

      Example
            Example RngIntRes_Dirichlet (H19E7)

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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013