Introduction

 
Ideals of Z
      ideal< R | a > : RngInt, RngIntElt -> RngIntRes
      Example RngIntRes_residue-ring (H20E1)

 
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, Map
            ResidueClassField(p) : RngIntElt -> FldFin, Map
            ResidueClassRing(Q) : RngIntEltFact -> RngIntRes
            Example RngIntRes_residue-ring (H20E2)

      Coercion
            Example RngIntRes_Coercion (H20E3)

      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 (H20E4)

 
Ideal Operations
      ideal< R | a₁, ..., a_r > : RngIntRes, RngIntResElt, ..., RngIntResElt -> RngIntRes
      GreatestCommonDivisor(a, b) : RngIntResElt, RngIntResElt -> RngIntResElt
      GreatestCommonDivisor(Q) : [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 (H20E5)
      Example RngIntRes_cyclic-unit-group (H20E6)

 
Dirichlet Characters

      Creation
            DirichletGroup(N) : RngIntElt -> GrpDrch
            FullDirichletGroup(N) : RngIntElt -> GrpDrch
            DirichletGroup(N,R) : RngIntElt, Rng -> GrpDrch
            DirichletGroup(N,R,z,r) : RngIntElt, Rng, RngElt, 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

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

      Attributes 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
            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 (H20E7)

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