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QUADRATIC FIELDS

 
Acknowledgements
 
Introduction
      Representation
 
Creation of Structures
 
Operations on Structures
      Ideal Class Group
      Norm Equations
 
Special Element Operations
      Greatest Common Divisors
      Modular Arithmetic
      Factorization
      Conjugates
      Other Element Functions
 
Special Functions for Ideals
 
Bibliography







DETAILS

 
Introduction

      Representation

 
Creation of Structures
      QuadraticField(m) : RngIntElt -> FldQuad
      EquationOrder(F) : FldQuad -> RngQuad
      MaximalOrder(F) : FldQuad -> RngQuad
      NumberField(O) : RngQuad -> FldQuad
      sub< O | f > : RngQuad, RngIntElt ->
      IsQuadratic(K) : FldNum -> BoolElt, FldQuad
      Example FldQuad_creation (H35E1)
      Example FldQuad_hom (H35E2)

 
Operations on Structures
      AssignNames(~F, [s]) : FldQuad, [ MonStgElt ]) ->
      Name(F, 1) : FldQuad, RngIntElt -> FldQuadElt
      FundamentalUnit(K) : FldQuad -> FldQuadElt
      Discriminant(K) : FldQuad -> RngIntElt
      Conductor(K) : FldQuad -> RngIntElt, [RngIntElt]
      Conductor(O) : RngQuad -> RngIntElt

      Ideal Class Group
            ClassGroup(K) : FldQuad -> GrpAb, Map
            ClassNumber(K) : FldQuad -> RngIntElt
            PicardGroup(O) : RngQuad -> GrpAb, Map
            Example FldQuad_quad_sieve (H35E3)
            QuadraticClassGroupTwoPart(K) : FldQuad -> GrpAb, Map
            Example FldQuad_qcgtp (H35E4)

      Norm Equations
            NormEquation(F, m) : FldQuad, RngIntElt -> BoolElt, SeqEnum
            Example FldQuad_norm-equation (H35E5)

 
Special Element Operations
      a mod b : RngQuadElt, RngQuadElt -> RngQuadElt

      Greatest Common Divisors
            Gcd(a, b) : RngQuadElt, RngQuadElt -> RngQuadElt
            Lcm(a, b) : RngQuadElt, RngQuadElt -> RngQuadElt

      Modular Arithmetic
            Modexp(a, e, n) : RngQuadElt, RngInt, RngQuadElt -> RngQuadElt

      Factorization
            Factorization(n) : RngQuadElt -> SeqEnum, Tup
            TrialDivision(n, B) : RngQuadElt, RngIntElt -> SeqEnum, SeqEnum, Tup

      Conjugates
            ComplexConjugate(a) : FldQuadElt -> FldQuadElt
            Conjugate(a) : FldQuadElt -> FldQuadElt

      Other Element Functions
            BiquadraticResidueSymbol(a, b) : RngQuadElt, RngQuadElt -> RngQuadElt
            Primary(a) : RngQuadElt -> RngQuadElt
            Example FldQuad_Represent (H35E6)

 
Special Functions for Ideals
      Content(I) : RngQuadFracIdl -> RngQuadFracIdl
      Conjugate(I) : RngQuadFracIdl -> RngQuadFracIdl
      Discriminant(I) : RngQuadFracIdl -> RngIntElt
      QuadraticForm(I) : RngQuadFracIdl -> QuadBinElt
      Ideal(f) : QuadBinElt -> RngQuadIdl
      Reduction(I) : RngQuadFracIdl -> RngQuadFracIdl

 
Bibliography

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