Quadratic Fields

• Introduction
  • Representation
• Creation of Structures
  • QuadraticField(m): RngIntElt → FldQuad
  • EquationOrder(F): FldQuad → RngQuad
  • MaximalOrder(F): FldQuad → RngQuad
  • IntegerRing(F): FldQuad → RngQuad
  • RingOfIntegers(F): FldQuad → RngQuad
  • NumberField(O): RngQuad → FldQuad
  • sub< O | f >: RngQuad, RngIntElt
  • IsQuadratic(K): FldNum → BoolElt, FldQuad
  • IsQuadratic(O): RngOrd → BoolElt, RngQuad
  • Example: Creating a Quadratic Field and Order
  • Example: Embedding a Quadratic Field in a Cyclotomic Field
• Operations on Structures
  • AssignNames(~F, [s]): FldQuad, [ MonStgElt ]
  • AssignNames(~O, [s]): RngQuad, [ MonStgElt ]
  • Name(F, 1): FldQuad, RngIntElt → FldQuadElt
  • Name(O, 1): RngQuad, RngIntElt → RngQuadElt
  • FundamentalUnit(K): FldQuad → FldQuadElt
  • FundamentalUnit(O): RngQuad → RngQuadElt
  • Discriminant(K): FldQuad → RngIntElt
  • Conductor(K): FldQuad → RngIntElt, [RngIntElt]
  • Conductor(O): RngQuad → RngIntElt
  • Ideal Class Group
    • ClassGroup(K): FldQuad → GrpAb, Map
    • ClassGroup(O): RngQuad → GrpAb, Map
    • ClassNumber(K): FldQuad → RngIntElt
    • ClassNumber(O): RngQuad → RngIntElt
    • PicardGroup(O): RngQuad → GrpAb, Map
    • PicardNumber(O): RngQuad → RngIntElt
    • Example: Class Groups by Sieving
    • QuadraticClassGroupTwoPart(K): FldQuad → GrpAb, Map
    • QuadraticClassGroupTwoPart(O): RngQuad → GrpAb, Map
    • QuadraticClassGroupTwoPart(d): RngIntElt → GrpAb, Map
    • Example: Two-Part of a Quadratic Class Group
  • Norm Equations
    • NormEquation(F, m): FldQuad, RngIntElt → BoolElt, SeqEnum
    • NormEquation(F, m: parameters): FldQuad, RngIntElt → BoolElt, SeqEnum
    • NormEquation(O, m): RngQuad, RngIntElt → BoolElt, SeqEnum
    • NormEquation(O, m: parameters): RngQuad, RngIntElt → BoolElt, SeqEnum
    • Example: Norm Equation in a Quadratic Order
• Special Element Operations
  • Division Algorithm
    • a div b: RngQuadElt, RngQuadElt → RngQuadElt
    • a mod b: RngQuadElt, RngQuadElt → RngQuadElt
    • GreatestCommonDivisor(a, b): RngQuadElt, RngQuadElt → RngQuadElt
    • Gcd(a, b): RngQuadElt, RngQuadElt → RngQuadElt
    • GCD(a, b): RngQuadElt, RngQuadElt → RngQuadElt
    • LeastCommonMultiple(a, b): RngQuadElt, RngQuadElt → RngQuadElt
    • Lcm(a, b): RngQuadElt, RngQuadElt → RngQuadElt
    • LCM(a, b): RngQuadElt, RngQuadElt → RngQuadElt
    • Modexp(a, e, n): RngQuadElt, RngInt, RngQuadElt → RngQuadElt
  • Factorization
    • Factorization(n): RngQuadElt → SeqEnum, Tup
    • Factorisation(n): RngQuadElt → SeqEnum, Tup
    • TrialDivision(n, B): RngQuadElt, RngIntElt → SeqEnum, SeqEnum, Tup
  • Conjugates
    • ComplexConjugate(a): FldQuadElt → FldQuadElt
    • ComplexConjugate(a): RngQuadElt → RngQuadElt
    • Conjugate(a): FldQuadElt → FldQuadElt
    • Conjugate(a): RngQuadElt → RngQuadElt
  • Other Element Functions
    • BiquadraticResidueSymbol(a, b): RngQuadElt, RngQuadElt → RngQuadElt
    • Primary(a): RngQuadElt → RngQuadElt
    • Example: Primes with 2 as a Biquadratic Residue
• 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