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RATIONAL FIELD

 
Acknowledgements
 
Introduction
      Representation
      Coercion
      Homomorphisms
 
Creation Functions
      Creation of Structures
      Creation of Elements
 
Structure Operations
      Related Structures
      Numerical Invariants
      Ring Predicates and Booleans
 
Element Operations
      Parent and Category
      Arithmetic Operators
      Numerator and Denominator
      Equality and Membership
      Predicates on Ring Elements
      Comparison
      Conjugates, Norm and Trace
      Absolute Value and Sign
      Rounding and Truncating
      Rational Reconstruction
      Valuation
      Sequence Conversions







DETAILS

 
Introduction

      Representation

      Coercion
            Example FldRat_Coercion (H20E1)

      Homomorphisms
            Example FldRat_homomorphism (H20E2)

 
Creation Functions

      Creation of Structures
            Rationals() : -> FldRat
            MaximalOrder(Q) : FldRat -> RngInt
            FieldOfFractions(Q) : FldRat -> FldRat
            Completion(Q, P) : FldRat, RngInt -> FldLoc, Map

      Creation of Elements
            a / b : RngIntElt, RngIntElt -> FldRatElt
            Q ! [a] : FldRat, RngElt -> FldRatElt
            Q ! [a, b] : FldRat, RngIntElt, RngIntElt -> FldRatElt
            Q ! a : FldRat, RngIntElt -> FldRatElt
            RootOfUnity(n, Q) : RngIntElt, FldRat -> FldRatElt
            Random(Q, m) : FldRat, RngIntElt -> FldRatElt

 
Structure Operations

      Related Structures
            IntegralBasis(Q) : FldRat -> [ FldRatElt ]
            MinimalField(q) : FldRatElt -> FldRat
            MinimalField(S) : SetEnum -> FldRat
            BaseField(Q) : FldRat -> FldRat
            Basis(Q) : FldRat -> [FldRatElt]
            UnitGroup(Q) : FldRat -> GrpAb, Map
            ClassGroup(Q) : FldRat -> GrpAb, Map
            AutomorphismGroup(Q) : FldRat -> GrpPerm, PowMapAut, Map
            Algebra(Q, Q) : FldRat, Fld -> AlgAss, Map
            VectorSpace(Q, Q) : FldRat, Fld -> ModTupFld, Map
            Decomposition(Q, p) : FldRat, RngIntElt -> []

      Numerical Invariants
            Conductor(Q) : FldRat -> RngIntElt
            Degree(Q) : FldRat -> RngIntElt
            Discriminant(Q) : FldRat -> RngIntElt
            DefiningPolynomial(Q) : FldRat -> RngUPolElt
            Signature(Q) : FldRat -> RngIntElt, RngIntElt

      Ring Predicates and Booleans

 
Element Operations

      Parent and Category

      Arithmetic Operators

      Numerator and Denominator
            Numerator(q) : FldRatElt -> RngIntElt
            Denominator(q) : FldRatElt -> RngIntElt
            Example FldRat_numerator (H20E3)

      Equality and Membership

      Predicates on Ring Elements
            IsIntegral(q) : FldRatElt -> BoolElt

      Comparison

      Conjugates, Norm and Trace
            ComplexConjugate(q) : FldRatElt -> FldRatElt
            Conjugate(q) : FldRatElt -> FldRatElt
            Norm(q) : FldRatElt -> FldRatElt
            Trace(q) : FldRatElt -> FldRatElt
            MinimalPolynomial(q) : FldRatElt -> RngUPolElt

      Absolute Value and Sign
            AbsoluteValue(q) : FldRatElt -> FldRatElt
            Sign(q) : FldRatElt -> RngIntElt
            Height(q) : FldRatElt -> RngIntElt

      Rounding and Truncating
            Ceiling(q) : FldRatElt -> RngIntElt
            Floor(q) : FldRatElt -> RngIntElt
            Round(q) : FldRatElt -> RngIntElt
            Truncate(q) : FldRatElt -> RngIntElt
            Qround(q, M) : FldRatElt, RngIntElt -> FldRatElt

      Rational Reconstruction
            RationalReconstruction(s) : RngIntResElt -> BoolElt, FldRatElt

      Valuation
            Valuation(x, p) : FldRatElt, RngIntElt -> RngIntElt, FldRatElt

      Sequence Conversions
            ElementToSequence(a) : FldRatElt -> [FldRatElt]

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Version: V2.19 of Mon Dec 17 14:40:36 EST 2012