Introduction

      Representation

      Coercion
            Example FldRat_Coercion (H21E1)

      Homomorphisms
            Example FldRat_homomorphism (H21E2)

 
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 (H21E3)

      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

      Continued Fractions
            ContinuedFraction(r) : FldRatElt -> [ RngIntElt ]
            ContinuedFractionValue(C) : [ RngIntElt ] -> FldRatElt
            HirzebruchJungContinuedFraction(r) : FldRatElt -> [ RngIntElt ]
            HirzebruchJungContinuedFractionValue(C) : [ 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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