Rational Field#
- Introduction
- Creation Functions
- Creation of Structures
Rationals() → FldRatRationalField() → FldRatMaximalOrder(Q): FldRat → RngIntIntegerRing(Q): FldRat → RngIntIntegerRing() → RngIntIntegers() → RngIntRingOfIntegers(Q): FldRat → RngIntFieldOfFractions(Q): FldRat → FldRatFieldOfFractions(Z): RngInt → FldRatCompletion(Q, P): FldRat, RngInt → FldLoc, Map
- Creation of Elements
a / b: RngIntElt, RngIntElt → FldRatEltQ ! [a]: FldRat, RngElt → FldRatEltQ ! [a, b]: FldRat, RngIntElt, RngIntElt → FldRatEltelt< Q | a, b >: FldRat, RngIntElt, RngIntElt → FldRatEltQ ! a: FldRat, RngIntElt → FldRatEltOne(Q): FldRat → FldRatEltIdentity(Q): FldRat → FldRatEltZero(Q): FldRat → FldRatEltRepresentative(Q): FldRat → FldRatEltRootOfUnity(n, Q): RngIntElt, FldRat → FldRatEltRandom(Q, m): FldRat, RngIntElt → FldRatElt
- Creation of Structures
- Structure Operations
- Related Structures
Category(Q): FldRat → CatParent(Q): FldRat → PowerStructurePrimeField(Q): FldRat → FldRatIntegralBasis(Q): FldRat → [ FldRatElt ]MinimalField(q): FldRatElt → FldRatMinimalField(S): SetEnum → FldRatBaseField(Q): FldRat → FldRatBasis(Q): FldRat → [FldRatElt]AbsoluteBasis(Q): FldRat → [FldRatElt]UnitGroup(Q): FldRat → GrpAb, MapClassGroup(Q): FldRat → GrpAb, MapAutomorphismGroup(Q): FldRat → GrpPerm, PowMapAut, MapAutomorphismGroup(Q, Q): FldRat, FldRat → GrpPerm, PowMapAut, MapAlgebra(Q, Q): FldRat, Fld → AlgAss, MapVectorSpace(Q, Q): FldRat, Fld → ModTupFld, MapDecomposition(Q, p): FldRat, RngIntElt → []Decomposition(Q, p): FldRat, Infty → []
- Numerical Invariants
Characteristic(Q): FldRat → RngIntEltConductor(Q): FldRat → RngIntEltDegree(Q): FldRat → RngIntEltAbsoluteDegree(Q): FldRat → RngIntEltDiscriminant(Q): FldRat → RngIntEltAbsoluteDiscriminant(Q): FldRat → RngIntEltDefiningPolynomial(Q): FldRat → RngUPolEltSignature(Q): FldRat → RngIntElt, RngIntElt
- Ring Predicates and Booleans
IsCommutative(Q): FldRat → BoolEltIsUnitary(Q): FldRat → BoolEltIsFinite(Q): FldRat → BoolEltIsOrdered(Q): FldRat → BoolEltIsField(Q): FldRat → BoolEltIsEuclideanDomain(Q): FldRat → BoolEltIsPID(Q): FldRat → BoolEltIsUFD(Q): FldRat → BoolEltIsDivisionRing(Q): FldRat → BoolEltIsEuclideanRing(Q): FldRat → BoolEltIsPrincipalIdealRing(Q): FldRat → BoolEltIsDomain(Q): FldRat → BoolEltQ eq R: FldRat, FldRat → BoolEltQ eq R: FldRat, RngInt → BoolEltQ ne R: FldRat, FldRat → BoolEltQ ne R: FldRat, RngInt → BoolElt
- Related Structures
- Element Operations
- Parent and Category
- Arithmetic Operators
+ a: FldRatElt → FldRatElt- a: FldRatElt → FldRatElta + b: FldRatElt, FldRatElt → FldRatElta - b: FldRatElt, FldRatElt → FldRatElta * b: FldRatElt, FldRatElt → FldRatElta ^ k: FldRatElt, RngIntElt → FldRatElta / b: FldRatElt, FldRatElt → FldRatElta +:= b: FldRatElt, FldRatElt → FldRatElta -:= b: FldRatElt, FldRatElt → FldRatElta *:= b: FldRatElt, FldRatElt → FldRatElta /:= b: FldRatElt, FldRatElt → FldRatElta ^:= k: FldRatElt, RngIntElt → FldRatElt
- Numerator and Denominator
- Equality and Membership
- Predicates on Ring Elements
IsIntegral(q): FldRatElt → BoolEltIsZero(a): FldRatElt → BoolEltIsOne(a): FldRatElt → BoolEltIsMinusOne(a): FldRatElt → BoolEltIsNilpotent(a): FldRatElt → BoolEltIsIdempotent(a): FldRatElt → BoolEltIsUnit(a): FldRatElt → BoolEltIsZeroDivisor(a): FldRatElt → BoolEltIsRegular(a): FldRatElt → BoolEltIsIrreducible(a): FldRatElt → BoolEltIsPrime(a): FldRatElt → BoolElt
- Comparison
a gt b: FldRatElt, FldRatElt → BoolElta ge b: FldRatElt, FldRatElt → BoolElta lt b: FldRatElt, FldRatElt → BoolElta le b: FldRatElt, FldRatElt → BoolEltMaximum(a, b): FldRatElt, FldRatElt → FldRatEltMaximum(Q): [FldRatElt] → FldRatEltMinimum(a, b): FldRatElt, FldRatElt → FldRatEltMinimum(Q): [FldRatElt] → FldRatElt
- Conjugates, Norm and Trace
- Absolute Value and Sign
- Rounding and Truncating
- Continued Fractions
ContinuedFraction(r): FldRatElt → [ RngIntElt ]ContinuedFractionValue(C): [ RngIntElt ] → FldRatEltHirzebruchJungContinuedFraction(r): FldRatElt → [ RngIntElt ]HJContinuedFraction(r): FldRatElt → [ RngIntElt ]HirzebruchJungContinuedFractionValue(C): [ RngIntElt ] → FldRatEltHJContinuedFractionValue(C): [ RngIntElt ] → FldRatElt
- Rational Reconstruction
- Valuation
- Sequence Conversions