Subsections

• Related Structures
• Numerical Invariants
• Ring Predicates and Booleans

Category(Q) : FldRat -> Cat

Parent(Q) : FldRat -> PowerStructure

PrimeField(Q) : FldRat -> FldRat

An integral basis for Q as a number field as a sequence of elements of Q (giving the sequence containing 1 for the rational field).

Return the least cyclotomic field containing the cyclotomic field element q; if q is rational this returns the rational field.

MinimalField(S) : SetEnum -> FldRat

Returns the minimal cyclotomic field containing the cyclotomic field elements in the enumerated set S; this will return the rational field if all elements of S are rational numbers.

The functions below are defined for the rational field Q mainly because it often arises as a degenerate case of quadratic or cyclotomic field constructions.

Characteristic(Q) : FldRat -> RngIntElt

The smallest positive integer n such that Q is contained in the cyclotomic field Q(zeta_n). For the rational field this is 1.

The degree of Q as a number field (which is 1 for the rational field).

The field discriminant of Q (which is 1 for the rational field).

An irreducible polynomial over Q a root of which generates Q as a number field (for the rational field this returns the linear polynomial x - 1).

IsCommutative(Q) : FldRat -> BoolElt

IsUnitary(Q) : FldRat -> BoolElt

IsFinite(Q) : FldRat -> BoolElt

IsOrdered(Q) : FldRat -> BoolElt

IsField(Q) : FldRat -> BoolElt

IsEuclideanDomain(Q) : FldRat -> BoolElt

IsPID(Q) : FldRat -> BoolElt

IsUFD(Q) : FldRat -> BoolElt

IsDivisionRing(Q) : FldRat -> BoolElt

IsEuclideanRing(Q) : FldRat -> BoolElt

IsPrincipalIdealRing(Q) : FldRat -> BoolElt

IsDomain(Q) : FldRat -> BoolElt

Q eq R : FldRat, FldRat -> BoolElt

Q eq R : FldRat, RngInt -> BoolElt

Q ne R : FldRat, FldRat -> BoolElt

Q ne R : FldRat, RngInt -> BoolElt