Structure Operations

Related Structures

Category(Q): FldRat -> Cat

Parent(Q): FldRat -> PowerStructure

PrimeField(Q): FldRat -> FldRat

IntegralBasis(Q): FldRat -> [ FldRatElt ]

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).

MinimalField(q): FldRatElt -> FldRat

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.

BaseField(Q): FldRat -> FldRat

In analogy to the number fields, returns the coefficient field of \(Q\) which will be \({\mathbb{Q}}\).

Basis(Q): FldRat -> [FldRatElt]

AbsoluteBasis(Q): FldRat -> [FldRatElt]

A basis for \(Q\) as a \({\mathbb{Q}}\)-vector space, i.e. \([1]\).

UnitGroup(Q): FldRat -> GrpAb, Map

The unit group of the maximal order of \({\mathbb{Q}}\) (i.e. of \({\mathbb{Z}}\)).

ClassGroup(Q): FldRat -> GrpAb, Map

The class group of the ring of integers \({\mathbb{Z}}\) of \({\mathbb{Q}}\) (which is trivial).

AutomorphismGroup(Q): FldRat -> GrpPerm, PowMapAut, Map

AutomorphismGroup(Q, Q): FldRat, FldRat -> GrpPerm, PowMapAut, Map

The group of \({\mathbb{Q}}\) automorphisms of \({\mathbb{Q}}\), ie. a trivial finitely presented group, the parent structure for \({\mathbb{Q}}\)-automorphisms and a map from the group to actual field automorphisms. In this case, of course the only \({\mathbb{Q}}\)-automorphism will be the identity.

Algebra(Q, Q): FldRat, Fld -> AlgAss, Map

The rational numbers form canonically an algebra. This function returns an associative \({\mathbb{Q}}\)-algebra isomorphic to \({\mathbb{Q}}\) and the map from the algebra to \({\mathbb{Q}}\).

VectorSpace(Q, Q): FldRat, Fld -> ModTupFld, Map

The rational numbers form canonically a vector space. This function returns a \({\mathbb{Q}}\)-vector space isomorphic to \({\mathbb{Q}}\) and the map from the vector space to \({\mathbb{Q}}\).

Decomposition(Q, p): FldRat, RngIntElt -> []

Decomposition(Q, p): FldRat, Infty -> []

For a prime \(p\) or for the “infinite prime” Infinity() compute the decomposition in \({\mathbb{Q}}\) as a number field. This returns a list of length one containing a 2-tuple describing the splitting behaviour: the first component contains \(p\) and the second it’s ramification degree, ie. 1.

Numerical Invariants

The functions below are defined for the rational field \({\mathbb{Q}}\) mainly because it often arises as a degenerate case of quadratic or cyclotomic field constructions. See the corresponding Chapters Quadratic Fields and Cyclotomic Fields for more.

Characteristic(Q): FldRat -> RngIntElt

Conductor(Q): FldRat -> RngIntElt

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

Degree(Q): FldRat -> RngIntElt

AbsoluteDegree(Q): FldRat -> RngIntElt

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

Discriminant(Q): FldRat -> RngIntElt

AbsoluteDiscriminant(Q): FldRat -> RngIntElt

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

DefiningPolynomial(Q): FldRat -> RngUPolElt

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

Signature(Q): FldRat -> RngIntElt, RngIntElt

The signature (number of real embeddings and pairs of complex embeddings) of \({\mathbb{Q}}\).

Ring Predicates and Booleans

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