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.
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.
In analogy to the number fields, returns the coefficient field of
Q which will be Q.
AbsoluteBasis(Q) : FldRat -> [FldRatElt]
A basis for Q as a {Q}-vector space, i.e. [1].
The unit group of the maximal order of Q (i.e. of Z).
The class group of the ring of integers Z of Q (which is trivial).
AutomorphismGroup(Q, Q) : FldRat, FldRat -> GrpPerm, PowMapAut, Map
The group of Q automorphisms of Q, ie. a trivial finitely
presented group, the parent structure for Q-automorphisms and a map
from the group to actual field automorphisms. In this case, of course
the only Q-automorphism will be the identity.
The field of the rational number form canonically an algebra. This function
returns an associative Q-algebra isomorphic to Q and the map
from the algebra to Q.
The field of the rational number form canonically a vector space.
This function
returns a Q-vector space isomorphic to Q and the map
from the vector space to Q.
Decomposition(Q, p) : FldRat, Infty -> []
For a prime p or for the "infinite prime" Infinity()
compute the decomposition in 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.
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(ζn). For the rational field this is 1.
AbsoluteDegree(Q) : FldRat -> RngIntElt
The degree of Q as a number field (which is 1 for the rational field).
AbsoluteDiscriminant(Q) : FldRat -> RngIntElt
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).
The signature (number of real embeddings and pairs of complex embeddings) of
Q.
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
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