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.
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).
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.
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.
The smallest positive integer n such that Q is contained in the cyclotomic field Q(ζn). 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).
The signature (number of real embeddings and pairs of complex embeddings) of Q.