60 Abelian Number Fields
 60.1 Construction of Abelian Number Fields
 60.2 Operations for Abelian Number Fields
 60.3 Integral Bases of Abelian Number Fields
 60.4 Galois Groups of Abelian Number Fields
 60.5 Gaussians

60 Abelian Number Fields

An abelian number field is a field in characteristic zero that is a finite dimensional normal extension of its prime field such that the Galois group is abelian. In GAP, one implementation of abelian number fields is given by fields of cyclotomic numbers (see Chapter 18). Note that abelian number fields can also be constructed with the more general AlgebraicExtension (67.1-1), a discussion of advantages and disadvantages can be found in 18.6. The functions described in this chapter have been developed for fields whose elements are in the filter IsCyclotomic (18.1-3), they may or may not work well for abelian number fields consisting of other kinds of elements.

Throughout this chapter, ? n subscript ? 𝑛 ?_{n} will denote the cyclotomic field generated by the field ? ? ? of rationals together with n 𝑛 n -th roots of unity.

In 60.1, constructors for abelian number fields are described, 60.2 introduces operations for abelian number fields, 60.3 deals with the vector space structure of abelian number fields, and 60.4 describes field automorphisms of abelian number fields,