15 Subfields of Cyclotomic Fields

The only number fields that GAP can handle at the moment are subfields of cyclotomic fields, e.g., Q(sqrt{5}) is a number field that is not cyclotomic but contained in the cyclotomic field Q_5 = Q(e^{frac{2pi i}{5}}). Although this means that GAP does not know arbitrary algebraic number fields but only those with abelian Galois group, here we call these fields number fields for short. The elements of number fields are called cyclotomics (see chapter Cyclotomics). Thus number fields are the domains (see chapter Domains) related to cyclotomics; they are special field records (see Field Records) which are needed to specify the field extension with respect to which e.g. the trace of a cyclotomic shall be computed.

In many situations cyclotomic fields need not be treated in a special way, except that there may be more efficient algorithms for them than for arbitrary number fields. For that, there are the global variables NumberFieldOps and CyclotomicFieldOps, both records which contain the field operations stored in FieldOps (see chapter Fields) and Domain Functions for Number Fields). If all necessary information about a function is already given in chapter Fields, this function is not described here; this is the case e.g. for Conjugates and related functions, like Trace and CharPol. Some functions, however, need further explanation, e.g., Coefficients for Number Fields tells more about Coefficients for number fields.

There are some functions which are different for cyclotomic fields and other number fields, e.g., the field constructors CF resp. NF. In such a situation, the special case is described in a section immediately following the section about the general case.

Besides the single number fields, there is another domain in GAP related to number fields, the domain Cyclotomics of all cyclotomics. Although this is an abstract field, namely the field Q^{ab}, Cyclotomics is not a field record. It is used by DefaultField, DefaultRing, Domain, Field and Ring (see DefaultField, DefaultRing, Domain, Field, Ring) which are mainly interested in the corresponding entries of Cyclotomics.operations since these functions know how to create fields resp. integral rings generated by some cyclotomics.

This chapter informs about:

characteristic functions (see IsNumberField, IsCyclotomicField),
field constructors (see Number Field Records, Cyclotomic Field Records),
(default) fields of cyclotomics (see DefaultField and Field for Cyclotomics), and (default) rings of cyclotomic integers (see DefaultRing and Ring for Cyclotomic Integers),
Galois groups of number fields (see GeneratorsPrimeResidues, GaloisGroup for Number Fields),
vector space bases (see ZumbroichBase, Integral Bases for Number Fields, NormalBaseNumberField) and coefficients (see Coefficients for Number Fields) and
overlaid functions in the operations records (see Domain Functions for Number Fields).

The external functions are in the file LIBNAME/"numfield.g"

Subsections

1. IsNumberField
2. IsCyclotomicField
3. Number Field Records
4. Cyclotomic Field Records
5. DefaultField and Field for Cyclotomics
6. DefaultRing and Ring for Cyclotomic Integers
7. GeneratorsPrimeResidues
8. GaloisGroup for Number Fields
9. ZumbroichBase
10. Integral Bases for Number Fields
11. NormalBaseNumberField
12. Coefficients for Number Fields
13. Domain Functions for Number Fields