# 13 Cyclotomics

GAP allows computations in abelian extension fields of the rational field Q, i.e., fields with abelian Galois group over Q. These fields are described in chapter Subfields of Cyclotomic Fields. They are subfields of cyclotomic fields Q_n = Q(e_n) where e_n = e^{frac{2pi i}{n}} is a primitive n--th root of unity. Their elements are called cyclotomics.

The internal representation of a cyclotomic does not refer to the smallest number field but the smallest cyclotomic field containing it (the so--called conductor). This is because it is easy to embed two cyclotomic fields in a larger one that contains both, i.e., there is a natural way to get the sum or the product of two arbitrary cyclotomics as element of a cyclotomic field. The disadvantage is that the arithmetical operations are too expensive to do arithmetics in number fields, e.g., calculations in a matrix ring over a number field. But it suffices to deal with irrationalities in character tables (see Character Tables). (And in fact, the comfortability of working with the natural embeddings is used there in many situations which did not actually afford it ldots)

All functions that take a field extension as ---possibly optional--- argument, e.g., `Trace` or `Coefficients` (see chapter Fields), are described in chapter Subfields of Cyclotomic Fields.

the representation of cyclotomics in GAP (see More about Cyclotomics),
integral elements of number fields (see Cyclotomic Integers, IntCyc, RoundCyc),
characteristic functions (see IsCyc, IsCycInt),
comparison and arithmetical operations of cyclotomics (see Comparisons of Cyclotomics, Operations for Cyclotomics),
functions concerning Galois conjugacy of cyclotomics (see GaloisCyc, StarCyc), or lists of them (see GaloisMat, RationalizedMat),
some special cyclotomics, as defined in CCN85 (see ATLAS irrationalities, Quadratic)

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

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GAP 3.4.4
April 1997