Introduction

Cyclotomic Fields (like the Quadratic Fields) are a subtype of the Number Fields ( FldNum). They have some extra functionality which is described below and use some more efficient implementations. Orders of cyclotomic fields form the category RngCyc and the fields themselves FldCyc. Functions for cyclotomic fields and orders which work generally for number fields, their orders and elements are listed in Chapter ORDERS AND ALGEBRAIC FIELDS.

There are two different representations of cyclotomic fields available:

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The "dense" representation: the field is conceptually represented as Q(x)/f(x) where f is a cyclotomic polynomial, i.e., the minimal polynomial of a primitive root of unity.

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The "sparse" representation: Let n=prod p_i^(r_i) be the factorisation of n into prime powers and n_i := p_i^(r_i). Then Q(zeta_n) = Q(zeta_(n_1), ..., zeta_(n_r)) and the field is represented as Q(x_1, ..., x_r)/< f_(n_1)(x_1), ..., f_(n_r)(x_r) >.