# 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 [Number Fields](../NumberFields/index-number-fields.md#fldnum-main).

There are two different representations of cyclotomic fields available:

- 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.

- 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 $Q(\zeta_n) = Q(\zeta_{n_1}, \ldots, \zeta_{n_r})$ and the field is represented as $Q(x_1, \ldots, x_r)/\langle f_{n_1}(x_1), \ldots, f_{n_r}(x_r)\rangle$.

As with the number fields, the non-simple representation, the issues are the same: the “sparse” representation allows for much larger fields – as long as the elements used have only few coefficients. The “dense” representation on the other hand has the asymptotically-fastest arithmetic.