### Sparse Representation for Cyclotomic Fields

Claus Fieker
Source: Experiment. Math. Volume 16, Issue 4 (2007), 493-500.

#### Abstract

Currently, all major implementations of cyclotomic fields as well as number fields are based on a dense model in which elements are represented either as dense polynomials in the generator of the field or as coefficient vectors with respect to a fixed basis. While this representation allows for the asymptotically fastest arithmetic for general elements, it is unsuitable for fields of degree greater than $10^4$ that arise in certain applications such as character theory for finite groups. We propose instead a sparse representation for cyclotomic fields that is particularly tailored to representation theory. We implemented our ideas in MAGMA and used it for fields of degree greater than $10^6$ over $\Q$

First Page:
Primary Subjects: 11-04
Secondary Subjects: 11R18, 11Y16
Full-text: Open access