## Experimental Mathematics

- Experiment. Math.
- Volume 16, Issue 4 (2007), 493-500.

### Sparse Representation for Cyclotomic Fields

#### 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 $\Q$

#### Article information

**Source**

Experiment. Math., Volume 16, Issue 4 (2007), 493-500.

**Dates**

First available in Project Euclid: 6 March 2008

**Permanent link to this document**

https://projecteuclid.org/euclid.em/1204836517

**Mathematical Reviews number (MathSciNet)**

MR2378488

**Zentralblatt MATH identifier**

1175.11074

**Subjects**

Primary: 11-04: Explicit machine computation and programs (not the theory of computation or programming)

Secondary: 11R18: Cyclotomic extensions 11Y16: Algorithms; complexity [See also 68Q25]

**Keywords**

Cyclotomic fields sparse representation

#### Citation

Fieker, Claus. Sparse Representation for Cyclotomic Fields. Experiment. Math. 16 (2007), no. 4, 493--500. https://projecteuclid.org/euclid.em/1204836517