Computation of Units in Number Fields

Abstract

We discuss three algorithms to find small norm elements in number fields. One of these algorithms is a continued fraction-like algorithm based on the LLL-reduction of positive definite quadratic forms as suggested by Beukers. The other two algorithms are adaptations of that algorithm. We discuss how to find units from these small norm elements and how to extract a system of independent units from that. We discuss properties of these algorithms and compare them to algorithms by Cohen, Diaz y Diaz, Olivier, by Buchmann, Peth\H{o} and by Pohst, Zassenhaus. We run tests on an implementation of these algorithms in Mathematica.