Computation of the Fundamental Units and the Regulator of a Cyclic Cubic Function Field

Abstract

This paper presents algorithms for computing the two fundamental units and the regulator of a cyclic cubic extension of a rational function field over a field of order {$q \equiv 1 \pmod{3}$}. The procedure is based on a method originally due to Voronoi that was recently adapted to purely cubic function fields of unit rank one. Our numerical examples show that the two fundamental units tend to have large degree, and frequently, the extension has a very small ideal class number.