Units in totally complex S3 fields

AbstractLet B be a totally complex number field, Galois over the rational field Q, with Galois group S3, the symmetric group on three elements. The group of units of B has torsion free rank 2. In this paper, we determine the various inequivalent representations that occur of S3 acting on the group of units and determine arithmetic criteria for deciding which representation occurs for a particular field. As a result, we can give a relatively simple computational procedure for determining a pair of fundamental units of B given a fundamental unit in a cubic subfield