A new method of computing fundamental units in algebraic number fields

Abstract

Two years ago [2] the authors suggested a new procedure for solving norm equations in algebraic number fields which was shown to be superior to the known ones. Several improvements of that method were developed in the meantime [I],[3]. Since units in algebraic number fields can be characterized as algebraic integers of norm +I it seemed only natural to use the basic idea of that procedure for determining fundamental units. However, solving norm equations already requires the knowledge of a full system of independent units and therefore our procedure had to be changed considerably. Together with some other ideas about the computation of units this led to a new efficient algorithm. It is proved that one of the two crucial steps, namely the computation of fundamental units from independent ones, is superior to the known methods, and also the construction of independent units can be expected to be faster.