Calculation of the regulator of a pure cubic field

Abstract

A description is given of a modified version of Voronoi’s algorithm for obtaining the regulator of a pure cubic field Q ( D 3 ) Q(\sqrt [3]{D}) . This new algorithm has the advantage of executing relatively rapidly for large values of D. It also eliminates a computational problem which occurs in almost all algorithms for finding units in algebraic number fields. This is the problem of performing calculations involving algebraic irrationals by using only approximations of these numbers. The algorithm was implemented on a computer and run on all values of D ( ⩽ 10 5 ) D\;( \leqslant {10^5}) such that the class number of Q ( D 3 ) Q\;(\sqrt [3]{D}) is not divisible by 3. Several tables summarizing the results of this computation are also presented.