Finding fundamental units in cubic fields

Abstract

This paper improves a method of Godwin (4) for finding a pair of fundamental units in a totally real cubic field. The determination of such a unit pair is a well known computational problem. There is an old algorithm (circa 1896) of Voronoi which solves this problem, but the algorithm is quite complicated (an account of it is given in the book of Delone and Faddeev ((3), chapter IV, part A)). The method of Godwin is, in principle, much simpler. However, this method also has its drawbacks (more is said about this in Section 4 below). Indeed, when Godwin's student Angell produced his large table (see (1)) of totally real cubic fields some 15 years after (4) appeared, Voronoi's algorithm was used to compute the pairs of fundamental units.