On the extreme points in a cubic order (Q2501405)

The Voronoi algorithm can be used to obtain a fundamental system of units of a cubic field. This is a periodic algorithm which involves an isotropic vector of a quadratic form for which one wants to exhibit an extremal point adjacent to 1. B. Adam proved that there are eight potential candidates for this extremal point. In the paper under review, this list of eight candidates is reduced to five. Moreover, the Voronoi algorithm is used to obtain a fundamental unit for a class of cubic fields \(Q(\omega)\) considered by C. Levesque and G. Rhin, namely when \(f(\omega)=0\) with \(f(X)=X^3-n^mX^e-(n-1)X-n^m\).