A method for finding units in cubic orders of a negative discriminant

Abstract

Among other extremely remarkable things in Hermite's famous letters to Jacobit we find a very brief outline of a method which can be used for the actual discovery of units in cubic orders. Much, however, remained to be done in order that Hermite's ideas could be embodied in a really practical method easily applicable to numerical examples. Zolotareff was the first to develop Hermite's suggestion in the case of a negative discriminant. In his little known but remarkable thesis On an indeterminate equation of the third degree published in Russian in 1869, Zolotareff developed a method for finding units in the order x+yO+?z2 where 0 is a root of the irreducible equation 03=A, based on Hermite's principle of continuous variables; that is, on the study of successive minima of a certain positive ternary form containing a continuous parameter. Zolotareff's most important contribution consisted in the peculiar manner of reducing the study of successive minima of a ternary quadratic form to a similar problem concerning binary forms. In itself Zolotareff's method is remarkable, but it requires further complements in order to give all the successive minima, as is strictly required by Hermite's principle, and these complements unfortunately detract much from its practical value. When studying Zolotareff's paper I noticed, however, that, by retaining his basic idea, but departing from Hermite's requirement to consider minima of a variable ternary form, one can build up a new method for finding units in cubic orders of a negative discriminant which can be applied to numerical examples with comparative ease. The best method hitherto known for that purpose is one given by Voronoi in 18961. I do not venture to say that the new method explained in this paper is more expedient. That may be decided only by application of both to numerous examples. 1. An order (or ring) in an algebraic field is a system of integers of that