On a problem of Eisenstein

1. Introduction. In 1844, a list of 11 open problems composed by Eisen-stein was published in Crelle's journal [7]. Among these problems, which are rather diverse in nature and precision, there are three that pertain to class groups of quadratic orders. Class groups of other orders were still unknown at that time, and Eisenstein's questions are couched in Gauss's language of quadratic forms. In various forms, they ask for " criteria " to recognize quadratic discriminants that yield class numbers divisible by an integer n, and to recognize for such discriminants the classes that are in the kernel or the image of the multiplication-by-n map. For n = 2, this is accomplished by Gauss's theory of genera and ambiguous forms. From our modern point of view, it is clear that, in the case of quadratic orders, one cannot hope for an immediate generalization of these results if one replaces n = 2 by n = 3, as is done by Eisenstein in his eighth problem. In fact, the behavior of the odd part of quadratic class groups is in many ways as intractable as it was in Eisenstein's days, and our knowledge of their " average behavior " is almost entirely conjectural [2]. The problem we will focus on in this paper is Eisenstein's fourth question , which can be seen as a weak version of the problem above for the case n = 3. In this question, we are asked to give a criterion to decide a priori whether the equation x