Class numbers of the simplest cubic fields

Source abstract

Using the "simplest cubic fields" of D. Shanks, we give a modified proof and an extension of a result of Uchida, showing how to obtain cyclic cubic fields with class number divisible by n , for any n . Using 2-descents on elliptic curves, we obtain precise information on the 2-Sylow subgroups of the class groups of these fields. A theorem of H. Heilbronn associates a set of quartic fields to the class group. We show how to obtain these fields via these elliptic curves.