Bicyclic Bicubic Fields

Abstract

There is an extensive body of literature on the bicyclic biquadratic fields. These fields provide the simplest examples of abelian noncyclic extensions ofQ.In sharp contrast, there is a dearth of literature on the bicyclic bicubic extensions of the rational numbers. These fields together with the abelian noncyclic octic extensions provide the next simplest abelian noncyclic extensions.In this article, we shall study abelian bicyclic bicubic extensions ofQof degree 9. Hasse [4, v-ix] has stated as important objectives: the computation of an integral basis, the determination of class number and the calculation of fundamental units for abelian fields. In this article, we will solve the first problem completely, and show that the solution to the unit problem leads to a solution of the class number problem. Moreover, we shall give a method for determining the unit group up to a subgroup which has index 1 or 3 and so determine the class number up to a factor of 3.