Solving Index Form Equations in Fields of Degree 9 with Cubic Subfields

We describe an efficient algorithm for solving index form equations in number fields of degree 9 which are composites of cubic fields with coprime discriminants. We develop the algorithm in detail for the case of complex cubic fields, but the main steps of the procedure are also applicable for other cases. Our most important tool is the main theorem of a recent paper of Gaál (1998a). In view of this result the index form equation in the ninth degree field implies relative index form equations over the subfields. In our case these equations are cubic relative Thue equations over cubic fields. The main purpose of the paper is to show that this approach is much more efficient than the direct method, which consists of reducing the index form equation to unit equations over the normal closure of the original field. At the end of the paper we describe our computational experience. Many ideas of the paper can be applied to develop fast algorithms for solving index form equations in other types of higher degree fields which are composites of subfields.