Some Higher Degree Fields

The resolution of index form equations becomes very difficult for higher degree fields. The method for general quintic fields is already time consuming; for sextic fields a general algorithm does not seem to be feasible, we developed methods for determining power integral bases only in sextic fields having subfields. The case of number fields of degree seven seems to be complicated, since these fields can not have subfields. Special number fields of degree seven (e.g., cyclic fields) can be considered by the methods we used so far.