Index form equations in quintic fields

The problem of determining power integral bases in algebraic number fields is equivalent to solving the corresponding index form equations. As is known (cf. Győry [25]), every index form equation can be reduced to an equation system consisting of unit equations in two variables over the normal closure of the original field. However, the unit rank of the normal closure is usually too large for practical use. In a recent paper Győry [27] succeeded in reducing index form equations to systems of unit equations in which the unknown units are elements of unit groups generated by much fewer generators. On the other hand, Wildanger [32] worked out an efficient enumeration algorithm that makes it feasible to solve unit equations even if the rank of the unit group is ten. Combining these developments we describe an algorithm to solve completely index form equations in quintic fields. The method is illustrated by numerical examples: we computed all power integral bases in totally real quintic fields with Galois group S5.