Calculating power integral bases by solving relative Thue equations

Abstract

In our recent paper \cite{girelthue} we gave an efficient algorithm to calculate "small" solutions of relative Thue equations (where "small" means an upper bound of type $10^{500}$ for the sizes of solutions). Here we apply this algorithm to calculating power integral bases in sextic fields with an imaginary quadratic subfield and to calculating relative power integral bases in pure quartic extensions of imaginary quadratic fields.  In both cases the crucial point of the calculation is the resolution of a relative Thue equation. We produce numerical data that were not known before.