Calculating power integral bases by solving relative Thue equations

TL;DR

This paper presents an efficient algorithm for solving relative Thue equations and applies it to compute power integral bases in specific algebraic number fields, producing new numerical data.

Contribution

It introduces a novel algorithm for small solutions of relative Thue equations and applies it to find power integral bases in sextic and quartic fields, generating previously unknown data.

Findings

Successfully computed power integral bases in new fields

Generated numerical data not previously available

Validated the effectiveness of the algorithm for complex cases

Abstract

In our recent paper 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.