Calculating power integral bases by using relative power integral bases

TL;DR

This paper explores the relationship between relative and absolute power integral bases in number fields, providing methods to compute absolute bases from relative ones, with applications to specific octic fields.

Contribution

It introduces a new approach to derive absolute power integral bases from relative bases, enhancing computational techniques in algebraic number theory.

Findings

Established a method to compute absolute bases from relative bases.

Applied the method to infinite families of octic fields with quadratic subfields.

Demonstrated the practical computation of power integral bases in specific cases.

Abstract

Let $M\subset K$ be number fields. We consider the relation of relative power integral bases of $K$ over $M$ with absolute power integral bases of $K$ over $Q$. We show how generators of absolute power integral bases can be calculated from generators of relative ones. We apply our ideas in infinite families of octic fields with quadratic subfields.