Calculating generators of power integral bases in sextic fields with a quadratic subfield: the general case

TL;DR

This work addresses the problem of finding generators of power integral bases in sextic fields with a quadratic subfield in the general case, where the relative basis over the subfield is arbitrary. It generalizes earlier approaches that assumed a special relative basis by expressing elements with a flexible relative basis $(1,(A\\alpha+B)/k,(C\\alpha^2+D\\alpha+E)/\\ell)$ and reducing to a norm equation on $Z_1-\\delta Z_2$; the reduction hinges on $I_{K/M}(\\gamma)=1$. A lattice-based reduction (LLL) yields explicit bounds on the coordinates, enabling an enumeration–test pipeline that checks $J(\\gamma)=1$ and isolates all generators with small coordinates. An illustrative trinomial example demonstrates the method, confirming monogenity and producing explicit generator families for the sextic field considered.

Abstract

In some previous works we gave algorithms for determining generators of power integral basis in sextic fields with a quadratic subfield, under certain restrictions. The purpose of the present paper is to extend those methods to the general case, when the relative integral basis of the sextic field over the quadratic subfield is of general form. This raises several technical difficulties, that we consider here.