Monogenity of pure quintic fields: the power of sieving

TL;DR

This paper introduces a fast sieving algorithm for finding generators of power integral bases in pure quintic fields, significantly reducing computational effort compared to previous methods.

Contribution

A novel sieving approach that decreases the number of exponents to test, enabling rapid computation of power integral bases in pure quintic fields.

Findings

The new method tests all small exponents within minutes.

The sieving technique is surprisingly efficient and broadly applicable.

The approach combines Baker's method, LLL-reduction, and a novel sieving idea.

Abstract

We provide a simple algorithm for calculating all generators of power integral bases in pure quintic fields. This procedure involves the usual standard elements like Baker's method, LLL-reduction. The main purpose of the paper is to introduce a new idea to considerably diminish the number of small exponents to be considered after the reduction step. This new idea allows to test all remaining small exponents within a few minutes, using an appropriate sieve method, which turns out to be surprisingly fast. This idea will be applicable in many similar cases.