Norm-Euclidean Galois fields

TL;DR

This paper establishes upper bounds on the discriminants of norm-Euclidean Galois number fields of prime degree greater than two, introduces an algorithm to identify candidates, and presents computational findings.

Contribution

It provides the first upper bounds on discriminants for these fields and offers a new algorithm for generating candidate fields up to a specified discriminant.

Findings

Derived upper bounds on discriminants for norm-Euclidean Galois fields with prime degree > 2

Developed a simple algorithm to generate candidate fields

Presented computational results validating the approach

Abstract

Let K be a Galois number field of prime degree $\ell$. Heilbronn showed that for a given $\ell$ there are only finitely many such fields that are norm-Euclidean. In the case of $\ell=2$ all such norm-Euclidean fields have been identified, but for $\ell\neq 2$, little else is known. We give the first upper bounds on the discriminants of such fields when $\ell>2$. Our methods lead to a simple algorithm which allows one to generate a list of candidate norm-Euclidean fields up to a given discriminant, and we provide some computational results.