The determination of norm-Euclidean cyclic cubic fields

TL;DR

This paper fully characterizes all norm-Euclidean cyclic cubic fields by improving bounds and computational methods, bridging the gap between theoretical estimates and explicit classification.

Contribution

It provides the first unconditional complete classification of norm-Euclidean cyclic cubic fields, refining bounds and computational techniques.

Findings

Exactly 13 such fields are identified unconditionally.

New explicit bounds for cubic non-residues are established.

Enhanced computational methods enable complete classification.

Abstract

It is known on the Generalised Riemann Hypothesis that there are precisely $13$ cyclic cubic fields that are norm-Euclidean. Unconditionally, there is a gap between analytic estimates which hold for all sufficiently large conductors and computational techniques. In this paper, we establish new results concerning explicit bounds for cubic non-residues and refine previous computational techniques, enabling us to completely characterise all norm-Euclidean cyclic cubic fields.