On Lenstra's criterion for norm-Euclideanity of number fields and properties of Dedekind zeta-functions

TL;DR

This paper analyzes Lenstra's criterion for norm-Euclideanity in number fields, demonstrating its ineffectiveness for large degrees under GRH and proposing a new, computationally testable condition involving Dedekind zeta-functions.

Contribution

It explicitly shows the criterion's limitations for large degree fields under GRH and introduces a new conjectural condition on Dedekind zeta-functions that can be computationally verified.

Findings

Criterion becomes ineffective for degrees ≥ 62238 under GRH.

A new condition on Dedekind zeta-functions can replace GRH assumptions.

This condition is computationally checkable for moderate degree fields.

Abstract

In 1977, Lenstra provided a criterion for norm-Euclideanity of number fields and noted that this criterion becomes ineffective for number fields of large enough degrees under the Generalised Riemann Hypothesis (GRH) for the Dedekind zeta-functions. In the first part of the paper we make Lenstra's observation explicit by proving that, under GRH, the criterion becomes ineffective for all number fields of degree $n \geq 62238$. This follows from combining the criterion assumption with the explicit lower bound for the discriminant of $K$ under GRH, and the (trivial) upper bound for the minimal proper ideal norm in $\mathcal{O}_K$. Unconditionally, the lower bound for the discriminant is too weak to lead to such a contradiction. However, we show that GRH can be replaced by another condition on the Dedekind zeta functions $\zeta_K$, a conjectural lower bound for $\zeta_K$ at a point to the right of $s = 1$. Combined with Zimmert's approach, this condition implies a different type of upper bound for the minimal proper ideal norm and again contradicts Lenstra's criterion for all $n$ large enough. The advantage of the new potential condition on $\zeta_K$ is that it can be computationally checked for number fields of not too large degrees.