On the $\zeta_3$-Pell equation

TL;DR

This paper investigates the distribution of algebraic integers in the cyclotomic field $K=\mathbb{Q}(\zeta_3)$ for which a specific norm equation involving cube roots is solvable, extending classical Pell equation results to a cyclotomic setting.

Contribution

It introduces a study of the solvability of a norm equation related to the $oldsymbol{ ext{zeta}_3}$-Pell equation in cyclotomic fields, generalizing Stevenhagen's conjecture.

Findings

Characterizes the distribution of solutions to the norm equation in $K$.

Provides insights into the generalization of the negative Pell equation in cyclotomic fields.

Abstract

Let $K = \mathbb{Q}(\zeta_3)$, where $\zeta_3$ is a primitive root of unity. In this paper we study the distribution of integers $\alpha \in \mathcal{O}_K$ for which the norm equation $N_{K(\sqrt[3]{\alpha})/K}(\mathbf{x}) = \zeta_3$ is solvable for integers $\mathbf{x} \in \mathcal{O}_{K(\sqrt[3]{\alpha})}$. The analogous question for $\zeta_2 = -1$ is the well-known negative Pell equation. We also address the natural generalization of Stevenhagen's conjecture on the negative Pell equation in this setting.