Paper detail

On the $ζ_3$-Pell equation

Let $K = \mathbb{Q}(ζ_3)$, where $ζ_3$ is a primitive root of unity. In this paper we study the distribution of integers $α\in \mathcal{O}_K$ for which the norm equation $N_{K(\sqrt[3]α)/K}(\mathbf{x}) = ζ_3$ is solvable for integers $\mathbf{x} \in \mathcal{O}_{K(\sqrt[3]α)}$. The analogous question for $ζ_2 = -1$ 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.