Stewart's Theorem revisited: suppressing the norm $\pm 1$ hypothesis

Abstract

Let $\gamma$ be an algebraic number of degree $2$ and not a root of unity. In this note we show that there exists a prime ideal $\mathfrak{p}$ of $\mathbb{Q}(\gamma)$ satisfying $\nu_\mathfrak{p}(\gamma^n-1)\ge 1$, such that the rational prime $p$ underlying $\mathfrak{p}$ grows quicker than $n$.