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

TL;DR

This paper revisits Stewart's Theorem in the context of algebraic number theory, demonstrating the existence of prime ideals with specific valuation properties that grow faster than the exponent, providing new insights into the norm $oxed{1}$ hypothesis.

Contribution

It introduces a novel approach to analyze prime ideals related to algebraic numbers of degree 2, challenging the traditional norm $oxed{1}$ hypothesis with explicit growth results.

Findings

Existence of prime ideals with valuations growing faster than $n$

Counterexamples to the norm $oxed{1}$ hypothesis in specific algebraic settings

Enhanced understanding of prime ideal distribution in quadratic fields

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 $n$.