On the Power of Integers and Conductors of Quadratic Fields

TL;DR

This paper investigates the properties of integers in quadratic fields, providing bounds on the order of elements modulo primes and conductors, with a focus on fundamental units and their behavior in various algebraic settings.

Contribution

It introduces new bounds for the order of algebraic integers modulo primes and conductors in quadratic fields, especially for fundamental units, using embeddings into matrix groups.

Findings

Derived bounds for the order of elements modulo primes.

Established bounds for the first integer n(f) related to conductors.

Analyzed the case when α is the fundamental unit of the quadratic field.

Abstract

We consider the integers $\alpha$ of the quadratic field $ \mathbb{Q} (\sqrt{d}$ $)$ where $d\in \Z$ is square-free and $d\equiv 1,2,3 \pmod 4$. Let $p$ be an odd prime. Using the embedding into $ \text{GL}(2,\mathbb{Z})$ we obtain bounds for the first $\nu\in\N$ such that $\alpha^\nu\equiv 1\pmod p.$ For the conductor $f$, we then study the first integer $n=n(f)$ such that $\alpha^n\in\mathcal{O}_{f}$. We obtain bounds for $n(f)$ and for $n(fp^{k})$. The most interesting case is that $\alpha$ is the fundamental unit of $ \mathbb{Q} (\sqrt{d}$ $)$.