On the Power of Integers and Conductors of Quadratic Fields

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}$ $)$.