On the Power of Integers and Conductors of Quadratic Fields

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