  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}$ $)$.
