On Sequences of Integers of Quadratic Fields and Relations with Artin’s Primitive Root Conjecture

I will consider the integers $\alpha$ of the quadratic field $ \mathbb{Q} (\sqrt[]{d})$ with $d$ is a square-free integer.  Using the embedding into $ \text{GL}(2,\mathbb{R})$ we obtain bounds for the smallest positive integer  $\nu$ such that $\alpha^\nu\equiv 1\bmod p.$ More generally, if $\mathcal{O}_{f}$ is a number ring of conductor $f$, we study the first integer $n=n(f)$ such that $\alpha^n\in\mathcal{O}_{f}$. We obtain bounds for $n(f)$ and for $n(fp^{k})$.
We allow any $\alpha$ of non-zero norm. The case where $\alpha$ is the fundamental unit in a real quadratic number field is
of special interest. We also study a certain probability distribution suggested by numerical results. In the second part of
my talk I will indicate how my results relate to Artin primitive root type problems over quadratic fields.