For $z\in\mathbb{Z}_{\neq 0}$, $n\in \mathbb{N}$ we know that there are at most $n$ solutions to $x^n\equiv 1\mod z$, for $x\in \mathbb{Z}/z\mathbb{Z}$ (there are precisely $n$ if $n|\phi(z)$).
Say we have a quadratic number field $\mathbb{Q}[\sqrt{d}]\supset \mathbb{Q}$ and an element $\alpha\in \mathbb{Q}[\sqrt{d}]$.
Question: What is the maximum number of solutions $x\in\mathcal{O}_{\mathbb{Q}[\sqrt{d}]}$ to $x^n=1\mod \alpha\mathcal{O}_{\mathbb{Q}[\sqrt{d}]}$ (that is the number of equivalence classes such that for all elements $x$ of the equivalence class, $x^n-1\in\alpha\mathcal{O}_{\mathbb{Q}[\sqrt{d}]}$)? When does this maximum number occur?