Let $P(x)\in\mathbb{Z}[x]$ be monic and irreducible over $\mathbb{Q}[x]$, and let $\theta$ be a root of $P(x)$. Let $K = \{a + b\theta\} \subseteq \mathbb{Z}[\theta]$. When is it the case that there are infinitely many $\alpha\in K$ s.t. $\text{Nm}(\alpha) = 1$?



Units of an extension of $\mathbb{Z}$
If and only if $\mathbb{Q}(\theta)$ is a real quadratic field.


In the imaginary quadratic case, and when $P$ has degree $1$, there are only finitely many units in the ring of integers of the field. When the degree of $P$ is at least $3$, the norm condition amounts to a Thue equation. In the real quadratic case, some power of the fundamental unit will be congruent to $1$ modulo the conductor ideal of the order and will thus be an element of the order along with its powers.


