Units from square-roots of rational numbers

Abstract

Let $D,Q$ be natural numbers, $(D,Q)=1$, such that $D/Q>1$ and $D/Q$ is not a square. Let $q$ be the smallest divisor of $Q$ such that $Q|q^2$. We show that the units $>1$ of the ring $\mathbb{Z}[\sqrt{D{q}^2/Q}]$ are connected with certain convergents of $\sqrt{D/Q}$. Among these units, the units of $\mathbb{Z}[\sqrt{DQ}]$ play a special role, inasmuch as they correspond to the convergents of $\sqrt{D/Q}$ that occur just before the end of each period. We also show that the last-mentioned units allow reading the (periodic) continued fraction expansion of certain quadratic irrationals from the (finite) continued fraction expansion of certain rational numbers.