Ideals and Continued Fractions

Throughout this chapter we will let $\mathcal{O} = \left[1, \omega \right]$ be the order of discriminant Δ in the quadratic field $\mathbb{K} = \mathbb{Q} \left(\sqrt{D}\right)$. If a is any ideal of $\mathcal{O}$, it is evident that its corresponding ideal class, [a], contains an infinitude of ideals. In order to deal with this difficulty in managing [a], we will restrict our attention to a finite subset of particular ideals of [a]. To this end we provide the following definitions.¹