Canonical number systems can be viewed as natural generalizations of radix representations of ordinary integers to algebraic integers. A slightly modified version of an algorithm of B. Kovács and A. Pethő is presented here for the determination of canonical number systems in orders of algebraic number fields. Using this algorithm canonical number systems of some quartic fields are computed.

Let $K=\mathbb{Q}\left(\omega \right)$, with ${\omega }^3=m$ a positive integer, be a pure cubic number field. We show that the elements $\alpha \in K^{\times }$ whose squares have the form $a-\omega$ for rational numbers $a$ form a group isomorphic to the group of rational points on the elliptic curve $E_m:y^2=x^3-m$. This result will allow us to construct unramified quadratic extensions of pure cubic number fields $K$.