Construction of number systems in algebraic number fields (Q2714145)

Let \(Q(\alpha)\) be a number field of degree \(n\), and \(\mathcal J\) the set of integers of \(Q(\alpha)\). Let \(F\) be a complete residue system \(\operatorname {mod} \beta\), such that \(0\in F\) and \(\beta\in \mathcal J\). The pair \(\{F,\beta\}\) is called a number system if each \(\gamma\in \mathcal J\) can be written in the form \(\gamma=\sum_{j=0}^{j=k}c_j\beta^j\) where \(c_j\in F\). The author proves that if \(\beta\) satisfies some inequalities then \(\{F,\beta\}\) is a number system with a suitable \(F\). The proof is based on an elegant algorithm. This result is connected with \textit{G. Steidl} [BIT 29, 563-571 (1989; Zbl 0685.12002)].