Number systems in integral domains, especially in orders of algebraic number fields. (English) Zbl 0760.11002

Let \(R\) be an integral domain, \(\alpha\in R\), \({\mathcal N}=\{n_ 1,\dots,n_ m\}\subset\mathbb Z\). \(\{\alpha,{\mathcal N}\}\) is called a number system in \(R\) if any \(\gamma\in R\) is uniquely representable in the form \[ \gamma=c_ 0+c_ 1\alpha+\cdots+c_ h\alpha^ h,\quad c_ j\in{\mathcal N} (j=0,1,\dots,h),\;c_ h\neq 0\text{ if } h\neq 0. \] The authors show, that there exists a number system in \(R\) if and only if either \(R=\mathbb Z[\alpha]\) with an algebraic number \(\alpha\), or \(R=F_ p[x]\) where \(F_ p\) is the field with \(p\) elements and \(x\) is transcendental over \(F_ p\). In both cases they characterize all number systems in \(R\). Moreover, the canonical number systems \(\{\alpha,{\mathcal N}_ 0(\alpha)\}\) are characterized in orders \({\mathcal O}\) of algebraic number fields \(K\), where \({\mathcal N}_ 0(\alpha)=\{0,1,\dots,| N_{K/\mathbb Q}(\alpha)|-1\}\), \(\alpha\in{\mathcal O}\). Finally, an algorithm is given to compute the canonical number systems in orders of algebraic number fields. Using this algorithm all canonical number systems are determined in the maximal orders of some totally real cubic number fields.
Reviewer: I.Gaál (Debrecen)


11R04 Algebraic numbers; rings of algebraic integers
11A63 Radix representation; digital problems
11R16 Cubic and quartic extensions