Number systems over orders of finite étale algebras

Kálmán Győry, Attila~Pethő, Jörg~Thuswaldner

Published 2018-10-23Version 1

Let $\K$ be an algebraic number field and $\Omega$ a finite \'etale $\K$-algebra. Denote by $\Z_{\Omega}$ the ring of integers of $\Omega$. Generalizing the recently introduced number systems over orders of number fields we introduce in this paper so-called \emph{\'etale number systems} over $\Z_{\Omega}$ (ENS for short). An ENS is a pair $(p,\DD)$, where $p\in \Z_{\Omega}[x]$ is monic and $\DD$ is a complete residue system modulo $p(0)$ in $\Z_{\Omega}$. ENS with finiteness property, {\it i.e.}, with the property that all elements of $\Z_{\Omega}/(p)$ have a representative belonging to $\DD[x],$ play an important role. We prove that it is algorithmically decidable whether or not an ENS admits the finiteness property. Under mild conditions we show that the pairs $(p(x+\alpha),\DD), p\in \Z_{\Omega}[x]$ are always ENS with finiteness property provided $\alpha\in \Z_{\Omega}$ is in some sense large enough, for example, if $\alpha$ is a sufficiently large rational integer. In the opposite direction we prove under different conditions that $(p(x-m),\DD)$ does not have the finiteness property for each large enough rational integer $m$. We obtain important relations between power integral bases and ENS in orders of \'etale $\Q$-algebras. The proofs depend on some general effective finiteness results of Evertse and Gy\H{o}ry on monogenic orders in \'etale algebras. The paper ends with some speculations on possible further generalizations.