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- W2896004734 abstract "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 $pin Z_{Omega}[x]$ 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 $(p(x+alpha),DD), pin Z_{Omega}[x]$ are always ENS with finiteness property provided $alphain Z_{Omega}$ 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 GyH{o}ry on monogenic orders in 'etale algebras. The paper ends with some speculations on possible further generalizations." @default.
- W2896004734 created "2018-10-26" @default.
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- W2896004734 date "2018-10-23" @default.
- W2896004734 modified "2023-10-18" @default.
- W2896004734 title "Number systems over orders of finite 'etale algebras" @default.
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