On polyadic numbers

V. G. Chirskii

Čebyševskij sbornik, Tome 24 (2023) no. 2, pp. 276-283

Résumé

The ring of polyadic numbers can be defined in several ways. One can introduce a metrizable topology on the ring of integers by counting the set of ideals $(m)$ by a complete system of neighborhoods of zero. The complete system of neighborhoods in the ring of integers is a collection of sets of the form $a+(m)$. The operations of addition and multiplication are continuous in this topology and the ring of integers with this topology is a topological ring. Completion of the resulting topological ring of integers - this is the ring of polyadic numbers. An equivalent definition is the inverse (projective) limit $$\lim_{\overleftarrow{m}}\mathbb{\mathrm{Z}}/m!\mathbb{\mathrm{Z}}.$$ Let's recall that the canonical decomposition of the polyadic number $\lambda$ has the form $$ \lambda= \sum_{n=0}^\infty a_{n}n!, a_{n}\in\mathbb{\mathrm{Z}}, 0\leq a_{n}\leq n.$$ This series converges in any field of $p-$ adic numbers $\mathbb{\mathrm{Q}}_p$ .Denoting the sum of this series in the field $\mathbb{\mathrm{Q}}_p$ with the symbol $\lambda^{(p)}$, we get that any polyadic number $\lambda$ can be considered as an element of the direct product of rings of integer $p-$ adic numbers $\mathbb{\mathrm{Z}}_p$ for all primes $p$. The converse statement is also true, meaning that the ring of polyadic integers coincides with this direct product. However, evidence of the latter claim could not be found. The purpose of this note is to fill this gap. In addition, some applications of polyadic numbers are described.

Détail

Keywords: polyadic number,direct product.