Laššák, Miroslav; Porubský, Štefan Fermat-Euler theorem in algebraic number fields. (English) Zbl 0877.11069 J. Number Theory 60, No. 2, 254-290 (1996). Let \(R\) be a finite commutative ring. The authors use the technique of idempotents in \(R\) to establish a generalization of the theorems of Fermat and Euler by determining minimal positive integers \(\nu\) and \(\mu\) such that for all \(x\in R\) one has \(x^{\mu+\nu}=x^\nu\). Their results are made more transparent in the case when \(R\) is a principal ideal ring. Several examples are given, including factor rings of \(\mathbb{Z}\) and the rings of integers of quadratic number fields and more generally of Dedekind domains with the finite norm property. Reviewer: W.Narkiewicz (Wrocław) Cited in 2 ReviewsCited in 6 Documents MSC: 11T30 Structure theory for finite fields and commutative rings (number-theoretic aspects) 11R04 Algebraic numbers; rings of algebraic integers 12E12 Equations in general fields 11R11 Quadratic extensions 13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations 13F10 Principal ideal rings Keywords:Fermat’s theorem; finite commutative rings; Dedekind domains; quadratic number fields PDFBibTeX XMLCite \textit{M. Laššák} and \textit{Š. Porubský}, J. Number Theory 60, No. 2, 254--290 (1996; Zbl 0877.11069) Full Text: DOI