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Fermat-Euler theorem in algebraic number fields. (English) Zbl 0877.11069

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.

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
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