Fermat–Euler Theorem in Algebraic Number Fields

Abstract

In this paper a (maximal) generalization of the classical Fermat–Euler theorem for finite commutative rings with identity is proved. Maximal means that we show how to extend the original Fermat–Euler theorem to all of the elements of such rings with the best possible choice of exponents. The proofs are based on an idempotent technique of Schwarz. The results are then applied to Dedekind's ringsRsatisfying the following finiteness condition:[formula]Further specialization of proved results to some special cases of Dedekind's ringsRdepends then upon a good detailed knowledge of the structure of the group of units of the corresponding residue class ringR/I. The most known prototypes of such rings are besides Znthe algebraic number fields. Amongst these the simplest cases represent the quadratic fields, including one of their oldest representatives, the ring of the Gaussian integers.