International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 71, Pages 4455-4464
doi:10.1155/S0161171203211534
On Dedekind's criterion and monogenicity over Dedekind rings
Department of Mathematics, Faculty of Sciences Dhar-Mahraz, University of Sidi Mohammed Ben Abdellah, Fes BP 1796, Morocco
Received 29 November 2002
Copyright © 2003 M. E. Charkani and O. Lahlou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We give a practical criterion characterizing the monogenicity of the integral closure of a Dedekind ring R, based on results on the resultant Res (p,pi) of the minimal polynomial p of a primitive integral element and of its irreducible factors pi modulo prime ideals of R. We obtain a generalization and an improvement of the Dedekind criterion (Cohen, 1996), and we give some applications in the case where R is a discrete valuation ring or the ring of integers of a number field, generalizing some well-known classical results.