On Dedekind's criterion and monogenicity over Dedekind rings

Abstract

We give a practical criterion characterizing the monogenicity of the integral closure of a Dedekind ringR, based on results on the resultantRes(p,pi)of the minimal polynomialpof a primitive integral element and of its irreducible factorspimodulo prime ideals ofR. We obtain a generalization and an improvement of the Dedekind criterion (Cohen, 1996), and we give some applications in the case whereRis a discrete valuation ring or the ring of integers of a number field, generalizing some well-known classical results.