On Dedekind's criterion and monogenicity over Dedekind rings

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.