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.