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 , P i ) of the minimal polynomial P of a primitive integral element and of its irreducible factors P i 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.Mathematics Subject Classification: 11Y40, 13A18, 13F30.
In this paper, we study an important well-known structure operation, namely, the semidirect product (or split extensions) which is a very useful tool to structure certain kinds of groups. More precisely, we study the isomorphism problem for semidirect products and then we determine how isomorphism of semidirect products and conjugacy of the images of the corresponding actions are related. As an application, for two positive integers [Formula: see text] and [Formula: see text], we compute the number of upper isomorphism classes of split extensions of an elementary abelian [Formula: see text]-group of order [Formula: see text] by an elementary abelian [Formula: see text]-group of order [Formula: see text]. Furthermore, we deal with split extensions where the kernel is a non-abelian [Formula: see text]-group of nilpotency class two and the quotient is an elementary abelian [Formula: see text]-group.
Let R be a Dedekind ring, K its quotient field, L a separable finite extension over K , and O L the integral closure of R in L. In this paper we provide a "practical" criterion that tests when a given α ∈ O L generates a power basis for O L over R (i.e. when O L = R[α]), improving significantly a result in this direction by M. Charkani and O. Lahlou. Applications in the context of cyclotomic, quadratic, biquadratic number fields, and some Dedekind rings are provided.
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