On Dedekind’s Criterion and Monogenicity over Dedekind Rings
نویسندگان
چکیده
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.
منابع مشابه
ϕ-ALMOST DEDEKIND RINGS AND $\Phi$-ALMOST DEDEKIND MODULES
The purpose of this paper is to introduce some new classes of rings and modules that are closely related to the classes of almost Dedekind domains and almost Dedekind modules. We introduce the concepts of $\phi$-almost Dedekind rings and $\Phi$-almost Dedekind modules and study some properties of this classes. In this paper we get some equivalent conditions for $\phi$-almost Dedekind rings and ...
متن کاملAxiomatics, methodology, and Dedekind’s theory of ideals
Richard Dedekind has had an incredible influence on modern mathematics, largely due to his methodological demands which are still valued by mathematicians today. Through an investigation of some of his works written between 1854 and 1877, I reveal a connection between these methodological demands and features of axiomatic reasoning that he employed. I discuss two foundational/philosophic works ...
متن کاملLogic Carnegie Mellon Pittsburgh , Pennsylvania 15213 Dedekind ’ s treatment of Galois theory
We present a translation of §§160–166 of Dedekind’s Supplement XI to Dirichlet’s Vorlesungen über Zahlentheorie, which contain an investigation of the subfields of C. In particular, Dedekind explores the lattice structure of these subfields, by studying isomorphisms between them. He also indicates how his ideas apply to Galois theory. After a brief introduction, we summarize the translated exce...
متن کاملLinking Numbers and Modular Cocycles
It is known that the 3-manifold SL(2,Z)\ SL(2,R) is diffeomorhic to the complement of the trefoil knot in S3. As is shown by E. Ghys the linking number of the trefoil with a modular knot associated to a hyperbolic conjugacy is related to the classical Dedekind symbol. These symbols arose historically in the transformation property of the logarithm of Dedekind’s eta function. In this paper we st...
متن کاملA NEW PROOF OF THE PERSISTENCE PROPERTY FOR IDEALS IN DEDEKIND RINGS AND PR¨UFER DOMAINS
In this paper, by using elementary tools of commutative algebra,we prove the persistence property for two especial classes of rings. In fact, thispaper has two main sections. In the first main section, we let R be a Dedekindring and I be a proper ideal of R. We prove that if I1, . . . , In are non-zeroproper ideals of R, then Ass1(Ik11 . . . Iknn ) = Ass1(Ik11 ) [ · · · [ Ass1(Iknn )for all k1,...
متن کامل