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 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.

Generating a power basis over a Dedekind ring

“…As an improvement of Theorem 2.5 of [5], we state in this section our main result (Theorem 3.1), which gives a test of whether an integral primitive element in an integral closure of a Dedekind ring is a PBG. …”

“…In [5] a version of such a generalization was established under some relatively restrictive hypotheses: Let R be a Dedekind ring, K its quotient field, L a finite separable extension of K , O L the integral closure of R in L, α a primitive element of L integral over R, and P = Irrd(α, K ). Assume that for every prime ideal p of R, the decomposition of P into monic irreducible factors in (R/p) [X] is of the form…”

“…. , r. If we let P i (X) ∈ R[ X] be a monic lift of P i (X), then it is shown in [5] that O L = R [α] if and only if ν p (Res(P , P i )) = deg(P i ) for every prime ideal p of R and for each i = 1, . .…”

“…In Section 3 we state our main result, Theorem 3.1, which gives yet a better generalization of the main result of [5] mentioned above by relaxing some of its conditions; namely, the selection of prime ideals and the requirement on the exponents e i . We prove Theorem 3.1 in Section 4.…”

“…We prove Theorem 3.1 in Section 4. In Remark 4.2, it is also indicated that the irreducible factorization of P (X) can also be avoided, which is in itself an improvement of [5].…”