Generating a power basis over a Dedekind ring

Abstract

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.

On the Integral Closedness of $R[α]$

“…Ershov, in [5], gave yet a generalization of this criterion to extensions of rings of valuation. This criterion had, and still have, important applications in many relevant areas such as (but not limited to) the study of prime ideal factorizations in Dedekind rings, the computation of discriminants of number fields, and the existence of integral power bases in extensions of Dedekind rings (see for instance [1], [7], [11], and [12]).…”

“…(COROLLARY 1.2) On the one hand, it is known that R[α] is integrally closed (i.e. S = R[α]) if and only if R p [α] is integrally closed for every nonzero prime ideal p of R (see [1]). On the other hand, the generalized discriminantindex formula "Disc R (F ) = Ind R (α) 2 D R (S)" was shown in [2].…”

Monogenity and Power Integral Bases: Recent Developments

“…• M. E. Charkani and A. Deajim [26] (see also A. Deajim and L. El Fadil [28]) x p − m over number fields • M. Sahmoudi and M. E. Charkani [148] considered relative pure cyclic extensions • A. Soullami, M. Sahmoudi and O. Boughaleb [150] x 3 n + ax 3 s − b over number fields • O. Boughaleb, A. Soullami and M. Sahmoudi [23] x p n + ax p s − b over number fields • H. Smith [152] relative radical extensions • S. K. Khanduja and B. Jhorar [138] give equivalent versions of Dedekind criterion in general rings • S. Arpin, S. Bozlee, L. Herr and H. Smith [5], [6] study monogenity of number rings from a modul-theoretic perspective • R. Sekigawa [149] constructs an infinite number of cyclic relative extensions of prime degree that are relative monogenic…”

Polynomial Index in Dedekind Rings

“…] R the the index of P (or of α) and denote it by Ind R (P ) (see [2] for instance). This notion of index is generalized to the case when R is a Dedekind domain (see [3]).…”