A Note on Generating a Power Basis over a Dedekind Ring

Abstract

In this note, we show that the result [1, Proposition 5.2] is inaccurate. We further give and prove the correct modification of such a result. Some applications are also given.

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…”

A Generalization of Eisenstein-Schönemann’s Irreducibility Criterion

“…Let L = K(α) be a simple extension generated by α ∈ K a root of a monic irreducible polynomial f ∈ R ν [x], where K is a fixed algebraic closure of K. By [13, Chapter I, Proposition 8.3], the Hensel's correspondence, given in [8], remains valid. By [4], it was suggested that we can replace K ν by K h . So, in order to describe all prime ideals of Z L lying above the maximal ideal (π), we need to factorize the polynomial f (x) into monic irreducible polynomials of K h [x].…”

On Indices of Septic Number Fields Defined by Trinomials x7 + ax + b

“…Theorem 6 (Theorem 1.1 of [27]). Let F(x) = r ∏ i=1 φ i l i be the factorization of F(x) into powers of monic irreducible coprime polynomials over F p .…”