Diophantine Equations and Power Integral Bases

Abstract

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On nonmonogenic number fields defined by

“…This problem has been widely studied and of interest to several mathematicians (cf. [1], [2], [4], [5], [6], [7], [8], [10], [11]). Let K be an algebraic number field generated by a complex root θ of a monic irreducible polynomial f (x) having degree n with coefficients from the ring Z of integers.…”

“…If there does not exist any such α ∈ Z K , then K is called non-monogenic. In 2016, Ahmad, Nakahara, and Husnine [1] proved that the sextic number field generated by m 1 6 is not monogenic if m ≡ 1 mod 4 and m ≡ ±1 mod 9. In 2017, Gaál and Remete [8] obtained some new results on monogenity of number fields generated by m 1 n with m a square free integer and 3 ≤ n ≤ 9 by applying the explicit form of the index equation.…”

A dynamical characterization for monogenity at every level of some infinite -towers

“…For ≥ 2, suppose that −1 = 1, that is disc (x n−1 ) = disc Q(x n−1 ). By Proposition 2.1 and by induction hypothesis we have 2 2 (0 Proof The fact that ℎ ℓ has degree ≥ 2 is immediate from its definition. It is enough to show that each has no repeated factor.…”

Determining Elements of Minimal Index in an Infinite Family of Totally Real Bicyclic Biquadratic Number Fields

“…It is a classical problem in algebraic number theory (cf. [3]) to decide if a number field K is monogene, that is if there exist an α ∈ Z K such that Z K = Z[α], where Z K is the ring of integers of K see [14], [3]. In this case {1, α, .…”

“…In case of quartic fields there exists a general method for determining power integral bases and elements of given index [5], [7], [3] which made possible also to investigate infinite parametric families of quartic fields, cf. [8].…”