On monogenity of certain pure number fields defined by xpr − m

Abstract

Let [Formula: see text] be a pure number field generated by a complex root [Formula: see text] of a monic irreducible polynomial [Formula: see text] where [Formula: see text] is a square free rational integer, [Formula: see text] is a rational prime integer, and [Formula: see text] is a positive integer. In this paper, we study the monogenity of [Formula: see text]. We prove that if [Formula: see text], then [Formula: see text] is monogenic. But if [Formula: see text] and [Formula: see text], then [Formula: see text] is not monogenic. Some illustrating examples are given.

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.…”

On power integral bases of certain pure number fields defined by $x^{3^r\cdot 7^s}-m$

“…Let K be a pure number field generated by a complex root α of a monic irreducible polynomial F (x) = x 3 r •7 s − m with m = ∓1 a square free integer, and r and s two positive integers. The case r = 0 or s = 0 has been studied in [4].…”

“…In [10,13,12,9,4,3,11], based on Newton polygon techniques, we studied the monogeneity of the pure number fields of degrees in the following list: 12, 18, 24, 36, p r , 3 k • 5 r , and 2 • 3 k . In this paper, based on Newton polygon techniques, we study the monogeneity of any pure number field K = Q(α) generated by a complex root of a monic irreducible polynomial F (x) = x 3 r •7 s − m, where m = ±1 is a square free integer, and r and s are positive integers.…”

On monogenity of certain number fields defined by $x^{2^r}+ax^m+b$

“…In [14], Gaál and Remete studied the monogenity of pure number fields K = Q( n √ m) for 3 ≤ n ≤ 9 and any square-free rational integer m = ±1. They also showed in [16] that if m ≡ 2 or 3 (mod 4) is a square-free rational integer, then the octic field K = Q(i, 4 √ m) is not monogenic. In [25],…”