On power integral bases of certain pure number fields defined by $$x^{42} - m$$

Abstract

Let K ¼ QðaÞ be a pure number field generated by a complex root a of a monic irreducible polynomial FðxÞ ¼ x 42 À m 2 Z½x, where m 6 ¼ AE1 is a square-free rational integer. In this paper, we study the monogenity of K. We prove that if m 6 1 ðmod4Þ, m 6 Ç1 ðmod9Þ, and m 6 2 fÇ1; 18; 19; 30; 31g ðmod49Þ, then K is monogenic. But, if m 1 ðmod4Þ, or m 1 ðmod9Þ, or m 1 ðmod49Þ, then K is not monogenic. Our results are illustrated by some examples.

A note ON MONOGENEITY of pure number fields

“…Recall also that in [19], Gassert studied the integral closedness of Z[α], where α is a complex root of a monic irreducible polynomial f (x) = x n − m ∈ Z[x]. Finally, we cite that in [7,8,9,10], based on Newton polygon techniques, El Fadil studied the monogeneity of pure number fields of degree 24, 12,, 36, and 2 • 3 k respectively. In this paper, we state some comments regarding Gassert's paper [19], we point out some errors in [19,Proposition 3.4], and so in the proof of [19,Proposition 3.5 and Theorem 1.2], we also improve [19, Theorem 1.1 and Theorem 1.2], and we conclude with Theorem 4.6, which gives sufficient conditions on f (x) = x n − m in order to have K not monogenic, where K is the number field generated by a complex root of f (x).…”

On monogenity of certain pure number fields defined by $x^{2^u\cdot 3^v\cdot 5^t}-m$

“…In [25], by applying the explicit form of the index equation, Gaál and Remete obtained deep new results on monogenity of number fields generated by m 1 n , with 3 ≤ n ≤ 9 and m ±1 a square free integer. In [3,4,18,8,9,10,11,12,13], based on Newton polygon's techniques, El Fadil et al studied the monogenity of some pure number fields. Also El Fadil, Chouli, and Kchit studied the monogenity of pure number fields defined by x 60 − m ∈ Z[x] with m a square free integer.…”

“…The cases uvt = 0 have been studied in [3,28,4,18]. Also the case u = 2 and t = v = 1 has been studied by in [15]. Our proofs are based on Newton's polygon techniques and on index divisors as introduced by Hensel as follows: The index of a field K is defined by i…”

On power integral bases of certain pure number fields defined by $X^{60}-m$

“…While Gaál's and Remete's techniques are based on the index calculation, Nakahara's methods are based on the existence of power relative integral bases of some special sub-fields. In [7,8,10,12,11,9], El Fadil et al used Newton polygon techniques to study the monogeneity of the pure number fields of degrees 6, 12, 18, 24, and 36. In this paper, Our purpose is for a square free integer m ±1 and F(x) = x 60 − m is an irreducible polynomail over Q, to study the monogeneity of the number field K = Q(α) generated by a complex root α of F(x).…”