On power integral bases for certain pure sextic fields

Abstract

In their paper [1], Shahzad Ahmad et al. given a characterization on any pure sextic number field Q(m1/6) with square-free integers m satisfying m 6 ±1 (mod 9) to have a power integral bases or do not. In this paper, for these results, we give a new easier proof than that given in [1]. We further investigate the cases m 1 (mod 4) independently to the satisfaction of m2 1 (mod 9), m 1 (mod 9), and the number fields defined by x2r3t−m, where r, t are two non-negative integers, and m is a square free integer are investigated. The proposed proofs are based on Dedekind’s criterion and on prime ideal factorization.

A note ON MONOGENEITY of pure number fields

“…The same authors showed in [2], that if m ≡ 1 (mod 4) and m ∓1 (mod 9), then the sextic number field generated by m 1 6 is not monogenic. In [6], based on prime ideal factorization, El Fadil showed that if m ≡ 1 (mod 4) or m 1 (mod 9), then the sextic number field generated by m 1 6 is not monogenic. In [18], by applying the explicit form of the index, Gaál and Remete obtained new results on monogeneity of the number fields generated by m 1 n for 3 ≤ n ≤ 9.…”

“…Moreover this comment is not used anywhere. (6) The paper finishes with an extra example, Example 3.6, in which the author considered the number field defined by an irreducible polynomial f (x) = x 6 3 − m and gcd(m, 6) = 1. The author claimed to caluclate the 2-adic and 3adic valuations of the index, given in two tables.…”

“…In [17], Gaál and Remete, calculated the elements of index 1, for which the coefficients with absolute value < 10 1000 in the integral basis, of pure quartic field generated by m 1 4 for 1 < m < 10 7 and m ≡ 2, 3 (mod 4). In [1], Ahmad, Nakahara, and Husnine proved that if m ≡ 2, 3 (mod 4) and m ∓1 (mod 9), then the sextic number field generated by m 1 6 is monogenic. The same authors showed in [2], that if m ≡ 1 (mod 4) and m ∓1 (mod 9), then the sextic number field generated by m 1 6 is not monogenic.…”

“…In [1], Ahmad, Nakahara, and Husnine proved that if m ≡ 2, 3 (mod 4) and m ∓1 (mod 9), then the sextic number field generated by m 1 6 is monogenic. The same authors showed in [2], that if m ≡ 1 (mod 4) and m ∓1 (mod 9), then the sextic number field generated by m 1 6 is not monogenic. In [6], based on prime ideal factorization, El Fadil showed that if m ≡ 1 (mod 4) or m 1 (mod 9), then the sextic number field generated by m 1 6 is not monogenic.…”

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

“…They also showed in [1] that if m ≡ 1 (mod 4) and m ≡ ∓1 (mod 9), then the sextic number field generated by m 1/6 is not monogenic. In [8], based on prime ideal factorization, we showed that if m ≡ 1 (mod 4) or m ≡ 1 (mod 9), then the sextic number field generated by m 1/6 is not monogenic. In [23], Hameed and Nakahara proved that if m ≡ 1 (mod 16), then the octic number field generated by m 1/8 is not monogenic, but if m ≡ 2, 3 (mod 4), then it is monogenic.…”