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

Abstract

Let K = Q(α) be a pure number field generated by a root α of a monic irreducible polynomial F(x) = x 2 u •3 v •5 t − m, with m ±1 a square free rational integer, u, v and t three positive integers. In this paper, we study the monogenity of K. We prove that if m 1 (mod 4), m ±1 (mod 9), and m {±1, ±7} (mod 25), then K is monogenic. But if m ≡ 1 (mod 4) or m ≡ 1 (mod 9) or m ≡ −1 (mod 9) and u = 2k for some odd integer k or u ≥ 2 and m ≡ 1 (mod 25) or m ≡ −1 (mod 25) and u = 2k for some odd integer k or u = v = 1 and m ≡ ±82 (mod 5 4 ), then K is not monogenic.