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

Abstract

Let K = Q(θ) be a number field generated by a complex root θ of a monic irreducible trinomial of type F (x) = x 2 r + ax m + b ∈ Z[x] and Z K its ring of integers. In this paper, we study the monogenity of K. More precisely, when m = 1, we provide some explicit conditions on a, b, and r for which K is not monogenic. We also construct a family of irreducible trinomials which are not monogenic, but their roots generate monogenic number fields. To illustrate our results, we give some computational examples.

Monogenity and Power Integral Bases: Recent Developments

“…• L. El Fadil [39] x 4 + ax + b • L. El Fadil and I. Gaál [54] x 4 + ax 2 + b • H. Smith [151] x 4 + ax + b, x 4 + cx 3 + d • L. Jones [128] showed that there exist exactly three distinct monogenic trinomials of the form x 4 + bx 2 + d with Galois C 4 • A. Jakhar, S. Kaur and S. Kumar [101] [11] x 2 r + ax m + b • H. Ben Yakkou and L. El Fadil [17] x n + ax + b, n = 5, 6, 3 k , 2 k • 3 ℓ , 2 k • 3 ℓ + 1 • A. Jakhar and S. Kumar [108] gave explicit conditions for the non-monogenity of x q s − ax − b • A. Jakhar [94] x p s − ax m − b • B. Jhorar and S. K. Khanduja [97] x n + ax + b, showed also that f ( [93] x n − ax m − b • A. Jakhar, S. Khanduja and N. Sangwan [99] x n + ax m + b • A. Jakhar, S. Khanduja and N. Sangwan [100] gave necessary and sufficient conditions in terms of a, b, m, n for a given prime p to divide I(ϑ) where ϑ is a root of x n + ax m + b • L. Jones [121] considered monogenic reciprocal trinomials of type x 2m + Ax m + 1 • L. Jones [114] showed that there are infinitely many primes p such that x 6 + px 3 + 1 is monogenic with Galois group D 6 • L. Jones [115] showed that x n + x + 1 is monogenic, if and only if its discriminant is squarefree • L. Jones and T. Phillips [130] showed that x n + ax + b is monogenic infinitely often • L. Jones and D. White [131]…”