On non-monogenic number fields defined by trinomials of type $x^n +ax^m+b$

Abstract

Let K = Q(θ) be a number field generated by a complex root θ of a monic irreducible trinomialIn this paper, we deal with the problem of the non-monogenity of K. More precisely, we provide some explicit conditions on a, b, n, and m for which K is not monogenic. As application, we show that there are infinite families of non-monogenic number fields defined by trinomials of degree n = 2 r • 3 k with r and k are positive integers. We also give two infinite families of non-monogenic number fields defined by trinomials of degree 6. Finally, we illustrate our results by giving some examples.

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

“…•3 k + ax m + b. Although formally it includes x 2 r + ax + b, all statements of Corollary 2.4 of [1] concerns the cases k ≥ 1, hence do not overlap with Theorem 2.5.…”

“…Recall that in [3] Ben Yakkou and El Fadil studied the non-monogenity of number fields defined by x n + ax + b. These results are generalized in [1] for number fields defined by x n + ax m + b. However, the obtained results cannot give a complete answer about the monogenity of number fields defined by x 2 r + ax + b.…”