On index divisors and monogenity of certain number fields defined by x12 + axm + b

In this paper, we study the monogenity of any number field defined by a monic irreducible trinomial F(x) = x(12) + ax(m) + b ? Z[x] with 1 = m = 11 an integer. For every integer m, we give sufficient conditions on a and b so that the field index i(K) is not trivial. In particular, if i(K) ? 1, then K is not monogenic. For m = 1, we give necessary and sufficient conditions on a and b, which characterize when a rational prime p divides the index i(K). For every prime divisor p of i(K), we also calculate the highest power p dividing i(K), in such a way we answer the problem 22 of Narkiewicz (Elementary and analytic theory of algebraic numbers, Springer Verlag, Auflag, 2004) for the number fields defined by trinomials x(12) + ax + b.