ON MONOGENITY OF CERTAIN NUMBER FIELDS DEFINED BY TRINOMIALS

Let K = Q(theta) be a number field generated by a complex root theta of a monic irreducible trinomial F(x) = xn + ax + b is an element of Z[x]. There is an extensive literature on monogenity of number fields defined by trinomials. For example, Gaal studied the multi-monogenity of sextic number fields defined by trinomials. Jhorar and Khanduja provide some explicit conditions on a, b and n for (1, theta, . . . , theta n-1) to be a power integral basis in K. But, if theta does not generate a power integral basis of ZK, then Jhorar's and Khanduja's results cannot answer the monogenity of K. In this paper, based on Newton polygon techniques, we deal with the problem of non-monogenity of K. More precisely, when theta does not generate a power integral basis of ZK, we give sufficient conditions on n, a and b for K to be not monogenic. For n is an element of {5, 6, 3r, 2k center dot 3r, 2s center dot 3k + 1}, we give explicitly some infinite families of these number fields that are not monogenic. Finally, we illustrate our results by some computational examples.