On monogenity of certain number fields defined by trinomials

Abstract

Let K = Q(θ) be a number field generated by a complex root θ of a monic irreducible trinomial F (x) = x + ax + b ∈ Z[x]. There is an extensive literature of monogenity of number fields defined by trinomials, Gaál studied the multi-monogenity of sextic number fields defined by trinomials. Jhorar and Khanduja studied the integral closedness of Z[θ]. But if Z[θ] is not integrally closed, then Jhorar and Khanduja’s results cannot answer on the monogenity of K. In this paper, based on Newton polygon techniques, we deal with the problem of monogenity of K. More precisely, when ZK 6= Z[θ], we give sufficient conditions on n, a and b for K to be not monogenic. For n ∈ {5, 6, 3, 2 · 3, 2 · 3 + 1}, we give explicitly some infinite families of these number fields that are not monogenic. Finally, we illustrate our results by some computational examples.