Non-monogenity of certain quintic number fields defined by trinomials

Let K be a number field generated by a complex root θ of an irreducible trinomial F(x) = x 5 + ax3 + b ∈ Z[x], where ab ̸= 0. In this paper, we provide some explicit conditions on a and b for which 2 divides the index of K. We give a necessary and sufficient condition for which 3 divides the index of K. As an application of our results, we provide some infinite parametric families of number fields which are non-monogenic.