On Index Divisors and Monogenity of Certain Octic Number Fields Defined by x⁸ + ax³ + b

For any octic number field K generated by a root α of a monic irreducible trinomial F(x) = x⁸ + ax³ + b ∈ ℤ[x] and for every rational prime p, we show when p divides the index of K. We also describe the prime power decomposition of the index i(K). In this way, we give a partial answer to Problem 22 of Narkiewicz [Elementary and Analytic Theory of Algebraic Numbers, Springer-Verlag, Berlin, Heidelberg (2004)] for this family of number fields. As an application of our results, we conclude that if i(K) ≠ 1, then K is not monogenic. We illustrate our results by some computational examples.