On Index Divisors and Monogenity of Certain Sextic Number Fields Defined by \(x^6+ax^5+b\)

The main goal of this paper is to provide a complete answer to the Problem 22 of Narkiewicz (2004) for any sextic number field K generated by a root of a monic irreducible trinomial \(F(x)=x^6+ax^5+b\in \mathbb {Z}[x]\) $F(x)=x^6+a{x}^5+b\in \mathbb{Z}[x]$ . Namely, we calculate the index of the field K. In particular, if \(i(K)\ne 1\) $i(K)\ne 1$ , then K is not mongenic. Finally, we illustrate our results by some computational examples.