On common index divisors and monogenity of septic number fields defined by trinomials of type \(x^7+ax^2+b\)

We study the index \(i(K)\) $i(K)$ of any septic number field \(K\) $K$ generatedby a root of an irreducible trinomial of type \(F(x)=x^7+ax^2+b \in \mathbb{Z}[x]\) $F(x)=x^7+a{x}^2+b\in \mathbb{Z}[x]$ . We showthat the unique prime which can divide \(i(K)\) $i(K)$ is \(2\) $2$ . Moreover, we give necessaryand sufficient conditions on \(a\) $a$ and \(b\) $b$ so that \(2\) $2$ is a common index divisor of \(K\) $K$ .Further, we show that \(i(K)=2\) $i(K)=2$ whenever \(2\) $2$ divides $i(K)$ $i(K)$ . In this way, we answercompletely Problem \(6\) $6$ and Problem \(22\) $22$ of Narkiewicz [34] for these families of number fields. As an application of our results, if \(2\) $2$ divides \(i(K)\) $i(K)$ , then the ring \(\mathcal{O}_K\) ${\mathcal{O}}_K$ of integers of \(K\) $K$ has no power integral basis. We illustrate our results bygiving some numerical examples.