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

Abstract

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