On index divisors and monogenity of certain octic number fields defined by \(x^8 + ax^3 + b\) (Q6654871)

Let \(K\) be the number field generated by a root of an irreducible polynomial \(F(x)=x^8+ax^3+b\in \mathbb Z[x]\). The author determines when the polynomial \(F(x)\) is monogenic (Theorem 2.4) and presents necessary and sufficient conditions for the divisibility of the field index \(i(K)\) by \(p\) for \(p\in\{2,3\}\) in terms of \(a,b\) (Theorems 2.1 and 2.2). It is also shown that \(i(K)=2^\alpha3^\beta\) with \(\alpha\ge0, 0\le\beta\le1\) (Corollary 2.1) and in some cases \(\alpha,\beta\) are determined.