Non-monogenity in a family of octic fields

Abstract

Let $m$ be a square-free integer, $m\equiv 2,3\pmod 4$. We show that the number field $K=\mathbb{Q} (i,\sqrt [4]{m})$ is non-monogene, that is, it does not admit any power integral bases of type $\{1,\alpha ,\ldots ,\alpha ^7\}$. In this infinite parametric family of Galois octic fields we construct an integral basis and show non-monogenity using congruence considerations only. Our method yields a new approach to consider monogenity or to prove non-monogenity in algebraic number fields which is applicable for parametric families of number fields. We calculate the index of elements as polynomials dependent upon the parameter, factor these polynomials, and consider systems of congruences according to the factors.