Index characterization and monogenity of septic number fields defined by x ⁷ + ax ⁴ + b

Abstract

In this paper, we calculate the index of any septic number field K generated by a root α of a monic irreducible trinomial F(x) = x ⁷ + ax ⁴ + b ∈ ℤ[x]. Our approach is based on Engstrom’s results and the factorization of any rational prime in K. In such a way we give a complete answer of Problem 22 of Narkiewicz ([28]) for this family of number fields. As an application of our results, if i(K) ≠ 1, then K is not monogenic. Also, we give generators of power integral bases in some cases where i(K) = 1. Our results are illustrated by some computational examples.