The index of the septic number field defined by $x^7+ax^5+b$

TL;DR

This paper investigates the index of septic number fields generated by specific trinomials, establishing that the index can only be 1, 2, or 4, and provides conditions for non-monogenicity with computational examples.

Contribution

It determines the possible values of the index for septic fields defined by $x^7+ax^5+b$ and addresses a problem posed by Narkiewicz, offering new criteria for non-monogenic fields.

Findings

Index of $K$ is in {1, 2, 4}.

Provides sufficient conditions for non-monogenicity.

Includes computational examples illustrating the results.

Abstract

Let $K $ be a septic number field generated by a complex root $\th$ of a monic irreducible trinomial $ F(x)= x^7+ax^5+b \in \Z[x]$. Let $i(K)$ be the index of $K$. In this paper, we show that $i(K) \in \{1, 2, 4\}$. In a such way, we answer to Problem $22$ of Narkiewicz \cite{Na} for these number fields. In particular, we provide sufficient conditions for which $K$ is non-monogenic. We illustrate our results by some computational examples.