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

Abstract

Let $K$ be a septic number field generated by a complex root $þ$ of a monic irreducible trinomial $F(x)=x^7+a{x}^5+b\in \mathbb{Z}[x]$. 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.

arXiv commenttext overlap with arXiv:2212.05029