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arXiv · 2206.14345

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

Hamid Ben Yakkou
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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.

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Hamid Ben Yakkou. 2022-06-29. The index of the septic number field defined by $x^7+ax^5+b$. https://doi.org/10.21136/mb.2024.0148-23

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