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On the index of the octic number field defined by \(x^8+ax+b\)

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Abstract

Let K be an octic number field generated by a complex root \(\theta\) of a monic irreducible trinomial \(F(x)= x^{8}+ax+b \in \mathbb{Z}[x]\), where a and b are two non-zero rational integers. Let \(i(K)\) be the index of K. We show that i(K) is either 1 or a power of 2. Further, assuming that \((a,b) \notin (32+64\mathbb{Z})\times {(16+64\mathbb{Z}) } \) and \((a,b) \notin (64\mathbb{Z})\times(112+128\mathbb{Z})\), we give necessary and sufficient conditions depending only on a and b so that 2 is a prime common index divisor of K. In particular, we provide sufficient conditions for which K is non-monogenic. In such a way our results extend a result proved in [5], when some sufficient conditions of the divisibility of i(K) by 2 are provided.

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  1. Faculty of Sciences, Sidi Mohamed ben Abdellah University, P.O. Box 1874, Dhar El Mahraz, Atlas-Fes, Morocco

    H. Ben Yakkou & B. Boudine

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  1. H. Ben Yakkou
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Ben Yakkou, H., Boudine, B. On the index of the octic number field defined by \(x^8+ax+b\). Acta Math. Hungar. 170, 585–607 (2023). https://doi.org/10.1007/s10474-023-01353-3

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  • DOI: https://doi.org/10.1007/s10474-023-01353-3

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