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On common index divisors and monogenity of septic number fields defined by trinomials of type \(x^7+ax^2+b\)

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Abstract

We study the index \(i(K)\) of any septic number field \(K\) generated by a root of an irreducible trinomial of type \(F(x)=x^7+ax^2+b \in \mathbb{Z}[x]\). We show that the unique prime which can divide \(i(K)\) is \(2\). Moreover, we give necessary and sufficient conditions on \(a\) and \(b\) so that \(2\) is a common index divisor of \(K\). Further, we show that \(i(K)=2\) whenever \(2\) divides \(i(K)\). In this way, we answer completely Problem \(6\) and Problem \(22\) of Narkiewicz [34] for these families of number fields. As an application of our results, if \(2\) divides \(i(K)\), then the ring \(\mathcal{O}_K\) of integers of \(K\) has no power integral basis. We illustrate our results by giving some numerical examples.

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Acknowledgements

I am grateful to the anonymous referee, whose valuable comments, important suggestions and precious remarks have tremendously improved the quality of this paper. I thank Professor Kálmán Győry for inspiring and insightful conversations about the general topic of this article. I thank him for encouragement, advice and guidance. Also, I thank Professor Lhoussain El Fadil who introduced me to work on monogenity of number fields.

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

    H. Ben Yakkou

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Ben Yakkou, H. On common index divisors and monogenity of septic number fields defined by trinomials of type \(x^7+ax^2+b\). Acta Math. Hungar. 172, 378–399 (2024). https://doi.org/10.1007/s10474-024-01409-y

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