Mathematica Bohemica

Mathematica Bohemica, first online, pp. 1-18

On common index divisors and monogenity of septic number fields defined by trinomials of type $x^7+ax^5+b$

Hamid Ben Yakkou

Received September 28, 2023. Published online November 5, 2024.

Abstract: Let $K $ be a septic number field generated by a root $\theta$ of an irreducible polynomial $ F(x)= x^7+ax^5+b \in\mathbb Z[x]$. In this paper, we explicitly characterize the index $i(K)$ of $K$. More precisely, for all $a$ and $b$, we show that $i(K) \in\{1, 2\}$. Our results answer completely to Problem 22 of W. Narkiewicz's book (2004) for these families of number fields. In particular, we provide sufficient conditions for which $K$ is not monogenic. We illustrate our results by some computational examples.

Keywords: monogenity; power integral basis; theorem of Ore; prime ideal factorization; common index divisor; Newton polygon

Classification MSC: 11R04, 11R16, 11R21, 11Y40

DOI: 10.21136/MB.2024.0148-23

PDF available at: Institute of Mathematics CAS

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Affiliations: Hamid Ben Yakkou, Sidi Mohamed Ben Abdellah University, Faculty of Sciences Dhar El Mahraz, P. O. Box 1874, 30050 Fez, Morocco, e-mail: beyakouhamid@gmail.com

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