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Article

Ben Yakkou, Hamid
On common index divisors and monogenity of septic number fields defined by trinomials of type $x^7+ax^5+b$. (English). Mathematica Bohemica, vol. 150 (2025), issue 2, pp. 245-262
MSC: 11R04, 11R16, 11R21, 11Y40 | DOI: 10.21136/MB.2024.0148-23
Keywords:
monogenity; power integral basis; theorem of Ore; prime ideal factorization; common index divisor; Newton polygon
Summary:
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.
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