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arXiv · 2203.07625

On non-monogenic number fields defined by trinomials of type $x^n +ax^m+b$

Hamid Ben Yakkou
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Abstract

Let $K=\Q(\theta)$ be a number field generated by a complex root $\th$ of a monic irreducible trinomial $F(x) = x^n+ax^{m}+b \in \Z[x]$. In this paper, we deal with the problem of the non-monogenity of $K$. More precisely, we provide some explicit conditions on $a$, $b$, $n$, and $m$ for which $K$ is not monogenic. As application, we show that there are infinite families of non-monogenic number fields defined by trinomials of degree $n=2^r\cdot3^k$ with $r$ and $k$ are positive integers. We also give two infinite families of non-monogenic number fields defined by trinomials of degree $6$. Finally, we illustrate our results by giving some examples.

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Hamid Ben Yakkou. 2022-03-15. On non-monogenic number fields defined by trinomials of type $x^n +ax^m+b$. https://doi.org/10.1216/rmj.2023.53.685

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