arXiv subjects

Hamid Ben Yakkou

Publications and source records attributed to Hamid Ben Yakkou.

8 recordsLinked to original sources

arXiv2025

On the monogenity of pure number fields: application to the existence of canonical number systems

Let $m$ be a rational integer with $m \neq 0, \pm 1$, and consider the pure number field $K = \mathbb{Q}(\sqrt[n]{m})$ with $n \ge 3$. Most papers discussing the monogenity of pure number fields focus exclusively on the case where $m$ is square-free. For every integer $n \ge 4$, the monogenity of number fields of degree $n$ is not completely characterized. For example, the monogenity of the pure quartic field $\mathbb{Q}(\sqrt[4]{m})$ is not yet fully described, even when $m$ is square-free (see the recent 2024 paper \cite{Nyul} by Arnóczki and Nyul). In this paper, based on a classical theorem of Ore concerning prime ideal decomposition in number fields \cite{MN92, O}, we study the monogenity of $K$ without assuming $m$ to be square-free. As an application, we present several examples related to canonical number systems (CNS). In particular, we observe that our results extend some of those presented in \cite{BFC, BF, HNHCNS}.

math.NT↗

arXiv2024

On common index divisors and monogenity of certain number fields defined by trinomials of type $x^{2^r}+ax^m+b$

Let $K = \Q(þ)$ be a number with $þ$ a root of an irreducible trinomial of type $ F(x)= x^{2^r}+ax^m+b \in \Z[x]$. In this paper, based on the $p$-adic Newton polygon techniques applied on decomposition of primes in number fields and the classical index theorem of Ore \cite{Narprime, O}, we study the monogenity of $K$. More precisely, we prove that if $a$ and $1+b$ are both divisible by $32$, then $K$ cannot be monogenic. For $m=1$, we provide explicit conditions on $a$, $b$ and $r$ for which $K$ is not monogenic. We also construct a family of irreducible trinomials which are not monogenic, but their roots generate monogenic number fields. To illustrate our results, we give some computational examples.

math.NT↗

arXiv2024

On indices and monogenity of quartic number fields defined by quadrinomials

Consider a quartic number field $K$ generated by a root of an irreducible quadrinomial of the form $ F(x)= x^4+ax^3+bx+c \in \Z[x]$. Let $i(K)$ denote the index of $K$. Engstrom \cite{Engstrom} established that $i(K)=2^u \cdot 3^v$ with $u \le 2$ and $v \le 1$. In this paper, we provide sufficient conditions on $a$, $b$ and $c$ for $i(K)$ to be divisible by $2$ or $3$, determining the exact corresponding values of $u$ and $v$ in each case. In particular, when $i(K) \neq 1$, $K$ cannot be monogenic. We also identify new infinite parametric families of monogenic quartic number fields generated by roots of non-monogenic quadrinomials. We illustrate our results by some computational examples. Our method is based on a theorem of Ore on the decomposition of primes in number fields \cite{Nar,O}.

math.NT↗

arXiv2023

The index of the septic number field defined by $x^7+ax^5+b$

Let $K $ be a septic number field generated by a complex root $þ$ of a monic irreducible trinomial $ F(x)= x^7+ax^5+b \in \Z[x]$. Let $i(K)$ be the index of $K$. In this paper, we show that $i(K) \in \{1, 2, 4\}$. In a such way, we answer to Problem $22$ of Narkiewicz \cite{Na} for these number fields. In particular, we provide sufficient conditions for which $K$ is non-monogenic. We illustrate our results by some computational examples.

math.NT↗

arXiv2023

On common index divisors and monogenity of of the nonic number field defined by a trinomial $x^9+ax+b$

Let $K $ be a nonic number field generated by a complex root $þ$ of a monic irreducible trinomial $ F(x)= x^9+ax+b \in \Z[x]$, where $ab \neq 0$. Let $i(K)$ be the index of $K$. A rational prime $p$ dividing $ i(K)$ is called a prime common index divisor of $K$. In this paper, for every rational prime $p$, we give necessary and sufficient conditions depending only $a$ and $b$ for which $p$ is a common index divisor of $K$. As application of our results we identify infinite parametric families of non-monogenic nonic numbers fields defined by such trinomials. At the end, some numerical examples illustrating our theoretical results are given.

math.NT↗

arXiv2022

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

Let $K=\Q(θ)$ be a number field generated by a complex root $þ$ 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.

math.NT↗

arXiv2021

On monogenity of certain number fields defined by trinomials

Let $K=\Q(θ)$ be a number field generated by a complex root $þ$ of a monic irreducible trinomial $F(x) = x^n+ax+b \in \Z[x]$. There is an extensive literature of monogenity of number fields defined by trinomials, Gaál studied the multi-monogenity of sextic number fields defined by trinomials. Jhorar and Khanduja studied the integral closedness of $\Z[þ]$. But if $ \Z[þ]$ is not integrally closed, then Jhorar and Khanduja's results cannot answer on the monogenity of $K$. In this paper, based on Newton polygon techniques, we deal with the problem of monogenity of $K$. More precisely, when $\Z_K \neq \Z[þ]$, we give sufficient conditions on $n$, $a$ and $b$ for $K$ to be not monogenic. For $n\in \{5, 6, 3^r, 2^k\cdot 3^r, 2^s\cdot 3^k+1\}$, we give explicitly some infinite families of these number fields that are not monogenic. Finally, we illustrate our results by some computational examples.

math.NT↗

arXiv2021

On monogenity of certain pure number fields defined by $x^{p^r}-m$

Let $K = \mathbb{Q} (α) $ be a pure number field generated by a complex root $α$ a monic irreducible polynomial $ F(x) = x^{p^r} -m$, with $ m \neq 1 $ is a square free rational integer, $p$ is a rational prime integer, and $r$ is a positive integer. In this paper, we study the monogenity of $K$. We prove that if {$ν_p(m^p-m)=1$}, then $K$ is monogenic. But if $r\ge p$ and {$ν_p(m^{p}-m)> p$}, then $K$ is not monogenic. Some illustrating examples are given.

math.NT↗