arXiv subjects

Toufik Zaïmi

Publications and source records attributed to Toufik Zaïmi.

3 recordsLinked to original sources

arXiv2021

On the trace of the integers of a number field

Let $Tr$ denote the trace $\mathbb{Z}$-module homomorphism defined on the ring $\mathcal{O}_{L} $ of the integers of a number field $L.$ We show that $Tr(\mathcal{O}_{L})\varsubsetneq \mathbb{Z}$ if and only if there is a prime factor $p$ of the degree of $L$ such that if $\wp _{1}^{e_{1}}...\wp _{s}^{e_{s}}$ is the prime factorization of the ideal $p\mathcal{O}_{L}$ in $\mathcal{O}_{L},$ then $p$ divides all powers $e_{1},...,e_{s}.$ Also, we prove that the equality $Tr(\mathcal{O}_{L})=\mathbb{Z}$ holds when $L$ is the compositum of certain number fields.

math.NT↗

arXiv2014

Comments on the height reducing property II

A complex number $α$ is said to satisfy the height reducing property if there is a finite set $F\subset \mathbb{Z}$ such that $\mathbb{Z}[α]=F[α]$, where $\mathbb{Z}$ is the ring of the rational integers. It is easy to see that $α$ is an algebraic number when it satisfies the height reducing property. We prove the relation $\operatorname{Card}(F)\geq \max\{2,\left\vert M_α(0)\right\vert \},$ where $M_α$ is the minimal polynomial of $α$ over the field of the rational numbers, and discuss the related optimal cases, for some classes of algebraic numbers $α$. In addition, we show that there is an algorithm to determine the minimal height polynomial of a given algebraic number, provided it has no conjugate of modulus one.

math.NT↗

arXiv2014

Characterization of the numbers which satisfy the height reducing property

Let $α$ be a complex number. We show that there is a finite subset $F$ of the ring of the rational integers $\mathbb{Z}$, such that $F\left[ α\right] =\mathbb{Z}\left[ α\right]$, if and only if $α$ is an algebraic number whose conjugates, over the field of the rationals, are all of modulus one, or all of modulus greater than one. This completes the answer to a question, on the numbers satisfying the height reducing property, posed in [3].

math.NT↗