 

M. Hbaib, F. Mahjoub, and F. Taktak
On the smallest Salem series in F_{q}((X^{1})) view print


Published: 
March 23, 2015. 
Keywords: 
Finite fields, formal power series, Salem series. 
Subject: 
11A55, 11R58. 


Abstract
The paper arose from the fact that the smallest element of the set of Salem numbers is not known. Indeed, it is not even known whether this set has a smallest element.
The aim of this paper is to prove that the minimal polynomial of the smallest Salem series of degree n in the field of formal power series over a finite field is given by P(Y)=Y^{n}XY^{n1}Y+X1, where we suppose that 1 is the least element of the finite field F_{q}* (as a finite total ordered set). Consequently, we are led to deduce that F _{q}((X^{1})) has no smallest Salem series.
Moreover, we will prove that the root of P(Y) of degree n=2^{s}+1 in F _{2m}((X^{1})) is well approximable.


Author information
Université de Sfax, Département de Mathématiques, Faculté des Sciences de Sfax, BP 802, 3000 Sfax, Tunisie.
mmmhbaib@gmail.com
faiza.mahjoub@yahoo.fr
fatmataktak@yahoo.fr

