New York Journal of Mathematics
Volume 21 (2015) 181-190


M. Hbaib, F. Mahjoub, and F. Taktak

On the smallest Salem series in Fq((X-1))

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

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)=Yn-XYn-1-Y+X-1, where we suppose that 1 is the least element of the finite field Fq* (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=2s+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.