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

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 n-1 - Y + X - 1, where we suppose that 1 is the least element of the finite field F-1 q (as a finite total ordered set). Consequently, we are led to deduce that F-1((X-1)) has no smallest Salem series. Moreover, we will prove that the root of P(Y ) of degree n = 2s + 1 in F-1m((X-1)) is well approximable. [ABSTRACT FROM AUTHOR]