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-XY^n-1-Y+X-1, 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 _2^m((X^-1)) is well approximable.