On the smallest Salem series in F q ( ( X − 1 ) )
نویسندگان
چکیده
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 Fq (as a finite total ordered set). Consequently, we are led to deduce that Fq((X)) has no smallest Salem series. Moreover, we will prove that the root of P (Y ) of degree n = 2 + 1 in F2m((X)) is well approximable.
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