A New Proof of the Ultrametric Hermite-Lindermann Theorem

Abstract

We propose a new proof of the Hermite-Lindeman Theorem in an ultrametric field by using classical properties of analytic functions. The proof remains valid in zero residue characteristic. Definitions and notations: The Archimedean absolute value of C is denoted by |. | ∞. Let IK be an algebraically closed complete ultrametric field of characteristic 0 and residue characteristic p. We denote by |. | the ultrametric absolute value of IK and we denote by Ω an algebraic closure of Q in IK. Given a ∈ IK and r > 0, we denote by d(a, r −) the disk {x ∈ IK |x−a| < r}, we denote by A(d(a, r −)) the IK-Banach algebra of power series f (x) = ∞ n=0 a n (x − a) n converging in d(a, r −) and for all s ∈]0, r[, we put |f |(s) = sup n∈IN |a n |r n. Given a ∈ Ω, we call denominator of a any strictly positive integer n such that na is integral over Z and we denote by den(a) the smallest denominator of a. Next, considering the conjugates a 2 , ..., a n of a over Q and putting a 1 = a, we put |a| = max{|a 1 | ∞ , ..., |a n | ∞ }. Next, we put s(a) = Log(max(|a|, den(a)).