On a result of James Ax

Abstract

James Ax has proved that when (K,V) is a Henselian rank one valued field which is perfect of characteristic not zero, then to each α in the algebraic closure K̄ of K there corresponds an element a ∈ K such that V̄(α − a) ≥ Δ(α), where Δ(α) = min{V̄(α′ − α): α′ runs over K-conjugates of α, V̄ is the extension of V to K̄}. In 1991, a counterexample was given to show that this result is false (cf. [J. Algebra140 (1991), 360-361]). In this paper, it is proved that the above result is true, but if and only if we have the additional hypothesis that (K,V) is a defectless valued field.