On a Result of James Ax Original Research Article

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 image of K there corresponds an element a set membership, variant K such that image(α − a) ≥ Δ(α), where Δ(α) = min{image(α′ − α): α′ runs over K-conjugates of α, image is the extension of V to image}. 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.