ON A RESULT OF AX,JAMES

James Ax has proved that when (K,V) is a Henselian rank one valued fie ld which is perfect of characteristic not zero, then to each alpha in the algebraic closure (K) over bar of K there corresponds an element a is an element of K such that (V) over bar(alpha - a) greater than or equal to Delta(alpha), where Delta(alpha) = min{(V) over bar(alpha' - alpha): alpha' runs over K-conjugates of alpha, (V) over bar is the ex tension of V to (K) over bar}. In 1991, a counterexample was given to show that this result is false (cf. [J. Algebra 140 (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. (C) 1995 Academic Press, Inc.