On a result of James Ax

Khanduja, S.K. (1995) On a result of James Ax Journal of Algebra, 172 (1). pp. 147-151. ISSN 0021-8693

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Official URL: http://www.sciencedirect.com/science/article/pii/S...

Related URL: http://dx.doi.org/10.1006/jabr.1995.1054


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.

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