On a Result of James Ax

✍ Scribed by S.K. Khanduja

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 182 KB
• Volume: 172
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1995.1054

✦ Synopsis

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 (\alpha) in the algebraic closure (\bar{K}) of (K) there corresponds an element (a \in K) such that (\bar{V}(\alpha-a) \geq \Delta(\alpha)), where (\Delta(\alpha)=\min \left{\bar{V}\left(\alpha^{\prime}-\alpha\right): \alpha^{\prime}\right.) runs over (K)-conjugates of (\alpha, \bar{V}) is the extension of (V) to (\widehat{K}) ). 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. io) 1995 Academic Press, Inc.