On finite tame extensions of valued fields

Abstract

Let v be a henselian valuation of arbitrary rank of a field K and v¯ be its unique prolongation to a fixed algebraic closure K¯ of K. For any α belonging to K¯\K, let Δ _K (α) (resp. ω _K (α)) denote the invariants defined to be the minimum (resp. maximum) of the set {v¯(α − α’) | α’ ≠ α runs over K-conjugates of α}. In 1998, while correcting a result of James Ax, Khanduja proved that every finite extension of (K, v) contained in K¯ is tame, if and only if to each α ε K¯\K, there corresponds a ε K such that v¯(α − a) ≥ Δ K (α). It was also shown that the analogue of the above result does not hold in general when K¯ is replaced by a finite extension of K (see J. Alg. 201, 1998, pp. 647–655). In this paper, similar characterizations of finite tame extensions are given, and some of the invariants associated with such an extension are explicitly determined.