On chains associated with elements algebraic over a henselian valued field

Abstract

Let v be a henselian valuation of a field K,and v̅ be the (unique) extension of v to a fixed algebraic closure of K̅. For an element θ∈K̅\K, a chain θ= θ₀, θ₁,...θ_mof elements of K̅ such that θ̅(θ_i−1−θ_i)=SUP{ v̅(θ_i−1−β)|[K(β):K]<[K(θ_i−1):K]} and θ_m ∈ K, is called a complete distinguished chain for θ with respect to (K, v). In 1995, Popescu and Zaharescu proved the existence of a complete distinguished chain for each θ∈K̅\K when (K, v) is a complete discrete rank one valued field (cf. [10]). In this paper, for a henselian valued field (K, v) of arbitrary rank, we characterize those elements θ∈K̅\K for which there exists a complete distinguished chain. It is shown that a complete distinguished chain for θ gives rise to several invariants associated to θ which are same for all the K-conjugates of θ.