On Chains Associated with Elements Algebraic over a Henselian Valued Field

• 来源期刊: 代数集刊（英文版）
• 关键词: valued fields non-archimedean valued fields valuations and their generalizations
• 年，卷（期）: 2005,（4）
• 页码范围: 607-616
• 页数: 10页
• 语种: 英文

摘要

Let v be a henselian valuation of a field K, and -v be the (unique) extension of vto a fixed algebraic closure -K of K. For an element θ∈ K \K, a chain θ = θ0, θ1,..., θm of elements of -K such that θi is of minimal degree over K with the property that -v(θi-1 - θi)= sup{-v(θi-1 - β) [ [K(β) : K] ＜ [K(θi-1) : K]} and θm ∈ 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∈ -K \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∈- K \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.