On construction of saturated distinguished chains

Abstract

Let υ be a Henselian valuation of arbitrary rank of a field K, and let ῡ be the (unique) extension of v to a fixed algebraic closure K̅of K. For an element α ε K̅\K, a chain α = α₀, α₁,…,α_r of elements of K̅,such that α_i is of minimum degree over K with the property that ῡ(α_i-1-α_i)= sup{ῡ(α_i-1-β)|[K(β): K] < [K(α_i-1):K]} and that α_r ε K , is called a saturated distinguished chain for α with respect to (K, υ). The notion of a saturated distinguished chain has been used to obtain results about the irreducible polynomials over any complete discrete rank one valued field K and to determine various arithmetic and metric invariants associated to elements of K̅(cf. [J. Number Theory, 52 (1995), 98–118.] and [J. Algebra, 266 (2003), 14–26]). In this paper, a method is described of constructing a saturated distinguished chain for α, and also determining explicitly some invariants associated to α, when the degree of the extension K (α)/K is not divisible by the characteristic of the residue field of υ.