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 Kof K. For an element α e K\K, a chain α = α 0 , α 1 ,…,α r of elements of K,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] i -1):K]} and that α r e 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 K(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.