On Construction of Saturated Distinguished Chains

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 K ¯ of K . For an element α ∈ 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 ] < [ K (α i −1 ) : K ]} and that α r ∈ 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.