Let v be a henselian valuation of a field K, and [Formula: see text] be the (unique) extension of v to a fixed algebraic closure [Formula: see text] of K. For an element [Formula: see text], a chain [Formula: see text] of elements of [Formula: see text] such that θi is of minimal degree over K with the property that [Formula: see text] 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 [Formula: see text] 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 [Formula: see text] 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.
Abstract. Let v be a henselian valuation of any rank of a field K and v be the unique extension of v to a fixed algebraic closure K of K. In 2005, it was studied properties of those pairs (θ, α) of elements ofwhere α is an element of smallest degree over K such thatSuch pairs are referred to as distinguished pairs. We use the concept of liftings of irreducible polynomials to give a different characterization of distinguished pairs.
Let v be a henselian valuation of arbitrary rank of a field K with value group G(K) and residue field R(K) and [Formula: see text] be the unique extension of v to a fixed algebraic closure [Formula: see text] of K with value group [Formula: see text]. It is known that a complete distinguished chain for an element θ belonging to [Formula: see text] with respect to (K, v) gives rise to several invariants associated to θ, including a chain of subgroups of [Formula: see text], a tower of fields, together with a sequence of elements belonging to [Formula: see text] which are the same for all K-conjugates of θ. These invariants satisfy some fundamental relations. In this paper, we deal with the converse: Given a chain of subgroups of [Formula: see text] containing G(K), a tower of extension fields of R(K), and a finite sequence of elements of [Formula: see text] satisfying certain properties, it is shown that there exists a complete distinguished chain for an element [Formula: see text] associated to these invariants. We use the notion of lifting of polynomials to construct it.
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