On Chains Associated with Elements Algebraic over a Henselian Valued Field

Abstract

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.

On Eisenstein–Dumas and Generalized Schönemann Polynomials

“…Since K(θ 1 )/K is a defectless extension in view of (Aghigh and Khanduja, 2005, Theorem 1.2) and it is given that G(K(θ 1 )) = G, we now obtain (9) using (10).…”

“…In 2005, Aghigh and Khanduja (cf. Aghigh and Khanduja, 2005) proved that if (K, v) is a henselian valued field of arbitrary rank, then an element θ belonging to K \ K has a saturated distinguished chain with respect to v if and only if K(θ) is a defectless extension of (K, v). A saturated distinguished chain for θ gives rise to several invariants associated with θ, some of which are given by Theorem B stated below which is proved in (cf.…”

“…A saturated distinguished chain for θ gives rise to several invariants associated with θ, some of which are given by Theorem B stated below which is proved in (cf. Aghigh and Khanduja, 2005).…”

On Brown’s constant associated with irreducible polynomials over henselian valued fields

“…As shown in [2,Theorem 2.4], the above supremum is attained by virtue of the hypothesis that K (θ )/K is defectless; indeed there exists α belonging to K such that (θ , α) is a distinguished pair. Let (θ , α) be a distinguished pair with g(x) the minimal polynomial of θ over K .…”

“…It is known that a simple extension K (θ ) of (K , v) is defectless if and only if θ has a complete distinguished chain with respect to v (cf. [2,Theorem 1.2]). …”