On finite tame extensions and metric invariants

Tame extensions form an important class of henselian valued field extensions. So far, several characterizations of them have been presented in terms of the metric invariants associated to algebraic elements. One of the most important of these metric invariants is the Krasner's constant. Let (K, v) be a henselian valued field of arbitrary rank, [Formula: see text] the algebraic closure of K, and ṽ the unique extension of v to [Formula: see text]. In [12], Khanduja shown that if for a separable element α ∈ [Formula: see text] \ K, there exists a ∈ K such that ṽ(α − a) ≥ ω K (α) (where ω K (α) is the Krasner's constant of α over K), then the extension K(α)/K is tame. Regarding the converse of the above result and assuming that the extension K(α)/K is tame, we give a characterization of those α ∈ [Formula: see text] \ K for which there exists a ∈ K satisfying ṽ(α−a) ≥ ω K (α). More precisely, it is shown that α has the foregoing property if and only if the length of its saturated distinguished chain is one. Then we deal with construction of distinguished pairs and determination of some invariants associated to them in terms of again the Krasner's constant. Finally, we apply the notions of saturated distinguished chains and metric invariants to give new methods to verify some already known results concerning tame extensions.