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 [Formula: see text] be a Henselian valued field of arbitrary rank, [Formula: see text] the algebraic closure of [Formula: see text], and [Formula: see text] the unique extension of [Formula: see text] to [Formula: see text]. In [12], Khanduja shown that if for a separable element [Formula: see text], there exists [Formula: see text] such that [Formula: see text] (where [Formula: see text] is the Krasner’s constant of [Formula: see text] over [Formula: see text]), then the extension [Formula: see text] is tame. Regarding the converse of the above result and assuming that the extension [Formula: see text] is tame, we give a characterization of those [Formula: see text] for which there exists [Formula: see text] satisfying [Formula: see text]. More precisely, it is shown that [Formula: see text] 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.