A formula for Tignol's constant

Abstract

Let (K, v) be a Henselian valued field and (L,w) be a finite separable extension of (K, v). In 2004, it was proved that the set A_L/K defined by A_L/K = {v(Tr_L/K(α)) − w(α) | α ∈ L, α ≠ = 0} has a minimum element if and only if [L : K] = ef where e,f are the ramification index and the residual degree of w/v, i.e., (L,w)/(K, v) is defectless. The constant minAL/K was first introduced by Tignol and is referred to as Tignol’s constant. In 2005, K. Ota and Khanduja gave a formula for minAL/K when (L,w)/(K, v) is an extension of local fields. In this paper, we give this formula when (L,w) is any finite separable defectless extension of a Henselian valued field of arbitrary rank and thereby generalize some well-known results of Dedekind regarding “different” of extensions of algebraic number fields and ramification of prime ideals.