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 ] = e f where e , f are the ramification index and the residual degree of w / v , i.e., ( L , w ) / ( K , v ) is defectless. The constant min A L / K was first introduced by Tignol and is referred to as Tignol's constant. In 2005, K. Ota and Khanduja gave a formula for min A L / 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.