On a theorem of Tignol for defectless extensions and its converse

Let (K,v) be a Henselian valued field of arbitrary rank. In 1990, Tignol proved that if (K′,v′)/(K,v) is a finite separable defectless extension of degree a prime number, then the set AK′/K={v(TrK′/K(α))−v′(α)|α∈K′,α≠0} has a minimum element. This paper extends Tignol's result to all finite separable extensions. Moreover a characterization of finite separable defectless extensions is given by showing that (K′,v′)/(K,v) is a defectless extension if and only if the set AK′/K has a minimum element. Our proof also leads to a new proof of the well-known result that each finite extension of a formally ℘-adic field (or more generally of a finitely ramified valued field) is defectless.