Generalization of Abhyankar's Lemma to henselian valued fields

Abstract

Abhyankar showed that for a finite tame extension $L_1/K$ and a finite extension $L_2/K$ of $\mathfrak{P}$-adic fields, the condition $[\nu L_1 : \nu K]$ divides $[\nu L_2 : \nu K]$ is sufficient to eliminate ramification, that is, $L_1 \cdot L_2 / L_2$ is unramified. In this paper, we show that the above condition is not sufficient in the case of an arbitrary henselian valued field. We construct a counterexample illustrating that fact. We also give a necessary and sufficient condition for the elimination of tame ramification of a henselian field after a finite extension of the base field.