Generalization of Abhyankar's Lemma to henselian valued fields

TL;DR

This paper extends Abhyankar's lemma to henselian valued fields, demonstrating that the divisibility condition on value group extensions is not sufficient for ramification elimination in this broader context, and providing a complete criterion.

Contribution

It generalizes Abhyankar's lemma to henselian valued fields, introduces a counterexample, and establishes a necessary and sufficient condition for tame ramification elimination.

Findings

Divisibility condition is insufficient in henselian fields.

Counterexample illustrating the failure of the condition.

A complete criterion for tame ramification elimination.

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.