Eliminating Tame Ramification: generalizations of Abhyankar's Lemma

TL;DR

This paper generalizes Abhyankar's Lemma to valued fields with rational rank 1 value groups, providing necessary and sufficient conditions for eliminating tame ramification in extensions.

Contribution

It extends the classical lemma to broader valued field contexts and establishes precise conditions for tame ramification elimination.

Findings

Generalization to valued fields with rational rank 1

Necessary and sufficient conditions for tame ramification elimination

Examples illustrating the theoretical results

Abstract

A basic version of Abhyankar's Lemma states that for two finite extensions $L$ and $F$ of a local field $K$, if $L|K$ is tamely ramified and if the ramification index of $L|K$ divides the ramification index of $F|K$, then the compositum $L.F$ is an unramified extension of $F$. In this paper, we generalize the result to valued fields with value groups of rational rank 1, and show that the latter condition is necessary. Replacing the condition on the ramification indices by the condition that the value group of $L$ be contained in that of $F$, we generalize the result further in order to give a necessary and sufficient condition for the elimination of tame ramification of an arbitrary extension $F|K$ by a suitable algebraic extension of the base field $K$. In addition, we derive more precise ramification theoretical statements and give several examples.