Vol. 307, No. 1, 2020

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Eliminating tame ramification: generalizations of Abhyankar's lemma

Arpan Dutta and Franz-Viktor Kuhlmann

Vol. 307 (2020), No. 1, 121–136
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.

Keywords
valuation, elimination of ramification, ramification theory, tame extension
Mathematical Subject Classification 2010
Primary: 12J20, 12J25, 13A18
Milestones
Received: 12 December 2019
Revised: 10 February 2020
Accepted: 12 February 2020
Published: 8 August 2020
Authors
Arpan Dutta
Department of Mathematics
IISER Mohali
SAS Nagar, Punjab
India
Franz-Viktor Kuhlmann
Institute of Mathematics
University of Szczecin
Poland