A basic version of Abhyankar’s lemma states that for two finite extensions
and
of a local
field
,
if
is tamely ramified and if the ramification index of
divides the ramification
index of
, then the
compositum
is an
unramified extension of
.
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
be contained
in that of
,
we generalize the result further in order to give a necessary and sufficient
condition for the elimination of tame ramification of an arbitrary extension
by a suitable algebraic extension of the base field
. In
addition, we derive more precise ramification theoretical statements and give several
examples.
Keywords
valuation, elimination of ramification, ramification
theory, tame extension
