This article is a natural continuation of our previous works [7] and [6]. In this article, we employ similar ideas as in [4] to provide an estimate of IC(K(X)|K, v) when (K(X)|K, v) is a valuation algebraic extension. Our central result is an analogue of [6, Theorem 1.3]. We further provide a natural construction of a complete sequence of key polynomials for v over K in the setting of valuation algebraic extensions.
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.
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