On the implicit constant fields and key polynomials for valuation algebraic extensions

Abstract

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.