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

Abstract

This article is a natural continuation of our previous works [Dutta 2021] and [Dutta 2022]. In this article, we employ similar ideas as in [Alexandru, Popescu and Zaharescu 1990] to provide an estimate of $IC(K(X)\mid K,v)$ when $(K(X)\mid K,v)$ is a valuation algebraic extension. Our central result is an analogue of [Dutta 2022, 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.