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

TL;DR

This paper extends valuation theory by estimating the implicit constant field in algebraic extensions and constructing key polynomials, advancing understanding of valuation algebraic extensions.

Contribution

It provides an estimate of the implicit constant field and constructs a complete sequence of key polynomials for valuation algebraic extensions, building on MacLane's ideas.

Findings

Estimated IC(K(X)|K,v) in algebraic extensions

Constructed a complete sequence of key polynomials

Extended valuation theory to algebraic extensions

Abstract

This article is a natural construction of our previous works. In this article, we employ similar ideas due to MacLane 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 the estimate obtained in the valuation transcendental case. We further provide a natural construction of a complete sequence of key polynomials for v over K in the setting of valuation algebraic extensions.