A characterization for the defect of rank one valued field extensions

TL;DR

This paper characterizes the defect in simple algebraic extensions of rank one valued fields using key polynomials, providing a classification of defect extensions of degree p as dependent or independent.

Contribution

It introduces a valuation-based characterization of defect in rank one valued field extensions, linking key polynomials to defect classification.

Findings

Provides a valuation-theoretic criterion for defect

Classifies degree p defect extensions as dependent or independent

Connects key polynomials with defect behavior

Abstract

In this paper we present a characterization for the defect of a simple algebraic extension of rank one valued fields using the key polynomials that define the valuation. As a particular example, this gives the classification of defect extensions of degree $p$ as dependent or independent presented by Kuhlmann.