On simple extensions generated by key polynomials

Abstract

Let $K^h$ be a Henselization of a valued field $(K,v),$ $w$ a valuation-transcendental extension of $v$ to $K[x],$ and $\varphi$ a key polynomial for $w$. In this paper, we prove that if $\varphi$ is irreducible over $K^h,$ then the simple extension $L|K,$ generated by a root of any key polynomial for $w,$ is unibranched and its defect is independent of the choice of key polynomial. As a consequence, we generalize some well-known results for Henselian valued fields to arbitrary valued fields. Moreover, we provide some criteria for $L|K$ to be unibranched and defectless in terms of Mac Lane-Vaquié chains, complete sequences of abstract key polynomials and saturated distinguished chains. As an application, we generalize a classical result of Ore about the existence of $p$-regular generators for number fields.