Valuations in algebraic field extensions

Abstract

Let $K\to L$ be an algebraic field extension and $\nu$ a valuation of $K$. The purpose of this paper is to describe the totality of extensions $\left\{\nu'\right\}$ of $\nu$ to $L$ using a refined version of MacLane's key polynomials. In the basic case when $L$ is a finite separable extension and $rk \nu=1$, we give an explicit description of the limit key polynomials (which can be viewed as a generalization of the Artin--Schreier polynomials). We also give a realistic upper bound on the order type of the set of key polynomials. Namely, we show that if $char K=0$ then the set of key polynomials has order type at most $\mathbb N$, while in the case $char K=p>0$ this order type is bounded above by $([\log_pn]+1)\omega$, where $n=[L:K]$. Our results provide a new point of view of the the well known formula $\sum\limits_{j=1}^se_jf_jd_j=n$ and the notion of defect.