Valuations in algebraic field extensions

Abstract

Let K → L be an algebraic field extension and ν a valuation of K. The purpose of this paper is to describe the totality of extensions { ν ′ } of ν to L using a refined version of MacLane's key polynomials. In the basic case when L is a finite separable extension and rk ν = 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 N , while in the case char K = p > 0 this order type is bounded above by ( [ log p n ] + 1 ) ω , where n = [ L : K ] . Our results provide a new point of view of the well-known formula ∑ j = 1 s e j f j d j = n and the notion of defect.