Valuations in algebraic field extensions

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 MacLaneʹs key polynomials. In the basic case when L is a finite separable extension and , 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 charK=0 then the set of key polynomials has order type at most , while in the case charK=p>0 this order type is bounded above by ([logpn]+1)ω, where n=[L:K]. Our results provide a new point of view of the well-known formula and the notion of defect.