Valuations in algebraic field extensions

By F. J. H. Govantes, M. Acosta, M. Spivakovsky. 2006

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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 rkν=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 charK=0 then the set of key polynomials has order type at most N, 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 ∑j=1sejfjdj=n and the notion of defect.

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ce0361c949b9a1125f9ed704f8e4d20662c6ed6a