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Extensions de valuation et polygone de Newton. (Extensions of a valuation and Newton polygon.) (French. English summary) Zbl 1170.13003
Let $$(K,\nu )$$ be a valued field and $$L=K(\theta)$$ a finite cyclic extension of $$K$$. The aim of this paper is to determine all valuations extending $$\nu$$. $$L$$ is isomorphic to $$K[x]/(P)$$, and any valuation of $$L$$ which extends $$\nu$$ defines a pseudo-valuation $$\zeta$$ on $$K[x]$$ whose kernel is the principal ideal $$(P)$$. The author associates to $$\zeta$$ a family of valuations on $$K[x]$$, called an admissible family, which is explicitly constructed by augmented valuations and limit augmented valuations. The author gives a necessary and sufficient condition for a valuation of $$K[x]$$ to belong to an admissible family associated to a pseudo-valuation $$\zeta$$ which corresponds to a valuation of $$L$$, this condition depends only on the polynomial $$P$$. The author determines all the valuations of $$L$$ which extend the valuation $$\nu$$ of $$K$$. In order to solve this problem the author defines the Newton polygon $${\mathcal PN}(P;\varphi;\mu)$$, associated to $$P$$, to a polynomial $$\varphi$$ and to a valuation $$\mu$$ of $$K[x]$$.

##### MSC:
 13A18 Valuations and their generalizations for commutative rings 12J10 Valued fields 14E15 Global theory and resolution of singularities (algebro-geometric aspects)
##### Keywords:
valuation; extension; Newton polygon
Full Text:
##### References:
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