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 \(K[x]\).


13A18 Valuations and their generalizations for commutative rings
12J10 Valued fields
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
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