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Extension of a valuation. (Extension d’une valuation.) (French) Zbl 1121.13006
The goal of this paper is to determine the extensions of a valuation $$\nu$$ on $$K$$, to a field $$L\supset K$$, where $$L$$ is either the quotient field of the polynomial ring $$K[x]$$ or $$L=K[x]/(G)$$, where $$G$$ is an irreducible polynomial. This problem was yet considered by Saunders Mac Lane, assuming that $$\nu$$ is a discrete valuation. The main point in the paper under review is that Vaquié considers any valuation, in this case appear some families of extended valuations $$S=(\mu_\alpha )_{\alpha \in A}$$, indexed by a totally ordered set $$A$$ which is not necessarily a countable set, and proves that any extended valuation can be obtained as a “limit” of an admissible family of some extended valuations lying on “simple” admissible families, which are non necessarily countable sets.
The notion of admissible family is a generalization of the notion of approximate roots of an irreducible polynomial $$f$$ in two variables. In the case of an analytically irreducible curve there exists only one extension, the admissible family is finite and corresponds to the approximative roots of $$f$$. He also proves that finding an admissible family for a valuation $$\mu$$ of $$K[x]$$ allows to find a system of generators of the graded algebra $$\text{ gr}_\mu K[x]$$, which plays an important role in the local uniformization problem.

##### MSC:
 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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##### References:
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