Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Extension d'une valuation

Author: Michel Vaquié
Journal: Trans. Amer. Math. Soc. 359 (2007), 3439-3481
MSC (2000): Primary 13A18; Secondary 12J10, 14E15
Published electronically: February 12, 2007
MathSciNet review: 2299463
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Abstract: We want to determine all the extensions of a valuation $ \nu$ of a field $ K$ to a cyclic extension $ L$ of $ K$, i.e. $ L=K(x)$ is the field of rational functions of $ x$ or $ L=K(\theta )$ is the finite separable extension generated by a root $ \theta$ of an irreducible polynomial $ G(x)$. In two articles from 1936, Saunders MacLane has introduced the notions of key polynomial and of augmented valuation for a given valuation $ \mu$ of $ K[x]$, and has shown how we can recover any extension to $ L$ of a discrete rank one valuation $ \nu$ of $ K$ by a countable sequence of augmented valuations $ \bigl (\mu _i\bigr ) _{i \in I}$, with $ I \subset \mathbb{N}$. The valuation $ \mu _i$ is defined by induction from the valuation $ \mu _{i-1}$, from a key polynomial $ \phi _i$ and from the value $ \gamma _i = \mu ( \phi _i )$.

In this article we study some properties of the augmented valuations and we generalize the results of MacLane to the case of any valuation $ \nu$ of $ K$. For this we need to introduce simple admissible families of augmented valuations $ {\mathcal A} = \bigl ( \mu _{\alpha} \bigr ) _{\alpha \in A}$, where $ A$ is not necessarily a countable set, and to define a limit key polynomial and limit augmented valuation for such families. Then, any extension $ \mu$ to $ L$ of a valuation $ \nu$ on $ K$ is again a limit of a family of augmented valuations.

We also get a ``factorization'' theorem which gives a description of the values $ ( \mu _{\alpha} (f))$ for any polynomial $ f$ in $ K[x]$.

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Additional Information

Michel Vaquié
Affiliation: Laboratoire Émile Picard, UMR 5580, Université Paul Sabatier, UFR MIG, 31062 Toulouse Cedex 9, France

Received by editor(s): March 29, 2004
Received by editor(s) in revised form: July 18, 2005
Published electronically: February 12, 2007
Article copyright: © Copyright 2007 American Mathematical Society