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 of a field to a cyclic extension of , i.e. is the field of rational functions of or is the finite separable extension generated by a root of an irreducible polynomial . In two articles from 1936, Saunders MacLane has introduced the notions of *key polynomial* and of *augmented valuation* for a given valuation of , and has shown how we can recover any extension to of a discrete rank one valuation of by a countable sequence of augmented valuations , with . The valuation is defined by induction from the valuation , from a key polynomial and from the value .

In this article we study some properties of the augmented valuations and we generalize the results of MacLane to the case of any valuation of . For this we need to introduce *simple admissible families* of augmented valuations , where is not necessarily a countable set, and to define a *limit key polynomial* and *limit augmented valuation* for such families. Then, any extension to of a valuation on is again a limit of a family of augmented valuations.

We also get a ``factorization'' theorem which gives a description of the values for any polynomial in .

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Additional Information

**Michel Vaquié**

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

Email:
vaquie@math.ups-tlse.fr

DOI:
http://dx.doi.org/10.1090/S0002-9947-07-04184-0

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