Extension d'une valuation
Author:
Michel Vaquié
Journal:
Trans. Amer. Math. Soc. 359 (2007), 34393481
MSC (2000):
Primary 13A18; Secondary 12J10, 14E15
Published electronically:
February 12, 2007
MathSciNet review:
2299463
Fulltext PDF Free Access
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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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 V. Alexandru, N. Popescu, A. Zaharescu: All valuations on . J. Math. Kyoto Univ. 30 (1990), 281296. MR 1068792 (92c:12011)
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 F.V. Kuhlmann: Valuation theoric and model theoric aspects of local uniformization, dans Resolution of Singularities, Progr. in math. 181, 2000. MR 1748629 (2001c:14001)
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 S. MacLane: A construction for absolute values in polynomial rings. Trans. Amer. Math. Soc. 40 (1936), 363395. MR 1501879
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 S. MacLane: A construction for prime ideals as absolute values of an algebraic field. Duke Math. J. 2 (1936), 492510. MR 1545943
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 P. PopescuPampu: Approximate roots, dans Valuation theory and its applications, Volume II, Fields Institute Comm. 33, 2003. MR 2018562 (2004k:14006)
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 L. Popescu, N. Popescu: On the residual transcendental extensions of a valuation. Key polynomials and augmented valuation. Tsukuba J. Math. 15 (1991), 5778. MR 1118582 (92h:12008)
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 [Sp]
 M. Spivakovsky: Resolution of singularities I: local uniformization, prépublication 1996.
 [Te]
 B. Teissier: Valuations, deformations and toric geometry, dans Valuation theory and its applications, Volume II, Fields Institute Comm. 33, 2003. MR 2018565 (2005m:14021)
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 M. Vaquié: Valuations, dans Resolution of Singularities, Progr. in math. 181, 2000. MR 1748635 (2001i:13005)
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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.upstlse.fr
DOI:
http://dx.doi.org/10.1090/S0002994707041840
PII:
S 00029947(07)041840
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
