Extension d'une valuation

Author:
Michel Vaquié

Journal:
Trans. Amer. Math. Soc. **359** (2007), 3439-3481

MSC (2000):
Primary 13A18; Secondary 12J10, 14E15

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

Published electronically:
February 12, 2007

MathSciNet review:
2299463

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Abstract | References | Similar Articles | Additional Information

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 .

**[A-M]**S.S. Abhyankar, T. Moh: Newton-Puiseux expension and generalized Tschirnhausen transformation. J. reine angew. Math.**260**(1973), 47-83,**261**(1973), 29-54. MR**0337955 (49:2724)****[A-P-Z]**V. Alexandru, N. Popescu, A. Zaharescu: All valuations on . J. Math. Kyoto Univ.**30**(1990), 281-296. MR**1068792 (92c:12011)****[Ka]**I. Kaplansky: Maximal fields with valuations. Duke Math. J.**9**(1942), 303-321. MR**0006161 (3:264d)****[K-G]**S.K. Khanduja, U. Garg: Rank valuations of . Mathematika**37**(1990), 97-105. MR**1067891 (91j:12016)****[Ku]**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)****[McL:1]**S. MacLane: A construction for absolute values in polynomial rings. Trans. Amer. Math. Soc.**40**(1936), 363-395. MR**1501879****[McL:2]**S. MacLane: A construction for prime ideals as absolute values of an algebraic field. Duke Math. J.**2**(1936), 492-510. MR**1545943****[Po]**P. Popescu-Pampu: Approximate roots, dans*Valuation theory and its applications, Volume II*, Fields Institute Comm. 33, 2003. MR**2018562 (2004k:14006)****[P-P]**L. Popescu, N. Popescu: On the residual transcendental extensions of a valuation. Key polynomials and augmented valuation. Tsukuba J. Math.**15**(1991), 57-78. MR**1118582 (92h:12008)****[P-V]**N. Popescu, C. Vraciu: On the extension of valuations on a field to - I et II. Rend. Sem. Mat. Univ. Padova**87**(1992), 151-168,**96**(1996),1-14. MR**1183907 (93i:12013)**; MR**1438285 (98d:12007)****[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)****[Va]**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.ups-tlse.fr

DOI:
https://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