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