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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Extension d’une valuation
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by Michel Vaquié PDF
Trans. Amer. Math. Soc. 359 (2007), 3439-3481 Request permission

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
  • Email: vaquie@math.ups-tlse.fr
  • Received by editor(s): March 29, 2004
  • Received by editor(s) in revised form: July 18, 2005
  • Published electronically: February 12, 2007
  • © Copyright 2007 American Mathematical Society
  • 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
  • MathSciNet review: 2299463