Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-25T18:54:03.806Z Has data issue: false hasContentIssue false
  • English
  • Français

NOTES ON EXTREMAL AND TAME VALUED FIELDS

Published online by Cambridge University Press:  29 June 2016

SYLVY ANSCOMBE  and
FRANZ-VIKTOR KUHLMANN
SYLVY ANSCOMBE
Affiliation:
JEREMIAH HORROCKS INSTITUTE LEIGHTON BUILDING LE7 UNIVERSITY OF CENTRAL LANCASHIRE PRESTON, PR1 2HE, UKE-mail:sanscombe@uclan.ac.uk
FRANZ-VIKTOR KUHLMANN
Affiliation:
INSTITUTE OF MATHEMATICS UNIVERSITY OF SILESIA UL. BANKOWA 14, 40-007KATOWICE POLANDE-mail:fvk@math.us.edu.pl
Get access Rights & Permissions [Opens in a new window]

Abstract

We extend the characterization of extremal valued fields given in [2] to the missing case of valued fields of mixed characteristic with perfect residue field. This leads to a complete characterization of the tame valued fields that are extremal. The key to the proof is a model theoretic result about tame valued fields in mixed characteristic. Further, we prove that in an extremal valued field of finite p-degree, the images of all additive polynomials have the optimal approximation property. This fact can be used to improve the axiom system that is suggested in [8] for the elementary theory of Laurent series fields over finite fields. Finally we give examples that demonstrate the problems we are facing when we try to characterize the extremal valued fields with imperfect residue fields. To this end, we describe several ways of constructing extremal valued fields; in particular, we show that in every ℵ1 saturated valued field the valuation is a composition of extremal valuations of rank 1.

Keywords

valuationpolynomialmodel theory of valued fieldsextremal valued fieldtame valued field
Type
Articles
Information
The Journal of Symbolic Logic , Volume 81 , Issue 2 , June 2016 , pp. 400 - 416
Copyright
Copyright © The Association for Symbolic Logic 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Anscombe, S. and Fehm, A., The existential theory of equicharacteristic henselian valued fields, http://arxiv.org/abs/1501.04522, 2015.Google Scholar
Azgin, S., Kuhlmann, F.-V., and Pop, F., Characterization of extremal valued fields . Proceedings of the American Mathematical Society, vol. 140 (2012), pp. 1535–1547.CrossRefGoogle Scholar
van den Dries, L. and Kuhlmann, F.-V., Images of additive polynomials in $F$ q ((t)) have the optimal approximation property . Canadian Mathematical Bulletin, vol. 45 (2002), pp. 71–79.Google Scholar
Denef, J. and Schoutens, H., On the decidability of the existential theory of Fp [[t]], Valuation Theory and Its Applications, Vol. II (Saskatoon, SK, 1999), pp. 43–60, Fields Institute Communications, vol. 33, American Mathematical Society, Providence, RI, 2003.Google Scholar
Durhan, S., Additive Polynomials over Perfect Fields , Valuation Theory in Interaction, Proceedings of the Second International Valuation Theory Conference, Segovia / ElEscorial, 2011, EMS Series of Congress Reports 2014.Google Scholar
Kaplansky, I., Maximal fields with valuations I . Duke Mathematical Journal, vol. 9 (1942), pp. 303–321.Google Scholar
Kuhlmann, F.-V., Quantifier elimination for henselian fields relative to additive and multiplicative congruences . Israel Journal of Mathematics, vol. 85 (1994), pp. 277–306.Google Scholar
Kuhlmann, F.-V., Elementary properties of power series fields over finite fields , this Journal, vol. 66 (2001), pp. 771–791.Google Scholar
Kuhlmann, F.-V., Value groups, residue fields and bad places of rational function fields . Transactions of the American Mathematical Society, vol. 356 (2004), pp. 4559–4600.Google Scholar
Kuhlmann, F.-V., On places of algebraic function fields in arbitrary characteristic . Advances in Mathematics, vol. 188 (2004), pp. 399–424.Google Scholar
Kuhlmann, F.-V., Dense subfields of Henselian fields, and integer parts , Logic in Tehran, Lecture Notes in Logic, vol. 26, pp. 204–226, Association of Symbolic Logic, La Jolla, CA, 2006.Google Scholar
Kuhlmann, F.-V., The algebra and model theory of tame valued fields . Journal für die Reine und Angewandte Mathematik, to appear. Preliminary version published in: Séminaire de Structures Algébriques Ordonnées, 81, Prépublications Paris 7 (2009).Google Scholar
Pop, F., Embedding problems over large fields . Annals of Mathematics, vol. 144 (1996), pp. 1–34.Google Scholar