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Elementary theory of valued fields with a valuation-preserving automorphism

Part of: Connections with logic Model theory

Published online by Cambridge University Press:  01 July 2010

Salih Azgin  and
Lou van den Dries
Salih Azgin
Affiliation:
Department of Mathematics and Statistics, McMaster University, 1280 Main Street, West Hamilton, Ontario L8S 4K1, Canada (sazgin@math.mcmaster.ca)
Lou van den Dries
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, IL 61801, USA (vddries@math.uiuc.edu)
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Abstract

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We consider valued fields with a value-preserving automorphism and improve on model-theoretic results by Bélair, Macintyre and Scanlon on these objects by dropping assumptions on the residue difference field. In the equicharacteristic 0 case we describe the induced structure on the value group and the residue difference field.

Keywords

valued difference fieldsmodel theoryelementary classification

MSC classification

Secondary: 03C60: Model-theoretic algebra 12L12: Model theory
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 10 , Issue 1 , January 2011 , pp. 1 - 35
Copyright
Copyright © Cambridge University Press 2010

References

1.Azgin, S., Model theory of valued difference fields, PhD thesis, University of Illinois at Urbana-Champaign (2007).Google Scholar
2.Azgin, S. and van den Dries, L., Equivalence of valued difference fields with a valuation preserving automorphism, preprint (arXiv:0902.0422v1; 2009).CrossRefGoogle Scholar
3.Bélair, L., Macintyre, A. and Scanlon, T., Model theory of Frobenius on Witt vectors, Am. J. Math. 129 (2007), 665–721.CrossRefGoogle Scholar
4.Cherlin, G., Model theoretic algebra—selected topics, Lecture Notes in Mathematics, Volume 521 (Springer, 1976).CrossRefGoogle Scholar
5.Cohn, R. M., Difference algebra (Interscience/John Wiley & Sons, 1965).Google Scholar
6.Joyal, A., δ-anneaux et vecteurs de Witt, C. R. Math.-Math. Rep. Acad. Sci. Canada 7 (1985), 177–182.Google Scholar
7.Kaplansky, I., Maximal fields with valuations, Duke Math. J. 9 (1942), 303–321.CrossRefGoogle Scholar
8.Kochen, S., The model theory of local fields, in Proc. Logic Conf., Kiel, 1974, Lecture Notes in Mathematics, Volume 499, pp. 384–425 (Springer, 1975).Google Scholar
9.Scanlon, T., A model complete theory of valued D-fields, J. Symb. Logic 65 (2001), 1758–1784.CrossRefGoogle Scholar
10.Scanlon, T., Quantifier elimination for the relative Frobenius, in Valuation Theory and Its Applications, Saskatoon, SK, 1999, Volume II, Fields Institute Communications, Volume 33 pp. 323–352 (American Mathematical Society, Providence, RI, 2003).Google Scholar
11.Serre, J. P., Local fields (Springer, 1979).CrossRefGoogle Scholar
12.Whaples, G., Galois cohomology of additive polynomials and nth power mappings of fields, Duke Math. J. 24 (1957), 143–150.CrossRefGoogle Scholar