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Lectures on the Model Theory of Valued Fields

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Model Theory in Algebra, Analysis and Arithmetic

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2111))

Abstract

The subject originates in the 1950s with Abraham Robinson when he established the model completeness of the theory of algebraically closed valued fields. In the 1960s Ax & Kochen and, independently, Ershov, proved a remarkable theorem on henselian valued fields, with applications to p-adic number theory. These results and their refinements and extensions remain important in more recent developments like motivic integration.

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Notes

  1. 1.

    One should of course regard \(\mathcal{L}_{\text{r}}\) and \(\mathcal{L}_{\text{v}}\) as disjoint.

References

  1. S. Abhyankar, On the valuations centered in a local domain. Am. J. Math. 78, 321–348 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  2. M. Artin, On the solutions of analytic equations. Invent. Math. 5, 277–291 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  3. M. Artin, Algebraic approximation of structures over complete local rings. Inst. Hautes Études Sci. Publ. Math. 36, 23–58 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  4. J. Ax, S. Kochen, Diophantine problems over local fields. I. Am. J. Math. 87, 605–630 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  5. J. Ax, S. Kochen, Diophantine problems over local fields. II. A complete set of axioms for p-adic number theory. Am. J. Math. 87, 631–648 (1965)

    Google Scholar 

  6. J. Ax, S. Kochen, Diophantine problems over local fields. III. Decidable fields. Ann. Math. 83, 437–456 (1966)

    MathSciNet  MATH  Google Scholar 

  7. S. Azgin, L. van den Dries, Elementary theory of valued fields with a valuation-preserving automorphism. J. Inst. Math. Jussieu 10, 1–35 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. L. Bélair, A. Macintyre, T. Scanlon, Model theory of the Frobenius on the Witt vectors. Am. J. Math. 129, 665–721 (2007)

    Article  MATH  Google Scholar 

  9. N. Bourbaki, Elements of Mathematics. Commutative Algebra (Hermann, Paris, 1972). Translated from the French

    Google Scholar 

  10. G. Cherlin, Model Theoretic Algebra—Selected Topics. Lecture Notes in Mathematics, vol. 521 (Springer, Berlin, 1976)

    Google Scholar 

  11. R. Cluckers, J. Nicaise, J. Sebag (eds.) Motivic Integration and Its Interactions with Model Theory and Non-Archimedean Geometry, Volume I, vol. 383 of London Mathematical Society Lecture Note Series (Cambridge University Press, Cambridge, 2011)

    Google Scholar 

  12. R. Cluckers, J. Nicaise, J. Sebag (eds.) Motivic integration and Its Interactions with Model Theory and Non-Archimedean Geometry. Volume II, vol. 384 of London Mathematical Society Lecture Note Series (Cambridge University Press, Cambridge, 2011)

    Google Scholar 

  13. P.J. Cohen, Decision procedures for real and p-adic fields. Comm. Pure Appl. Math. 22, 131–151 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  14. F. Delon, Quelques propriétés des corps valués en théorie des modeles. PhD thesis, Université Paris VII, 1982

    Google Scholar 

  15. J. Denef, The rationality of the Poincaré series associated to the p-adic points on a variety. Invent. Math. 77, 1–23 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  16. J. Denef, F. Loeser, Motivic integration and the Grothendieck group of pseudo-finite fields. In Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pp. 13–23 (Higher Ed. Press, Beijing, 2002)

    Google Scholar 

  17. J. Denef, L. van den Dries, p-adic and real subanalytic sets. Ann. Math. 128, 79–138 (1988)

    Google Scholar 

  18. J. Denef, p-adic semi-algebraic sets and cell decomposition. J. Reine Angew. Math. 369, 154–166 (1986)

    Google Scholar 

  19. A.J. Engler, A. Prestel, Valued Fields. Springer Monographs in Mathematics (Springer, Berlin, 2005)

    Google Scholar 

  20. Ju.L. Ershov, On elementary theories of local fields. Algebra i Logika Sem. 4, 5–30 (1965)

    MATH  Google Scholar 

  21. Ju.L. Ershov, On elementary theory of maximal normalized fields. Algebra i Logika Sem. 4, 31–70 (1965)

    MATH  Google Scholar 

  22. Ju.L. Ershov, On the elementary theory of maximal normed fields. II. Algebra i Logika Sem. 5, 5–40 (1966)

    MathSciNet  MATH  Google Scholar 

  23. Ju.L. Ershov, On the elementary theory of maximal normed fields. III. Algebra i Logika Sem. 6, 31–38 (1967)

    MATH  Google Scholar 

  24. J. Fresnel, M. van der Put, Géométrie Analytique Rigide et Applications, vol. 18 of Progress in Mathematics (Birkhäuser Boston, Boston, 1981)

    Google Scholar 

  25. M.J. Greenberg, Lectures on Forms in Many Variables (W. A. Benjamin, New York-Amsterdam, 1969)

    Google Scholar 

  26. H. Hahn, Über die nichtarchimedischen Grössensysteme. S.-B. Akad. Wiss. Wien, Math.-naturw. Kl. Abt. IIa 116, 601–655 (1907)

    Google Scholar 

  27. D. Haskell, E. Hrushovski, D. Macpherson, Definable sets in algebraically closed valued fields: elimination of imaginaries. J. Reine Angew. Math. 597, 175–236 (2006)

    MathSciNet  MATH  Google Scholar 

  28. D. Haskell, E. Hrushovski, D. Macpherson, Stable Domination and Independence in Algebraically Closed Valued Fields, vol. 30 of Lecture Notes in Logic (Association for Symbolic Logic, Chicago, 2008)

    Google Scholar 

  29. J. Holly, Definable equivalence relations and disc spaces of algebraically closed valued fields. PhD thesis, University of Illinois, 1992

    Google Scholar 

  30. E. Hrushovski, D. Kazhdan, Integration in valued fields. In Algebraic geometry and number theory, vol. 253 of Progr. Math., pp. 261–405 (Birkhäuser Boston, Boston, 2006)

    Google Scholar 

  31. I. Kaplansky, Maximal fields with valuations. Duke Math. J. 9, 303–321 (1942)

    Article  MathSciNet  Google Scholar 

  32. S. Kochen, The model theory of local fields. In 1xsy) ISILC Logic Conference (Proc. Internat. Summer Inst. and Logic Colloq., Kiel, 1974), pp. 384–425. Lecture Notes in Math., vol. 499 (Springer, Berlin, 1975)

    Google Scholar 

  33. W. Krull, Allgemeine Bewertungstheorie. J. Reine Angew. Math. 167, 160–196 (1932)

    Google Scholar 

  34. S. Lang, Algebra, 2nd edn (Addison-Wesley Publishing Company Advanced Book Program, Reading, 1984)

    MATH  Google Scholar 

  35. L. Lipshitz, Z. Robinson, Rings of separated power series and quasi-affinoid geometry. Astérisque 264, vi+171 (2000)

    Google Scholar 

  36. A. Macintyre, On definable subsets of p-adic fields. J. Symbolic Logic 41, 605–610 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  37. A. Macintyre, K. McKenna, L. van den Dries, Elimination of quantifiers in algebraic structures. Adv. Math. 47, 74–87 (1983)

    Article  MATH  Google Scholar 

  38. M. Nagata, Local Rings. Interscience Tracts in Pure and Applied Mathematics, No. 13 (Interscience Publishers a division of Wiley, New York-London, 1962)

    Google Scholar 

  39. A. Ostrowski, Untersuchungen zur arithmetischen Theorie der Körper. Math. Z. 39, 269–404 (1935)

    Article  MathSciNet  Google Scholar 

  40. J. Pas, Uniform p-adic cell decomposition and local zeta functions. J. Reine Angew. Math. 399, 137–172 (1989)

    MathSciNet  MATH  Google Scholar 

  41. A. Prestel, P. Roquette, Formally p-adic Fields, vol. 1050 of Lecture Notes in Mathematics (Springer, Berlin, 1984)

    Google Scholar 

  42. P. Ribenboim, Théorie des Valuations, vol. 1964 of Séminaire de Mathématiques Supérieures, No. 9 (Été). Deuxième édition multigraphiée (Les Presses de l’Université de Montréal, Montreal, 1968)

    Google Scholar 

  43. A. Robinson, Complete Theories (North-Holland Publishing, Amsterdam, 1956)

    MATH  Google Scholar 

  44. P. Roquette, History of valuation theory. I. In Valuation Theory and Its Applications, Vol. I (Saskatoon, SK, 1999), vol. 32 of Fields Inst. Commun., pp. 291–355 (American Mathematical Society, Providence, 2002)

    Google Scholar 

  45. T. Scanlon, A model complete theory of valued D-fields. J. Symbolic Logic 65, 1758–1784 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  46. T. Scanlon, Analytic difference rings. In International Congress of Mathematicians. Vol. II, pp. 71–92 (Eur. Math. Soc., Zürich, 2006)

    Google Scholar 

  47. O.F.G. Schilling, The Theory of Valuations. Mathematical Surveys, No. 4 (American Mathematical Society, New York, 1950)

    Google Scholar 

  48. J.-P. Serre, Corps Locaux. Publications de l’Institut de Mathématique de l’Université de Nancago, VIII. Actualités Sci. Indust., No. 1296 (Hermann, Paris, 1962)

    Google Scholar 

  49. J.-P. Serre, Cours d’arithmétique, vol. 2 of Collection SUP: “Le Mathématicien” (Presses Universitaires de France, Paris, 1970)

    Google Scholar 

  50. J.R. Shoenfield, Mathematical Logic (Addison-Wesley Publishing, Reading, London-Don Mills, Ont., 1967)

    MATH  Google Scholar 

  51. G. Terjanian, Un contre-exemple à une conjecture d’Artin. C. R. Acad. Sci. Paris Sér. A-B 262, A612 (1966)

    MathSciNet  Google Scholar 

  52. J. van der Hoeven, Transseries and Real Differential Algebra, vol. 1888 of Lecture Notes in Mathematics (Springer, Berlin, 2006)

    Google Scholar 

  53. O. Zariski, P. Samuel, Commutative Algebra. Vol. II The University Series in Higher Mathematics (D. Van Nostrand, Princeton, Toronto-London-New York, 1960)

    Google Scholar 

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  1. Department of Mathematics, University of Illinois, Urbana, IL, USA

    Lou van den Dries

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van den Dries, L. (2014). Lectures on the Model Theory of Valued Fields. In: Model Theory in Algebra, Analysis and Arithmetic. Lecture Notes in Mathematics(), vol 2111. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54936-6_4

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