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Infinite finitely generated fields are biinterpretable with $ {\mathbb{N}}$

Author: Thomas Scanlon
Journal: J. Amer. Math. Soc. 21 (2008), 893-908
MSC (2000): Primary 12L12; Secondary 03C60
Published electronically: February 6, 2008
Erratum: J. Amer. Math. Soc. 24 (2011), 917
MathSciNet review: 2393432
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Abstract: Using the work of several other mathematicians, principally the results of Poonen refining the work of Pop that algebraic independence is definable within the class of finitely generated fields and of Rumely that the ring of rational integers is uniformly interpreted in global fields, and a theorem on the definability of valuations on function fields of curves, we show that each infinite finitely generated field considered in the ring language is parametrically biinterpretable with $ \mathbb{N}$. As a consequence, for any finitely generated field there is a first-order sentence in the language of rings which is true in that field but false in every other finitely generated field and, hence, Pop's conjecture that elementarily equivalent finitely generated fields are isomorphic is true.

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  • 1. M. ASCHENBRENNER, Ideal membership in polynomial rings over the integers, J. Amer. Math. Soc. 17 (2004), no. 2, 407-441. MR 2051617 (2005c:13032)
  • 2. M. AUSLANDER and A. BRUMER, Brauer groups of discrete valuation rings, Nederl. Akad. Wetensch. Proc. Ser. A 71 (Indag. Math. 30) (1968), 286-296. MR 0228471 (37:4051)
  • 3. J.-L. DURET, Sur la théorie élémentaire des corps de fonctions, J. Symbolic Logic 51 (1986), no. 4, 948-956. MR 865921 (88d:03066)
  • 4. J.-L. DURET, Équivalence élémentaire et isomorphisme des corps de courbe sur un corps algébriquement clos, J. Symbolic Logic 57 (1992), no. 3, 808-823. MR 1187449 (94g:03073)
  • 5. D. K. FADDEEV, Simple algebras over a field of algebraic functions of one variable. (Russian) Trudy Mat. Inst. Steklov., v. 38, pp. 321-344. Izdat. Akad. Nauk SSSR, Moscow, 1951. MR 0047632 (13:905c)
  • 6. M. FRIED and M. JARDEN, Field arithmetic, Second edition. Ergebniße der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 11. Springer-Verlag, Berlin, 2005. xxiv+780 pp. MR 2102046 (2005k:12003)
  • 7. P. GILLE and T. SZAMUELY, Central simple algebras and Galois cohomology, Cambridge Studies in Advanced Mathematics, 101. Cambridge University Press, Cambridge, 2006. xii+343 pp. MR 2266528 (2007k:16033)
  • 8. W. HODGES, Model theory, Encyclopedia of Mathematics and its Applications, 42, Cambridge University Press, Cambridge, 1993. MR 1221741 (94e:03002)
  • 9. R. KAYE, Models of Peano Arithmetic, Oxford Logic Guides 15, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1991. MR 1098499 (92k:03034)
  • 10. A. KHELIF, Bi-interprétabilité et structures QFA: étude de groupes résolubles et des anneaux commutatifs, C. R. Acad. Sci. Paris Sér. I Math., 345 (2007), no. 2, 59-61. MR 2343552
  • 11. T.-Y. LAM, A first course in noncommutative rings, Graduate Texts in Mathematics 131 Springer-Verlag, New York, 1991. MR 1125071 (92f:16001)
  • 12. Y. MATIYASEVICH, Hilbert's Tenth Problem, translated from the 1993 Russian original by the author with a foreword by Martin Davis, Foundations of Computing Series, MIT Press, Cambridge, MA, 1993. MR 1244324 (94m:03002b)
  • 13. A. NIES, Describing groups, Bulletin of Symbolic Logic 13 (2007), no. 3, 305-339. MR 2359909
  • 14. B. POONEN, Uniform first-order definitions in finitely generated fields, Duke Math. J. 138 (2007), no. 1, 1-22. MR 2309154
  • 15. F. POP, Elementary equivalence versus isomorphism, Invent. Math. 150 (2002), no. 2, 385-408. MR 1933588 (2003i:12016)
  • 16. J. ROBINSON, The undecidability of algebraic rings and fields, Proc. Amer. Math. Soc. 10 (1959), 950-957. MR 0112842 (22:3691)
  • 17. R. ROBINSON, Undecidable rings, Trans. Amer. Math. Soc. 70 (1951), 137-159. MR 0041081 (12:791b)
  • 18. R. RUMELY, Undecidability and definability for the theory of global fields, Trans. Amer. Math. Soc. 262 (1980), no. 1, 195-217. MR 583852 (81m:03053)
  • 19. J.-P. SERRE, Local fields, translated from the French by Marvin Jay Greenberg. Graduate Texts in Mathematics, 67, Springer-Verlag, New York-Berlin, 1979. viii+241 pp. MR 554237 (82e:12016)

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Additional Information

Thomas Scanlon
Affiliation: Department of Mathematics, University of California, Berkeley, Evans Hall, Berkeley, California 94720-3840

Received by editor(s): October 4, 2006
Published electronically: February 6, 2008
Additional Notes: The author was partially supported by NSF CAREER grant DMS-0450010
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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