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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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Infinite finitely generated fields are biinterpretable with ${\mathbb N}$
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by Thomas Scanlon
J. Amer. Math. Soc. 21 (2008), 893-908
Published electronically: February 6, 2008

Erratum: J. Amer. Math. Soc. 24 (2011), 917-917.


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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Bibliographic Information
  • Thomas Scanlon
  • Affiliation: Department of Mathematics, University of California, Berkeley, Evans Hall, Berkeley, California 94720-3840
  • MR Author ID: 626736
  • ORCID: 0000-0003-2501-679X
  • Email:
  • 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
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 21 (2008), 893-908
  • MSC (2000): Primary 12L12; Secondary 03C60
  • DOI:
  • MathSciNet review: 2393432