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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), 917-917.
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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Additional 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

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.