Infinite finitely generated fields are biinterpretable with ℕ

Scanlon, Thomas¹

¹Department of Mathematics, University of California, Berkeley, Evans Hall, Berkeley, California 94720-3840

Journal of the American Mathematical Society, Tome 21 (2008) no. 3, pp. 893-908

Résumé

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.

Détail

DOI : 10.1090/S0894-0347-08-00598-5

Bibliographie

[1] Aschenbrenner, Matthias Ideal membership in polynomial rings over the integers J. Amer. Math. Soc. 2004 407 441

[2] Auslander, Maurice, Brumer, Armand Brauer groups of discrete valuation rings Indag. Math. 1968 286 296

[3] Duret, Jean-Louis Sur la théorie élémentaire des corps de fonctions J. Symbolic Logic 1986 948 956

[4] Duret, Jean-Louis Équivalence élémentaire et isomorphisme des corps de courbe sur un corps algébriquement clos J. Symbolic Logic 1992 808 823

[5] Faddeev, D. K. Simple algebras over a field of algebraic functions of one variable Trudy Mat. Inst. Steklov. 1951 321 344

[6] Fried, Michael D., Jarden, Moshe Field arithmetic 2005

[7] Gille, Philippe, Szamuely, Tamás Central simple algebras and Galois cohomology 2006

[8] Hodges, Wilfrid Model theory 1993

[9] Kaye, Richard Models of Peano arithmetic 1991

[10] Khelif, Anatole Bi-interprétabilité et structures QFA: étude de groupes résolubles et des anneaux commutatifs C. R. Math. Acad. Sci. Paris 2007 59 61

[11] Lam, T. Y. A first course in noncommutative rings 1991

[12] Matiyasevich, Yu. V. Desyataya problema Gil′berta 1993 224

[13] Nies, André Describing groups Bull. Symbolic Logic 2007 305 339

[14] Poonen, Bjorn Uniform first-order definitions in finitely generated fields Duke Math. J. 2007 1 22

[15] Pop, Florian Elementary equivalence versus isomorphism Invent. Math. 2002 385 408

[16] Robinson, Julia The undecidability of algebraic rings and fields Proc. Amer. Math. Soc. 1959 950 957

[17] Robinson, Raphael M. Undecidable rings Trans. Amer. Math. Soc. 1951 137 159

[18] Rumely, R. S. Undecidability and definability for the theory of global fields Trans. Amer. Math. Soc. 1980 195 217

[19] Serre, Jean-Pierre Local fields 1979