Infinite finitely generated fields are biinterpretable with ℕ

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 N \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.

The Logical Complexity of Finitely Generated Commutative Rings

“…Combining this theorem with Proposition 2.28 immediately yields: Although Theorem 3.1 can be deduced from the main result of [38] (and is unaffected by the error therein), we prefer to start from scratch and give a self-contained proof of this fact.…”

“…Influenced by [38] and realizing that not all finitely generated commutative rings are bi-interpretable with N, in 2006 the first-named author became interested in algebraically characterizing those which are. The corollary above was announced by the second-named author in [12], where a proof based on the main result of [38] was suggested. In his Ph.…”

“…In his Ph. D. thesis [24], the third-named author later gave a proof of this corollary circumventing the flaws of [38].…”

Interprétation de l'arithmétique dans certains groupes de permutations affines par morceaux d'un intervalle

“…Thomas Scanlon [31] a récemmentétabli la bi-interprétabilité entre l'arithmétique et tout corps commutatif de type fini et a utilisé cela pour démontrer que tout tel corps est catégoriquement finiment axiomatisé dans la classe de tous corps commutatifs de type fini, et ainsi que la conjecture de Pop est vrai : deux corps commutatifs de type fini sontélémentairementéquivalents si et seulement si ils sont isomorphes. * Nous n'allons pas montrer cela ici.…”

Bi-interprétabilité et structures QFA : étude de groupes résolubles et des anneaux commutatifs

“…La preuve utilise un résultat de Scanlon [7] qui dit qu'un corps commutatif de type fini est bi-interprétable avec l'anneau des entiers. Pour un anneau commutatif intègre de type fini, on montre cette bi-interprétabilité via le corps des fractions.…”