INFINITE FINITELY GENERATED FIELDS ARE BIINTERPRETABLE WITHN

摘要

Pop conjectured that if K and L are two finitely generated fields having the same first-order theories in the language of rings, then they must be isomorphic [15].The best known result towards this conjecture is that if K = L, then there is an embedding K → L making L into a finite extension of K and vice versa [15].The key to the proof of this theorem is that the transcendence degree is encoded in the elementary theory of a finitely generated field. We make essential use of a refinement of this result due to Poonen that algebraic dependence is first-order definable within the class of finitely generated fields [14]