Uniform first-order definitions in finitely generated fields

摘要

We prove that there is a first-order sentence in the language of rings that is true for all finitely generated fields of characteristic 0 and false for all fields of characteristic greater than 0. We also prove that for each n epsilon N, there is a first-order formula psi(n) (x(1),..., x(n)) that when interpreted in a finitely generated field K is true for elements x(1),..., x(n) epsilon K if and only if the elements are algebraically dependent over the prime field in K.