Algebraic invariants for first order equivalence between fields

Question

I know that every two models of the theory $ACF$ (namely two algebraic closed fields) with the same characteristic are elementary equivalent. But what about generic fields? Are there any algebraic invariants such that you can assert that $K \equiv L$ if and only if $K$ has the same invariant of $L$?

Answer 1

You're essentially asking for a classification of the completions of the theory of fields by nice algebraic invariants of their models. We certainly don't have anything close to an understanding of this problem in general. The best I can do is answer the question for a few well-behaved classes of fields.

• Two finite fields are elementarily equivalent (in fact isomorphic) if and only if they have the same cardinality.
• Two separably closed fields are elementarily equivalent if and only if they have the same characteristic and degree of imperfection.
• Any two real closed fields are elementarily equivalent.
• Two pseudo-finite fields are elementarily equivalent if and only if their subfields of elements algebraic over the prime field are isomorphic.
• More generally, I believe the complete theories of pseudo-algebraically closed (PAC) fields are also classified. You can probably find this in the book Field Arithmetic by Fried and Jarden.