Joe Shipman asks:
> What is interesting about the real and p-adic fields is that they are
> elementarily equivalent to their algebraic subfields (that is, to the
> subfields consisting of those elements which satisfy a polynomial
> equation with integer coefficients).
> What model-theoretic property of these fields is responsible for this
>
One answer is that the real field is model complete
in the language of fields (while having quantifier elimination in
slightly richer languages). So by the Tarski-Vaught test
the set of algebraic elements will be an elementary submodel.
To say a bit more about your original question.
Tom Scanlon (building on work of Pop, Poonen and others) has
recently proved that if K is a finitely generated field, there is a
sentence describing K up to isomorphism among the finitely generated
fields. In particular, this proves Pop's conjecture that elementarily
equivalent finitely generated fields are isomorphic.
Dave Marker
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