[FOM] Shipman's field question

Dave Marker marker at math.uic.edu
Wed Oct 25 11:18:29 EDT 2006

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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