[FOM] Need reference for results in Field Theory

[joeshipman@aol.com]

Thanks to Waaldijk and Zahidi for illuminating the complexities here. 
I'd like to reframe my query, since the general situation is more 
complicated than I expected and I don't really need the answer in the 
general situation.

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 
phenomenon? In other words, how can one tell, for a given field, 
whether it satisfies the same sentences as the subfield of its 
algebraic numbers? (In characteristic p, of course this means algebraic 
over the prime subfield, and can still be represented by satisfying a 
polynomial with integer coefficients.)

To generalize this beyond fields to other structures is a bit tricky, 
because the "algebraic subfield" of a field is not the same as the 
"definable substructure" of a structure -- in the language of fields we 
cannot define individual algebraic numbers, just finite sets of 
conjugate algebraic numbers. (For the real numbers, if we have < in the 
language we can define individual algebraic numbers, and for the 
p-adics we can do it if we have the valuation available.)

What is the official model-theoretic term for the elements of a 
structure which belong to finite definable sets? That would seem to be 
a better analogue of "algebraic" than "definable" is.

-- Joe Shipman
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