[FOM] Need reference for results in Field Theory
joeshipman at aol.com
Fri Oct 20 12:40:30 EDT 2006
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
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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