[FOM] Need reference for results in Field Theory
Baldwin, John T.
jbaldwin at uic.edu
Fri Oct 20 16:13:52 EDT 2006
> -- Joe Shipman asked
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.
>
answer the algebraic closure of the empty set.
In any strongly minimal theory (simplest sort of aleph_1 categorical
theory) the algebraic closure of the empty set, if infinite, is an
elementary submodel of the universe.
This is a sufficient condition for Joe's other question.
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?
Of course it is not the answer as the examples he gives are not strongly
minimal. 0-minimal would also suffice I think (catching the reals)
But in general this seems an intriguing issue.
>
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