[FOM] Shipman's field question

Thomas Forster T.Forster at dpmms.cam.ac.uk
Wed Oct 25 16:44:18 EDT 2006

I recently spent an evening with a friend who had an Executive Toy in the
form of ball-and-stick gadgets for making toy models of chemicals.  It
occurred to me that the question of whether or not these structures are
rigid ( = not deformable) is not obviously trivial.  Is this something on
which the (decidable) first-order theory of the reals as an ordered field
has anything amusing to say...?  


On Wed, 25 Oct 2006, Dave Marker wrote:

> 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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