[FOM] Shipman's field question

[JoeShipman@aol.com]

-----Original Message-----
>From: marker at math.uic.edu

>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.
 
That's great, just the result I need.

Can there be two nonisomorphic  countable fields which have the same 
finitely generated subfields? (This  question is slightly imprecise, 
because an easy way out would be if they  have the same isomorphism 
classes of finitely generated subfields but  different numbers of some 
given isomorphism class, if there is such an easy  way out modify my 
question in the obvious way).

Is the answer  different for char 0 and positive characteristic? (In positive 
characteristic I  can see non-separability creating difficulties....)

--  JS


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