[FOM] Need reference for results in Field Theory

[joeshipman@aol.com]

I need to reformulate my second query, it is very easy to disprove.

What I am really asking is whether there is an easily describable set 
of representatives of the elementary equivalence classes of fields (of 
characteristic 0).

One candidate for this is the set of all pure transcendental extensions 
(of transcendence degree 0, 1, 2, ... , aleph_zero) of isomorphism 
classes of subfields of the algebraic numbers.

So a reformulated query is:

2a) Are any of those fields elementarily equivalent to each other?
2b) Is any field of characteristic 0 inequivalent to all of those 
fields?

-- JS

-----Original Message-----
From: joeshipman at aol.com
To: fom at cs.nyu.edu
Sent: Sat, 14 Oct 2006 10:24 AM
Subject: [FOM] Need reference for results in Field Theory

  Who first proved the following theorem? (I assume the answer isn't
myself!):

If K and L are nonisomorphic subfields of the algebraic numbers, then
there is a polynomial with integer coefficients which has a root in one
of them and no root in the other.

Also, who proved this one (or is it false)?

Any field of characteristic 0 is elementarily equivalent to a subfield
of the algebraic numbers.

-- JS

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