[FOM] Need reference for results in Field Theory

[joeshipman@aol.com]

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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