On Ø-definable elements in a field
We develop an arithmetic characterization of elements in a field which are first-order definable by a parameter-free existential formula in the language of rings. As applications we show that in fields containing any algebraically closed field only the elements of the prime field are existentially ∅-definable. On the other hand, many finitely generated extensins of Q contain existentially ∅-definable elements which are transcendental over Q. Finally, we show that all transcendental elements in...