Resumen de On Ø-definable elements in a field

Apoloniusz Tyszka

• We develop an arithmetic characterization of elements in a field which are firstorder definable by a parameterfree existential formula in the language of rings. As applications we show that in fields containing an algebraically closed field only the elements of the prime field are existentially ;definable. On the other hand, many finitely generated extensions of Q contain existentially ;definable elements which are transcendental over Q. Finally, we show that all transcendental elements inRhaving a recursive approximation by rationals, are definable in R(t), and the same holds when one replaces R by any Pythagorean subfield of R