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 R having a recursive approximation by rationals, are definable in R(t), and the same holds when one replaces R by any Pythagorean subfield of R.
@article{urn:eudml:doc:41797,
title = {On \O -definable elements in a field},
journal = {Collectanea Mathematica},
volume = {58},
year = {2007},
pages = {73-84},
zbl = {1126.03040},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41797}
}
Tyszka, Apoloniusz. On Ø-definable elements in a field. Collectanea Mathematica, Tome 58 (2007) pp. 73-84. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41797/