On Ø-definable elements in a field

Apoloniusz Tyszka

Collectanea Mathematica (2007)

  • Volume: 58, Issue: 1, page 73-84
  • ISSN: 0010-0757

Abstract

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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.

How to cite

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Tyszka, Apoloniusz. "On Ø-definable elements in a field." Collectanea Mathematica 58.1 (2007): 73-84. <http://eudml.org/doc/41797>.

@article{Tyszka2007,
abstract = {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.},
author = {Tyszka, Apoloniusz},
journal = {Collectanea Mathematica},
keywords = {existentially definable element; transcendental element; finitely generated field extension; Mordell-Faltings theorem; recursively approximable real number},
language = {eng},
number = {1},
pages = {73-84},
title = {On Ø-definable elements in a field},
url = {http://eudml.org/doc/41797},
volume = {58},
year = {2007},
}

TY - JOUR
AU - Tyszka, Apoloniusz
TI - On Ø-definable elements in a field
JO - Collectanea Mathematica
PY - 2007
VL - 58
IS - 1
SP - 73
EP - 84
AB - 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.
LA - eng
KW - existentially definable element; transcendental element; finitely generated field extension; Mordell-Faltings theorem; recursively approximable real number
UR - http://eudml.org/doc/41797
ER -

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