# On Ø-definable elements in a field

Collectanea Mathematica (2007)

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

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topTyszka, 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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