Tyszka, Apoloniusz On \(\emptyset\)-definable elements in a field. (English) Zbl 1126.03040 Collect. Math. 58, No. 1, 73-84 (2007). An arithmetic characterization of elements in a field which are definable by an existential formula without parameters is given. These elements form a subfield \(\overline{K}\) of the given field \(K\). It is shown that \(\overline{K}\) is the prime subfield \(P\) of \(K\) whenever the algebraic closure of \(P\) is contained in \(K\). By contrast, it is shown that for many finitely generated fields \(K\) of characteristic \(0\), \(\overline{K}\) is transcendental over the field \(\mathbb{Q}\) of rationals. Finally it is shown that all transcendental real numbers which are recursively approximable by rationals are definable in the field \(\mathbb{R}(t)\), and the same holds with \(\mathbb{R}\) replaced by any Pythagorean subfield of \(\mathbb{R}\). Reviewer: Serban A. Basarab (Bucureşti) MSC: 03C60 Model-theoretic algebra 12L12 Model theory of fields Keywords:existentially definable element; transcendental element; finitely generated field extension; Mordell-Faltings theorem; recursively approximable real number PDFBibTeX XMLCite \textit{A. Tyszka}, Collect. Math. 58, No. 1, 73--84 (2007; Zbl 1126.03040) Full Text: arXiv EuDML