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On \(\emptyset\)-definable elements in a field. (English) Zbl 1126.03040
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 \(\mathbb{R}\).
MSC:
03C60 Model-theoretic algebra
12L12 Model theory of fields
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