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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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