Defining transcendentals in function fields.(English)Zbl 1015.03041

There are two notions of “relative algebraic closure” of a field $$K$$ in a field extension $$F/K$$: the field-theoretic relative algebraic closure $\overline{K} \cap F = \{x\in F \mid f(x)=0\text{ for some } f\in K[T] \}$ and the model-theoretic relative algebraic closure $\text{acl}_{F}(K):=\{ x\in F \mid x\in A \text{ for some finite }K\text{-definable }A\subseteq F \}$ (where “$$K$$-definable” means definable by a first-order formula in the language of fields with parameters from $$K$$).
Clearly, $$\overline{K}\cap F\subseteq \text{acl}_F(K)$$; the present paper provides examples of field extensions $$F/K$$ where this is a proper inclusion (i.e., where the two notions of relative algebraic closure of $$K$$ in $$F$$ do not coincide, or, equivalently, where $$F$$ contains $$K$$-definable transcendentals over $$K$$).

MSC:

 03C60 Model-theoretic algebra 12L12 Model theory of fields
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References:

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