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On the characterization of Hilbertian fields. (English) Zbl 1217.12003
A field $$K$$ is Hilbertian if the following property holds for every irreducible polynomial $$f(T,X)\in K[T,X]$$ that is separable in $$X$$:
(*) for every nonzero $$p(T)\in K[T]$$, there exists $$a\in K$$ such that $$p(a)\neq 0$$ and $$f(a,X)$$ is irreducible.
An a priori weaker property for $$K$$ just requires that (*) holds for every absolutely irreducible $$f(T,X)\in K[T,X]$$ that is separable in $$X$$. One then says that $$K$$ is $$R$$-Hilbertian.
The main result in the present paper is that, for an arbitrary field $$K$$, being Hilbertian is equivalent to being $$R$$-Hilbertian. This answers a question posed by P. Dèbes and D. Haran in [Acta Arith. 88, No. 3, 269–287 (1999; Zbl 0933.12002)], where they obtained the same conclusion assuming that $$K$$ was a PAC field.

##### MSC:
 12E25 Hilbertian fields; Hilbert’s irreducibility theorem 12F10 Separable extensions, Galois theory 12E30 Field arithmetic
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