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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.

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