Bary-Soroker, Lior On the characterization of Hilbertian fields. (English) Zbl 1217.12003 Int. Math. Res. Not. 2008, Article ID rnn089, 10 p. (2008). 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. Reviewer: Bernat Plans (Barcelona) Cited in 4 Documents MSC: 12E25 Hilbertian fields; Hilbert’s irreducibility theorem 12F10 Separable extensions, Galois theory 12E30 Field arithmetic Keywords:Hilbertian field; irreducibility specialization property; regular extension Citations:Zbl 0933.12002 PDFBibTeX XMLCite \textit{L. Bary-Soroker}, Int. Math. Res. Not. 2008, Article ID rnn089, 10 p. (2008; Zbl 1217.12003) Full Text: DOI arXiv