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Hilbertian fields under separable algebraic extensions. (English) Zbl 0933.12003
Through an investigation of M. Fried’s proof of Weissauer’s Theorem [see M. Fried and M. Jarden, Field Arithmetic, Springer (1986; Zbl 0625.12001)], the author develops a group theoretical argument that enables him to exhibit a quite general sufficient condition for an algebraic separable extension \(M\) of an hilbertian field \(K\) to be hilbertian.
This new criterion can be used to prove all the cases mentioned in M. Jarden and A. Lubotzky [J. Lond. Math. Soc. (2) 46, 205-227 (1992; Zbl 0724.12005)] where it is known that an extension \(M\) of an hilbertian field \(K\) is hilbertian.
As a consequence of this criterion, the main result of the paper states that, if \(K\) is an hilbertian field, \(M_1,M_2\) are two Galois extensions of \(K\), and \(M\) is an intermediate field of \(M_1M_2/K\) such that \(M\not\subseteq M_1\) and \(M\not\subseteq M_2\), then \(M\) is hilbertian.

12E25 Hilbertian fields; Hilbert’s irreducibility theorem
12F12 Inverse Galois theory
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