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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 {\it M. Fried} and {\it 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 {\it M. Jarden} and {\it 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.

12E25Hilbertian fields; Hilbert’s irreducibility theorem
12F12Inverse Galois theory
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