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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.

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