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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 ${M}_{1}{M}_{2}/K$ such that $M\text{⊈}{M}_{1}$ and $M\text{⊈}{M}_{2}$, then $M$ is hilbertian.

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