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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 MM 1 and MM 2 , then M is hilbertian.


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