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 of an hilbertian field 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 of an hilbertian field is hilbertian.
As a consequence of this criterion, the main result of the paper states that, if is an hilbertian field, are two Galois extensions of , and is an intermediate field of such that and , then is hilbertian.