Haran's diamond theorem

Let K be a Hilbertian field and L a separable extension of K. Assume there exist two Galois extensions N and M of K such that L is contained in the compositum NM, but is contained in neither N nor M. Then L is Hilbertian.

The name of the theorem comes from the pictured diagram of fields, and was coined by Jarden.

This theorem was firstly proved using non-standard methods by Weissauer.

Let K be a Hilbertian field and N, M two Galois extensions of K. Assume that neither contains the other.

This theorem has a very nice consequence: Since the field of rational numbers, Q is Hilbertian (Hilbert's irreducibility theorem), we get that the algebraic closure of Q is not the compositum of two proper Galois extensions.