High Quality Content by WIKIPEDIA articles! In mathematics, the Haran diamond theorem gives a general sufficient condition for a separable extension of a Hilbertian field to be Hilbertian. 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 following diagram of fields, and was coined by Jarden. This theorem was firstly proved using non-standard methods by Weissauer. It was reproved by Fried using standard methods. The latter proof lead Haran to his diamond theorem. Weissauer's theorem. Let K be a Hilbertian field, N a Galois extension of K, and L a finite proper extension of N. Then L is Hilbertian.

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