Haran's Diamond Theorem

In mathematics, the Haran diamond theorem gives a general sufficient condition for a separable extension of a Hilbertian field to be Hilbertian.

Statement of the 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.

Some corollaries

Weissauer's theorem

This theorem was firstly proved using non-standard methods by Weissauer. It was reproved by Fried using standard methods. The latter proof led 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.

;Proof using the diamond theorem

If ''L'' is finite over ''K'', it is Hilbertian; hence we assume that ''L/K'' is infinite. Let ''x'' be a primitive element for ''L/N'', i.e., ''L'' = ''N''(''x'').

Let ''M'' be the Galois closure of ''K''(''x''). Then all the assumptions of the diamond theorem are satisfied, hence ''L'' is Hilbertian.

Haran–Jarden condition

Another, preceding to the diamond theorem, sufficient permanence condition was given by Haran–Jarden:
Theorem.
Let ''K'' be a Hilbertian field and ''N'', ''M'' two Galois extensions of ''K''. Assume that neither contains the other. Then their compositum ''NM'' is Hilbertian.

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.

References

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{{DEFAULTSORT:Haran's Diamond Theorem
Galois theory
Theorems in algebra
Number theory