In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Haran diamond theorem gives a general sufficient condition for a separable extension of a
Hilbertian field
In mathematics, a thin set in the sense of Serre, named after Jean-Pierre Serre, is a certain kind of subset constructed in algebraic geometry over a given field ''K'', by allowed operations that are in a definite sense 'unlikely'. The two fundame ...
to be Hilbertian.
Statement of the diamond theorem
Let ''K'' be a
Hilbertian field
In mathematics, a thin set in the sense of Serre, named after Jean-Pierre Serre, is a certain kind of subset constructed in algebraic geometry over a given field ''K'', by allowed operations that are in a definite sense 'unlikely'. The two fundame ...
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
In number theory, Hilbert's irreducibility theorem, conceived by David Hilbert in 1892, states that every finite set of irreducible polynomials in a finite number of variables and having rational number coefficients admit a common specialization o ...
), 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