Haran's diamond theorem
Encyclopedia
In mathematics
, the Haran diamond theorem gives a general sufficient condition for a separable extension of a Hilbertian field to be Hilbertian.
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.
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.
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, 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 extensionsN 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.
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
Hilbert's irreducibility theorem
In number theory, Hilbert's irreducibility theorem, conceived by David Hilbert, states that every finite number of irreducible polynomials in a finite number of variables and having rational number coefficients admit a common specialization of a proper subset of the variables to rational numbers...
), we get that the algebraic closure of Q is not the compositum of two proper Galois extensions.