“Haran's diamond theorem”的意思、由来-开放百科全书

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Haran's diamond theorem

释义

Statement of the diamond theorem
Some corollaries
Weissauer's theorem Haran–Jarden condition
References

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

{{citation

| last = Haran | first = Dan
| doi = 10.1007/s002220050325
| issue = 1
| journal = Inventiones Mathematicae
| mr = 1702139 | zbl=0933.12003
| pages = 113–126
| title = Hilbertian fields under separable algebraic extensions
| volume = 137
| year = 1999}}.

{{citation

| last1 = Fried | first1 = Michael D.
| last2 = Jarden | first2 = Moshe
| edition = 3rd revised
| isbn = 978-3-540-77269-9
| location = Berlin
| mr = 2445111 | zbl=1145.12001
| publisher = Springer-Verlag
| series = Ergebnisse der Mathematik und ihrer Grenzgebiete. 3 Folge
| title = Field Arithmetic
| volume = 11
| year = 2008}}.{{DEFAULTSORT:Haran's Diamond Theorem}}

3 : Galois theory|Theorems in algebra|Number theory

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