Hilbert’s irreducibility theorem
In this entry, $K$ is a field of characteristic zero and $V$ is an irreducible^{} algebraic variety over $K$.
Definition 1.
A variety^{} $V$ satisfies the Hilbert property over $K$ if $V\mathit{}\mathrm{(}K\mathrm{)}$ is not a thin algebraic set.
Definition 2.
A field $K$ is said to be Hilbertian if there exists an irreducible variety $V\mathrm{/}K$ of $\mathrm{dim}\mathit{}V\mathrm{\ge}\mathrm{1}$ which has the Hilbert property.
Theorem (Hilbert’s irreducibility theorem).
A number field^{} $K$ is Hilbertian. In particular, for every $n$, the affine space ${\mathrm{A}}^{n}\mathit{}\mathrm{(}K\mathrm{)}$ has the Hilbert property over $K$.
However, the field of real numbers $\mathbb{R}$ and the field of $p$-adic rationals ${\mathbb{Q}}_{p}$ are not Hilbertian.
References
- 1 J.-P. Serre, Topics in Galois Theory^{}, Research Notes in Mathematics, Jones and Barlett Publishers, London.
Title | Hilbert’s irreducibility theorem |
---|---|
Canonical name | HilbertsIrreducibilityTheorem |
Date of creation | 2013-03-22 15:14:16 |
Last modified on | 2013-03-22 15:14:16 |
Owner | alozano (2414) |
Last modified by | alozano (2414) |
Numerical id | 5 |
Author | alozano (2414) |
Entry type | Theorem |
Classification | msc 12E25 |
Synonym | Hilbertian |
Defines | Hilbert property |
Defines | Hilbertian field |