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 $\dim \mathit{}V\mathrm{\ge }\mathrm{1}$ 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 $ℝ$ and the field of $p$-adic rationals ${\mathbb{Q}}_p$ are not Hilbertian.

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