# 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 $V(K)$ is not a thin algebraic set.

###### Definition 2.

A field $K$ is said to be Hilbertian if there exists an irreducible variety $V/K$ of $\dim V\geq 1$ which has the Hilbert property.

###### Theorem (Hilbert’s irreducibility theorem).

A number field $K$ is Hilbertian. In particular, for every $n$, the affine space $\mathbb{A}^{n}(K)$ 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 HilbertsIrreducibilityTheorem 2013-03-22 15:14:16 2013-03-22 15:14:16 alozano (2414) alozano (2414) 5 alozano (2414) Theorem msc 12E25 Hilbertian Hilbert property Hilbertian field