Hilbert’s irreducibility theorem

In this entry, K is a field of characteristic zero and V is an irreduciblePlanetmathPlanetmath algebraic variety over K.

Definition 1.

A varietyPlanetmathPlanetmath 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 dimV1 which has the Hilbert property.

Theorem (Hilbert’s irreducibility theorem).

A number fieldMathworldPlanetmath K is Hilbertian. In particular, for every n, the affine space An(K) has the Hilbert property over K.

However, the field of real numbers and the field of p-adic rationals p are not Hilbertian.


  • 1 J.-P. Serre, Topics in Galois TheoryMathworldPlanetmath, 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