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.

单词	HilbertsIrreducibilityTheorem
释义	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.
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