Hilbert's irreducibility theorem

In number theory, Hilbert's irreducibility theorem, conceived by David Hilbert in 1892, states that every finite set of irreducible polynomials in a finite number of variables and having rational number coefficients admit a common specialization of a proper subset of the variables to rational numbers such that all the polynomials remain irreducible.

Hilbert's irreducibility theorem.

Let be irreducible polynomials in the ring Then there exists an r-tuple of rational numbers (a1, ..., ar) such that are irreducible in the ring Remarks.

Hilbert's irreducibility theorem has numerous applications in number theory and algebra.

For example: It has been reformulated and generalized extensively, by using the language of algebraic geometry.