Hilbert's irreducibility theorem

From Wikipedia, the free encyclopedia

In mathematics, Hilbert's irreducibility theorem, conceived by David Hilbert, states that an irreducible polynomial in two variables and having rational number coefficients will remain irreducible as a polynomial in one variable, when a rational number is substituted for the other variable, in infinitely many ways.

More formally, writing P(X, Y) for the polynomial, there will be infinitely many choices of a rational number q, such that

P(q, Y)

is also irreducible.

This result has applications, in particular, to the inverse Galois problem. It is also used as a step in the Andrew Wiles proof of Fermat's last theorem.

It has been reformulated and generalised extensively, by using the language of algebraic geometry. See thin set (Serre).