Hilbert's irreducibility theorem

In mathematics, Hilbert's irreducibility theorem is a result of David Hilbert, stating 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.