Hilbert's irreducibility theorem
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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. See thin set (Serre).