RÉSUMÉ

We show how a combination of group theory, finiteness theorems of Siegel and Faltings on points on curves, and the theory of function fields gives sharpenings of Hilbert's irreducibility theorem over number fields $k$ like the following: Let $f(t,X)$ be an irreducible polynomial in $X$ over $k(t)$. Then $f(t_0,X)$ is irreducible for all but finitely many $t_0$ in the ring of integers of $k$, provided that the Galois group of $f$ over $k(t)$ is doubly transitive on the roots of $f$ and the curve $f(T,X)=0$ is not rational.

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