Hilbert's irreducibility theorem

Remarks

The set of good $a$ is large: one may take $a \in \Z^r$, and there are infinitely many such $a$. Classical proofs expand the roots of $f$ as series in $t$ near infinity and show that rational specialisations giving a factor are rare. Example: $x^2 - t$ is irreducible in $\Q[t, x]$; the value $t = 2$ gives the irreducible $x^2 - 2$, while $t = 4$ gives $(x - 2)(x + 2)$. So irreducibility can fail at some values, and the theorem only says that it survives at many. A stronger form preserves the Galois group: if an irreducible $f \in \Q(t)[x]$ has Galois group $G$ over $\Q(t)$, then $f(a, x)$ has Galois group $G$ over $\Q$ for infinitely many $a \in \Q$. The analogue fails over $\R$: $x^2 - t^2 - 1$ is irreducible in $\R[t, x]$, but for every $a \in \R$ the polynomial $x^2 - a^2 - 1$ has the real roots $\pm \sqrt{a^2 + 1}$.

Going deeper

For a student

Fields with this property are called Hilbertian fields; number fields are Hilbertian, while finite fields and algebraically closed fields are not. The theorem is the main tool to pass from Galois groups over $\Q(t)$ to Galois groups over $\Q$. For example, the generic polynomial with Galois group the full symmetric group $S_n$ over $\Q(t_1, \dots, t_n)$ specialises to polynomials over $\Q$ with Galois group $S_n$. This is the starting point of many attacks on the inverse Galois problem.