  Let $P(T,X)$ be an irreducible polynomial in two variables with rational
coefficients. It follows from Hilbert's Irreducibility Theorem that for most
rational numbers $t$ the specialized polynomial $P(t,X)$ is irreducible and has
the same Galois group as $P$. We discuss here a method for obtaining an
explicit description of the set of exceptional numbers $t$, i.e., those for
which $P(t,X)$ is either reducible or has a different Galois group than $P$. To
illustrate the method we determine the exceptional specializations of two
polynomials of degrees four and six.
