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Let $K$ be a number field, and suppose $\lambda (x,t)\in K[x,t]$ is irreducible over $K\left(t\right)$. Using algebraic geometry and group theory, we describe conditions under which the $K$-exceptional set of $\lambda$, i.e. the set of $\alpha \in K$ for which the specialized polynomial $\lambda (x,\alpha )$ is $K$-reducible, is finite. We give three applications of the methods we develop. First, we show that for any fixed $n\ge 10$, all but finitely many $K$-specializations of the degree $n$ generalized Laguerre polynomial $L_n^{\left(t\right)}\left(x\right)$ are $K$-irreducible and have Galois group $S_n$. Second, we study specializations...