Specializations of one-parameter families of polynomials

Let K be a number field, and let lambda(x,t)\in K[x, t] be irreducible over K(t). Using algebraic geometry and group theory, we study the set of alpha\in K for which the specialized polynomial lambda(x,alpha) is K-reducible. We apply this to show that for any fixed n>=10 and for any number field K, all but finitely many K-specializations of the degree n generalized Laguerre polynomial are K-irreducible and have Galois group S_n. In conjunction with the theory of complex multiplication, we also show that for any K and for any n>=53, all but finitely many of the K-specializations of the modular equation Phi_n(x, t) are K-irreducible and have Galois group containing PSL_2(Z/n).