Specializations of one-parameter families of polynomials
by
Siman Wong
University of Massachusetts
Coauthors: Farshid Hajir (UMass)

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 PSL₂(\Z/n).

Date received: March 23, 2004