Specializations of one-parameter families of polynomials

Abstract

Let K be a number field, and let λ(x, t) ∈ K[x, t] be irreducible over K(t). Using algebraic geometry and group theory, we study the set of α ∈ K for which the specialized polynomial λ(x, α) 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 L (t) n (x) 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 Φ n (x, t) are K-irreducible and have Galois group containing P SL 2 (Z/n).

On the Galois group of generalized Laguerre polynomials

“…[10], [11], and Hajir-Wong [12]. In particular, we have the following theorem of Filaseta and Lam [4] on the irreducibility of GLP.…”

“…In the latter work, Theorem 2.2 is used to prove that A n is contained in the Galois group of L (−n−r) n for all large enough (with respect to n) positive integers r. 2. Using a different set of techniques, the following companion to Theorem 1.1 is proved in [12]: If we fix n ≥ 5 and a number field K, then for all but finitely many α ∈ K, L (α) n (x) is irreducible and has Galois group A n or S n over K. For each n ≤ 4, infinitely many Q-reducible specializations exist, and for n = 4, there are infinitely many specializations which are irreducible but have D 4 -Galois group over Q, cf. [11,Section 6].…”

Algebraic properties of a family of Jacobi polynomials

“…We have shown in Section 3 that the Galois group of K /Q(y) is S n . We adopt the notation of [7]. Let E be an intermediate field of K /Q(y), let E = Gal(K /E), and let X E be the smooth curve with function field E. Following [7, thm.…”

“…In a recent work, Hajir-Wong [7] describe a method for studying the exceptional set for a one-parameter family F (t) n (x) ∈ Q[x, t] of polynomials, i.e. the set of α ∈ Q for which F (α) n (x) is reducible.…”

“…Let X denote the smooth curve corresponding to the Galois closure K of P n (x, y) over Q(y). Following [7], we will show:…”