Extension of Constants, Rigidity, and the Chowla-Zassenhaus Conjecture

Abstract

Let ƒ ∈ Q[y] be a polynomial of degree n over the rationals. Assume ƒ is indecomposable and consider the splitting field Ωƒ of ƒ(y) − x over Q(x). Denote the constants of Ωƒ by Q̂ƒ. Then, Q̂ƒ ⊂ Q(ζn) where ζn is a primitive nth root of 1. When n = p, a prime, and ƒ = xp (cyclic polynomial), Q̂ƒ = Q(ζp). When ƒ = Tp, the pth Chebychev polynomial, Q̂ƒ = Q(ζp + ζ−1p). Cohen raised the following question. If Q̂ƒ is nontrivial (ƒ has nontrivial extension of constants), is it then true that ƒ is linearly equivalent over Q to a cyclic or Chebychev polynomial? We show this is false for each non-square odd integer n. This uses elementary group theory and the Branch Cycle Argument. Such ƒ also give counterexamples to a conjecture of Chowla and Zassenhaus: For all sufficiently large p (dependent on the degree of ƒ), ƒ(x) − b is irreducible for some b ∈ Fp. That is, we show for these particular ƒ′s, for infinitely many p, there is no b ∈ Fp so that ƒ(x) − b is irreducible over Fp. Also, for these p, there is no b ∈ Fp so that ƒ(x) − b splits completely over Z/p. Further, using Mller′s classification of geometric monodromy groups of polynomials we show n must be odd for such counterexamples. These are (An, Sn) realizations by polynomials over Q. More delicate examples require rigidity applied to non-Galois covers. These contrast the arithmetic of covers with and without using braid operations on branch cycle descriptions. Braid operations describe four families of covers that include the renowned Davenport polynomials of degree 7.Q̂ƒ