Schinzel’s Problem: Imprimitive Covers and the Monodromy Method

(1.2) the ranges of f and g are identical on almost all OK/p. The most trivial cases are where g(x) = f(ax + b) for some a, b ∈ Q̄, the algebraic numbers. When K = Q, mostly that relation forces a, b∈K. For example this holds when f is indecomposable (not a composite of lower degree polynomials). With the indecomposability assumption, solutions to Davenport’s and Schinzel’s problems were essentially the same (solved in [Fr73, Thm. 1]; see [Fr11, Thm. 4.1]). Cases where a, b are not in K are important to Davenport’s problem, but not to Schinzel’s. Though Schinzel’s problem is our main focus, in §2.4 the indecomposable case reappears in Prob. 1.3, our case of Schinzel’s problem. The dihedral group, Dn, with n even, the example of §1.3, will aid a reader unaccustomed to branch cycles. Compare our goals with the §1.4 conjecture.