On the Galois groups of the iterates of x²+1

Extract

§1. Introduction. In [1], Odoni discusses the iterates of the polynomial x² +1 and their Galois groups over the rationals (a problem initially proposed by J. McKay). Setting f¹,(x) = x²+1 and fⁿ(x) =f¹(f_n-1(X)) for n ≥ 2, write K_n for the splitting field of f_n(x) over and Ω_n = Gal (K_n/). Then Odoni proves that Ω_n is isomorphic to a subgroup of [C₂]ⁿ, the nth wreath power of the cyclic group C² of order 2, and gives a simple rational criterion for Ω_n [C₂]ⁿ to hold. In this note we describe a computer implementation of Odoni's criterion, and state the result that Ω_n [C₂]ⁿ for n ≤ 5 × 10⁷.