Finitely ramified iterated extensions

Abstract

Let K be a number field, t a parameter, F = K(t), and '(x) ∈ K(x) a polyno- mial of degree d ≥ 2. The polynomialn(x, t) = ' ◦n (x)−t ∈ F(x), where ' ◦n = '◦'◦� � �◦ ' is the n-fold iterate of ', is absolutely irreducible over F; we compute a recursion for its discriminant. Let F' be the field obtained by adjoining to F all roots (in a fixed F) of �n(x, t) for all n ≥ 1; its Galois group Gal(F'/F) is the iterated monodromy group of '. The iterated extension F' is finitely ramified over F if and only if ' is post-critically finite. We show that, moreover, for post-critically finite ', every specialization of F'/F at t = t0 ∈ K is finitely ramified over K, pointing to the possibility of studying Galois groups of number fields with restricted ramification via tree representations associated to iterated monodromy groups of post-critically finite polynomials. We discuss the wildness of rami- fication in some of these representations, describe prime decomposition in terms of certain finite graphs, and also give some examples of monogene number fields that arise from the construction.