2004/08/12 by Wayne Aitken, Aitken, Wayne, Farshid Hajir +3 · 5 citations
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Coding theory and cryptography #Polynomial and algebraic computation #math.DS #math.NT #msc:11R32 #msc:37F10
paper · pdf · doi:10.48550/arxiv.math/0408170
19 pages
arxiv created 2004/08/12 · arxiv updated 2009/12/01
Let K be a number field, t a parameter, F=K(t) and f in K[x] a polynomial of degree d. The polynomial Pn(x,t)= fn(x) - t in F[x] where fn is the n-fold iterate of f, is absolutely irreducible over F; we compute a recursion for its discriminant. Let L=L(f) be the field obtained by adjoining to F all roots, in a fixed algebraic closure, of Pn for all n; its Galois group Gal(L/F) is the iterated monodromy group of f. The iterated extension L/F is finitely ramified if and only if f is post-critically finite (pcf). We show that, moreover, for pcf polynomials f, every specialization of L/F at t=t0 in K is finitely ramified over K, pointing to the possibility of studying Galois groups with restricted ramification via tree representations associated to iterated monodromy groups of pcf polynomials. We discuss the wildness of ramification in some of these representations, describe prime decomposition in terms of certain finite graphs, and also give some examples of monogene number fields.