The image of an arboreal Galois representation. (English) Zbl 1167.11011

Boston, Nigel; Jones, Rafe

Pure Appl. Math. Q. 5, No. 1, 213-225 (2009).

For a polynomial \(f\in \mathbb Q[X]\) denote by \(G(f)\) the Galois group of the field generated by all roots of all iterates of \(f\). If \(\deg f=d\), then the set of these roots may be identified with vertices of the \(d\)-ary rooted tree \(T\) and this leads to a continuous homomorphism \(\text{Gal}(\overline{\mathbb Q}/\mathbb Q)\to \operatorname{Aut}(T)\) whose image is \(G(f)\). The authors determine \(G(f)\) for the generic quadratic polynomial \(f(x)=(x-t)^2+t+m\in \mathbb Q(t)[x]\) (\(m\in \mathbb Z\), \(m\neq-1\)), conjecture that the group \(G(f)\) for a specialization of \(f\) is usually a subgroup of a finite index in \(G(f)\) and prove this in some special cases. They formulate also other interesting conjectures.

Reviewer: Władysław Narkiewicz (Wrocław)

MSC:

11C08	Polynomials in number theory
11R32	Galois theory

Keywords:

Galois group; Galois representations; quadratic polynomial; iterated polynomials; arboreal representations