The image of an arboreal Galois representation
type: Item
instance of: scholarly article
title: The image of an arboreal Galois representation
zbMATH Open document ID: 1167.11011
author: Nigel Boston
author: Rafe Jones
published in: Pure and Applied Mathematics Quarterly
publication date: 2009-04-14T00:00:00Z
review text: 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 \(\text{Gal}(\overline{\mathbb Q}/\mathbb Q)\to \Aut(T)\) 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.
reviewed by: Władysław Narkiewicz
Mathematics Subject Classification ID: 11C08
Mathematics Subject Classification ID: 11R32
zbMATH DE Number: 5543148
zbMATH Keywords: Galois group
zbMATH Keywords: Galois representations
zbMATH Keywords: quadratic polynomial
zbMATH Keywords: iterated polynomials
zbMATH Keywords: arboreal representations
MaRDI profile type: Publication
DOI: 10.4310/PAMQ.2009.V5.N1.A6
description: scientific article; zbMATH DE number 5543148