Polynomials with Surjective Arboreal Galois Representations Exist in Every Degree

TL;DR

This paper proves Odoni's conjecture that for every degree, there exists a polynomial over a number field with surjective arboreal Galois representations, meaning the Galois groups of iterates match the automorphism groups of regular n-branching trees.

Contribution

It confirms Odoni's conjecture for number fields, establishing the existence of polynomials with surjective arboreal Galois representations in every degree.

Findings

Existence of such polynomials for all degrees over number fields.

Galois groups of iterates are isomorphic to automorphism groups of regular trees.

Supports Odoni's conjecture in the context of number fields.

Abstract

Let~$E$ be a Hilbertian field of characteristic~$0$. R.W.K. Odoni conjectured that for every positive integer~$n$ there exists a polynomial~$f\in E[X]$ of degree~$n$ such that each iterate~$f^{\circ{k}}$ of~$f$ is irreducible and the Galois group of the splitting field of~$f^{\circ k}$ is isomorphic to the automorphism group of a regular,~$n$-branching tree of height~$k.$ We prove this conjecture when~$E$ is a number field.