Odoni's conjecture on arboreal Galois representations is false

TL;DR

The paper disproves Odoni's conjecture by showing that for Hilbertian fields, there is no polynomial with a surjective arboreal Galois representation, challenging previous assumptions in number theory.

Contribution

It provides a counterexample to Odoni's conjecture, demonstrating that the expected surjectivity of arboreal Galois representations does not always hold.

Findings

Odoni's conjecture is false for Hilbertian fields.

No polynomial has a surjective arboreal Galois representation in this setting.

The result impacts understanding of Galois actions on preimage trees.

Abstract

Suppose $f \in K[x]$ is a polynomial. The absolute Galois group of $K$ acts on the preimage tree $\mathrm{T}$ of $0$ under $f$. The resulting homomorphism $\phi_f: \mathrm{Gal}_K \to \mathrm{Aut} \mathrm{T}$ is called the arboreal Galois representation. Odoni conjectured that for all Hilbertian fields $K$ there exists a polynomial $f$ for which $\phi_f$ is surjective. We show that this conjecture is false.