# 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.

## Key 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.