Odoni's conjecture for number fields

TL;DR

This paper proves Odoni's conjecture for certain degrees and number fields, showing the existence of polynomials with arboreal Galois groups as iterated wreath products of symmetric groups.

Contribution

It establishes Odoni's conjecture for even degrees over any number field and for odd degrees over number fields with odd degree extension.

Findings

Proves Odoni's conjecture for even degrees over all number fields.

Proves Odoni's conjecture for odd degrees over number fields with odd degree extension.

Confirms the structure of arboreal Galois groups in these cases.

Abstract

Let $K$ be a number field, and let $d\geq 2$. A conjecture of Odoni (stated more generally for characteristic zero Hilbertian fields $K$) posits that there is a monic polynomial $f\in K[x]$ of degree $d$, and a point $x_0\in K$, such that for every $n\geq 0$, the so-called arboreal Galois group $Gal(K(f^{-n}(x_0))/K)$ is an $n$-fold wreath product of the symmetric group $S_d$. In this paper, we prove Odoni's conjecture when $d$ is even and $K$ is an arbitrary number field, and also when both $d$ and $[K:Q]$ are odd.