Odoni’s conjecture on arboreal Galois representations is false

Dittmann, Philip; Kadets, Borys

doi:10.1090/proc/15920

Odoni’s conjecture on arboreal Galois representations is false
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by Philip Dittmann and Borys Kadets PDF

Proc. Amer. Math. Soc. 150 (2022), 3335-3343 Request permission

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 $\phi _f\colon \operatorname {Gal}_K \to \operatorname {Aut} \mathrm {T}$ 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.

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