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arXiv · 2012.03076

Odoni's conjecture on arboreal Galois representations is false

Philip Dittmann·Borys Kadets
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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.

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Philip Dittmann, Borys Kadets. 2020-12-05. Odoni's conjecture on arboreal Galois representations is false. https://doi.org/10.1090/proc/15920

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