## Abstract

Suppose f ∈ K [ x ] f \in K[x] is a polynomial. The absolute Galois group of K K acts on the preimage tree T \mathrm {T} of 0 0 under f f . The resulting homomorphism ϕ f : Gal K → Aut T \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 K there exists a polynomial f f for which ϕ f \phi _f is surjective. We show that this conjecture is false.

Original language | English |
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Journal | Proceedings of the American Mathematical Society |

DOIs | |

Publication status | Published - 1 Apr 2022 |