ODONI'S CONJECTURE ON ARBOREAL GALOIS REPRESENTATIONS IS FALSE

Philip Dittmann; Borys Kadets

doi:10.1090/proc/15920

ODONI'S CONJECTURE ON ARBOREAL GALOIS REPRESENTATIONS IS FALSE

Philip Dittmann, Borys Kadets

Research output: Contribution to journal › Article › peer-review

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Abstract

Suppose f ∈ K[x] is a polynomial. The absolute Galois group of K acts on the preimage tree T of 0 under f. The resulting homomorphism φf : GalK → Aut T is called the arboreal Galois representation. Odoni conjectured that for all Hilbertian fields K there exists a polynomial f for which φf is surjective. We show that this conjecture is false.

Original language	English
Pages (from-to)	3335-3343
Number of pages	9
Journal	Proceedings of the American Mathematical Society
Volume	150
Issue number	8
DOIs	https://doi.org/10.1090/proc/15920
State	Published - 1 Aug 2022
Externally published	Yes

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© 2022 American Mathematical Society.

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abstract = "Suppose f ∈ K[x] is a polynomial. The absolute Galois group of K acts on the preimage tree T of 0 under f. The resulting homomorphism φf : GalK → Aut T is called the arboreal Galois representation. Odoni conjectured that for all Hilbertian fields K there exists a polynomial f for which φf is surjective. We show that this conjecture is false.",

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ODONI'S CONJECTURE ON ARBOREAL GALOIS REPRESENTATIONS IS FALSE

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