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Odoni's conjecture for number fields

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Robert L. Benedetto, Jamie Juul

Number Theory · Submitted 6 Mar 2018

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim 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.

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arXiv ID1803.01987 (version 1)
Pith IDOJQLHYZX, Pith's permanent record of this paper and its review
CitationJamie Juul, Robert L. Benedetto. "Odoni's conjecture for number fields." arXiv:1803.01987 (2018). https://pith.science/paper/OJQLHYZX
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,158 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.77 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.13 · compares with prior results 0.12 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

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.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Let $K$ be a number field, and let $d\geq 2$. background 1.00 0.05 0.01 unclear 0.52
s1 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 $S_d$. background 1.00 0.03 0.04 speculative 0.77
s2 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. result 0.78 0.97 0.97 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original 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.

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How the literature uses it

3 arXiv papers cite this in 3 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 2 sentences without a confident label

Cites it as background 1 paper
  • Benedetto and Juul [ref] found examples in even degrees $d$ over any number field $K$, and in odd degrees when $K$ does not contain $(√(d), √(d-2))$.

    Arboreal Cantor actions · Introduction · 0.81

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