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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Towards van der Waerden’s conjecture
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by Sam Chow and Rainer Dietmann HTML | PDF
Trans. Amer. Math. Soc. 376 (2023), 2739-2785 Request permission

Abstract:

How often is a quintic polynomial solvable by radicals? We establish that the number of such polynomials, monic and irreducible with integer coefficients in $[-H,H]$, is $O(H^{3.91})$. More generally, we show that if $n \geqslant 3$ and $n \notin \{ 7, 8, 10 \}$ then there are $O(H^{n-1.017})$ monic, irreducible polynomials of degree $n$ with integer coefficients in $[-H,H]$ and Galois group not containing $A_n$. Save for the alternating group and degrees $7,8,10$, this establishes a 1936 conjecture of van der Waerden.
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Additional Information
  • Sam Chow
  • Affiliation: Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, United Kingdom
  • MR Author ID: 1077989
  • ORCID: 0000-0001-7651-4831
  • Email: Sam.Chow@warwick.ac.uk
  • Rainer Dietmann
  • Affiliation: Department of Mathematics, Royal Holloway, University of London, Egham TW20 0EX, United Kingdom
  • MR Author ID: 667217
  • Email: Rainer.Dietmann@rhul.ac.uk
  • Received by editor(s): July 12, 2021
  • Received by editor(s) in revised form: April 27, 2022, and August 2, 2022
  • Published electronically: January 24, 2023
  • Additional Notes: The first author was supported by EPSRC Fellowship Grant EP/S00226X/2, and by the Swedish Research Council under grant no. 2016-06596.
  • © Copyright 2023 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 376 (2023), 2739-2785
  • MSC (2020): Primary 11R32; Secondary 11C08, 11D45, 11G35
  • DOI: https://doi.org/10.1090/tran/8784
  • MathSciNet review: 4557880