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Small generators of abelian number fields

  • Martin Widmer ORCID logo EMAIL logo
Published/Copyright: April 24, 2024
Forum Mathematicum
From the journal Forum Mathematicum Volume 37 Issue 2

Abstract

We show that for each abelian number field K of sufficiently large degree d there exists an element α ∈ K with K = ℚ ⁢ ( α ) and absolute Weil height H ⁢ ( α ) ≪ d | Δ K | 1 2 ⁢ d , where Δ K denotes the discriminant of K. This answers a question of Ruppert from 1998 in the case of abelian extensions of sufficiently large degree. We also show that the exponent 1 2 ⁢ d is best-possible when d is even.

Keywords: Weil height; small generators; Ruppert’s conjecture; abelian number fields; Linnik’s theorem
MSC 2020: 11R04; 11G50; 11R29; 11R06

Communicated by Philipp Habegger

Acknowledgements

It is my pleasure to thank Glyn Harman, for pointing out that a lower bound as in Lemma 3 can be derived by an easy adaptation of the usual proof of Linnik’s theorem, and to Marc Technau, for showing that Lemma 2 follows easily from [8, Proposition 18.5]. I am also grateful to Shabnam Akhtari and Jeffrey Vaaler for many fruitful and interesting discussions on Ruppert’s questions and beyond. I also would like to thank the reviewer for many helpful comments that improved the exposition of this article.

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Received: 2023-12-19
Revised: 2024-03-01
Published Online: 2024-04-24
Published in Print: 2025-02-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Triangles with one fixed side–length, a Furstenberg-type problem, and incidences in finite vector spaces
  3. Existence and multiplicity of solutions for fractional Schrödinger-p-Kirchhoff equations in ℝ N
  4. Estimates of Picard modular cusp forms
  5. Some q-supercongruences from a q-analogue of Watson's 3 F 2 summation
  6. Existence of strong solutions for one-dimensional reflected mixed stochastic delay differential equations
  7. Free groups generated by two unipotent maps
  8. Degenerate Schrödinger--Kirchhoff {(p,N)}-Laplacian problem with singular Trudinger--Moser nonlinearity in ℝ N
  9. Colored multizeta values in positive characteristic
  10. Simultaneous nonvanishing of central L-values with large level
  11. Laplace convolutions of weighted averages of arithmetical functions
  12. Cohomological properties of maximal pro-p Galois groups that are preserved under profinite completion
  13. On the geometric trace of a generalized Selberg trace formula
  14. Elementary properties of free lattices
  15. Weighted estimates for product singular integral operators in Journé’s class on RD-spaces
  16. Small generators of abelian number fields
  17. Pointwise convergence and nonlinear smoothing of the generalized Zakharov–Kuznetsov equation
  18. Weighted bilinear multiplier theorems in Dunkl setting via singular integrals
https://doi.org/10.1515/forum-2023-0467
Keywords for this article
Weil height; small generators; Ruppert’s conjecture; abelian number fields; Linnik’s theorem

Articles in the same Issue

  1. Frontmatter
  2. Triangles with one fixed side–length, a Furstenberg-type problem, and incidences in finite vector spaces
  3. Existence and multiplicity of solutions for fractional Schrödinger-p-Kirchhoff equations in ℝ N
  4. Estimates of Picard modular cusp forms
  5. Some q-supercongruences from a q-analogue of Watson's 3 F 2 summation
  6. Existence of strong solutions for one-dimensional reflected mixed stochastic delay differential equations
  7. Free groups generated by two unipotent maps
  8. Degenerate Schrödinger--Kirchhoff {(p,N)}-Laplacian problem with singular Trudinger--Moser nonlinearity in ℝ N
  9. Colored multizeta values in positive characteristic
  10. Simultaneous nonvanishing of central L-values with large level
  11. Laplace convolutions of weighted averages of arithmetical functions
  12. Cohomological properties of maximal pro-p Galois groups that are preserved under profinite completion
  13. On the geometric trace of a generalized Selberg trace formula
  14. Elementary properties of free lattices
  15. Weighted estimates for product singular integral operators in Journé’s class on RD-spaces
  16. Small generators of abelian number fields
  17. Pointwise convergence and nonlinear smoothing of the generalized Zakharov–Kuznetsov equation
  18. Weighted bilinear multiplier theorems in Dunkl setting via singular integrals
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