---
_id: '65358'
abstract:
- lang: eng
  text: We determine the asymptotic growth of extensions of local function fields
    of characteristic p counted by discriminant, where the Galois group is a subgroup
    of the affine group AGL_1(p). More general, we solve the corresponding counting
    problems for all groups which arise in a tower of a cyclic extension of order
    p over a cyclic extension of degree d coprime to p. This in particular give answers
    for certain non-abelian groups including S_3, dihedral groups of order 2p, and
    many Frobenius groups.
author:
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
- first_name: Raphael
  full_name: Müller, Raphael
  id: '55246'
  last_name: Müller
citation:
  ama: Klüners J, Müller R. Counting Frobenius extensions over local function fields.
    <i>arXiv:260402152</i>. Published online 2026.
  apa: Klüners, J., &#38; Müller, R. (2026). Counting Frobenius extensions over local
    function fields. In <i>arXiv:2604.02152</i>.
  bibtex: '@article{Klüners_Müller_2026, title={Counting Frobenius extensions over
    local function fields}, journal={arXiv:2604.02152}, author={Klüners, Jürgen and
    Müller, Raphael}, year={2026} }'
  chicago: Klüners, Jürgen, and Raphael Müller. “Counting Frobenius Extensions over
    Local Function Fields.” <i>ArXiv:2604.02152</i>, 2026.
  ieee: J. Klüners and R. Müller, “Counting Frobenius extensions over local function
    fields,” <i>arXiv:2604.02152</i>. 2026.
  mla: Klüners, Jürgen, and Raphael Müller. “Counting Frobenius Extensions over Local
    Function Fields.” <i>ArXiv:2604.02152</i>, 2026.
  short: J. Klüners, R. Müller, ArXiv:2604.02152 (2026).
date_created: 2026-04-07T08:13:59Z
date_updated: 2026-04-07T08:14:45Z
external_id:
  arxiv:
  - '2604.02152'
language:
- iso: eng
publication: arXiv:2604.02152
status: public
title: Counting Frobenius extensions over local function fields
type: preprint
user_id: '82981'
year: '2026'
...
---
_id: '60874'
abstract:
- lang: eng
  text: Given a number field, it is an important question in algorithmic number theory
    to determine all its subfields. If the search is restricted to abelian subfields,
    one can try to determine them by using class field theory. For this, it is necessary
    to know the ramified primes. We show that the ramified primes of the subfield
    can be computed efficiently. Using this information we give algorithms to determine
    all the quadratic and the cyclic cubic subfields of the initial field. The approach
    generalises to cyclic subfields of prime degree. In the case of quadratic subfields,
    our approach is much faster than other methods.
article_number: '100039'
author:
- first_name: Andreas-Stephan
  full_name: Elsenhans, Andreas-Stephan
  last_name: Elsenhans
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
citation:
  ama: Elsenhans A-S, Klüners J. Computing quadratic subfields of number fields. <i>Journal
    of Computational Algebra</i>. 2025;15. doi:<a href="https://doi.org/10.1016/j.jaca.2025.100039">10.1016/j.jaca.2025.100039</a>
  apa: Elsenhans, A.-S., &#38; Klüners, J. (2025). Computing quadratic subfields of
    number fields. <i>Journal of Computational Algebra</i>, <i>15</i>, Article 100039.
    <a href="https://doi.org/10.1016/j.jaca.2025.100039">https://doi.org/10.1016/j.jaca.2025.100039</a>
  bibtex: '@article{Elsenhans_Klüners_2025, title={Computing quadratic subfields of
    number fields}, volume={15}, DOI={<a href="https://doi.org/10.1016/j.jaca.2025.100039">10.1016/j.jaca.2025.100039</a>},
    number={100039}, journal={Journal of Computational Algebra}, publisher={Elsevier
    BV}, author={Elsenhans, Andreas-Stephan and Klüners, Jürgen}, year={2025} }'
  chicago: Elsenhans, Andreas-Stephan, and Jürgen Klüners. “Computing Quadratic Subfields
    of Number Fields.” <i>Journal of Computational Algebra</i> 15 (2025). <a href="https://doi.org/10.1016/j.jaca.2025.100039">https://doi.org/10.1016/j.jaca.2025.100039</a>.
  ieee: 'A.-S. Elsenhans and J. Klüners, “Computing quadratic subfields of number
    fields,” <i>Journal of Computational Algebra</i>, vol. 15, Art. no. 100039, 2025,
    doi: <a href="https://doi.org/10.1016/j.jaca.2025.100039">10.1016/j.jaca.2025.100039</a>.'
  mla: Elsenhans, Andreas-Stephan, and Jürgen Klüners. “Computing Quadratic Subfields
    of Number Fields.” <i>Journal of Computational Algebra</i>, vol. 15, 100039, Elsevier
    BV, 2025, doi:<a href="https://doi.org/10.1016/j.jaca.2025.100039">10.1016/j.jaca.2025.100039</a>.
  short: A.-S. Elsenhans, J. Klüners, Journal of Computational Algebra 15 (2025).
date_created: 2025-08-05T07:01:39Z
date_updated: 2025-08-05T07:10:25Z
doi: 10.1016/j.jaca.2025.100039
external_id:
  arxiv:
  - '1907.13383'
intvolume: '        15'
language:
- iso: eng
publication: Journal of Computational Algebra
publication_identifier:
  issn:
  - 2772-8277
publication_status: published
publisher: Elsevier BV
status: public
title: Computing quadratic subfields of number fields
type: journal_article
user_id: '82981'
volume: 15
year: '2025'
...
---
_id: '55192'
abstract:
- lang: eng
  text: "We describe the group of $\\mathbb Z$-linear automorphisms of the ring of\r\nintegers
    of a number field $K$ that preserve the set $V_{K,k}$ of $k$th\r\npower-free integers:
    every such map is the composition of a field automorphism\r\nand the multiplication
    by a unit.\r\n  We show that those maps together with translations generate the
    extended\r\nsymmetry group of the shift space $\\mathbb D_{K,k}$ associated to
    $V_{K,k}$.\r\nMoreover, we show that no two such dynamical systems $\\mathbb D_{K,k}$
    and\r\n$\\mathbb D_{L,l}$ are topologically conjugate and no one is a factor system
    of\r\nanother.\r\n  We generalize the concept of $k$th power-free integers to
    sieves and study\r\nthe resulting admissible shift spaces."
author:
- first_name: Fabian
  full_name: Gundlach, Fabian
  id: '100450'
  last_name: Gundlach
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
citation:
  ama: Gundlach F, Klüners J. Symmetries of power-free integers in number fields and
    their shift  spaces. <i>arXiv:240708438</i>. Published online 2024.
  apa: Gundlach, F., &#38; Klüners, J. (2024). Symmetries of power-free integers in
    number fields and their shift  spaces. In <i>arXiv:2407.08438</i>.
  bibtex: '@article{Gundlach_Klüners_2024, title={Symmetries of power-free integers
    in number fields and their shift  spaces}, journal={arXiv:2407.08438}, author={Gundlach,
    Fabian and Klüners, Jürgen}, year={2024} }'
  chicago: Gundlach, Fabian, and Jürgen Klüners. “Symmetries of Power-Free Integers
    in Number Fields and Their Shift  Spaces.” <i>ArXiv:2407.08438</i>, 2024.
  ieee: F. Gundlach and J. Klüners, “Symmetries of power-free integers in number fields
    and their shift  spaces,” <i>arXiv:2407.08438</i>. 2024.
  mla: Gundlach, Fabian, and Jürgen Klüners. “Symmetries of Power-Free Integers in
    Number Fields and Their Shift  Spaces.” <i>ArXiv:2407.08438</i>, 2024.
  short: F. Gundlach, J. Klüners, ArXiv:2407.08438 (2024).
date_created: 2024-07-12T08:16:37Z
date_updated: 2024-07-12T08:19:11Z
external_id:
  arxiv:
  - '2407.08438'
language:
- iso: eng
publication: arXiv:2407.08438
status: public
title: Symmetries of power-free integers in number fields and their shift  spaces
type: preprint
user_id: '82981'
year: '2024'
...
---
_id: '55554'
abstract:
- lang: eng
  text: "We discuss various connections between ideal classes, divisors, Picard and\r\nChow
    groups of one-dimensional noetherian domains. As a result of these, we\r\ngive
    a method to compute Chow groups of orders in global fields and show that\r\nthere
    are infinitely many number fields which contain orders with trivial Chow\r\ngroups."
article_number: '89'
author:
- first_name: Markus
  full_name: Kirschmer, Markus
  id: '82258'
  last_name: Kirschmer
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
citation:
  ama: Kirschmer M, Klüners J. Chow groups of one-dimensional noetherian domains.
    <i>Research in Number Theory</i>. 2024;10(4). doi:<a href="https://doi.org/10.1007/s40993-024-00579-6">10.1007/s40993-024-00579-6</a>
  apa: Kirschmer, M., &#38; Klüners, J. (2024). Chow groups of one-dimensional noetherian
    domains. <i>Research in Number Theory</i>, <i>10</i>(4), Article 89. <a href="https://doi.org/10.1007/s40993-024-00579-6">https://doi.org/10.1007/s40993-024-00579-6</a>
  bibtex: '@article{Kirschmer_Klüners_2024, title={Chow groups of one-dimensional
    noetherian domains}, volume={10}, DOI={<a href="https://doi.org/10.1007/s40993-024-00579-6">10.1007/s40993-024-00579-6</a>},
    number={489}, journal={Research in Number Theory}, publisher={Springer Science
    and Business Media LLC}, author={Kirschmer, Markus and Klüners, Jürgen}, year={2024}
    }'
  chicago: Kirschmer, Markus, and Jürgen Klüners. “Chow Groups of One-Dimensional
    Noetherian Domains.” <i>Research in Number Theory</i> 10, no. 4 (2024). <a href="https://doi.org/10.1007/s40993-024-00579-6">https://doi.org/10.1007/s40993-024-00579-6</a>.
  ieee: 'M. Kirschmer and J. Klüners, “Chow groups of one-dimensional noetherian domains,”
    <i>Research in Number Theory</i>, vol. 10, no. 4, Art. no. 89, 2024, doi: <a href="https://doi.org/10.1007/s40993-024-00579-6">10.1007/s40993-024-00579-6</a>.'
  mla: Kirschmer, Markus, and Jürgen Klüners. “Chow Groups of One-Dimensional Noetherian
    Domains.” <i>Research in Number Theory</i>, vol. 10, no. 4, 89, Springer Science
    and Business Media LLC, 2024, doi:<a href="https://doi.org/10.1007/s40993-024-00579-6">10.1007/s40993-024-00579-6</a>.
  short: M. Kirschmer, J. Klüners, Research in Number Theory 10 (2024).
date_created: 2024-08-06T07:03:20Z
date_updated: 2024-11-05T09:46:04Z
doi: 10.1007/s40993-024-00579-6
external_id:
  arxiv:
  - '2208.14688'
intvolume: '        10'
issue: '4'
language:
- iso: eng
publication: Research in Number Theory
publication_identifier:
  issn:
  - 2522-0160
  - 2363-9555
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Chow groups of one-dimensional noetherian domains
type: journal_article
user_id: '82981'
volume: 10
year: '2024'
...
---
_id: '57218'
abstract:
- lang: eng
  text: "We arrange the orders in an algebraic number field in a tree. This tree can\r\nbe
    used to enumerate all orders of bounded index in the maximal order as well\r\nas
    the orders over some given order."
author:
- first_name: Markus
  full_name: Kirschmer, Markus
  last_name: Kirschmer
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
citation:
  ama: Kirschmer M, Klüners J. Enumerating orders in number fields. <i>arXiv:241108568</i>.
    Published online 2024.
  apa: Kirschmer, M., &#38; Klüners, J. (2024). Enumerating orders in number fields.
    In <i>arXiv:2411.08568</i>.
  bibtex: '@article{Kirschmer_Klüners_2024, title={Enumerating orders in number fields},
    journal={arXiv:2411.08568}, author={Kirschmer, Markus and Klüners, Jürgen}, year={2024}
    }'
  chicago: Kirschmer, Markus, and Jürgen Klüners. “Enumerating Orders in Number Fields.”
    <i>ArXiv:2411.08568</i>, 2024.
  ieee: M. Kirschmer and J. Klüners, “Enumerating orders in number fields,” <i>arXiv:2411.08568</i>.
    2024.
  mla: Kirschmer, Markus, and Jürgen Klüners. “Enumerating Orders in Number Fields.”
    <i>ArXiv:2411.08568</i>, 2024.
  short: M. Kirschmer, J. Klüners, ArXiv:2411.08568 (2024).
date_created: 2024-11-19T07:49:07Z
date_updated: 2024-11-19T07:49:48Z
external_id:
  arxiv:
  - '2411.08568'
language:
- iso: eng
publication: arXiv:2411.08568
status: public
title: Enumerating orders in number fields
type: preprint
user_id: '82981'
year: '2024'
...
---
_id: '49372'
author:
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
- first_name: Jiuya
  full_name: Wang, Jiuya
  last_name: Wang
citation:
  ama: Klüners J, Wang J. Idélic Approach in Enumerating Heisenberg Extensions. <i>La
    Matematica</i>. Published online 2023. doi:<a href="https://doi.org/10.1007/s44007-023-00067-w">10.1007/s44007-023-00067-w</a>
  apa: Klüners, J., &#38; Wang, J. (2023). Idélic Approach in Enumerating Heisenberg
    Extensions. <i>La Matematica</i>. <a href="https://doi.org/10.1007/s44007-023-00067-w">https://doi.org/10.1007/s44007-023-00067-w</a>
  bibtex: '@article{Klüners_Wang_2023, title={Idélic Approach in Enumerating Heisenberg
    Extensions}, DOI={<a href="https://doi.org/10.1007/s44007-023-00067-w">10.1007/s44007-023-00067-w</a>},
    journal={La Matematica}, publisher={Springer Science and Business Media LLC},
    author={Klüners, Jürgen and Wang, Jiuya}, year={2023} }'
  chicago: Klüners, Jürgen, and Jiuya Wang. “Idélic Approach in Enumerating Heisenberg
    Extensions.” <i>La Matematica</i>, 2023. <a href="https://doi.org/10.1007/s44007-023-00067-w">https://doi.org/10.1007/s44007-023-00067-w</a>.
  ieee: 'J. Klüners and J. Wang, “Idélic Approach in Enumerating Heisenberg Extensions,”
    <i>La Matematica</i>, 2023, doi: <a href="https://doi.org/10.1007/s44007-023-00067-w">10.1007/s44007-023-00067-w</a>.'
  mla: Klüners, Jürgen, and Jiuya Wang. “Idélic Approach in Enumerating Heisenberg
    Extensions.” <i>La Matematica</i>, Springer Science and Business Media LLC, 2023,
    doi:<a href="https://doi.org/10.1007/s44007-023-00067-w">10.1007/s44007-023-00067-w</a>.
  short: J. Klüners, J. Wang, La Matematica (2023).
date_created: 2023-12-01T09:23:59Z
date_updated: 2023-12-06T09:50:43Z
department:
- _id: '102'
doi: 10.1007/s44007-023-00067-w
language:
- iso: eng
publication: La Matematica
publication_identifier:
  issn:
  - 2730-9657
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Idélic Approach in Enumerating Heisenberg Extensions
type: journal_article
user_id: '21202'
year: '2023'
...
---
_id: '34839'
abstract:
- lang: eng
  text: We describe the relations among the ℓ-torsion conjecture, a conjecture of
    Malle giving an upper bound for the number of extensions, and the discriminant
    multiplicity conjecture. We prove that the latter two conjectures are equivalent
    in some sense. Altogether, the three conjectures are equivalent for the class
    of solvable groups. We then prove the ℓ-torsion conjecture for ℓ-groups and the
    other two conjectures for nilpotent groups.
author:
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
- first_name: Jiuya
  full_name: Wang, Jiuya
  last_name: Wang
citation:
  ama: Klüners J, Wang J. ℓ-torsion bounds for the class group of number fields with
    an ℓ-group as Galois group. <i>Proceedings of the American Mathematical Society</i>.
    2022;150(7):2793-2805. doi:<a href="https://doi.org/10.1090/proc/15882">10.1090/proc/15882</a>
  apa: Klüners, J., &#38; Wang, J. (2022). ℓ-torsion bounds for the class group of
    number fields with an ℓ-group as Galois group. <i>Proceedings of the American
    Mathematical Society</i>, <i>150</i>(7), 2793–2805. <a href="https://doi.org/10.1090/proc/15882">https://doi.org/10.1090/proc/15882</a>
  bibtex: '@article{Klüners_Wang_2022, title={ℓ-torsion bounds for the class group
    of number fields with an ℓ-group as Galois group}, volume={150}, DOI={<a href="https://doi.org/10.1090/proc/15882">10.1090/proc/15882</a>},
    number={7}, journal={Proceedings of the American Mathematical Society}, publisher={American
    Mathematical Society (AMS)}, author={Klüners, Jürgen and Wang, Jiuya}, year={2022},
    pages={2793–2805} }'
  chicago: 'Klüners, Jürgen, and Jiuya Wang. “ℓ-Torsion Bounds for the Class Group
    of Number Fields with an ℓ-Group as Galois Group.” <i>Proceedings of the American
    Mathematical Society</i> 150, no. 7 (2022): 2793–2805. <a href="https://doi.org/10.1090/proc/15882">https://doi.org/10.1090/proc/15882</a>.'
  ieee: 'J. Klüners and J. Wang, “ℓ-torsion bounds for the class group of number fields
    with an ℓ-group as Galois group,” <i>Proceedings of the American Mathematical
    Society</i>, vol. 150, no. 7, pp. 2793–2805, 2022, doi: <a href="https://doi.org/10.1090/proc/15882">10.1090/proc/15882</a>.'
  mla: Klüners, Jürgen, and Jiuya Wang. “ℓ-Torsion Bounds for the Class Group of Number
    Fields with an ℓ-Group as Galois Group.” <i>Proceedings of the American Mathematical
    Society</i>, vol. 150, no. 7, American Mathematical Society (AMS), 2022, pp. 2793–805,
    doi:<a href="https://doi.org/10.1090/proc/15882">10.1090/proc/15882</a>.
  short: J. Klüners, J. Wang, Proceedings of the American Mathematical Society 150
    (2022) 2793–2805.
date_created: 2022-12-22T10:47:01Z
date_updated: 2023-03-06T08:47:42Z
department:
- _id: '102'
doi: 10.1090/proc/15882
external_id:
  arxiv:
  - '2003.12161 '
intvolume: '       150'
issue: '7'
keyword:
- Applied Mathematics
- General Mathematics
language:
- iso: eng
page: 2793-2805
publication: Proceedings of the American Mathematical Society
publication_identifier:
  issn:
  - 0002-9939
  - 1088-6826
publication_status: published
publisher: American Mathematical Society (AMS)
status: public
title: ℓ-torsion bounds for the class group of number fields with an ℓ-group as Galois
  group
type: journal_article
user_id: '93826'
volume: 150
year: '2022'
...
---
_id: '34835'
abstract:
- lang: eng
  text: 'We prove an upper bound for the asymptotics of counting functions of number
    fields with nilpotent Galois groups. '
author:
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
citation:
  ama: Klüners J. The asymptotics of nilpotent Galois groups. <i>Acta Arithmetica</i>.
    2022;204(2):165-184. doi:<a href="https://doi.org/10.4064/aa211207-16-5">10.4064/aa211207-16-5</a>
  apa: Klüners, J. (2022). The asymptotics of nilpotent Galois groups. <i>Acta Arithmetica</i>,
    <i>204</i>(2), 165–184. <a href="https://doi.org/10.4064/aa211207-16-5">https://doi.org/10.4064/aa211207-16-5</a>
  bibtex: '@article{Klüners_2022, title={The asymptotics of nilpotent Galois groups},
    volume={204}, DOI={<a href="https://doi.org/10.4064/aa211207-16-5">10.4064/aa211207-16-5</a>},
    number={2}, journal={Acta Arithmetica}, publisher={Institute of Mathematics, Polish
    Academy of Sciences}, author={Klüners, Jürgen}, year={2022}, pages={165–184} }'
  chicago: 'Klüners, Jürgen. “The Asymptotics of Nilpotent Galois Groups.” <i>Acta
    Arithmetica</i> 204, no. 2 (2022): 165–84. <a href="https://doi.org/10.4064/aa211207-16-5">https://doi.org/10.4064/aa211207-16-5</a>.'
  ieee: 'J. Klüners, “The asymptotics of nilpotent Galois groups,” <i>Acta Arithmetica</i>,
    vol. 204, no. 2, pp. 165–184, 2022, doi: <a href="https://doi.org/10.4064/aa211207-16-5">10.4064/aa211207-16-5</a>.'
  mla: Klüners, Jürgen. “The Asymptotics of Nilpotent Galois Groups.” <i>Acta Arithmetica</i>,
    vol. 204, no. 2, Institute of Mathematics, Polish Academy of Sciences, 2022, pp.
    165–84, doi:<a href="https://doi.org/10.4064/aa211207-16-5">10.4064/aa211207-16-5</a>.
  short: J. Klüners, Acta Arithmetica 204 (2022) 165–184.
date_created: 2022-12-22T10:08:23Z
date_updated: 2023-03-06T08:48:33Z
department:
- _id: '102'
doi: 10.4064/aa211207-16-5
external_id:
  arxiv:
  - '2011.04325 '
intvolume: '       204'
issue: '2'
keyword:
- Algebra and Number Theory
language:
- iso: eng
page: 165-184
publication: Acta Arithmetica
publication_identifier:
  issn:
  - 0065-1036
  - 1730-6264
publication_status: published
publisher: Institute of Mathematics, Polish Academy of Sciences
status: public
title: The asymptotics of nilpotent Galois groups
type: journal_article
user_id: '93826'
volume: 204
year: '2022'
...
---
_id: '34840'
abstract:
- lang: eng
  text: 'In this paper we obtain a complete list of imaginary n-quadratic fields with
    class groups of exponent 3 and 5 under ERH for every positive integer n where
    an n-quadratic field is a number field of degree 2ⁿ represented as the composite
    of n quadratic fields. '
author:
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
- first_name: Toru
  full_name: Komatsu, Toru
  last_name: Komatsu
citation:
  ama: Klüners J, Komatsu T. Imaginary multiquadratic number fields with class group
    of exponent $3$ and $5$. <i>Mathematics of Computation</i>. 2021;90(329):1483-1497.
    doi:<a href="https://doi.org/10.1090/mcom/3609">10.1090/mcom/3609</a>
  apa: Klüners, J., &#38; Komatsu, T. (2021). Imaginary multiquadratic number fields
    with class group of exponent $3$ and $5$. <i>Mathematics of Computation</i>, <i>90</i>(329),
    1483–1497. <a href="https://doi.org/10.1090/mcom/3609">https://doi.org/10.1090/mcom/3609</a>
  bibtex: '@article{Klüners_Komatsu_2021, title={Imaginary multiquadratic number fields
    with class group of exponent $3$ and $5$}, volume={90}, DOI={<a href="https://doi.org/10.1090/mcom/3609">10.1090/mcom/3609</a>},
    number={329}, journal={Mathematics of Computation}, publisher={American Mathematical
    Society (AMS)}, author={Klüners, Jürgen and Komatsu, Toru}, year={2021}, pages={1483–1497}
    }'
  chicago: 'Klüners, Jürgen, and Toru Komatsu. “Imaginary Multiquadratic Number Fields
    with Class Group of Exponent $3$ and $5$.” <i>Mathematics of Computation</i> 90,
    no. 329 (2021): 1483–97. <a href="https://doi.org/10.1090/mcom/3609">https://doi.org/10.1090/mcom/3609</a>.'
  ieee: 'J. Klüners and T. Komatsu, “Imaginary multiquadratic number fields with class
    group of exponent $3$ and $5$,” <i>Mathematics of Computation</i>, vol. 90, no.
    329, pp. 1483–1497, 2021, doi: <a href="https://doi.org/10.1090/mcom/3609">10.1090/mcom/3609</a>.'
  mla: Klüners, Jürgen, and Toru Komatsu. “Imaginary Multiquadratic Number Fields
    with Class Group of Exponent $3$ and $5$.” <i>Mathematics of Computation</i>,
    vol. 90, no. 329, American Mathematical Society (AMS), 2021, pp. 1483–97, doi:<a
    href="https://doi.org/10.1090/mcom/3609">10.1090/mcom/3609</a>.
  short: J. Klüners, T. Komatsu, Mathematics of Computation 90 (2021) 1483–1497.
date_created: 2022-12-22T10:48:44Z
date_updated: 2023-03-06T08:57:45Z
department:
- _id: '102'
doi: 10.1090/mcom/3609
external_id:
  arxiv:
  - 2004.03308v2
intvolume: '        90'
issue: '329'
keyword:
- Applied Mathematics
- Computational Mathematics
- Algebra and Number Theory
language:
- iso: eng
page: 1483-1497
publication: Mathematics of Computation
publication_identifier:
  issn:
  - 0025-5718
  - 1088-6842
publication_status: published
publisher: American Mathematical Society (AMS)
status: public
title: Imaginary multiquadratic number fields with class group of exponent $3$ and
  $5$
type: journal_article
user_id: '93826'
volume: 90
year: '2021'
...
---
_id: '34842'
abstract:
- lang: eng
  text: Let D<0 be a fundamental discriminant and denote by E(D) the exponent of the
    ideal class group Cl(D) of K=ℚ(√D). Under the assumption that no Siegel zeros
    exist we compute all such D with E(D) dividing 8. We compute all D with |D| ≤
    3.1⋅10²⁰ such that E(D) ≤ 8.
author:
- first_name: Andreas-Stephan
  full_name: Elsenhans, Andreas-Stephan
  last_name: Elsenhans
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
- first_name: Florin
  full_name: Nicolae, Florin
  last_name: Nicolae
citation:
  ama: Elsenhans A-S, Klüners J, Nicolae F. Imaginary quadratic number fields with
    class groups of small exponent. <i>Acta Arithmetica</i>. 2020;193(3):217-233.
    doi:<a href="https://doi.org/10.4064/aa180220-20-3">10.4064/aa180220-20-3</a>
  apa: Elsenhans, A.-S., Klüners, J., &#38; Nicolae, F. (2020). Imaginary quadratic
    number fields with class groups of small exponent. <i>Acta Arithmetica</i>, <i>193</i>(3),
    217–233. <a href="https://doi.org/10.4064/aa180220-20-3">https://doi.org/10.4064/aa180220-20-3</a>
  bibtex: '@article{Elsenhans_Klüners_Nicolae_2020, title={Imaginary quadratic number
    fields with class groups of small exponent}, volume={193}, DOI={<a href="https://doi.org/10.4064/aa180220-20-3">10.4064/aa180220-20-3</a>},
    number={3}, journal={Acta Arithmetica}, publisher={Institute of Mathematics, Polish
    Academy of Sciences}, author={Elsenhans, Andreas-Stephan and Klüners, Jürgen and
    Nicolae, Florin}, year={2020}, pages={217–233} }'
  chicago: 'Elsenhans, Andreas-Stephan, Jürgen Klüners, and Florin Nicolae. “Imaginary
    Quadratic Number Fields with Class Groups of Small Exponent.” <i>Acta Arithmetica</i>
    193, no. 3 (2020): 217–33. <a href="https://doi.org/10.4064/aa180220-20-3">https://doi.org/10.4064/aa180220-20-3</a>.'
  ieee: 'A.-S. Elsenhans, J. Klüners, and F. Nicolae, “Imaginary quadratic number
    fields with class groups of small exponent,” <i>Acta Arithmetica</i>, vol. 193,
    no. 3, pp. 217–233, 2020, doi: <a href="https://doi.org/10.4064/aa180220-20-3">10.4064/aa180220-20-3</a>.'
  mla: Elsenhans, Andreas-Stephan, et al. “Imaginary Quadratic Number Fields with
    Class Groups of Small Exponent.” <i>Acta Arithmetica</i>, vol. 193, no. 3, Institute
    of Mathematics, Polish Academy of Sciences, 2020, pp. 217–33, doi:<a href="https://doi.org/10.4064/aa180220-20-3">10.4064/aa180220-20-3</a>.
  short: A.-S. Elsenhans, J. Klüners, F. Nicolae, Acta Arithmetica 193 (2020) 217–233.
date_created: 2022-12-22T10:51:13Z
date_updated: 2023-03-06T10:19:53Z
department:
- _id: '102'
doi: 10.4064/aa180220-20-3
external_id:
  arxiv:
  - '1803.02056 '
intvolume: '       193'
issue: '3'
keyword:
- Algebra and Number Theory
language:
- iso: eng
page: 217-233
publication: Acta Arithmetica
publication_identifier:
  issn:
  - 0065-1036
  - 1730-6264
publication_status: published
publisher: Institute of Mathematics, Polish Academy of Sciences
status: public
title: Imaginary quadratic number fields with class groups of small exponent
type: journal_article
user_id: '93826'
volume: 193
year: '2020'
...
---
_id: '34841'
abstract:
- lang: eng
  text: "We give an exact formula for the number of G-extensions of local function
    fields Fq((t)) for finite abelian groups G up to a conductor bound. As an application
    we give a lower bound for the corresponding counting problem by discriminant.\r\n"
author:
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
- first_name: Raphael
  full_name: Müller, Raphael
  last_name: Müller
citation:
  ama: Klüners J, Müller R. The conductor density of local function fields with abelian
    Galois group. <i>Journal of Number Theory</i>. 2020;212:311-322. doi:<a href="https://doi.org/10.1016/j.jnt.2019.11.007">10.1016/j.jnt.2019.11.007</a>
  apa: Klüners, J., &#38; Müller, R. (2020). The conductor density of local function
    fields with abelian Galois group. <i>Journal of Number Theory</i>, <i>212</i>,
    311–322. <a href="https://doi.org/10.1016/j.jnt.2019.11.007">https://doi.org/10.1016/j.jnt.2019.11.007</a>
  bibtex: '@article{Klüners_Müller_2020, title={The conductor density of local function
    fields with abelian Galois group}, volume={212}, DOI={<a href="https://doi.org/10.1016/j.jnt.2019.11.007">10.1016/j.jnt.2019.11.007</a>},
    journal={Journal of Number Theory}, publisher={Elsevier BV}, author={Klüners,
    Jürgen and Müller, Raphael}, year={2020}, pages={311–322} }'
  chicago: 'Klüners, Jürgen, and Raphael Müller. “The Conductor Density of Local Function
    Fields with Abelian Galois Group.” <i>Journal of Number Theory</i> 212 (2020):
    311–22. <a href="https://doi.org/10.1016/j.jnt.2019.11.007">https://doi.org/10.1016/j.jnt.2019.11.007</a>.'
  ieee: 'J. Klüners and R. Müller, “The conductor density of local function fields
    with abelian Galois group,” <i>Journal of Number Theory</i>, vol. 212, pp. 311–322,
    2020, doi: <a href="https://doi.org/10.1016/j.jnt.2019.11.007">10.1016/j.jnt.2019.11.007</a>.'
  mla: Klüners, Jürgen, and Raphael Müller. “The Conductor Density of Local Function
    Fields with Abelian Galois Group.” <i>Journal of Number Theory</i>, vol. 212,
    Elsevier BV, 2020, pp. 311–22, doi:<a href="https://doi.org/10.1016/j.jnt.2019.11.007">10.1016/j.jnt.2019.11.007</a>.
  short: J. Klüners, R. Müller, Journal of Number Theory 212 (2020) 311–322.
date_created: 2022-12-22T10:50:03Z
date_updated: 2025-06-13T08:18:30Z
department:
- _id: '102'
doi: 10.1016/j.jnt.2019.11.007
external_id:
  arxiv:
  - '1904.02573 '
intvolume: '       212'
keyword:
- Algebra and Number Theory
language:
- iso: eng
page: 311-322
publication: Journal of Number Theory
publication_identifier:
  issn:
  - 0022-314X
publication_status: published
publisher: Elsevier BV
status: public
title: The conductor density of local function fields with abelian Galois group
type: journal_article
user_id: '82981'
volume: 212
year: '2020'
...
---
_id: '34843'
abstract:
- lang: eng
  text: "A polynomial time algorithm to find generators of the lattice of all subfields
    of a given number field was given in van Hoeij et al. (2013).\r\n\r\nThis article
    reports on a massive speedup of this algorithm. This is primary achieved by our
    new concept of Galois-generating subfields. In general this is a very small set
    of subfields that determine all other subfields in a group-theoretic way. We compute
    them by targeted calls to the method from van Hoeij et al. (2013). For an early
    termination of these calls, we give a list of criteria that imply that further
    calls will not result in additional subfields.\r\n\r\nFinally, we explain how
    we use subfields to get a good starting group for the computation of Galois groups."
author:
- first_name: Andreas-Stephan
  full_name: Elsenhans, Andreas-Stephan
  last_name: Elsenhans
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
citation:
  ama: Elsenhans A-S, Klüners J. Computing subfields of number fields and applications
    to Galois group computations. <i>Journal of Symbolic Computation</i>. 2018;93:1-20.
    doi:<a href="https://doi.org/10.1016/j.jsc.2018.04.013">10.1016/j.jsc.2018.04.013</a>
  apa: Elsenhans, A.-S., &#38; Klüners, J. (2018). Computing subfields of number fields
    and applications to Galois group computations. <i>Journal of Symbolic Computation</i>,
    <i>93</i>, 1–20. <a href="https://doi.org/10.1016/j.jsc.2018.04.013">https://doi.org/10.1016/j.jsc.2018.04.013</a>
  bibtex: '@article{Elsenhans_Klüners_2018, title={Computing subfields of number fields
    and applications to Galois group computations}, volume={93}, DOI={<a href="https://doi.org/10.1016/j.jsc.2018.04.013">10.1016/j.jsc.2018.04.013</a>},
    journal={Journal of Symbolic Computation}, publisher={Elsevier BV}, author={Elsenhans,
    Andreas-Stephan and Klüners, Jürgen}, year={2018}, pages={1–20} }'
  chicago: 'Elsenhans, Andreas-Stephan, and Jürgen Klüners. “Computing Subfields of
    Number Fields and Applications to Galois Group Computations.” <i>Journal of Symbolic
    Computation</i> 93 (2018): 1–20. <a href="https://doi.org/10.1016/j.jsc.2018.04.013">https://doi.org/10.1016/j.jsc.2018.04.013</a>.'
  ieee: 'A.-S. Elsenhans and J. Klüners, “Computing subfields of number fields and
    applications to Galois group computations,” <i>Journal of Symbolic Computation</i>,
    vol. 93, pp. 1–20, 2018, doi: <a href="https://doi.org/10.1016/j.jsc.2018.04.013">10.1016/j.jsc.2018.04.013</a>.'
  mla: Elsenhans, Andreas-Stephan, and Jürgen Klüners. “Computing Subfields of Number
    Fields and Applications to Galois Group Computations.” <i>Journal of Symbolic
    Computation</i>, vol. 93, Elsevier BV, 2018, pp. 1–20, doi:<a href="https://doi.org/10.1016/j.jsc.2018.04.013">10.1016/j.jsc.2018.04.013</a>.
  short: A.-S. Elsenhans, J. Klüners, Journal of Symbolic Computation 93 (2018) 1–20.
date_created: 2022-12-22T10:52:18Z
date_updated: 2023-03-06T09:05:51Z
department:
- _id: '102'
doi: 10.1016/j.jsc.2018.04.013
external_id:
  arxiv:
  - '1610.06837 '
intvolume: '        93'
keyword:
- Computational Mathematics
- Algebra and Number Theory
language:
- iso: eng
page: 1-20
publication: Journal of Symbolic Computation
publication_identifier:
  issn:
  - 0747-7171
publication_status: published
publisher: Elsevier BV
status: public
title: Computing subfields of number fields and applications to Galois group computations
type: journal_article
user_id: '93826'
volume: 93
year: '2018'
...
---
_id: '34844'
abstract:
- lang: eng
  text: 'Let k be a number field, K/k a finite Galois extension with Galois group
    G, χ a faithful character of G. We prove that the Artin L-function L(s,χ,K/k)
    determines the Galois closure of K over $\ℚ$. In the special case $k=\ℚ$ it also
    determines the character χ. '
author:
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
- first_name: Florin
  full_name: Nicolae, Florin
  last_name: Nicolae
citation:
  ama: Klüners J, Nicolae F. Are number fields determined by Artin L-functions? <i>Journal
    of Number Theory</i>. 2016;167:161-168. doi:<a href="https://doi.org/10.1016/j.jnt.2016.03.023">10.1016/j.jnt.2016.03.023</a>
  apa: Klüners, J., &#38; Nicolae, F. (2016). Are number fields determined by Artin
    L-functions? <i>Journal of Number Theory</i>, <i>167</i>, 161–168. <a href="https://doi.org/10.1016/j.jnt.2016.03.023">https://doi.org/10.1016/j.jnt.2016.03.023</a>
  bibtex: '@article{Klüners_Nicolae_2016, title={Are number fields determined by Artin
    L-functions?}, volume={167}, DOI={<a href="https://doi.org/10.1016/j.jnt.2016.03.023">10.1016/j.jnt.2016.03.023</a>},
    journal={Journal of Number Theory}, publisher={Elsevier BV}, author={Klüners,
    Jürgen and Nicolae, Florin}, year={2016}, pages={161–168} }'
  chicago: 'Klüners, Jürgen, and Florin Nicolae. “Are Number Fields Determined by
    Artin L-Functions?” <i>Journal of Number Theory</i> 167 (2016): 161–68. <a href="https://doi.org/10.1016/j.jnt.2016.03.023">https://doi.org/10.1016/j.jnt.2016.03.023</a>.'
  ieee: 'J. Klüners and F. Nicolae, “Are number fields determined by Artin L-functions?,”
    <i>Journal of Number Theory</i>, vol. 167, pp. 161–168, 2016, doi: <a href="https://doi.org/10.1016/j.jnt.2016.03.023">10.1016/j.jnt.2016.03.023</a>.'
  mla: Klüners, Jürgen, and Florin Nicolae. “Are Number Fields Determined by Artin
    L-Functions?” <i>Journal of Number Theory</i>, vol. 167, Elsevier BV, 2016, pp.
    161–68, doi:<a href="https://doi.org/10.1016/j.jnt.2016.03.023">10.1016/j.jnt.2016.03.023</a>.
  short: J. Klüners, F. Nicolae, Journal of Number Theory 167 (2016) 161–168.
date_created: 2022-12-22T10:52:47Z
date_updated: 2023-03-06T10:44:22Z
department:
- _id: '102'
doi: 10.1016/j.jnt.2016.03.023
external_id:
  arxiv:
  - '1509.06883 '
intvolume: '       167'
keyword:
- Algebra and Number Theory
language:
- iso: eng
page: 161-168
publication: Journal of Number Theory
publication_identifier:
  issn:
  - 0022-314X
publication_status: published
publisher: Elsevier BV
status: public
title: Are number fields determined by Artin L-functions?
type: journal_article
user_id: '93826'
volume: 167
year: '2016'
...
---
_id: '34845'
abstract:
- lang: eng
  text: Computational Galois theory, in particular the problem of computing the Galois
    group of a given polynomial, is a very old problem. Currently, the best algorithmic
    solution is Stauduhar’s method. Computationally, one of the key challenges in
    the application of Stauduhar’s method is to find, for a given pair of groups H<G,
    a G-relative H-invariant, that is a multivariate polynomial F that is H-invariant,
    but not G-invariant. While generic, theoretical methods are known to find such
    F, in general they yield impractical answers. We give a general method for computing
    invariants of large degree which improves on previous known methods, as well as
    various special invariants that are derived from the structure of the groups.
    We then apply our new invariants to the task of computing the Galois groups of
    polynomials over the rational numbers, resulting in the first practical degree
    independent algorithm.
author:
- first_name: Claus
  full_name: Fieker, Claus
  last_name: Fieker
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
citation:
  ama: Fieker C, Klüners J. Computation of Galois groups of rational polynomials.
    <i>LMS Journal of Computation and Mathematics</i>. 2014;17(1):141-158. doi:<a
    href="https://doi.org/10.1112/s1461157013000302">10.1112/s1461157013000302</a>
  apa: Fieker, C., &#38; Klüners, J. (2014). Computation of Galois groups of rational
    polynomials. <i>LMS Journal of Computation and Mathematics</i>, <i>17</i>(1),
    141–158. <a href="https://doi.org/10.1112/s1461157013000302">https://doi.org/10.1112/s1461157013000302</a>
  bibtex: '@article{Fieker_Klüners_2014, title={Computation of Galois groups of rational
    polynomials}, volume={17}, DOI={<a href="https://doi.org/10.1112/s1461157013000302">10.1112/s1461157013000302</a>},
    number={1}, journal={LMS Journal of Computation and Mathematics}, publisher={Wiley},
    author={Fieker, Claus and Klüners, Jürgen}, year={2014}, pages={141–158} }'
  chicago: 'Fieker, Claus, and Jürgen Klüners. “Computation of Galois Groups of Rational
    Polynomials.” <i>LMS Journal of Computation and Mathematics</i> 17, no. 1 (2014):
    141–58. <a href="https://doi.org/10.1112/s1461157013000302">https://doi.org/10.1112/s1461157013000302</a>.'
  ieee: 'C. Fieker and J. Klüners, “Computation of Galois groups of rational polynomials,”
    <i>LMS Journal of Computation and Mathematics</i>, vol. 17, no. 1, pp. 141–158,
    2014, doi: <a href="https://doi.org/10.1112/s1461157013000302">10.1112/s1461157013000302</a>.'
  mla: Fieker, Claus, and Jürgen Klüners. “Computation of Galois Groups of Rational
    Polynomials.” <i>LMS Journal of Computation and Mathematics</i>, vol. 17, no.
    1, Wiley, 2014, pp. 141–58, doi:<a href="https://doi.org/10.1112/s1461157013000302">10.1112/s1461157013000302</a>.
  short: C. Fieker, J. Klüners, LMS Journal of Computation and Mathematics 17 (2014)
    141–158.
date_created: 2022-12-22T10:53:44Z
date_updated: 2023-03-06T09:43:56Z
department:
- _id: '102'
doi: 10.1112/s1461157013000302
external_id:
  arxiv:
  - '1211.3588'
intvolume: '        17'
issue: '1'
keyword:
- Computational Theory and Mathematics
- General Mathematics
language:
- iso: eng
page: 141-158
publication: LMS Journal of Computation and Mathematics
publication_identifier:
  issn:
  - 1461-1570
publication_status: published
publisher: Wiley
status: public
title: Computation of Galois groups of rational polynomials
type: journal_article
user_id: '93826'
volume: 17
year: '2014'
...
---
_id: '34847'
abstract:
- lang: eng
  text: 'Let G be a wreath product of the form C₂ ≀ H, where C₂ is the cyclic group
    of order 2. Under mild conditions for H we determine the asymptotic behavior of
    the counting functions for number fields K/k with Galois group G and bounded discriminant.
    Those counting functions grow linearly with the norm of the discriminant and this
    result coincides with a conjecture of Malle. Up to a constant factor these groups
    have the same asymptotic behavior as the conjectured one for symmetric groups. '
author:
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
citation:
  ama: Klüners J. The Distribution of Number Fields with Wreath Products as Galois
    Groups . <i>International Journal of Number Theory</i>. 2012;08(03):845-858. doi:<a
    href="https://doi.org/10.1142/s1793042112500492">10.1142/s1793042112500492</a>
  apa: Klüners, J. (2012). The Distribution of Number Fields with Wreath Products
    as Galois Groups . <i>International Journal of Number Theory</i>, <i>08</i>(03),
    845–858. <a href="https://doi.org/10.1142/s1793042112500492">https://doi.org/10.1142/s1793042112500492</a>
  bibtex: '@article{Klüners_2012, title={The Distribution of Number Fields with Wreath
    Products as Galois Groups }, volume={08}, DOI={<a href="https://doi.org/10.1142/s1793042112500492">10.1142/s1793042112500492</a>},
    number={03}, journal={International Journal of Number Theory}, publisher={World
    Scientific Pub Co Pte Lt}, author={Klüners, Jürgen}, year={2012}, pages={845–858}
    }'
  chicago: 'Klüners, Jürgen. “The Distribution of Number Fields with Wreath Products
    as Galois Groups .” <i>International Journal of Number Theory</i> 08, no. 03 (2012):
    845–58. <a href="https://doi.org/10.1142/s1793042112500492">https://doi.org/10.1142/s1793042112500492</a>.'
  ieee: 'J. Klüners, “The Distribution of Number Fields with Wreath Products as Galois
    Groups ,” <i>International Journal of Number Theory</i>, vol. 08, no. 03, pp.
    845–858, 2012, doi: <a href="https://doi.org/10.1142/s1793042112500492">10.1142/s1793042112500492</a>.'
  mla: Klüners, Jürgen. “The Distribution of Number Fields with Wreath Products as
    Galois Groups .” <i>International Journal of Number Theory</i>, vol. 08, no. 03,
    World Scientific Pub Co Pte Lt, 2012, pp. 845–58, doi:<a href="https://doi.org/10.1142/s1793042112500492">10.1142/s1793042112500492</a>.
  short: J. Klüners, International Journal of Number Theory 08 (2012) 845–858.
date_created: 2022-12-22T10:55:47Z
date_updated: 2023-03-02T14:10:38Z
department:
- _id: '102'
doi: 10.1142/s1793042112500492
external_id:
  arxiv:
  - '1108.5597 '
intvolume: '         8'
issue: '03'
language:
- iso: eng
page: 845-858
publication: International Journal of Number Theory
publication_identifier:
  issn:
  - 1793-0421
  - 1793-7310
publication_status: published
publisher: World Scientific Pub Co Pte Lt
status: public
title: 'The Distribution of Number Fields with Wreath Products as Galois Groups '
type: journal_article
user_id: '21202'
volume: '08'
year: '2012'
...
---
_id: '34885'
abstract:
- lang: eng
  text: We prove that the distribution of the values of the 4-rank of ideal class
    groups of quadratic fields is not affected when it is weighted by a divisor type
    function. We then give several applications concerning a new lower bound of the
    sums of class numbers of real quadratic fields with discriminant less than a bound
    tending to infinity and several questions of P. Sarnak concerning reciprocal geodesics.
author:
- first_name: Étienne
  full_name: Fouvry, Étienne
  last_name: Fouvry
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
citation:
  ama: Fouvry É, Klüners J. Weighted Distribution of the 4-rank of Class Groups and
    Applications. <i>International Mathematics Research Notices</i>. 2011;2011(16):3618-3656.
    doi:<a href="https://doi.org/10.1093/imrn/rnq223">10.1093/imrn/rnq223</a>
  apa: Fouvry, É., &#38; Klüners, J. (2011). Weighted Distribution of the 4-rank of
    Class Groups and Applications. <i>International Mathematics Research Notices</i>,
    <i>2011</i>(16), 3618–3656. <a href="https://doi.org/10.1093/imrn/rnq223">https://doi.org/10.1093/imrn/rnq223</a>
  bibtex: '@article{Fouvry_Klüners_2011, title={Weighted Distribution of the 4-rank
    of Class Groups and Applications}, volume={2011}, DOI={<a href="https://doi.org/10.1093/imrn/rnq223">10.1093/imrn/rnq223</a>},
    number={16}, journal={International Mathematics Research Notices}, publisher={Oxford
    University Press (OUP)}, author={Fouvry, Étienne and Klüners, Jürgen}, year={2011},
    pages={3618–3656} }'
  chicago: 'Fouvry, Étienne, and Jürgen Klüners. “Weighted Distribution of the 4-Rank
    of Class Groups and Applications.” <i>International Mathematics Research Notices</i>
    2011, no. 16 (2011): 3618–56. <a href="https://doi.org/10.1093/imrn/rnq223">https://doi.org/10.1093/imrn/rnq223</a>.'
  ieee: 'É. Fouvry and J. Klüners, “Weighted Distribution of the 4-rank of Class Groups
    and Applications,” <i>International Mathematics Research Notices</i>, vol. 2011,
    no. 16, pp. 3618–3656, 2011, doi: <a href="https://doi.org/10.1093/imrn/rnq223">10.1093/imrn/rnq223</a>.'
  mla: Fouvry, Étienne, and Jürgen Klüners. “Weighted Distribution of the 4-Rank of
    Class Groups and Applications.” <i>International Mathematics Research Notices</i>,
    vol. 2011, no. 16, Oxford University Press (OUP), 2011, pp. 3618–56, doi:<a href="https://doi.org/10.1093/imrn/rnq223">10.1093/imrn/rnq223</a>.
  short: É. Fouvry, J. Klüners, International Mathematics Research Notices 2011 (2011)
    3618–3656.
date_created: 2022-12-23T09:08:00Z
date_updated: 2023-03-06T09:07:46Z
department:
- _id: '102'
doi: 10.1093/imrn/rnq223
intvolume: '      2011'
issue: '16'
keyword:
- General Mathematics
language:
- iso: eng
page: 3618-3656
publication: International Mathematics Research Notices
publication_identifier:
  issn:
  - 1687-0247
  - 1073-7928
publication_status: published
publisher: Oxford University Press (OUP)
status: public
title: Weighted Distribution of the 4-rank of Class Groups and Applications
type: journal_article
user_id: '93826'
volume: 2011
year: '2011'
...
---
_id: '34846'
abstract:
- lang: eng
  text: Given a field extension K/k of degree n we are interested in finding the subfields
    of K containing k. There can be more than polynomially many subfields. We introduce
    the notion of generating subfields, a set of up to n subfields whose intersections
    give the rest. We provide an efficient algorithm which uses linear algebra in
    k or lattice reduction along with factorization in any extension of K. Implementations
    show that previously difficult cases can now be handled.
author:
- first_name: Mark
  full_name: van Hoeij, Mark
  last_name: van Hoeij
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
- first_name: Andrew
  full_name: Novocin, Andrew
  last_name: Novocin
citation:
  ama: van Hoeij M, Klüners J, Novocin A. Generating subfields. <i>Journal of Symbolic
    Computation</i>. 2011;52:17-34. doi:<a href="https://doi.org/10.1016/j.jsc.2012.05.010">10.1016/j.jsc.2012.05.010</a>
  apa: van Hoeij, M., Klüners, J., &#38; Novocin, A. (2011). Generating subfields.
    <i>Journal of Symbolic Computation</i>, <i>52</i>, 17–34. <a href="https://doi.org/10.1016/j.jsc.2012.05.010">https://doi.org/10.1016/j.jsc.2012.05.010</a>
  bibtex: '@article{van Hoeij_Klüners_Novocin_2011, title={Generating subfields},
    volume={52}, DOI={<a href="https://doi.org/10.1016/j.jsc.2012.05.010">10.1016/j.jsc.2012.05.010</a>},
    journal={Journal of Symbolic Computation}, publisher={Elsevier BV}, author={van
    Hoeij, Mark and Klüners, Jürgen and Novocin, Andrew}, year={2011}, pages={17–34}
    }'
  chicago: 'Hoeij, Mark van, Jürgen Klüners, and Andrew Novocin. “Generating Subfields.”
    <i>Journal of Symbolic Computation</i> 52 (2011): 17–34. <a href="https://doi.org/10.1016/j.jsc.2012.05.010">https://doi.org/10.1016/j.jsc.2012.05.010</a>.'
  ieee: 'M. van Hoeij, J. Klüners, and A. Novocin, “Generating subfields,” <i>Journal
    of Symbolic Computation</i>, vol. 52, pp. 17–34, 2011, doi: <a href="https://doi.org/10.1016/j.jsc.2012.05.010">10.1016/j.jsc.2012.05.010</a>.'
  mla: van Hoeij, Mark, et al. “Generating Subfields.” <i>Journal of Symbolic Computation</i>,
    vol. 52, Elsevier BV, 2011, pp. 17–34, doi:<a href="https://doi.org/10.1016/j.jsc.2012.05.010">10.1016/j.jsc.2012.05.010</a>.
  short: M. van Hoeij, J. Klüners, A. Novocin, Journal of Symbolic Computation 52
    (2011) 17–34.
date_created: 2022-12-22T10:54:15Z
date_updated: 2023-03-06T09:46:15Z
department:
- _id: '102'
doi: 10.1016/j.jsc.2012.05.010
intvolume: '        52'
keyword:
- Computational Mathematics
- Algebra and Number Theory
language:
- iso: eng
page: 17-34
publication: Journal of Symbolic Computation
publication_identifier:
  issn:
  - 0747-7171
publication_status: published
publisher: Elsevier BV
status: public
title: Generating subfields
type: journal_article
user_id: '93826'
volume: 52
year: '2011'
...
---
_id: '34886'
abstract:
- lang: eng
  text: We give asymptotic upper and lower bounds for the number of squarefree d (0
    < d ≤ X) such that the equation x² − dy²= −1 is solvable. These estimates, as
    usual, can equivalently be interpreted in terms of real quadratic fields with
    a fundamental unit with norm −1 and give strong evidence in the direction of a
    conjecture due to P. Stevenhagen.
author:
- first_name: Étienne
  full_name: Fouvry, Étienne
  last_name: Fouvry
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
citation:
  ama: Fouvry É, Klüners J. On the negative Pell equation. <i>Annals of Mathematics</i>.
    2010;172(3):2035-2104. doi:<a href="https://doi.org/10.4007/annals.2010.172.2035">10.4007/annals.2010.172.2035</a>
  apa: Fouvry, É., &#38; Klüners, J. (2010). On the negative Pell equation. <i>Annals
    of Mathematics</i>, <i>172</i>(3), 2035–2104. <a href="https://doi.org/10.4007/annals.2010.172.2035">https://doi.org/10.4007/annals.2010.172.2035</a>
  bibtex: '@article{Fouvry_Klüners_2010, title={On the negative Pell equation}, volume={172},
    DOI={<a href="https://doi.org/10.4007/annals.2010.172.2035">10.4007/annals.2010.172.2035</a>},
    number={3}, journal={Annals of Mathematics}, publisher={Annals of Mathematics},
    author={Fouvry, Étienne and Klüners, Jürgen}, year={2010}, pages={2035–2104} }'
  chicago: 'Fouvry, Étienne, and Jürgen Klüners. “On the Negative Pell Equation.”
    <i>Annals of Mathematics</i> 172, no. 3 (2010): 2035–2104. <a href="https://doi.org/10.4007/annals.2010.172.2035">https://doi.org/10.4007/annals.2010.172.2035</a>.'
  ieee: 'É. Fouvry and J. Klüners, “On the negative Pell equation,” <i>Annals of Mathematics</i>,
    vol. 172, no. 3, pp. 2035–2104, 2010, doi: <a href="https://doi.org/10.4007/annals.2010.172.2035">10.4007/annals.2010.172.2035</a>.'
  mla: Fouvry, Étienne, and Jürgen Klüners. “On the Negative Pell Equation.” <i>Annals
    of Mathematics</i>, vol. 172, no. 3, Annals of Mathematics, 2010, pp. 2035–104,
    doi:<a href="https://doi.org/10.4007/annals.2010.172.2035">10.4007/annals.2010.172.2035</a>.
  short: É. Fouvry, J. Klüners, Annals of Mathematics 172 (2010) 2035–2104.
date_created: 2022-12-23T09:09:02Z
date_updated: 2023-03-06T09:50:37Z
department:
- _id: '102'
doi: 10.4007/annals.2010.172.2035
intvolume: '       172'
issue: '3'
keyword:
- Statistics
- Probability and Uncertainty
- Mathematics (miscellaneous)
language:
- iso: eng
page: 2035-2104
publication: Annals of Mathematics
publication_identifier:
  issn:
  - 0003-486X
publication_status: published
publisher: Annals of Mathematics
status: public
title: On the negative Pell equation
type: journal_article
user_id: '93826'
volume: 172
year: '2010'
...
---
_id: '34888'
abstract:
- lang: eng
  text: We call a positive square-free integer d special, if d is not divisible by
    primes congruent to 3 mod 4. We show that the period of the expansion of in continued
    fractions is asymptotically more often odd than even, when we restrict to special
    integers. We note that this period is always even for a non-special square-free
    integer d. It is well known that the above period is odd if and only if the negative
    Pell equation x²−dy²=−1 is solvable. The latter problem is solvable if and only
    if the narrow and the ordinary class groups of ℚ(√d) are equal. In a prior work
    we fully described the asymptotics of the 4-ranks of those class groups. Here
    we get the first non-trivial results about the asymptotic behavior of the 8-rank
    of the narrow class group. For example, we show that more than 76% of the quadratic
    fields ℚ(√d), where d is special, have the property that the 8-rank of the narrow
    class group is zero.
author:
- first_name: Étienne
  full_name: Fouvry, Étienne
  last_name: Fouvry
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
citation:
  ama: Fouvry É, Klüners J. The parity of the period of the continued fraction of
    d. <i>Proceedings of the London Mathematical Society</i>. 2010;101(2):337-391.
    doi:<a href="https://doi.org/10.1112/plms/pdp057">10.1112/plms/pdp057</a>
  apa: Fouvry, É., &#38; Klüners, J. (2010). The parity of the period of the continued
    fraction of d. <i>Proceedings of the London Mathematical Society</i>, <i>101</i>(2),
    337–391. <a href="https://doi.org/10.1112/plms/pdp057">https://doi.org/10.1112/plms/pdp057</a>
  bibtex: '@article{Fouvry_Klüners_2010, title={The parity of the period of the continued
    fraction of d}, volume={101}, DOI={<a href="https://doi.org/10.1112/plms/pdp057">10.1112/plms/pdp057</a>},
    number={2}, journal={Proceedings of the London Mathematical Society}, publisher={Wiley},
    author={Fouvry, Étienne and Klüners, Jürgen}, year={2010}, pages={337–391} }'
  chicago: 'Fouvry, Étienne, and Jürgen Klüners. “The Parity of the Period of the
    Continued Fraction of d.” <i>Proceedings of the London Mathematical Society</i>
    101, no. 2 (2010): 337–91. <a href="https://doi.org/10.1112/plms/pdp057">https://doi.org/10.1112/plms/pdp057</a>.'
  ieee: 'É. Fouvry and J. Klüners, “The parity of the period of the continued fraction
    of d,” <i>Proceedings of the London Mathematical Society</i>, vol. 101, no. 2,
    pp. 337–391, 2010, doi: <a href="https://doi.org/10.1112/plms/pdp057">10.1112/plms/pdp057</a>.'
  mla: Fouvry, Étienne, and Jürgen Klüners. “The Parity of the Period of the Continued
    Fraction of d.” <i>Proceedings of the London Mathematical Society</i>, vol. 101,
    no. 2, Wiley, 2010, pp. 337–91, doi:<a href="https://doi.org/10.1112/plms/pdp057">10.1112/plms/pdp057</a>.
  short: É. Fouvry, J. Klüners, Proceedings of the London Mathematical Society 101
    (2010) 337–391.
date_created: 2022-12-23T09:22:49Z
date_updated: 2023-03-06T10:16:54Z
department:
- _id: '102'
doi: 10.1112/plms/pdp057
intvolume: '       101'
issue: '2'
keyword:
- General Mathematics
language:
- iso: eng
page: 337-391
publication: Proceedings of the London Mathematical Society
publication_identifier:
  issn:
  - 0024-6115
publication_status: published
publisher: Wiley
related_material:
  link:
  - relation: confirmation
    url: https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=fc7ab412993fd2bf0069de42fbef1ecc69137755
status: public
title: The parity of the period of the continued fraction of d
type: journal_article
user_id: '93826'
volume: 101
year: '2010'
...
---
_id: '34887'
abstract:
- lang: eng
  text: 'Let d be a nonsquare positive integer. We give the value of the natural probability
    that the narrow ideal class groups of the quadratic fields ℚ(√d) and ℚ(√−d) have
    the same 4-ranks. '
author:
- first_name: Étienne
  full_name: Fouvry, Étienne
  last_name: Fouvry
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
citation:
  ama: Fouvry É, Klüners J. On the Spiegelungssatz for the 4-rank. <i>Algebra &#38;amp;
    Number Theory</i>. 2010;4(5):493-508. doi:<a href="https://doi.org/10.2140/ant.2010.4.493">10.2140/ant.2010.4.493</a>
  apa: Fouvry, É., &#38; Klüners, J. (2010). On the Spiegelungssatz for the 4-rank.
    <i>Algebra &#38;amp; Number Theory</i>, <i>4</i>(5), 493–508. <a href="https://doi.org/10.2140/ant.2010.4.493">https://doi.org/10.2140/ant.2010.4.493</a>
  bibtex: '@article{Fouvry_Klüners_2010, title={On the Spiegelungssatz for the 4-rank},
    volume={4}, DOI={<a href="https://doi.org/10.2140/ant.2010.4.493">10.2140/ant.2010.4.493</a>},
    number={5}, journal={Algebra &#38;amp; Number Theory}, publisher={Mathematical
    Sciences Publishers}, author={Fouvry, Étienne and Klüners, Jürgen}, year={2010},
    pages={493–508} }'
  chicago: 'Fouvry, Étienne, and Jürgen Klüners. “On the Spiegelungssatz for the 4-Rank.”
    <i>Algebra &#38;amp; Number Theory</i> 4, no. 5 (2010): 493–508. <a href="https://doi.org/10.2140/ant.2010.4.493">https://doi.org/10.2140/ant.2010.4.493</a>.'
  ieee: 'É. Fouvry and J. Klüners, “On the Spiegelungssatz for the 4-rank,” <i>Algebra
    &#38;amp; Number Theory</i>, vol. 4, no. 5, pp. 493–508, 2010, doi: <a href="https://doi.org/10.2140/ant.2010.4.493">10.2140/ant.2010.4.493</a>.'
  mla: Fouvry, Étienne, and Jürgen Klüners. “On the Spiegelungssatz for the 4-Rank.”
    <i>Algebra &#38;amp; Number Theory</i>, vol. 4, no. 5, Mathematical Sciences Publishers,
    2010, pp. 493–508, doi:<a href="https://doi.org/10.2140/ant.2010.4.493">10.2140/ant.2010.4.493</a>.
  short: É. Fouvry, J. Klüners, Algebra &#38;amp; Number Theory 4 (2010) 493–508.
date_created: 2022-12-23T09:10:12Z
date_updated: 2023-03-06T10:18:14Z
department:
- _id: '102'
doi: 10.2140/ant.2010.4.493
intvolume: '         4'
issue: '5'
keyword:
- Algebra and Number Theory
language:
- iso: eng
page: 493-508
publication: Algebra &amp; Number Theory
publication_identifier:
  issn:
  - 1937-0652
publication_status: published
publisher: Mathematical Sciences Publishers
status: public
title: On the Spiegelungssatz for the 4-rank
type: journal_article
user_id: '93826'
volume: 4
year: '2010'
...