---
_id: '34889'
abstract:
- lang: eng
  text: We prove that van Hoeij’s original algorithm to factor univariate polynomials
    over the rationals runs in polynomial time, as well as natural variants. In particular,
    our approach also yields polynomial time complexity results for bivariate polynomials
    over a finite field.
author:
- first_name: Karim
  full_name: Belabas, Karim
  last_name: Belabas
- 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: Allan
  full_name: Steel, Allan
  last_name: Steel
citation:
  ama: Belabas K, van Hoeij M, Klüners J, Steel A. Factoring polynomials over global
    fields. <i>Journal de Théorie des Nombres de Bordeaux</i>. 2009;21(1):15-39. doi:<a
    href="https://doi.org/10.5802/jtnb.655">10.5802/jtnb.655</a>
  apa: Belabas, K., van Hoeij, M., Klüners, J., &#38; Steel, A. (2009). Factoring
    polynomials over global fields. <i>Journal de Théorie Des Nombres de Bordeaux</i>,
    <i>21</i>(1), 15–39. <a href="https://doi.org/10.5802/jtnb.655">https://doi.org/10.5802/jtnb.655</a>
  bibtex: '@article{Belabas_van Hoeij_Klüners_Steel_2009, title={Factoring polynomials
    over global fields}, volume={21}, DOI={<a href="https://doi.org/10.5802/jtnb.655">10.5802/jtnb.655</a>},
    number={1}, journal={Journal de Théorie des Nombres de Bordeaux}, publisher={Cellule
    MathDoc/CEDRAM}, author={Belabas, Karim and van Hoeij, Mark and Klüners, Jürgen
    and Steel, Allan}, year={2009}, pages={15–39} }'
  chicago: 'Belabas, Karim, Mark van Hoeij, Jürgen Klüners, and Allan Steel. “Factoring
    Polynomials over Global Fields.” <i>Journal de Théorie Des Nombres de Bordeaux</i>
    21, no. 1 (2009): 15–39. <a href="https://doi.org/10.5802/jtnb.655">https://doi.org/10.5802/jtnb.655</a>.'
  ieee: 'K. Belabas, M. van Hoeij, J. Klüners, and A. Steel, “Factoring polynomials
    over global fields,” <i>Journal de Théorie des Nombres de Bordeaux</i>, vol. 21,
    no. 1, pp. 15–39, 2009, doi: <a href="https://doi.org/10.5802/jtnb.655">10.5802/jtnb.655</a>.'
  mla: Belabas, Karim, et al. “Factoring Polynomials over Global Fields.” <i>Journal
    de Théorie Des Nombres de Bordeaux</i>, vol. 21, no. 1, Cellule MathDoc/CEDRAM,
    2009, pp. 15–39, doi:<a href="https://doi.org/10.5802/jtnb.655">10.5802/jtnb.655</a>.
  short: K. Belabas, M. van Hoeij, J. Klüners, A. Steel, Journal de Théorie Des Nombres
    de Bordeaux 21 (2009) 15–39.
date_created: 2022-12-23T09:33:37Z
date_updated: 2023-03-06T09:09:56Z
department:
- _id: '102'
doi: 10.5802/jtnb.655
external_id:
  arxiv:
  - 'math/0409510 '
intvolume: '        21'
issue: '1'
keyword:
- Algebra and Number Theory
language:
- iso: eng
page: 15-39
publication: Journal de Théorie des Nombres de Bordeaux
publication_identifier:
  issn:
  - 1246-7405
publication_status: published
publisher: Cellule MathDoc/CEDRAM
status: public
title: Factoring polynomials over global fields
type: journal_article
user_id: '93826'
volume: 21
year: '2009'
...
---
_id: '35959'
abstract:
- lang: eng
  text: In this survey, we report about a new algorithm for factoring polynomials
    due to Mark van Hoeij. The main idea is that the combinatorial problem that occurs
    in the Zassenhaus algorithm is reduced to a very special knapsack problem. In
    case of rational polynomials, this knapsack problem can be very efficiently solved
    by the LLL algorithm. This gives a polynomial time algorithm, which also works
    very well in practice.
author:
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
citation:
  ama: 'Klüners J. The van Hoeij Algorithm for Factoring Polynomials. In: <i>The LLL
    Algorithm</i>. Springer Berlin Heidelberg; 2009. doi:<a href="https://doi.org/10.1007/978-3-642-02295-1_8">10.1007/978-3-642-02295-1_8</a>'
  apa: Klüners, J. (2009). The van Hoeij Algorithm for Factoring Polynomials. In <i>The
    LLL Algorithm</i>. Springer Berlin Heidelberg. <a href="https://doi.org/10.1007/978-3-642-02295-1_8">https://doi.org/10.1007/978-3-642-02295-1_8</a>
  bibtex: '@inbook{Klüners_2009, place={Berlin, Heidelberg}, title={The van Hoeij
    Algorithm for Factoring Polynomials}, DOI={<a href="https://doi.org/10.1007/978-3-642-02295-1_8">10.1007/978-3-642-02295-1_8</a>},
    booktitle={The LLL Algorithm}, publisher={Springer Berlin Heidelberg}, author={Klüners,
    Jürgen}, year={2009} }'
  chicago: 'Klüners, Jürgen. “The van Hoeij Algorithm for Factoring Polynomials.”
    In <i>The LLL Algorithm</i>. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009.
    <a href="https://doi.org/10.1007/978-3-642-02295-1_8">https://doi.org/10.1007/978-3-642-02295-1_8</a>.'
  ieee: 'J. Klüners, “The van Hoeij Algorithm for Factoring Polynomials,” in <i>The
    LLL Algorithm</i>, Berlin, Heidelberg: Springer Berlin Heidelberg, 2009.'
  mla: Klüners, Jürgen. “The van Hoeij Algorithm for Factoring Polynomials.” <i>The
    LLL Algorithm</i>, Springer Berlin Heidelberg, 2009, doi:<a href="https://doi.org/10.1007/978-3-642-02295-1_8">10.1007/978-3-642-02295-1_8</a>.
  short: 'J. Klüners, in: The LLL Algorithm, Springer Berlin Heidelberg, Berlin, Heidelberg,
    2009.'
date_created: 2023-01-11T09:48:17Z
date_updated: 2023-03-06T09:10:34Z
department:
- _id: '102'
doi: 10.1007/978-3-642-02295-1_8
language:
- iso: eng
place: Berlin, Heidelberg
publication: The LLL Algorithm
publication_identifier:
  isbn:
  - '9783642022944'
  - '9783642022951'
  issn:
  - 1619-7100
publication_status: published
publisher: Springer Berlin Heidelberg
related_material:
  link:
  - relation: confirmation
    url: https://www.researchgate.net/profile/Juergen-Klueners/publication/226764840_The_van_Hoeij_Algorithm_for_Factoring_Polynomials/links/00463532f2216a64ae000000/The-van-Hoeij-Algorithm-for-Factoring-Polynomials.pdf?origin=publication_detail
status: public
title: The van Hoeij Algorithm for Factoring Polynomials
type: book_chapter
user_id: '93826'
year: '2009'
...
---
_id: '34895'
abstract:
- lang: eng
  text: 'We obtain strong information on the asymptotic behaviour of the counting
    function for nilpotent Galois extensions with bounded discriminant of arbitrary
    number fields. This extends previous investigations for the case of abelian groups.
    In particular, the result confirms a conjecture by the second author on this function
    for arbitrary groups in the nilpotent case. We further prove compatibility of
    the conjecture with taking wreath products with the cyclic group of order 2 and
    give examples in degree up to 8. '
author:
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
- first_name: G.
  full_name: Malle, G.
  last_name: Malle
citation:
  ama: Klüners J, Malle G. Counting nilpotent Galois extensions. <i>Journal für die
    reine und angewandte Mathematik (Crelles Journal)</i>. 2006;2004(572):1-26. doi:<a
    href="https://doi.org/10.1515/crll.2004.050">10.1515/crll.2004.050</a>
  apa: Klüners, J., &#38; Malle, G. (2006). Counting nilpotent Galois extensions.
    <i>Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)</i>, <i>2004</i>(572),
    1–26. <a href="https://doi.org/10.1515/crll.2004.050">https://doi.org/10.1515/crll.2004.050</a>
  bibtex: '@article{Klüners_Malle_2006, title={Counting nilpotent Galois extensions},
    volume={2004}, DOI={<a href="https://doi.org/10.1515/crll.2004.050">10.1515/crll.2004.050</a>},
    number={572}, journal={Journal für die reine und angewandte Mathematik (Crelles
    Journal)}, publisher={Walter de Gruyter GmbH}, author={Klüners, Jürgen and Malle,
    G.}, year={2006}, pages={1–26} }'
  chicago: 'Klüners, Jürgen, and G. Malle. “Counting Nilpotent Galois Extensions.”
    <i>Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)</i> 2004,
    no. 572 (2006): 1–26. <a href="https://doi.org/10.1515/crll.2004.050">https://doi.org/10.1515/crll.2004.050</a>.'
  ieee: 'J. Klüners and G. Malle, “Counting nilpotent Galois extensions,” <i>Journal
    für die reine und angewandte Mathematik (Crelles Journal)</i>, vol. 2004, no.
    572, pp. 1–26, 2006, doi: <a href="https://doi.org/10.1515/crll.2004.050">10.1515/crll.2004.050</a>.'
  mla: Klüners, Jürgen, and G. Malle. “Counting Nilpotent Galois Extensions.” <i>Journal
    Für Die Reine Und Angewandte Mathematik (Crelles Journal)</i>, vol. 2004, no.
    572, Walter de Gruyter GmbH, 2006, pp. 1–26, doi:<a href="https://doi.org/10.1515/crll.2004.050">10.1515/crll.2004.050</a>.
  short: J. Klüners, G. Malle, Journal Für Die Reine Und Angewandte Mathematik (Crelles
    Journal) 2004 (2006) 1–26.
date_created: 2022-12-23T09:50:49Z
date_updated: 2023-03-06T09:11:16Z
department:
- _id: '102'
doi: 10.1515/crll.2004.050
external_id:
  arxiv:
  - math/0112318
intvolume: '      2004'
issue: '572'
keyword:
- Applied Mathematics
- General Mathematics
language:
- iso: eng
page: 1-26
publication: Journal für die reine und angewandte Mathematik (Crelles Journal)
publication_identifier:
  issn:
  - 0075-4102
  - 1435-5345
publication_status: published
publisher: Walter de Gruyter GmbH
status: public
title: Counting nilpotent Galois extensions
type: journal_article
user_id: '93826'
volume: 2004
year: '2006'
...
---
_id: '34891'
abstract:
- lang: eng
  text: 'We study the asymptotics conjecture of Malle for dihedral groups Dℓ of order
    2ℓ, where ℓ is an odd prime. We prove the expected lower bound for those groups.
    For the upper bounds we show that there is a connection to class groups of quadratic
    number fields. The asymptotic behavior of those class groups is predicted by the
    Cohen--Lenstra heuristics. Under the assumption of this heuristic we are able
    to prove the expected upper bounds. '
author:
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
citation:
  ama: Klüners J. Asymptotics of number fields and the Cohen–Lenstra heuristics. <i>Journal
    de Théorie des Nombres de Bordeaux</i>. 2006;18(3):607-615. doi:<a href="https://doi.org/10.5802/jtnb.561">10.5802/jtnb.561</a>
  apa: Klüners, J. (2006). Asymptotics of number fields and the Cohen–Lenstra heuristics.
    <i>Journal de Théorie Des Nombres de Bordeaux</i>, <i>18</i>(3), 607–615. <a href="https://doi.org/10.5802/jtnb.561">https://doi.org/10.5802/jtnb.561</a>
  bibtex: '@article{Klüners_2006, title={Asymptotics of number fields and the Cohen–Lenstra
    heuristics}, volume={18}, DOI={<a href="https://doi.org/10.5802/jtnb.561">10.5802/jtnb.561</a>},
    number={3}, journal={Journal de Théorie des Nombres de Bordeaux}, publisher={Cellule
    MathDoc/CEDRAM}, author={Klüners, Jürgen}, year={2006}, pages={607–615} }'
  chicago: 'Klüners, Jürgen. “Asymptotics of Number Fields and the Cohen–Lenstra Heuristics.”
    <i>Journal de Théorie Des Nombres de Bordeaux</i> 18, no. 3 (2006): 607–15. <a
    href="https://doi.org/10.5802/jtnb.561">https://doi.org/10.5802/jtnb.561</a>.'
  ieee: 'J. Klüners, “Asymptotics of number fields and the Cohen–Lenstra heuristics,”
    <i>Journal de Théorie des Nombres de Bordeaux</i>, vol. 18, no. 3, pp. 607–615,
    2006, doi: <a href="https://doi.org/10.5802/jtnb.561">10.5802/jtnb.561</a>.'
  mla: Klüners, Jürgen. “Asymptotics of Number Fields and the Cohen–Lenstra Heuristics.”
    <i>Journal de Théorie Des Nombres de Bordeaux</i>, vol. 18, no. 3, Cellule MathDoc/CEDRAM,
    2006, pp. 607–15, doi:<a href="https://doi.org/10.5802/jtnb.561">10.5802/jtnb.561</a>.
  short: J. Klüners, Journal de Théorie Des Nombres de Bordeaux 18 (2006) 607–615.
date_created: 2022-12-23T09:37:01Z
date_updated: 2023-03-06T09:12:04Z
department:
- _id: '102'
doi: 10.5802/jtnb.561
external_id:
  arxiv:
  - 'math/0512260 '
intvolume: '        18'
issue: '3'
keyword:
- Algebra and Number Theory
language:
- iso: eng
page: 607-615
publication: Journal de Théorie des Nombres de Bordeaux
publication_identifier:
  issn:
  - 1246-7405
publication_status: published
publisher: Cellule MathDoc/CEDRAM
status: public
title: Asymptotics of number fields and the Cohen–Lenstra heuristics
type: journal_article
user_id: '93826'
volume: 18
year: '2006'
...
---
_id: '34890'
abstract:
- lang: eng
  text: 'We prove that the 4-rank of class groups of quadratic number fields behaves
    as predicted in an extension due to Gerth of the Cohen–Lenstra heuristics. '
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 4-rank of class groups of quadratic number fields.
    <i>Inventiones mathematicae</i>. 2006;167(3):455-513. doi:<a href="https://doi.org/10.1007/s00222-006-0021-2">10.1007/s00222-006-0021-2</a>
  apa: Fouvry, É., &#38; Klüners, J. (2006). On the 4-rank of class groups of quadratic
    number fields. <i>Inventiones Mathematicae</i>, <i>167</i>(3), 455–513. <a href="https://doi.org/10.1007/s00222-006-0021-2">https://doi.org/10.1007/s00222-006-0021-2</a>
  bibtex: '@article{Fouvry_Klüners_2006, title={On the 4-rank of class groups of quadratic
    number fields}, volume={167}, DOI={<a href="https://doi.org/10.1007/s00222-006-0021-2">10.1007/s00222-006-0021-2</a>},
    number={3}, journal={Inventiones mathematicae}, publisher={Springer Science and
    Business Media LLC}, author={Fouvry, Étienne and Klüners, Jürgen}, year={2006},
    pages={455–513} }'
  chicago: 'Fouvry, Étienne, and Jürgen Klüners. “On the 4-Rank of Class Groups of
    Quadratic Number Fields.” <i>Inventiones Mathematicae</i> 167, no. 3 (2006): 455–513.
    <a href="https://doi.org/10.1007/s00222-006-0021-2">https://doi.org/10.1007/s00222-006-0021-2</a>.'
  ieee: 'É. Fouvry and J. Klüners, “On the 4-rank of class groups of quadratic number
    fields,” <i>Inventiones mathematicae</i>, vol. 167, no. 3, pp. 455–513, 2006,
    doi: <a href="https://doi.org/10.1007/s00222-006-0021-2">10.1007/s00222-006-0021-2</a>.'
  mla: Fouvry, Étienne, and Jürgen Klüners. “On the 4-Rank of Class Groups of Quadratic
    Number Fields.” <i>Inventiones Mathematicae</i>, vol. 167, no. 3, Springer Science
    and Business Media LLC, 2006, pp. 455–513, doi:<a href="https://doi.org/10.1007/s00222-006-0021-2">10.1007/s00222-006-0021-2</a>.
  short: É. Fouvry, J. Klüners, Inventiones Mathematicae 167 (2006) 455–513.
date_created: 2022-12-23T09:36:15Z
date_updated: 2023-03-06T09:12:30Z
department:
- _id: '102'
doi: 10.1007/s00222-006-0021-2
intvolume: '       167'
issue: '3'
keyword:
- General Mathematics
language:
- iso: eng
page: 455-513
publication: Inventiones mathematicae
publication_identifier:
  issn:
  - 0020-9910
  - 1432-1297
publication_status: published
publisher: Springer Science and Business Media LLC
related_material:
  link:
  - relation: confirmation
    url: https://math.uni-paderborn.de/fileadmin/mathematik/AG-Computeralgebra/Publications-klueners/ranks.pdf
status: public
title: On the 4-rank of class groups of quadratic number fields
type: journal_article
user_id: '93826'
volume: 167
year: '2006'
...
---
_id: '35958'
abstract:
- lang: eng
  text: We establish a link between some heuristic asymptotic formulas (due to Cohen
    and Lenstra) concerning the moments of the p–part of the class groups of quadratic
    fields and formulas giving the frequency of the values of the p–rank of these
    class groups.
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. Cohen–Lenstra Heuristics of Quadratic Number Fields.
    In: <i>Lecture Notes in Computer Science</i>. Springer Berlin Heidelberg; 2006.
    doi:<a href="https://doi.org/10.1007/11792086_4">10.1007/11792086_4</a>'
  apa: Fouvry, É., &#38; Klüners, J. (2006). Cohen–Lenstra Heuristics of Quadratic
    Number Fields. In <i>Lecture Notes in Computer Science</i>. Springer Berlin Heidelberg.
    <a href="https://doi.org/10.1007/11792086_4">https://doi.org/10.1007/11792086_4</a>
  bibtex: '@inbook{Fouvry_Klüners_2006, place={Berlin, Heidelberg}, title={Cohen–Lenstra
    Heuristics of Quadratic Number Fields}, DOI={<a href="https://doi.org/10.1007/11792086_4">10.1007/11792086_4</a>},
    booktitle={Lecture Notes in Computer Science}, publisher={Springer Berlin Heidelberg},
    author={Fouvry, Étienne and Klüners, Jürgen}, year={2006} }'
  chicago: 'Fouvry, Étienne, and Jürgen Klüners. “Cohen–Lenstra Heuristics of Quadratic
    Number Fields.” In <i>Lecture Notes in Computer Science</i>. Berlin, Heidelberg:
    Springer Berlin Heidelberg, 2006. <a href="https://doi.org/10.1007/11792086_4">https://doi.org/10.1007/11792086_4</a>.'
  ieee: 'É. Fouvry and J. Klüners, “Cohen–Lenstra Heuristics of Quadratic Number Fields,”
    in <i>Lecture Notes in Computer Science</i>, Berlin, Heidelberg: Springer Berlin
    Heidelberg, 2006.'
  mla: Fouvry, Étienne, and Jürgen Klüners. “Cohen–Lenstra Heuristics of Quadratic
    Number Fields.” <i>Lecture Notes in Computer Science</i>, Springer Berlin Heidelberg,
    2006, doi:<a href="https://doi.org/10.1007/11792086_4">10.1007/11792086_4</a>.
  short: 'É. Fouvry, J. Klüners, in: Lecture Notes in Computer Science, Springer Berlin
    Heidelberg, Berlin, Heidelberg, 2006.'
date_created: 2023-01-11T09:46:47Z
date_updated: 2023-03-06T09:13:15Z
department:
- _id: '102'
doi: 10.1007/11792086_4
language:
- iso: eng
place: Berlin, Heidelberg
publication: Lecture Notes in Computer Science
publication_identifier:
  isbn:
  - '9783540360759'
  - '9783540360766'
  issn:
  - 0302-9743
  - 1611-3349
publication_status: published
publisher: Springer Berlin Heidelberg
related_material:
  link:
  - relation: confirmation
    url: https://www.researchgate.net/profile/Juergen-Klueners/publication/221451567_Cohen-Lenstra_Heuristics_of_Quadratic_Number_Fields/links/0a85e53298eaed8777000000/Cohen-Lenstra-Heuristics-of-Quadratic-Number-Fields.pdf?origin=publication_detail
status: public
title: Cohen–Lenstra Heuristics of Quadratic Number Fields
type: book_chapter
user_id: '93826'
year: '2006'
...
---
_id: '34892'
abstract:
- lang: eng
  text: 'We prove that the number of quartic S4--extensions of the rationals of given
    discriminant d is $O_\eps(d^{1/2+\eps})$ for all $\eps>0$. For a prime number
    p we derive that the dimension of the space of octahedral modular forms of weight
    1 and conductor p or p² is bounded above by O(p¹/²log(p)²). '
author:
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
citation:
  ama: Klüners J. The number of S₄-fields with given discriminant. <i>Acta Arithmetica</i>.
    2006;122(2):185-194. doi:<a href="https://doi.org/10.4064/aa122-2-3">10.4064/aa122-2-3</a>
  apa: Klüners, J. (2006). The number of S₄-fields with given discriminant. <i>Acta
    Arithmetica</i>, <i>122</i>(2), 185–194. <a href="https://doi.org/10.4064/aa122-2-3">https://doi.org/10.4064/aa122-2-3</a>
  bibtex: '@article{Klüners_2006, title={The number of S₄-fields with given discriminant},
    volume={122}, DOI={<a href="https://doi.org/10.4064/aa122-2-3">10.4064/aa122-2-3</a>},
    number={2}, journal={Acta Arithmetica}, publisher={Institute of Mathematics, Polish
    Academy of Sciences}, author={Klüners, Jürgen}, year={2006}, pages={185–194} }'
  chicago: 'Klüners, Jürgen. “The Number of S₄-Fields with given Discriminant.” <i>Acta
    Arithmetica</i> 122, no. 2 (2006): 185–94. <a href="https://doi.org/10.4064/aa122-2-3">https://doi.org/10.4064/aa122-2-3</a>.'
  ieee: 'J. Klüners, “The number of S₄-fields with given discriminant,” <i>Acta Arithmetica</i>,
    vol. 122, no. 2, pp. 185–194, 2006, doi: <a href="https://doi.org/10.4064/aa122-2-3">10.4064/aa122-2-3</a>.'
  mla: Klüners, Jürgen. “The Number of S₄-Fields with given Discriminant.” <i>Acta
    Arithmetica</i>, vol. 122, no. 2, Institute of Mathematics, Polish Academy of
    Sciences, 2006, pp. 185–94, doi:<a href="https://doi.org/10.4064/aa122-2-3">10.4064/aa122-2-3</a>.
  short: J. Klüners, Acta Arithmetica 122 (2006) 185–194.
date_created: 2022-12-23T09:40:25Z
date_updated: 2023-03-06T09:52:41Z
department:
- _id: '102'
doi: 10.4064/aa122-2-3
external_id:
  arxiv:
  - math/0411484
intvolume: '       122'
issue: '2'
keyword:
- Algebra and Number Theory
language:
- iso: eng
page: 185-194
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 number of S₄-fields with given discriminant
type: journal_article
user_id: '93826'
volume: 122
year: '2006'
...
---
_id: '34894'
abstract:
- lang: eng
  text: 'In this Note we give a counter example to a conjecture of Malle which predicts
    the asymptotic behavior of the counting functions for field extensions with given
    Galois group and bounded discriminant. '
author:
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
citation:
  ama: Klüners J. A counter example to Malle’s conjecture on the asymptotics of discriminants.
    <i>Comptes Rendus Mathematique</i>. 2005;340(6):411-414. doi:<a href="https://doi.org/10.1016/j.crma.2005.02.010">10.1016/j.crma.2005.02.010</a>
  apa: Klüners, J. (2005). A counter example to Malle’s conjecture on the asymptotics
    of discriminants. <i>Comptes Rendus Mathematique</i>, <i>340</i>(6), 411–414.
    <a href="https://doi.org/10.1016/j.crma.2005.02.010">https://doi.org/10.1016/j.crma.2005.02.010</a>
  bibtex: '@article{Klüners_2005, title={A counter example to Malle’s conjecture on
    the asymptotics of discriminants}, volume={340}, DOI={<a href="https://doi.org/10.1016/j.crma.2005.02.010">10.1016/j.crma.2005.02.010</a>},
    number={6}, journal={Comptes Rendus Mathematique}, publisher={Elsevier BV}, author={Klüners,
    Jürgen}, year={2005}, pages={411–414} }'
  chicago: 'Klüners, Jürgen. “A Counter Example to Malle’s Conjecture on the Asymptotics
    of Discriminants.” <i>Comptes Rendus Mathematique</i> 340, no. 6 (2005): 411–14.
    <a href="https://doi.org/10.1016/j.crma.2005.02.010">https://doi.org/10.1016/j.crma.2005.02.010</a>.'
  ieee: 'J. Klüners, “A counter example to Malle’s conjecture on the asymptotics of
    discriminants,” <i>Comptes Rendus Mathematique</i>, vol. 340, no. 6, pp. 411–414,
    2005, doi: <a href="https://doi.org/10.1016/j.crma.2005.02.010">10.1016/j.crma.2005.02.010</a>.'
  mla: Klüners, Jürgen. “A Counter Example to Malle’s Conjecture on the Asymptotics
    of Discriminants.” <i>Comptes Rendus Mathematique</i>, vol. 340, no. 6, Elsevier
    BV, 2005, pp. 411–14, doi:<a href="https://doi.org/10.1016/j.crma.2005.02.010">10.1016/j.crma.2005.02.010</a>.
  short: J. Klüners, Comptes Rendus Mathematique 340 (2005) 411–414.
date_created: 2022-12-23T09:44:05Z
date_updated: 2023-03-06T09:15:10Z
department:
- _id: '102'
doi: 10.1016/j.crma.2005.02.010
external_id:
  arxiv:
  - 'math/0411486 '
intvolume: '       340'
issue: '6'
keyword:
- General Mathematics
language:
- iso: eng
page: 411-414
publication: Comptes Rendus Mathematique
publication_identifier:
  issn:
  - 1631-073X
publication_status: published
publisher: Elsevier BV
status: public
title: A counter example to Malle's conjecture on the asymptotics of discriminants
type: journal_article
user_id: '93826'
volume: 340
year: '2005'
...
---
_id: '34893'
abstract:
- lang: eng
  text: Let K be a global field and O be an order of K. We develop algorithms for
    the computation of the unit group of residue class rings for ideals O in . As
    an application we show how to compute the unit group and the Picard group of O
    provided that we are able to compute the unit group and class group of the maximal
    order O of K.
author:
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
- first_name: Sebastian
  full_name: Pauli, Sebastian
  last_name: Pauli
citation:
  ama: Klüners J, Pauli S. Computing residue class rings and Picard groups of orders.
    <i>Journal of Algebra</i>. 2005;292(1):47-64. doi:<a href="https://doi.org/10.1016/j.jalgebra.2005.04.013">10.1016/j.jalgebra.2005.04.013</a>
  apa: Klüners, J., &#38; Pauli, S. (2005). Computing residue class rings and Picard
    groups of orders. <i>Journal of Algebra</i>, <i>292</i>(1), 47–64. <a href="https://doi.org/10.1016/j.jalgebra.2005.04.013">https://doi.org/10.1016/j.jalgebra.2005.04.013</a>
  bibtex: '@article{Klüners_Pauli_2005, title={Computing residue class rings and Picard
    groups of orders}, volume={292}, DOI={<a href="https://doi.org/10.1016/j.jalgebra.2005.04.013">10.1016/j.jalgebra.2005.04.013</a>},
    number={1}, journal={Journal of Algebra}, publisher={Elsevier BV}, author={Klüners,
    Jürgen and Pauli, Sebastian}, year={2005}, pages={47–64} }'
  chicago: 'Klüners, Jürgen, and Sebastian Pauli. “Computing Residue Class Rings and
    Picard Groups of Orders.” <i>Journal of Algebra</i> 292, no. 1 (2005): 47–64.
    <a href="https://doi.org/10.1016/j.jalgebra.2005.04.013">https://doi.org/10.1016/j.jalgebra.2005.04.013</a>.'
  ieee: 'J. Klüners and S. Pauli, “Computing residue class rings and Picard groups
    of orders,” <i>Journal of Algebra</i>, vol. 292, no. 1, pp. 47–64, 2005, doi:
    <a href="https://doi.org/10.1016/j.jalgebra.2005.04.013">10.1016/j.jalgebra.2005.04.013</a>.'
  mla: Klüners, Jürgen, and Sebastian Pauli. “Computing Residue Class Rings and Picard
    Groups of Orders.” <i>Journal of Algebra</i>, vol. 292, no. 1, Elsevier BV, 2005,
    pp. 47–64, doi:<a href="https://doi.org/10.1016/j.jalgebra.2005.04.013">10.1016/j.jalgebra.2005.04.013</a>.
  short: J. Klüners, S. Pauli, Journal of Algebra 292 (2005) 47–64.
date_created: 2022-12-23T09:41:06Z
date_updated: 2023-03-06T09:55:09Z
department:
- _id: '102'
doi: 10.1016/j.jalgebra.2005.04.013
intvolume: '       292'
issue: '1'
keyword:
- Algebra and Number Theory
language:
- iso: eng
page: 47-64
publication: Journal of Algebra
publication_identifier:
  issn:
  - 0021-8693
publication_status: published
publisher: Elsevier BV
status: public
title: Computing residue class rings and Picard groups of orders
type: journal_article
user_id: '93826'
volume: 292
year: '2005'
...
---
_id: '42807'
author:
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
citation:
  ama: Klüners J. <i>Über die Asymptotik von Zahlkörpern mit vorgegebener Galoisgruppe
    (Habilitation)</i>. Shaker Verlag; 2005.
  apa: Klüners, J. (2005). <i>Über die Asymptotik von Zahlkörpern mit vorgegebener
    Galoisgruppe (Habilitation)</i>. Shaker Verlag.
  bibtex: '@book{Klüners_2005, place={Universiät Kassel}, title={Über die Asymptotik
    von Zahlkörpern mit vorgegebener Galoisgruppe (Habilitation)}, publisher={Shaker
    Verlag}, author={Klüners, Jürgen}, year={2005} }'
  chicago: 'Klüners, Jürgen. <i>Über die Asymptotik von Zahlkörpern mit vorgegebener
    Galoisgruppe (Habilitation)</i>. Universiät Kassel: Shaker Verlag, 2005.'
  ieee: 'J. Klüners, <i>Über die Asymptotik von Zahlkörpern mit vorgegebener Galoisgruppe
    (Habilitation)</i>. Universiät Kassel: Shaker Verlag, 2005.'
  mla: Klüners, Jürgen. <i>Über die Asymptotik von Zahlkörpern mit vorgegebener Galoisgruppe
    (Habilitation)</i>. Shaker Verlag, 2005.
  short: J. Klüners, Über die Asymptotik von Zahlkörpern mit vorgegebener Galoisgruppe
    (Habilitation), Shaker Verlag, Universiät Kassel, 2005.
date_created: 2023-03-07T09:05:29Z
date_updated: 2023-04-11T08:13:38Z
department:
- _id: '102'
extern: '1'
language:
- iso: ger
page: '114'
place: Universiät Kassel
publication_identifier:
  isbn:
  - 978-3-8322-4003-5
publisher: Shaker Verlag
status: public
title: Über die Asymptotik von Zahlkörpern mit vorgegebener Galoisgruppe (Habilitation)
type: misc
user_id: '93826'
year: '2005'
...
---
_id: '34896'
abstract:
- lang: eng
  text: We apply class field theory to the computation of the minimal discriminants
    for certain solvable groups. In particular, we apply our techniques to small Frobenius
    groups and all imprimitive degree 8 groups such that the corresponding fields
    have only a degree 2 and no degree 4 subfield.
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. Minimal discriminants for fields with small Frobenius
    groups as Galois groups. <i>Journal of Number Theory</i>. 2003;99(2):318-337.
    doi:<a href="https://doi.org/10.1016/s0022-314x(02)00071-9">10.1016/s0022-314x(02)00071-9</a>
  apa: Fieker, C., &#38; Klüners, J. (2003). Minimal discriminants for fields with
    small Frobenius groups as Galois groups. <i>Journal of Number Theory</i>, <i>99</i>(2),
    318–337. <a href="https://doi.org/10.1016/s0022-314x(02)00071-9">https://doi.org/10.1016/s0022-314x(02)00071-9</a>
  bibtex: '@article{Fieker_Klüners_2003, title={Minimal discriminants for fields with
    small Frobenius groups as Galois groups}, volume={99}, DOI={<a href="https://doi.org/10.1016/s0022-314x(02)00071-9">10.1016/s0022-314x(02)00071-9</a>},
    number={2}, journal={Journal of Number Theory}, publisher={Elsevier BV}, author={Fieker,
    Claus and Klüners, Jürgen}, year={2003}, pages={318–337} }'
  chicago: 'Fieker, Claus, and Jürgen Klüners. “Minimal Discriminants for Fields with
    Small Frobenius Groups as Galois Groups.” <i>Journal of Number Theory</i> 99,
    no. 2 (2003): 318–37. <a href="https://doi.org/10.1016/s0022-314x(02)00071-9">https://doi.org/10.1016/s0022-314x(02)00071-9</a>.'
  ieee: 'C. Fieker and J. Klüners, “Minimal discriminants for fields with small Frobenius
    groups as Galois groups,” <i>Journal of Number Theory</i>, vol. 99, no. 2, pp.
    318–337, 2003, doi: <a href="https://doi.org/10.1016/s0022-314x(02)00071-9">10.1016/s0022-314x(02)00071-9</a>.'
  mla: Fieker, Claus, and Jürgen Klüners. “Minimal Discriminants for Fields with Small
    Frobenius Groups as Galois Groups.” <i>Journal of Number Theory</i>, vol. 99,
    no. 2, Elsevier BV, 2003, pp. 318–37, doi:<a href="https://doi.org/10.1016/s0022-314x(02)00071-9">10.1016/s0022-314x(02)00071-9</a>.
  short: C. Fieker, J. Klüners, Journal of Number Theory 99 (2003) 318–337.
date_created: 2022-12-23T09:53:23Z
date_updated: 2023-03-06T09:19:16Z
department:
- _id: '102'
doi: 10.1016/s0022-314x(02)00071-9
intvolume: '        99'
issue: '2'
keyword:
- Algebra and Number Theory
language:
- iso: eng
page: 318-337
publication: Journal of Number Theory
publication_identifier:
  issn:
  - 0022-314X
publication_status: published
publisher: Elsevier BV
status: public
title: Minimal discriminants for fields with small Frobenius groups as Galois groups
type: journal_article
user_id: '93826'
volume: 99
year: '2003'
...
---
_id: '35954'
abstract:
- lang: eng
  text: 'Let {\ASIE K}\,/{\small \ℚ}({\ASIE t \!}) be a finite extension. We describe
    algorithms for computing subfields and automorphisms of {\ASIE K}\,/{\small \ℚ}({\ASIE
    t }\!). As an application we give an algorithm for finding decompositions of rational
    functions in {\small \ℚ(α)}. We also present an algorithm which decides if an
    extension {\ASIE L}\,/{\small \ℚ}({\ASIE t \!}) is a subfield of {\ASIE K}. In
    case [{\ASIE K : \;}{\small\ℚ}({\ASIE t \!})] = [{\ASIE L : \;}{\small \ℚ}({\ASIE
    t \!})] we obtain a {\small \ℚ}({\ASIE t \!})-isomorphism test. Furthermore, we
    describe an algorithm which computes subfields of the normal closure of {\ASIE
    K}\,/{\small \ℚ}({\ASIE t \!}).'
author:
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
citation:
  ama: Klüners J. Algorithms for function fields. <i>Experiment Math </i>. 2002;11(2):171-181.
  apa: Klüners, J. (2002). Algorithms for function fields. <i>Experiment. Math. </i>,
    <i>11</i>(2), 171–181.
  bibtex: '@article{Klüners_2002, title={Algorithms for function fields}, volume={11},
    number={2}, journal={Experiment. Math. }, publisher={Elsevier BV}, author={Klüners,
    Jürgen}, year={2002}, pages={171–181} }'
  chicago: 'Klüners, Jürgen. “Algorithms for Function Fields.” <i>Experiment. Math.
    </i> 11, no. 2 (2002): 171–81.'
  ieee: J. Klüners, “Algorithms for function fields,” <i>Experiment. Math. </i>, vol.
    11, no. 2, pp. 171–181, 2002.
  mla: Klüners, Jürgen. “Algorithms for Function Fields.” <i>Experiment. Math. </i>,
    vol. 11, no. 2, Elsevier BV, 2002, pp. 171–81.
  short: J. Klüners, Experiment. Math.  11 (2002) 171–181.
date_created: 2023-01-11T09:45:40Z
date_updated: 2023-03-06T10:26:58Z
department:
- _id: '102'
intvolume: '        11'
issue: '2'
keyword:
- algorithms
- decompositions
- Galois groups
- subfields
language:
- iso: eng
page: 171-181
publication: 'Experiment. Math. '
publication_status: published
publisher: Elsevier BV
related_material:
  link:
  - relation: confirmation
    url: https://projecteuclid.org/journals/experimental-mathematics/volume-11/issue-2/Algorithms-for-function-fields/em/1062621213.full
status: public
title: Algorithms for function fields
type: journal_article
user_id: '93826'
volume: 11
year: '2002'
...
---
_id: '34897'
abstract:
- lang: eng
  text: This paper announces the creation of a database for number fields. It describes
    the contents and the methods of access, indicates the origin of the polynomials,
    and formulates the aims of this collection of fields.
author:
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
- first_name: Gunter
  full_name: Malle, Gunter
  last_name: Malle
citation:
  ama: Klüners J, Malle G. A Database for Field Extensions of the Rationals. <i>LMS
    Journal of Computation and Mathematics</i>. 2001;4:182-196. doi:<a href="https://doi.org/10.1112/s1461157000000851">10.1112/s1461157000000851</a>
  apa: Klüners, J., &#38; Malle, G. (2001). A Database for Field Extensions of the
    Rationals. <i>LMS Journal of Computation and Mathematics</i>, <i>4</i>, 182–196.
    <a href="https://doi.org/10.1112/s1461157000000851">https://doi.org/10.1112/s1461157000000851</a>
  bibtex: '@article{Klüners_Malle_2001, title={A Database for Field Extensions of
    the Rationals}, volume={4}, DOI={<a href="https://doi.org/10.1112/s1461157000000851">10.1112/s1461157000000851</a>},
    journal={LMS Journal of Computation and Mathematics}, publisher={Wiley}, author={Klüners,
    Jürgen and Malle, Gunter}, year={2001}, pages={182–196} }'
  chicago: 'Klüners, Jürgen, and Gunter Malle. “A Database for Field Extensions of
    the Rationals.” <i>LMS Journal of Computation and Mathematics</i> 4 (2001): 182–96.
    <a href="https://doi.org/10.1112/s1461157000000851">https://doi.org/10.1112/s1461157000000851</a>.'
  ieee: 'J. Klüners and G. Malle, “A Database for Field Extensions of the Rationals,”
    <i>LMS Journal of Computation and Mathematics</i>, vol. 4, pp. 182–196, 2001,
    doi: <a href="https://doi.org/10.1112/s1461157000000851">10.1112/s1461157000000851</a>.'
  mla: Klüners, Jürgen, and Gunter Malle. “A Database for Field Extensions of the
    Rationals.” <i>LMS Journal of Computation and Mathematics</i>, vol. 4, Wiley,
    2001, pp. 182–96, doi:<a href="https://doi.org/10.1112/s1461157000000851">10.1112/s1461157000000851</a>.
  short: J. Klüners, G. Malle, LMS Journal of Computation and Mathematics 4 (2001)
    182–196.
date_created: 2022-12-23T09:56:22Z
date_updated: 2023-03-02T09:53:08Z
department:
- _id: '102'
doi: 10.1112/s1461157000000851
external_id:
  arxiv:
  - math/0102232
intvolume: '         4'
keyword:
- Computational Theory and Mathematics
- General Mathematics
language:
- iso: eng
page: 182-196
publication: LMS Journal of Computation and Mathematics
publication_identifier:
  issn:
  - 1461-1570
publication_status: published
publisher: Wiley
status: public
title: A Database for Field Extensions of the Rationals
type: journal_article
user_id: '93826'
volume: 4
year: '2001'
...
---
_id: '34900'
abstract:
- lang: eng
  text: We describe methods for the computation of Galois groups of univariate polynomials
    over the rationals which we have implemented up to degree 15. These methods are
    based on Stauduhar’s algorithm. All computations are done in unramified p -adic
    extensions. For imprimitive groups we give an improvement using subfields. In
    the primitive case we use known subgroups of the Galois group together with a
    combination of Stauduhar’s method and the absolute resolvent method.
author:
- first_name: Katharina
  full_name: Geissler, Katharina
  last_name: Geissler
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
citation:
  ama: Geissler K, Klüners J. Galois Group Computation for Rational Polynomials. <i>Journal
    of Symbolic Computation</i>. 2000;30(6):653-674. doi:<a href="https://doi.org/10.1006/jsco.2000.0377">10.1006/jsco.2000.0377</a>
  apa: Geissler, K., &#38; Klüners, J. (2000). Galois Group Computation for Rational
    Polynomials. <i>Journal of Symbolic Computation</i>, <i>30</i>(6), 653–674. <a
    href="https://doi.org/10.1006/jsco.2000.0377">https://doi.org/10.1006/jsco.2000.0377</a>
  bibtex: '@article{Geissler_Klüners_2000, title={Galois Group Computation for Rational
    Polynomials}, volume={30}, DOI={<a href="https://doi.org/10.1006/jsco.2000.0377">10.1006/jsco.2000.0377</a>},
    number={6}, journal={Journal of Symbolic Computation}, publisher={Elsevier BV},
    author={Geissler, Katharina and Klüners, Jürgen}, year={2000}, pages={653–674}
    }'
  chicago: 'Geissler, Katharina, and Jürgen Klüners. “Galois Group Computation for
    Rational Polynomials.” <i>Journal of Symbolic Computation</i> 30, no. 6 (2000):
    653–74. <a href="https://doi.org/10.1006/jsco.2000.0377">https://doi.org/10.1006/jsco.2000.0377</a>.'
  ieee: 'K. Geissler and J. Klüners, “Galois Group Computation for Rational Polynomials,”
    <i>Journal of Symbolic Computation</i>, vol. 30, no. 6, pp. 653–674, 2000, doi:
    <a href="https://doi.org/10.1006/jsco.2000.0377">10.1006/jsco.2000.0377</a>.'
  mla: Geissler, Katharina, and Jürgen Klüners. “Galois Group Computation for Rational
    Polynomials.” <i>Journal of Symbolic Computation</i>, vol. 30, no. 6, Elsevier
    BV, 2000, pp. 653–74, doi:<a href="https://doi.org/10.1006/jsco.2000.0377">10.1006/jsco.2000.0377</a>.
  short: K. Geissler, J. Klüners, Journal of Symbolic Computation 30 (2000) 653–674.
date_created: 2022-12-23T09:58:16Z
date_updated: 2023-03-06T09:58:06Z
department:
- _id: '102'
doi: 10.1006/jsco.2000.0377
intvolume: '        30'
issue: '6'
keyword:
- Computational Mathematics
- Algebra and Number Theory
language:
- iso: eng
page: 653-674
publication: Journal of Symbolic Computation
publication_identifier:
  issn:
  - 0747-7171
publication_status: published
publisher: Elsevier BV
status: public
title: Galois Group Computation for Rational Polynomials
type: journal_article
user_id: '93826'
volume: 30
year: '2000'
...
---
_id: '34901'
abstract:
- lang: eng
  text: Let L = K(α) be an Abelian extension of degree n of a number field K, given
    by the minimal polynomial of α over K. We describe an algorithm for computing
    the local Artin map associated with the extension L / K at a finite or infinite
    prime v of K. We apply this algorithm to decide if a nonzero a ∈ K is a norm from
    L, assuming that L / K is cyclic.
author:
- first_name: Vincenzo
  full_name: Acciaro, Vincenzo
  last_name: Acciaro
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
citation:
  ama: Acciaro V, Klüners J. Computing Local Artin Maps, and Solvability of Norm Equations.
    <i>Journal of Symbolic Computation</i>. 2000;30(3):239-252. doi:<a href="https://doi.org/10.1006/jsco.2000.0361">10.1006/jsco.2000.0361</a>
  apa: Acciaro, V., &#38; Klüners, J. (2000). Computing Local Artin Maps, and Solvability
    of Norm Equations. <i>Journal of Symbolic Computation</i>, <i>30</i>(3), 239–252.
    <a href="https://doi.org/10.1006/jsco.2000.0361">https://doi.org/10.1006/jsco.2000.0361</a>
  bibtex: '@article{Acciaro_Klüners_2000, title={Computing Local Artin Maps, and Solvability
    of Norm Equations}, volume={30}, DOI={<a href="https://doi.org/10.1006/jsco.2000.0361">10.1006/jsco.2000.0361</a>},
    number={3}, journal={Journal of Symbolic Computation}, publisher={Elsevier BV},
    author={Acciaro, Vincenzo and Klüners, Jürgen}, year={2000}, pages={239–252} }'
  chicago: 'Acciaro, Vincenzo, and Jürgen Klüners. “Computing Local Artin Maps, and
    Solvability of Norm Equations.” <i>Journal of Symbolic Computation</i> 30, no.
    3 (2000): 239–52. <a href="https://doi.org/10.1006/jsco.2000.0361">https://doi.org/10.1006/jsco.2000.0361</a>.'
  ieee: 'V. Acciaro and J. Klüners, “Computing Local Artin Maps, and Solvability of
    Norm Equations,” <i>Journal of Symbolic Computation</i>, vol. 30, no. 3, pp. 239–252,
    2000, doi: <a href="https://doi.org/10.1006/jsco.2000.0361">10.1006/jsco.2000.0361</a>.'
  mla: Acciaro, Vincenzo, and Jürgen Klüners. “Computing Local Artin Maps, and Solvability
    of Norm Equations.” <i>Journal of Symbolic Computation</i>, vol. 30, no. 3, Elsevier
    BV, 2000, pp. 239–52, doi:<a href="https://doi.org/10.1006/jsco.2000.0361">10.1006/jsco.2000.0361</a>.
  short: V. Acciaro, J. Klüners, Journal of Symbolic Computation 30 (2000) 239–252.
date_created: 2022-12-23T09:58:48Z
date_updated: 2023-03-06T09:57:34Z
department:
- _id: '102'
doi: 10.1006/jsco.2000.0361
intvolume: '        30'
issue: '3'
keyword:
- Computational Mathematics
- Algebra and Number Theory
language:
- iso: eng
page: 239-252
publication: Journal of Symbolic Computation
publication_identifier:
  issn:
  - 0747-7171
publication_status: published
publisher: Elsevier BV
status: public
title: Computing Local Artin Maps, and Solvability of Norm Equations
type: journal_article
user_id: '93826'
volume: 30
year: '2000'
...
---
_id: '34899'
abstract:
- lang: eng
  text: We describe methods for the construction of polynomials with certain types
    of Galois groups. As an application we deduce that all transitive groups G up
    to degree 15 occur as Galois groups of regular extensions of ℚ (t), and in each
    case compute a polynomial f ∈ ℚ [ x ] with Gal(f)  = G.
author:
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
- first_name: Gunter
  full_name: Malle, Gunter
  last_name: Malle
citation:
  ama: Klüners J, Malle G. Explicit Galois Realization of Transitive Groups of Degree
    up to 15. <i>Journal of Symbolic Computation</i>. 2000;30(6):675-716. doi:<a href="https://doi.org/10.1006/jsco.2000.0378">10.1006/jsco.2000.0378</a>
  apa: Klüners, J., &#38; Malle, G. (2000). Explicit Galois Realization of Transitive
    Groups of Degree up to 15. <i>Journal of Symbolic Computation</i>, <i>30</i>(6),
    675–716. <a href="https://doi.org/10.1006/jsco.2000.0378">https://doi.org/10.1006/jsco.2000.0378</a>
  bibtex: '@article{Klüners_Malle_2000, title={Explicit Galois Realization of Transitive
    Groups of Degree up to 15}, volume={30}, DOI={<a href="https://doi.org/10.1006/jsco.2000.0378">10.1006/jsco.2000.0378</a>},
    number={6}, journal={Journal of Symbolic Computation}, publisher={Elsevier BV},
    author={Klüners, Jürgen and Malle, Gunter}, year={2000}, pages={675–716} }'
  chicago: 'Klüners, Jürgen, and Gunter Malle. “Explicit Galois Realization of Transitive
    Groups of Degree up to 15.” <i>Journal of Symbolic Computation</i> 30, no. 6 (2000):
    675–716. <a href="https://doi.org/10.1006/jsco.2000.0378">https://doi.org/10.1006/jsco.2000.0378</a>.'
  ieee: 'J. Klüners and G. Malle, “Explicit Galois Realization of Transitive Groups
    of Degree up to 15,” <i>Journal of Symbolic Computation</i>, vol. 30, no. 6, pp.
    675–716, 2000, doi: <a href="https://doi.org/10.1006/jsco.2000.0378">10.1006/jsco.2000.0378</a>.'
  mla: Klüners, Jürgen, and Gunter Malle. “Explicit Galois Realization of Transitive
    Groups of Degree up to 15.” <i>Journal of Symbolic Computation</i>, vol. 30, no.
    6, Elsevier BV, 2000, pp. 675–716, doi:<a href="https://doi.org/10.1006/jsco.2000.0378">10.1006/jsco.2000.0378</a>.
  short: J. Klüners, G. Malle, Journal of Symbolic Computation 30 (2000) 675–716.
date_created: 2022-12-23T09:57:28Z
date_updated: 2023-03-06T10:48:05Z
department:
- _id: '102'
doi: 10.1006/jsco.2000.0378
intvolume: '        30'
issue: '6'
keyword:
- Computational Mathematics
- Algebra and Number Theory
language:
- iso: eng
page: 675-716
publication: Journal of Symbolic Computation
publication_identifier:
  issn:
  - 0747-7171
publication_status: published
publisher: Elsevier BV
status: public
title: Explicit Galois Realization of Transitive Groups of Degree up to 15
type: journal_article
user_id: '93826'
volume: 30
year: '2000'
...
---
_id: '34898'
abstract:
- lang: eng
  text: We compute a polynomial with Galois group SL₂(11) over ℚ. Furthermore we prove
    that SL₂(11) is the Galois group of a regular extension of ℚ (t).
author:
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
citation:
  ama: Klüners J. A Polynomial with Galois GroupSL2(11). <i>Journal of Symbolic Computation</i>.
    2000;30(6):733-737. doi:<a href="https://doi.org/10.1006/jsco.2000.0380">10.1006/jsco.2000.0380</a>
  apa: Klüners, J. (2000). A Polynomial with Galois GroupSL2(11). <i>Journal of Symbolic
    Computation</i>, <i>30</i>(6), 733–737. <a href="https://doi.org/10.1006/jsco.2000.0380">https://doi.org/10.1006/jsco.2000.0380</a>
  bibtex: '@article{Klüners_2000, title={A Polynomial with Galois GroupSL2(11)}, volume={30},
    DOI={<a href="https://doi.org/10.1006/jsco.2000.0380">10.1006/jsco.2000.0380</a>},
    number={6}, journal={Journal of Symbolic Computation}, publisher={Elsevier BV},
    author={Klüners, Jürgen}, year={2000}, pages={733–737} }'
  chicago: 'Klüners, Jürgen. “A Polynomial with Galois GroupSL2(11).” <i>Journal of
    Symbolic Computation</i> 30, no. 6 (2000): 733–37. <a href="https://doi.org/10.1006/jsco.2000.0380">https://doi.org/10.1006/jsco.2000.0380</a>.'
  ieee: 'J. Klüners, “A Polynomial with Galois GroupSL2(11),” <i>Journal of Symbolic
    Computation</i>, vol. 30, no. 6, pp. 733–737, 2000, doi: <a href="https://doi.org/10.1006/jsco.2000.0380">10.1006/jsco.2000.0380</a>.'
  mla: Klüners, Jürgen. “A Polynomial with Galois GroupSL2(11).” <i>Journal of Symbolic
    Computation</i>, vol. 30, no. 6, Elsevier BV, 2000, pp. 733–37, doi:<a href="https://doi.org/10.1006/jsco.2000.0380">10.1006/jsco.2000.0380</a>.
  short: J. Klüners, Journal of Symbolic Computation 30 (2000) 733–737.
date_created: 2022-12-23T09:56:52Z
date_updated: 2023-03-06T10:48:40Z
department:
- _id: '102'
doi: 10.1006/jsco.2000.0380
intvolume: '        30'
issue: '6'
keyword:
- Computational Mathematics
- Algebra and Number Theory
language:
- iso: eng
page: 733-737
publication: Journal of Symbolic Computation
publication_identifier:
  issn:
  - 0747-7171
publication_status: published
publisher: Elsevier BV
status: public
title: A Polynomial with Galois GroupSL2(11)
type: journal_article
user_id: '93826'
volume: 30
year: '2000'
...
---
_id: '34902'
abstract:
- lang: eng
  text: We present a new polynomial decomposition which generalizes the functional
    and homogeneous bivariate decomposition of irreducible monic polynomials in one
    variable over the rationals. With these decompositions it is possible to calculate
    the roots of an imprimitive polynomial by solving polynomial equations of lower
    degree.
author:
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
citation:
  ama: Klüners J. On Polynomial Decompositions. <i>Journal of Symbolic Computation</i>.
    1999;27(3):261-269. doi:<a href="https://doi.org/10.1006/jsco.1998.0252">10.1006/jsco.1998.0252</a>
  apa: Klüners, J. (1999). On Polynomial Decompositions. <i>Journal of Symbolic Computation</i>,
    <i>27</i>(3), 261–269. <a href="https://doi.org/10.1006/jsco.1998.0252">https://doi.org/10.1006/jsco.1998.0252</a>
  bibtex: '@article{Klüners_1999, title={On Polynomial Decompositions}, volume={27},
    DOI={<a href="https://doi.org/10.1006/jsco.1998.0252">10.1006/jsco.1998.0252</a>},
    number={3}, journal={Journal of Symbolic Computation}, publisher={Elsevier BV},
    author={Klüners, Jürgen}, year={1999}, pages={261–269} }'
  chicago: 'Klüners, Jürgen. “On Polynomial Decompositions.” <i>Journal of Symbolic
    Computation</i> 27, no. 3 (1999): 261–69. <a href="https://doi.org/10.1006/jsco.1998.0252">https://doi.org/10.1006/jsco.1998.0252</a>.'
  ieee: 'J. Klüners, “On Polynomial Decompositions,” <i>Journal of Symbolic Computation</i>,
    vol. 27, no. 3, pp. 261–269, 1999, doi: <a href="https://doi.org/10.1006/jsco.1998.0252">10.1006/jsco.1998.0252</a>.'
  mla: Klüners, Jürgen. “On Polynomial Decompositions.” <i>Journal of Symbolic Computation</i>,
    vol. 27, no. 3, Elsevier BV, 1999, pp. 261–69, doi:<a href="https://doi.org/10.1006/jsco.1998.0252">10.1006/jsco.1998.0252</a>.
  short: J. Klüners, Journal of Symbolic Computation 27 (1999) 261–269.
date_created: 2022-12-23T10:01:15Z
date_updated: 2023-03-06T09:21:29Z
department:
- _id: '102'
doi: 10.1006/jsco.1998.0252
intvolume: '        27'
issue: '3'
keyword:
- Computational Mathematics
- Algebra and Number Theory
language:
- iso: eng
page: 261-269
publication: Journal of Symbolic Computation
publication_identifier:
  issn:
  - 0747-7171
publication_status: published
publisher: Elsevier BV
status: public
title: On Polynomial Decompositions
type: journal_article
user_id: '93826'
volume: 27
year: '1999'
...
---
_id: '35941'
abstract:
- lang: eng
  text: "Let L = ℚ(α) be an abelian number field of degree n. Most\r\nalgorithms for
    computing the lattice of subfields of L require the computation\r\nof all the
    conjugates of α. This is usually achieved by factoring the minimal\r\npolynomial
    mα(x) of α over L. In practice, the existing algorithms for factoring\r\npolynomials
    over algebraic number fields can handle only problems of moderate\r\nsize. In
    this paper we describe a fast probabilistic algorithm for computing\r\nthe conjugates
    of α, which is based on p-adic techniques. Given mα(x) and a\r\nrational prime
    p which does not divide the discriminant disc(mα(x)) of mα(x),\r\nthe algorithm
    computes the Frobenius automorphism of p in time polynomial\r\nin the size of
    p and in the size of mα(x). By repeatedly applying the algorithm\r\nto randomly
    chosen primes it is possible to compute all the conjugates of α."
author:
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
- first_name: Vincenzo
  full_name: Acciaro, Vincenzo
  last_name: Acciaro
citation:
  ama: Klüners J, Acciaro V. Computing Automorphisms of Abelian Number Fields. <i>Mathematics
    of Computation</i>. 1999;68(227):1179-1186.
  apa: Klüners, J., &#38; Acciaro, V. (1999). Computing Automorphisms of Abelian Number
    Fields. <i>Mathematics of Computation</i>, <i>68</i>(227), 1179–1186.
  bibtex: '@article{Klüners_Acciaro_1999, title={Computing Automorphisms of Abelian
    Number Fields}, volume={68}, number={227}, journal={Mathematics of Computation},
    publisher={American Mathematical Society (AMS)}, author={Klüners, Jürgen and Acciaro,
    Vincenzo}, year={1999}, pages={1179–1186} }'
  chicago: 'Klüners, Jürgen, and Vincenzo Acciaro. “Computing Automorphisms of Abelian
    Number Fields.” <i>Mathematics of Computation</i> 68, no. 227 (1999): 1179–86.'
  ieee: J. Klüners and V. Acciaro, “Computing Automorphisms of Abelian Number Fields,”
    <i>Mathematics of Computation</i>, vol. 68, no. 227, pp. 1179–1186, 1999.
  mla: Klüners, Jürgen, and Vincenzo Acciaro. “Computing Automorphisms of Abelian
    Number Fields.” <i>Mathematics of Computation</i>, vol. 68, no. 227, American
    Mathematical Society (AMS), 1999, pp. 1179–86.
  short: J. Klüners, V. Acciaro, Mathematics of Computation 68 (1999) 1179–1186.
date_created: 2023-01-11T09:31:21Z
date_updated: 2023-03-06T10:28:52Z
department:
- _id: '102'
intvolume: '        68'
issue: '227'
language:
- iso: eng
page: 1179-1186
publication: Mathematics of Computation
publication_identifier:
  issn:
  - 1088-6842
  - 0025-5718
publication_status: published
publisher: American Mathematical Society (AMS)
related_material:
  link:
  - relation: confirmation
    url: https://www.ams.org/journals/mcom/1999-68-227/S0025-5718-99-01084-4/S0025-5718-99-01084-4.pdf
status: public
title: Computing Automorphisms of Abelian Number Fields
type: journal_article
user_id: '93826'
volume: 68
year: '1999'
...
---
_id: '34905'
abstract:
- lang: eng
  text: "Let ℚ(α) be an algebraic number field given by the\r\nminimal polynomial
    f of α. We want to determine all subfields\r\nℚ(β) ⊂ Q(α) of given degree. It
    is convenient to describe each\r\nsubfield by a pair (g, h) ∈ Z [t] x ℚ[t] such
    that g is the minimal\r\npolynomial of β = h(α) . There is a bijection between
    the block\r\nsystems of the Galois group of f and the subfields of ℚ(α). These\r\nblock
    systems are computed using cyclic subgroups of the Galois\r\ngroup which we get
    from the Dedekind criterion. When a block\r\nsystem is known we compute the corresponding
    subfield using p-\r\nadic methods. We give a detailed description for all parts
    of the\r\nalgorithm."
author:
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
citation:
  ama: Klüners J. On computing subfields. A detailed description of the algorithm
    . <i>Journal de Theorie des Nombres de Bordeaux</i>. 1998;10(2):243-271. doi:<a
    href="https://jtnb.centre-mersenne.org/item/?id=JTNB_1998__10_2_243_0">https://jtnb.centre-mersenne.org/item/?id=JTNB_1998__10_2_243_0</a>
  apa: Klüners, J. (1998). On computing subfields. A detailed description of the algorithm
    . <i>Journal de Theorie Des Nombres de Bordeaux</i>, <i>10</i>(2), 243–271. <a
    href="https://jtnb.centre-mersenne.org/item/?id=JTNB_1998__10_2_243_0">https://jtnb.centre-mersenne.org/item/?id=JTNB_1998__10_2_243_0</a>
  bibtex: '@article{Klüners_1998, title={On computing subfields. A detailed description
    of the algorithm }, volume={10}, DOI={<a href="https://jtnb.centre-mersenne.org/item/?id=JTNB_1998__10_2_243_0">https://jtnb.centre-mersenne.org/item/?id=JTNB_1998__10_2_243_0</a>},
    number={2}, journal={Journal de Theorie des Nombres de Bordeaux}, publisher={Elsevier
    BV}, author={Klüners, Jürgen}, year={1998}, pages={243–271} }'
  chicago: 'Klüners, Jürgen. “On Computing Subfields. A Detailed Description of the
    Algorithm .” <i>Journal de Theorie Des Nombres de Bordeaux</i> 10, no. 2 (1998):
    243–71. <a href="https://jtnb.centre-mersenne.org/item/?id=JTNB_1998__10_2_243_0">https://jtnb.centre-mersenne.org/item/?id=JTNB_1998__10_2_243_0</a>.'
  ieee: 'J. Klüners, “On computing subfields. A detailed description of the algorithm
    ,” <i>Journal de Theorie des Nombres de Bordeaux</i>, vol. 10, no. 2, pp. 243–271,
    1998, doi: <a href="https://jtnb.centre-mersenne.org/item/?id=JTNB_1998__10_2_243_0">https://jtnb.centre-mersenne.org/item/?id=JTNB_1998__10_2_243_0</a>.'
  mla: Klüners, Jürgen. “On Computing Subfields. A Detailed Description of the Algorithm
    .” <i>Journal de Theorie Des Nombres de Bordeaux</i>, vol. 10, no. 2, Elsevier
    BV, 1998, pp. 243–71, doi:<a href="https://jtnb.centre-mersenne.org/item/?id=JTNB_1998__10_2_243_0">https://jtnb.centre-mersenne.org/item/?id=JTNB_1998__10_2_243_0</a>.
  short: J. Klüners, Journal de Theorie Des Nombres de Bordeaux 10 (1998) 243–271.
date_created: 2022-12-23T10:24:43Z
date_updated: 2023-03-06T10:34:22Z
doi: https://jtnb.centre-mersenne.org/item/?id=JTNB_1998__10_2_243_0
intvolume: '        10'
issue: '2'
language:
- iso: eng
page: 243-271
publication: Journal de Theorie des Nombres de Bordeaux
publication_status: published
publisher: Elsevier BV
related_material:
  link:
  - relation: confirmation
    url: http://www.numdam.org/item/JTNB_1998__10_2_243_0/
status: public
title: 'On computing subfields. A detailed description of the algorithm '
type: journal_article
user_id: '93826'
volume: 10
year: '1998'
...