---
_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: '42794'
abstract:
- lang: eng
  text: We exhibit a practical algorithm for solving the constructive membership problem
    for discrete free subgroups of rank 2 in PSL₂(R) or SL₂(R). This algorithm, together
    with methods for checking whether a two-generator subgroup of PSL₂(R) or SL₂(R)
    is discrete and free, have been implemented in Magma for groups defined over real
    algebraic number fields.
author:
- first_name: B.
  full_name: Eick, B.
  last_name: Eick
- first_name: Markus
  full_name: Kirschmer, Markus
  id: '82258'
  last_name: Kirschmer
- first_name: C.
  full_name: Leedham-Green, C.
  last_name: Leedham-Green
citation:
  ama: Eick B, Kirschmer M, Leedham-Green C. The constructive membership problem for
    discrete free subgroups of rank 2 of SL₂(R). <i>LMS Journal of Computation and
    Mathematics</i>. 2014;17(1):345-359. doi:<a href="https://doi.org/10.1112/s1461157014000047">10.1112/s1461157014000047</a>
  apa: Eick, B., Kirschmer, M., &#38; Leedham-Green, C. (2014). The constructive membership
    problem for discrete free subgroups of rank 2 of SL₂(R). <i>LMS Journal of Computation
    and Mathematics</i>, <i>17</i>(1), 345–359. <a href="https://doi.org/10.1112/s1461157014000047">https://doi.org/10.1112/s1461157014000047</a>
  bibtex: '@article{Eick_Kirschmer_Leedham-Green_2014, title={The constructive membership
    problem for discrete free subgroups of rank 2 of SL₂(R)}, volume={17}, DOI={<a
    href="https://doi.org/10.1112/s1461157014000047">10.1112/s1461157014000047</a>},
    number={1}, journal={LMS Journal of Computation and Mathematics}, publisher={Wiley},
    author={Eick, B. and Kirschmer, Markus and Leedham-Green, C.}, year={2014}, pages={345–359}
    }'
  chicago: 'Eick, B., Markus Kirschmer, and C. Leedham-Green. “The Constructive Membership
    Problem for Discrete Free Subgroups of Rank 2 of SL₂(R).” <i>LMS Journal of Computation
    and Mathematics</i> 17, no. 1 (2014): 345–59. <a href="https://doi.org/10.1112/s1461157014000047">https://doi.org/10.1112/s1461157014000047</a>.'
  ieee: 'B. Eick, M. Kirschmer, and C. Leedham-Green, “The constructive membership
    problem for discrete free subgroups of rank 2 of SL₂(R),” <i>LMS Journal of Computation
    and Mathematics</i>, vol. 17, no. 1, pp. 345–359, 2014, doi: <a href="https://doi.org/10.1112/s1461157014000047">10.1112/s1461157014000047</a>.'
  mla: Eick, B., et al. “The Constructive Membership Problem for Discrete Free Subgroups
    of Rank 2 of SL₂(R).” <i>LMS Journal of Computation and Mathematics</i>, vol.
    17, no. 1, Wiley, 2014, pp. 345–59, doi:<a href="https://doi.org/10.1112/s1461157014000047">10.1112/s1461157014000047</a>.
  short: B. Eick, M. Kirschmer, C. Leedham-Green, LMS Journal of Computation and Mathematics
    17 (2014) 345–359.
date_created: 2023-03-07T08:30:15Z
date_updated: 2023-04-04T09:31:17Z
department:
- _id: '102'
doi: 10.1112/s1461157014000047
extern: '1'
intvolume: '        17'
issue: '1'
keyword:
- Computational Theory and Mathematics
- General Mathematics
language:
- iso: eng
page: 345-359
publication: LMS Journal of Computation and Mathematics
publication_identifier:
  issn:
  - 1461-1570
publication_status: published
publisher: Wiley
status: public
title: The constructive membership problem for discrete free subgroups of rank 2 of
  SL₂(R)
type: journal_article
user_id: '93826'
volume: 17
year: '2014'
...
---
_id: '42796'
abstract:
- lang: eng
  text: "We give an enumeration of all positive definite primitive Z-lattices in dimension
    n ≥ 3 whose genus consists of a single isometry class. This is achieved by using
    bounds obtained from the Smith–Minkowski–Siegel mass formula to computationally
    construct the square-free determinant lattices with this property, and then repeatedly
    calculating pre-images under a mapping first introduced by G. L. Watson.\r\n\r\nWe
    hereby complete the classification of single-class genera in dimensions 4 and
    5 and correct some mistakes in Watson’s classifications in other dimensions. A
    list of all single-class primitive Z-lattices has been compiled and incorporated
    into the Catalogue of Lattices."
author:
- first_name: David
  full_name: Lorch, David
  last_name: Lorch
- first_name: Markus
  full_name: Kirschmer, Markus
  id: '82258'
  last_name: Kirschmer
citation:
  ama: Lorch D, Kirschmer M. Single-class genera of positive integral lattices. <i>LMS
    Journal of Computation and Mathematics</i>. 2013;16:172-186. doi:<a href="https://doi.org/10.1112/s1461157013000107">10.1112/s1461157013000107</a>
  apa: Lorch, D., &#38; Kirschmer, M. (2013). Single-class genera of positive integral
    lattices. <i>LMS Journal of Computation and Mathematics</i>, <i>16</i>, 172–186.
    <a href="https://doi.org/10.1112/s1461157013000107">https://doi.org/10.1112/s1461157013000107</a>
  bibtex: '@article{Lorch_Kirschmer_2013, title={Single-class genera of positive integral
    lattices}, volume={16}, DOI={<a href="https://doi.org/10.1112/s1461157013000107">10.1112/s1461157013000107</a>},
    journal={LMS Journal of Computation and Mathematics}, publisher={Wiley}, author={Lorch,
    David and Kirschmer, Markus}, year={2013}, pages={172–186} }'
  chicago: 'Lorch, David, and Markus Kirschmer. “Single-Class Genera of Positive Integral
    Lattices.” <i>LMS Journal of Computation and Mathematics</i> 16 (2013): 172–86.
    <a href="https://doi.org/10.1112/s1461157013000107">https://doi.org/10.1112/s1461157013000107</a>.'
  ieee: 'D. Lorch and M. Kirschmer, “Single-class genera of positive integral lattices,”
    <i>LMS Journal of Computation and Mathematics</i>, vol. 16, pp. 172–186, 2013,
    doi: <a href="https://doi.org/10.1112/s1461157013000107">10.1112/s1461157013000107</a>.'
  mla: Lorch, David, and Markus Kirschmer. “Single-Class Genera of Positive Integral
    Lattices.” <i>LMS Journal of Computation and Mathematics</i>, vol. 16, Wiley,
    2013, pp. 172–86, doi:<a href="https://doi.org/10.1112/s1461157013000107">10.1112/s1461157013000107</a>.
  short: D. Lorch, M. Kirschmer, LMS Journal of Computation and Mathematics 16 (2013)
    172–186.
date_created: 2023-03-07T08:34:28Z
date_updated: 2023-04-04T07:57:04Z
department:
- _id: '102'
doi: 10.1112/s1461157013000107
extern: '1'
intvolume: '        16'
keyword:
- Computational Theory and Mathematics
- General Mathematics
language:
- iso: eng
page: 172-186
publication: LMS Journal of Computation and Mathematics
publication_identifier:
  issn:
  - 1461-1570
publication_status: published
publisher: Wiley
status: public
title: Single-class genera of positive integral lattices
type: journal_article
user_id: '93826'
volume: 16
year: '2013'
...
---
_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'
...