[{"issue":"1","year":"2014","date_created":"2022-12-22T10:53:44Z","publisher":"Wiley","title":"Computation of Galois groups of rational polynomials","publication":"LMS Journal of Computation and Mathematics","abstract":[{"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.","lang":"eng"}],"external_id":{"arxiv":["1211.3588"]},"language":[{"iso":"eng"}],"keyword":["Computational Theory and Mathematics","General Mathematics"],"publication_status":"published","publication_identifier":{"issn":["1461-1570"]},"citation":{"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>.","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>","short":"C. Fieker, J. Klüners, LMS Journal of Computation and Mathematics 17 (2014) 141–158.","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>.","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} }"},"page":"141-158","intvolume":"        17","author":[{"first_name":"Claus","last_name":"Fieker","full_name":"Fieker, Claus"},{"first_name":"Jürgen","full_name":"Klüners, Jürgen","id":"21202","last_name":"Klüners"}],"volume":17,"date_updated":"2023-03-06T09:43:56Z","doi":"10.1112/s1461157013000302","type":"journal_article","status":"public","user_id":"93826","department":[{"_id":"102"}],"_id":"34845"},{"type":"journal_article","status":"public","user_id":"93826","department":[{"_id":"102"}],"_id":"42794","extern":"1","publication_status":"published","publication_identifier":{"issn":["1461-1570"]},"citation":{"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>.","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} }","short":"B. Eick, M. Kirschmer, C. Leedham-Green, LMS Journal of Computation and Mathematics 17 (2014) 345–359.","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>."},"page":"345-359","intvolume":"        17","author":[{"first_name":"B.","last_name":"Eick","full_name":"Eick, B."},{"first_name":"Markus","full_name":"Kirschmer, Markus","id":"82258","last_name":"Kirschmer"},{"first_name":"C.","full_name":"Leedham-Green, C.","last_name":"Leedham-Green"}],"volume":17,"date_updated":"2023-04-04T09:31:17Z","doi":"10.1112/s1461157014000047","publication":"LMS Journal of Computation and Mathematics","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."}],"language":[{"iso":"eng"}],"keyword":["Computational Theory and Mathematics","General Mathematics"],"issue":"1","year":"2014","date_created":"2023-03-07T08:30:15Z","publisher":"Wiley","title":"The constructive membership problem for discrete free subgroups of rank 2 of SL₂(R)"},{"intvolume":"        16","page":"172-186","citation":{"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>","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.","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} }","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>.","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>.","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>"},"publication_identifier":{"issn":["1461-1570"]},"publication_status":"published","doi":"10.1112/s1461157013000107","date_updated":"2023-04-04T07:57:04Z","volume":16,"author":[{"first_name":"David","last_name":"Lorch","full_name":"Lorch, David"},{"last_name":"Kirschmer","full_name":"Kirschmer, Markus","id":"82258","first_name":"Markus"}],"status":"public","type":"journal_article","extern":"1","_id":"42796","department":[{"_id":"102"}],"user_id":"93826","year":"2013","title":"Single-class genera of positive integral lattices","publisher":"Wiley","date_created":"2023-03-07T08:34:28Z","abstract":[{"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.","lang":"eng"}],"publication":"LMS Journal of Computation and Mathematics","keyword":["Computational Theory and Mathematics","General Mathematics"],"language":[{"iso":"eng"}]},{"publisher":"Wiley","date_created":"2022-12-23T09:56:22Z","title":"A Database for Field Extensions of the Rationals","year":"2001","external_id":{"arxiv":["math/0102232"]},"keyword":["Computational Theory and Mathematics","General Mathematics"],"language":[{"iso":"eng"}],"publication":"LMS Journal of Computation and Mathematics","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."}],"date_updated":"2023-03-02T09:53:08Z","author":[{"last_name":"Klüners","id":"21202","full_name":"Klüners, Jürgen","first_name":"Jürgen"},{"first_name":"Gunter","last_name":"Malle","full_name":"Malle, Gunter"}],"volume":4,"doi":"10.1112/s1461157000000851","publication_status":"published","publication_identifier":{"issn":["1461-1570"]},"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>","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>.","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>.","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} }","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.","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>"},"intvolume":"         4","page":"182-196","_id":"34897","user_id":"93826","department":[{"_id":"102"}],"type":"journal_article","status":"public"}]