[{"title":"Counting Frobenius extensions over local function fields","author":[{"last_name":"Klüners","full_name":"Klüners, Jürgen","id":"21202","first_name":"Jürgen"},{"first_name":"Raphael","last_name":"Müller","full_name":"Müller, Raphael","id":"55246"}],"date_created":"2026-04-07T08:13:59Z","date_updated":"2026-04-07T08:14:45Z","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>.","short":"J. Klüners, R. Müller, ArXiv:2604.02152 (2026).","mla":"Klüners, Jürgen, and Raphael Müller. “Counting Frobenius Extensions over Local Function Fields.” <i>ArXiv:2604.02152</i>, 2026.","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} }","ieee":"J. Klüners and R. Müller, “Counting Frobenius extensions over local function fields,” <i>arXiv:2604.02152</i>. 2026.","chicago":"Klüners, Jürgen, and Raphael Müller. “Counting Frobenius Extensions over Local Function Fields.” <i>ArXiv:2604.02152</i>, 2026."},"year":"2026","language":[{"iso":"eng"}],"user_id":"82981","external_id":{"arxiv":["2604.02152"]},"_id":"65358","status":"public","abstract":[{"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.","lang":"eng"}],"publication":"arXiv:2604.02152","type":"preprint"},{"external_id":{"arxiv":["1907.13383"]},"language":[{"iso":"eng"}],"publication":"Journal of Computational Algebra","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."}],"publisher":"Elsevier BV","date_created":"2025-08-05T07:01:39Z","title":"Computing quadratic subfields of number fields","year":"2025","_id":"60874","user_id":"82981","article_number":"100039","type":"journal_article","status":"public","date_updated":"2025-08-05T07:10:25Z","volume":15,"author":[{"full_name":"Elsenhans, Andreas-Stephan","last_name":"Elsenhans","first_name":"Andreas-Stephan"},{"full_name":"Klüners, Jürgen","id":"21202","last_name":"Klüners","first_name":"Jürgen"}],"doi":"10.1016/j.jaca.2025.100039","publication_identifier":{"issn":["2772-8277"]},"publication_status":"published","intvolume":"        15","citation":{"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} }","short":"A.-S. Elsenhans, J. Klüners, Journal of Computational Algebra 15 (2025).","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>.","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>","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>","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>.","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>."}},{"language":[{"iso":"eng"}],"external_id":{"arxiv":["2407.08438"]},"_id":"55192","user_id":"82981","abstract":[{"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.","lang":"eng"}],"status":"public","type":"preprint","publication":"arXiv:2407.08438","title":"Symmetries of power-free integers in number fields and their shift  spaces","date_updated":"2024-07-12T08:19:11Z","date_created":"2024-07-12T08:16:37Z","author":[{"last_name":"Gundlach","id":"100450","full_name":"Gundlach, Fabian","first_name":"Fabian"},{"first_name":"Jürgen","id":"21202","full_name":"Klüners, Jürgen","last_name":"Klüners"}],"year":"2024","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.","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.","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} }","short":"F. Gundlach, J. Klüners, ArXiv:2407.08438 (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>."}},{"article_number":"89","_id":"55554","user_id":"82981","status":"public","type":"journal_article","doi":"10.1007/s40993-024-00579-6","date_updated":"2024-11-05T09:46:04Z","volume":10,"author":[{"first_name":"Markus","last_name":"Kirschmer","id":"82258","full_name":"Kirschmer, Markus"},{"last_name":"Klüners","full_name":"Klüners, Jürgen","id":"21202","first_name":"Jürgen"}],"intvolume":"        10","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>","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>.","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} }","short":"M. Kirschmer, J. Klüners, Research in Number Theory 10 (2024).","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>.","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>"},"publication_identifier":{"issn":["2522-0160","2363-9555"]},"publication_status":"published","language":[{"iso":"eng"}],"external_id":{"arxiv":["2208.14688"]},"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."}],"publication":"Research in Number Theory","title":"Chow groups of one-dimensional noetherian domains","publisher":"Springer Science and Business Media LLC","date_created":"2024-08-06T07:03:20Z","year":"2024","issue":"4"},{"year":"2024","citation":{"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).","bibtex":"@article{Kirschmer_Klüners_2024, title={Enumerating orders in number fields}, journal={arXiv:2411.08568}, author={Kirschmer, Markus and Klüners, Jürgen}, year={2024} }","apa":"Kirschmer, M., &#38; Klüners, J. (2024). Enumerating orders in number fields. In <i>arXiv:2411.08568</i>.","ieee":"M. Kirschmer and J. Klüners, “Enumerating orders in number fields,” <i>arXiv:2411.08568</i>. 2024.","chicago":"Kirschmer, Markus, and Jürgen Klüners. “Enumerating Orders in Number Fields.” <i>ArXiv:2411.08568</i>, 2024.","ama":"Kirschmer M, Klüners J. Enumerating orders in number fields. <i>arXiv:241108568</i>. Published online 2024."},"title":"Enumerating orders in number fields","date_updated":"2024-11-19T07:49:48Z","date_created":"2024-11-19T07:49:07Z","author":[{"last_name":"Kirschmer","full_name":"Kirschmer, Markus","first_name":"Markus"},{"last_name":"Klüners","id":"21202","full_name":"Klüners, Jürgen","first_name":"Jürgen"}],"abstract":[{"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.","lang":"eng"}],"status":"public","publication":"arXiv:2411.08568","type":"preprint","language":[{"iso":"eng"}],"external_id":{"arxiv":["2411.08568"]},"_id":"57218","user_id":"82981"},{"publisher":"Springer Science and Business Media LLC","date_updated":"2023-12-06T09:50:43Z","author":[{"last_name":"Klüners","id":"21202","full_name":"Klüners, Jürgen","first_name":"Jürgen"},{"first_name":"Jiuya","full_name":"Wang, Jiuya","last_name":"Wang"}],"date_created":"2023-12-01T09:23:59Z","title":"Idélic Approach in Enumerating Heisenberg Extensions","doi":"10.1007/s44007-023-00067-w","publication_status":"published","publication_identifier":{"issn":["2730-9657"]},"year":"2023","citation":{"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} }","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).","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>","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>.","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>."},"_id":"49372","user_id":"21202","department":[{"_id":"102"}],"language":[{"iso":"eng"}],"type":"journal_article","publication":"La Matematica","status":"public"},{"publication_status":"published","publication_identifier":{"issn":["0002-9939","1088-6826"]},"citation":{"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>.","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>.","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>","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.","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} }"},"page":"2793-2805","intvolume":"       150","author":[{"first_name":"Jürgen","last_name":"Klüners","id":"21202","full_name":"Klüners, Jürgen"},{"first_name":"Jiuya","full_name":"Wang, Jiuya","last_name":"Wang"}],"volume":150,"date_updated":"2023-03-06T08:47:42Z","doi":"10.1090/proc/15882","type":"journal_article","status":"public","user_id":"93826","department":[{"_id":"102"}],"_id":"34839","issue":"7","year":"2022","date_created":"2022-12-22T10:47:01Z","publisher":"American Mathematical Society (AMS)","title":"ℓ-torsion bounds for the class group of number fields with an ℓ-group as Galois group","publication":"Proceedings of the American Mathematical Society","abstract":[{"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.","lang":"eng"}],"external_id":{"arxiv":["2003.12161 "]},"language":[{"iso":"eng"}],"keyword":["Applied Mathematics","General Mathematics"]},{"department":[{"_id":"102"}],"user_id":"93826","_id":"34835","status":"public","type":"journal_article","doi":"10.4064/aa211207-16-5","volume":204,"author":[{"first_name":"Jürgen","last_name":"Klüners","full_name":"Klüners, Jürgen","id":"21202"}],"date_updated":"2023-03-06T08:48:33Z","intvolume":"       204","page":"165-184","citation":{"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} }","short":"J. Klüners, Acta Arithmetica 204 (2022) 165–184.","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>.","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>","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>."},"publication_identifier":{"issn":["0065-1036","1730-6264"]},"publication_status":"published","language":[{"iso":"eng"}],"keyword":["Algebra and Number Theory"],"external_id":{"arxiv":["2011.04325 "]},"abstract":[{"text":"We prove an upper bound for the asymptotics of counting functions of number fields with nilpotent Galois groups. ","lang":"eng"}],"publication":"Acta Arithmetica","title":"The asymptotics of nilpotent Galois groups","date_created":"2022-12-22T10:08:23Z","publisher":"Institute of Mathematics, Polish Academy of Sciences","year":"2022","issue":"2"},{"doi":"10.1090/mcom/3609","author":[{"first_name":"Jürgen","id":"21202","full_name":"Klüners, Jürgen","last_name":"Klüners"},{"full_name":"Komatsu, Toru","last_name":"Komatsu","first_name":"Toru"}],"volume":90,"date_updated":"2023-03-06T08:57:45Z","citation":{"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>.","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>.","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>","short":"J. Klüners, T. Komatsu, Mathematics of Computation 90 (2021) 1483–1497.","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} }","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>.","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>"},"page":"1483-1497","intvolume":"        90","publication_status":"published","publication_identifier":{"issn":["0025-5718","1088-6842"]},"user_id":"93826","department":[{"_id":"102"}],"_id":"34840","status":"public","type":"journal_article","title":"Imaginary multiquadratic number fields with class group of exponent $3$ and $5$","date_created":"2022-12-22T10:48:44Z","publisher":"American Mathematical Society (AMS)","year":"2021","issue":"329","language":[{"iso":"eng"}],"keyword":["Applied Mathematics","Computational Mathematics","Algebra and Number Theory"],"external_id":{"arxiv":["2004.03308v2"]},"abstract":[{"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. ","lang":"eng"}],"publication":"Mathematics of Computation"},{"publication":"Acta Arithmetica","abstract":[{"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.","lang":"eng"}],"external_id":{"arxiv":["1803.02056 "]},"keyword":["Algebra and Number Theory"],"language":[{"iso":"eng"}],"issue":"3","year":"2020","publisher":"Institute of Mathematics, Polish Academy of Sciences","date_created":"2022-12-22T10:51:13Z","title":"Imaginary quadratic number fields with class groups of small exponent","type":"journal_article","status":"public","_id":"34842","user_id":"93826","department":[{"_id":"102"}],"publication_status":"published","publication_identifier":{"issn":["0065-1036","1730-6264"]},"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>","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>.","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} }","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.","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>"},"intvolume":"       193","page":"217-233","date_updated":"2023-03-06T10:19:53Z","author":[{"first_name":"Andreas-Stephan","last_name":"Elsenhans","full_name":"Elsenhans, Andreas-Stephan"},{"first_name":"Jürgen","last_name":"Klüners","id":"21202","full_name":"Klüners, Jürgen"},{"full_name":"Nicolae, Florin","last_name":"Nicolae","first_name":"Florin"}],"volume":193,"doi":"10.4064/aa180220-20-3"},{"type":"journal_article","status":"public","department":[{"_id":"102"}],"user_id":"82981","_id":"34841","publication_identifier":{"issn":["0022-314X"]},"publication_status":"published","page":"311-322","intvolume":"       212","citation":{"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>.","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>.","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>","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} }","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.","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>"},"volume":212,"author":[{"first_name":"Jürgen","full_name":"Klüners, Jürgen","id":"21202","last_name":"Klüners"},{"full_name":"Müller, Raphael","last_name":"Müller","first_name":"Raphael"}],"date_updated":"2025-06-13T08:18:30Z","doi":"10.1016/j.jnt.2019.11.007","publication":"Journal of Number Theory","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"}],"external_id":{"arxiv":["1904.02573 "]},"language":[{"iso":"eng"}],"keyword":["Algebra and Number Theory"],"year":"2020","date_created":"2022-12-22T10:50:03Z","publisher":"Elsevier BV","title":"The conductor density of local function fields with abelian Galois group"},{"title":"Computing subfields of number fields and applications to Galois group computations","date_created":"2022-12-22T10:52:18Z","publisher":"Elsevier BV","year":"2018","language":[{"iso":"eng"}],"keyword":["Computational Mathematics","Algebra and Number Theory"],"external_id":{"arxiv":["1610.06837 "]},"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."}],"publication":"Journal of Symbolic Computation","doi":"10.1016/j.jsc.2018.04.013","volume":93,"author":[{"first_name":"Andreas-Stephan","full_name":"Elsenhans, Andreas-Stephan","last_name":"Elsenhans"},{"first_name":"Jürgen","id":"21202","full_name":"Klüners, Jürgen","last_name":"Klüners"}],"date_updated":"2023-03-06T09:05:51Z","intvolume":"        93","page":"1-20","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>","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>.","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} }","short":"A.-S. Elsenhans, J. Klüners, Journal of Symbolic Computation 93 (2018) 1–20.","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>.","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>"},"publication_identifier":{"issn":["0747-7171"]},"publication_status":"published","department":[{"_id":"102"}],"user_id":"93826","_id":"34843","status":"public","type":"journal_article"},{"type":"journal_article","status":"public","_id":"34844","user_id":"93826","department":[{"_id":"102"}],"publication_status":"published","publication_identifier":{"issn":["0022-314X"]},"citation":{"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} }","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.","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>.","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>"},"intvolume":"       167","page":"161-168","date_updated":"2023-03-06T10:44:22Z","author":[{"id":"21202","full_name":"Klüners, Jürgen","last_name":"Klüners","first_name":"Jürgen"},{"full_name":"Nicolae, Florin","last_name":"Nicolae","first_name":"Florin"}],"volume":167,"doi":"10.1016/j.jnt.2016.03.023","publication":"Journal of Number Theory","abstract":[{"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. ","lang":"eng"}],"external_id":{"arxiv":["1509.06883 "]},"keyword":["Algebra and Number Theory"],"language":[{"iso":"eng"}],"year":"2016","publisher":"Elsevier BV","date_created":"2022-12-22T10:52:47Z","title":"Are number fields determined by Artin L-functions?"},{"status":"public","type":"journal_article","_id":"34845","user_id":"93826","department":[{"_id":"102"}],"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>","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.","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} }","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>"},"intvolume":"        17","page":"141-158","publication_status":"published","publication_identifier":{"issn":["1461-1570"]},"doi":"10.1112/s1461157013000302","date_updated":"2023-03-06T09:43:56Z","author":[{"first_name":"Claus","last_name":"Fieker","full_name":"Fieker, Claus"},{"last_name":"Klüners","id":"21202","full_name":"Klüners, Jürgen","first_name":"Jürgen"}],"volume":17,"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"}],"publication":"LMS Journal of Computation and Mathematics","keyword":["Computational Theory and Mathematics","General Mathematics"],"language":[{"iso":"eng"}],"external_id":{"arxiv":["1211.3588"]},"year":"2014","issue":"1","title":"Computation of Galois groups of rational polynomials","publisher":"Wiley","date_created":"2022-12-22T10:53:44Z"},{"_id":"34847","user_id":"21202","department":[{"_id":"102"}],"status":"public","type":"journal_article","doi":"10.1142/s1793042112500492","date_updated":"2023-03-02T14:10:38Z","author":[{"full_name":"Klüners, Jürgen","id":"21202","last_name":"Klüners","first_name":"Jürgen"}],"volume":"08","citation":{"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} }","short":"J. Klüners, International Journal of Number Theory 08 (2012) 845–858.","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>.","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>","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>","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>."},"page":"845-858","intvolume":"         8","publication_status":"published","publication_identifier":{"issn":["1793-0421","1793-7310"]},"language":[{"iso":"eng"}],"external_id":{"arxiv":["1108.5597 "]},"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. "}],"publication":"International Journal of Number Theory","title":"The Distribution of Number Fields with Wreath Products as Galois Groups ","publisher":"World Scientific Pub Co Pte Lt","date_created":"2022-12-22T10:55:47Z","year":"2012","issue":"03"},{"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."}],"publication":"International Mathematics Research Notices","keyword":["General Mathematics"],"language":[{"iso":"eng"}],"year":"2011","issue":"16","title":"Weighted Distribution of the 4-rank of Class Groups and Applications","publisher":"Oxford University Press (OUP)","date_created":"2022-12-23T09:08:00Z","status":"public","type":"journal_article","_id":"34885","user_id":"93826","department":[{"_id":"102"}],"citation":{"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} }","short":"É. Fouvry, J. Klüners, International Mathematics Research Notices 2011 (2011) 3618–3656.","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>.","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>","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>","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>.","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>."},"page":"3618-3656","intvolume":"      2011","publication_status":"published","publication_identifier":{"issn":["1687-0247","1073-7928"]},"doi":"10.1093/imrn/rnq223","date_updated":"2023-03-06T09:07:46Z","author":[{"first_name":"Étienne","full_name":"Fouvry, Étienne","last_name":"Fouvry"},{"last_name":"Klüners","full_name":"Klüners, Jürgen","id":"21202","first_name":"Jürgen"}],"volume":2011},{"keyword":["Computational Mathematics","Algebra and Number Theory"],"language":[{"iso":"eng"}],"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."}],"publication":"Journal of Symbolic Computation","title":"Generating subfields","publisher":"Elsevier BV","date_created":"2022-12-22T10:54:15Z","year":"2011","_id":"34846","user_id":"93826","department":[{"_id":"102"}],"status":"public","type":"journal_article","doi":"10.1016/j.jsc.2012.05.010","date_updated":"2023-03-06T09:46:15Z","author":[{"last_name":"van Hoeij","full_name":"van Hoeij, Mark","first_name":"Mark"},{"first_name":"Jürgen","full_name":"Klüners, Jürgen","id":"21202","last_name":"Klüners"},{"first_name":"Andrew","last_name":"Novocin","full_name":"Novocin, Andrew"}],"volume":52,"citation":{"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>.","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} }","short":"M. van Hoeij, J. Klüners, A. Novocin, Journal of Symbolic Computation 52 (2011) 17–34.","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>."},"page":"17-34","intvolume":"        52","publication_status":"published","publication_identifier":{"issn":["0747-7171"]}},{"publication":"Annals of Mathematics","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."}],"keyword":["Statistics","Probability and Uncertainty","Mathematics (miscellaneous)"],"language":[{"iso":"eng"}],"issue":"3","year":"2010","publisher":"Annals of Mathematics","date_created":"2022-12-23T09:09:02Z","title":"On the negative Pell equation","type":"journal_article","status":"public","_id":"34886","user_id":"93826","department":[{"_id":"102"}],"publication_status":"published","publication_identifier":{"issn":["0003-486X"]},"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>","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>.","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>.","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>","short":"É. Fouvry, J. Klüners, Annals of Mathematics 172 (2010) 2035–2104.","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>.","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} }"},"intvolume":"       172","page":"2035-2104","date_updated":"2023-03-06T09:50:37Z","author":[{"full_name":"Fouvry, Étienne","last_name":"Fouvry","first_name":"Étienne"},{"last_name":"Klüners","full_name":"Klüners, Jürgen","id":"21202","first_name":"Jürgen"}],"volume":172,"doi":"10.4007/annals.2010.172.2035"},{"abstract":[{"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.","lang":"eng"}],"publication":"Proceedings of the London Mathematical Society","language":[{"iso":"eng"}],"keyword":["General Mathematics"],"year":"2010","issue":"2","title":"The parity of the period of the continued fraction of d","date_created":"2022-12-23T09:22:49Z","publisher":"Wiley","status":"public","type":"journal_article","user_id":"93826","department":[{"_id":"102"}],"_id":"34888","citation":{"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} }","short":"É. Fouvry, J. Klüners, Proceedings of the London Mathematical Society 101 (2010) 337–391.","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>.","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>","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>","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>."},"page":"337-391","intvolume":"       101","related_material":{"link":[{"url":"https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=fc7ab412993fd2bf0069de42fbef1ecc69137755","relation":"confirmation"}]},"publication_status":"published","publication_identifier":{"issn":["0024-6115"]},"doi":"10.1112/plms/pdp057","author":[{"first_name":"Étienne","full_name":"Fouvry, Étienne","last_name":"Fouvry"},{"full_name":"Klüners, Jürgen","id":"21202","last_name":"Klüners","first_name":"Jürgen"}],"volume":101,"date_updated":"2023-03-06T10:16:54Z"},{"publisher":"Mathematical Sciences Publishers","date_updated":"2023-03-06T10:18:14Z","author":[{"last_name":"Fouvry","full_name":"Fouvry, Étienne","first_name":"Étienne"},{"first_name":"Jürgen","full_name":"Klüners, Jürgen","id":"21202","last_name":"Klüners"}],"date_created":"2022-12-23T09:10:12Z","volume":4,"title":"On the Spiegelungssatz for the 4-rank","doi":"10.2140/ant.2010.4.493","publication_status":"published","publication_identifier":{"issn":["1937-0652"]},"issue":"5","year":"2010","citation":{"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>.","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} }","short":"É. Fouvry, J. Klüners, Algebra &#38;amp; Number Theory 4 (2010) 493–508.","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>","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>","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>.","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>."},"page":"493-508","intvolume":"         4","_id":"34887","user_id":"93826","department":[{"_id":"102"}],"keyword":["Algebra and Number Theory"],"language":[{"iso":"eng"}],"type":"journal_article","publication":"Algebra &amp; Number Theory","abstract":[{"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. ","lang":"eng"}],"status":"public"}]