Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-24T13:43:06.376Z Has data issue: false hasContentIssue false
  • English
  • Français

A database of number fields

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  01 December 2014

John W. Jones  and
David P. Roberts
John W. Jones
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, PO Box 871804, Tempe, AZ 85287, USA email [email protected]
David P. Roberts
Affiliation:
Division of Science and Mathematics, University of Minnesota Morris, Morris, MN 56267, USA email [email protected]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We describe an online database of number fields which accompanies this paper. The database centers on complete lists of number fields with prescribed invariants. Our description here focuses on summarizing tables and connections to theoretical issues of current interest.

MSC classification

Primary: 11R21: Other number fields 11R32: Galois theory
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 17 , Issue 1 , 2014 , pp. 595 - 618
Copyright
© The Author(s) 2014 

References

Belabas, K., ‘cubic v.1.2’, http://www.math.u-bordeaux1.fr/∼kbelabas/research/software/cubic-1.2.tgz.Google Scholar
Belabas, K., ‘On quadratic fields with large 3-rank’, Math. Comp. 73 (2004) no. 248, 2061–2074 (electronic); MR 2059751 (2005c:11132).CrossRefGoogle Scholar
Bergé, A.-M., Martinet, J. and Olivier, M., ‘The computation of sextic fields with a quadratic subfield’, Math. Comp. 54 (1990) no. 190, 869–884; MR 1011438 (90k:11169).Google Scholar
Bhargava, M., ‘The density of discriminants of quartic rings and fields’, Ann. of Math. (2) 162 (2005) no. 2, 1031–1063; MR 2183288 (2006m:11163).Google Scholar
Bhargava, M., ‘Mass formulae for extensions of local fields, and conjectures on the density of number field discriminants’, Int. Math. Res. Not. IMRN 2007 (2007), doi:10.1093/imrn/rnm052; MR 2354798 (2009e:11220).CrossRefGoogle Scholar
Bhargava, M., ‘The density of discriminants of quintic rings and fields’, Ann. of Math. (2) 172 (2010) no. 3, 1559–1591; MR 2745272 (2011k:11152).CrossRefGoogle Scholar
Bordeaux tables of number fields, http://pari.math.u-bordeaux.fr/pub/pari/packages/nftables/.Google Scholar
Bosman, J., ‘A polynomial with Galois group SL2(F16)’, LMS J. Comput. Math. 10 (2007) 1461–1570 (electronic); MR 2365691 (2008k:12008).CrossRefGoogle Scholar
Boston, N. and Ellenberg, J. S., ‘Random pro-p groups, braid groups, and random tame Galois groups’, Groups Geom. Dyn. 5 (2011) no. 2, 265–280; MR 2782173 (2012b:11172).Google Scholar
Boston, N. and Perry, D., ‘Maximal 2-extensions with restricted ramification’, J. Algebra 232 (2000) no. 2, 664–672; MR 1792749 (2001k:12005).Google Scholar
Butler, G. and McKay, J., ‘The transitive groups of degree up to eleven’, Comm. Algebra 11 (1983) no. 8, 863–911.Google Scholar
Dahmen, S. R., ‘Classical and modular methods applied to diophantine equations’, PhD Thesis, University of Utrecht, 2008.Google Scholar
Driver, E. D. and Jones, J. W., ‘Minimum discriminants of imprimitive decic fields’, Exp. Math. 19 (2010) no. 4, 475–479; MR 2778659 (2012a:11169).CrossRefGoogle Scholar
Hoelscher, J. L., ‘Infinite class field towers’, Math. Ann. 344 (2009) no. 4, 923–928; MR 2507631 (2010h:11185).CrossRefGoogle Scholar
Hulek, K., Kloosterman, R. and Schütt, M., ‘Modularity of Calabi–Yau varieties’, Global aspects of complex geometry (Springer, Berlin, 2006) 271–309; MR 2264114 (2007g:11052).Google Scholar
Hunter, J., ‘The minimum discriminants of quintic fields’, Proc. Glasgow Math. Assoc. 3 (1957) 57–67; MR 0091309 (19,944b).CrossRefGoogle Scholar
Jones, J. W. and Roberts, D. P., ‘Artin $L$ -functions with small conductor’, in preparation.Google Scholar
Jones, J. W. and Roberts, D. P., ‘Sextic number fields with discriminant (−1) j 2 a 3 b ’, Number theory (Ottawa, ON, 1996) , CRM Proceedings & Lecture Notes 19 (American Mathematical Society, Providence, RI, 1999) 141–172; MR 2000b:11142.Google Scholar
Jones, J. W. and Roberts, D. P., ‘Septic fields with discriminant ± 2 a 3 b ’, Math. Comp. 72 (2003) no. 244, 1975–1985 (electronic); MR 1986816 (2004e:11119).Google Scholar
Jones, J. W. and Roberts, D. P., ‘A database of local fields’, J. Symbolic Comput. 41 (2006) no. 1, 80–97; website: http://math.asu.edu/∼jj/localfields.Google Scholar
Jones, J. W. and Roberts, D. P., ‘Galois number fields with small root discriminant’, J. Number Theory 122 (2007) no. 2, 379–407; MR 2292261 (2008e:11140).Google Scholar
Jones, J. W. and Roberts, D. P., ‘Number fields ramified at one prime’, Algorithmic number theory , Lecture Notes in Computational Science 5011 (Springer, Berlin, 2008) 226–239; MR 2467849 (2010b:11152).Google Scholar
Jones, J. W. and Roberts, D. P., ‘The tame–wild principle for discriminant relations for number fields’, Algebra Number Theory 8 (2014) no. 3, 609–645.CrossRefGoogle Scholar
Jones, J. W. and Wallington, R., ‘Number fields with solvable Galois groups and small Galois root discriminants’, Math. Comp. 81 (2012) no. 277, 555–567.Google Scholar
Klüners, J. and Malle, G., ‘A database for field extensions of the rationals’, LMS J. Comput. Math. 4 (2001) 182–196 (electronic); MR 2003i:11184.Google Scholar
Koch, H., Galois theory of p-extensions , Springer Monographs in Mathematics (Springer, Berlin, 2002). With a foreword by I. R. Shafarevich, translated from the 1970 German original by Franz Lemmermeyer, with a postscript by the author and Lemmermeyer; MR 1930372 (2003f:11181).Google Scholar
Malle, G., ‘On the distribution of Galois groups’, J. Number Theory 92 (2002) no. 2, 315–329; MR 1884706 (2002k:12010).CrossRefGoogle Scholar
Malle, G. and Matzat, B. H., Inverse Galois theory , Springer Monographs in Mathematics (Springer, Berlin, 1999); MR 1711577 (2000k:12004).Google Scholar
Martinet, J., ‘Petits discriminants des corps de nombres’, Number theory days, 1980 (Exeter, 1980) , London Mathematical Society Lecture Note Series 56 (Cambridge University Press, Cambridge, 1982) 151–193; MR 84g:12009.Google Scholar
Odlyzko, A. M., ‘Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results’, Sém. Théor. Nombres Bordeaux (2) 2 (1990) no. 1, 119–141; MR 1061762 (91i:11154).Google Scholar
Ono, K. and Taguchi, Y., ‘2-adic properties of certain modular forms and their applications to arithmetic functions’, Int. J. Number Theory 1 (2005) no. 1, 75–101; MR 2172333 (2006e:11057).Google Scholar
Pohst, M., ‘On the computation of number fields of small discriminants including the minimum discriminants of sixth degree fields’, J. Number Theory 14 (1982) no. 1, 99–117; MR 644904 (83g:12009).CrossRefGoogle Scholar
Roberts, D. P., ‘Chebyshev covers and exceptional number fields’, in preparation. http://facultypages.morris.umn.edu/∼roberts/.Google Scholar
Roberts, D. P., ‘Wild partitions and number theory’, J. Integer Seq. 10 (2007) no. 6, Article 07.6.6;MR 2335791 (2009b:11206).Google Scholar
Schaeffer, G. J., ‘The Hecke stability method and ethereal forms’, PhD Thesis, University of California, Berkeley, CA (ProQuest, UMI Dissertations Publishing, Ann Arbor, MI, 2012); MR 3093915.Google Scholar
Schwarz, A., Pohst, M. and Diaz y Diaz, F., ‘A table of quintic number fields’, Math. Comp. 63 (1994) no. 207, 361–376; MR 1219705 (94i:11108).Google Scholar
The GAP Group, GAP—Groups, Algorithms, and Programming, version 4.4, 2006 (http://www.gap-system.org).Google Scholar
The PARI Group, Bordeaux, Pari/gp, version 2.6.2, 2013.Google Scholar
Voight, J., ‘Tables of totally real number fields’, http://www.math.dartmouth.edu/∼jvoight/nf-tables/index.html.Google Scholar
Voight, J., ‘Enumeration of totally real number fields of bounded root discriminant’, Algorithmic number theory , Lecture Notes in Computer Science 5011 (Springer, Berlin, 2008) 268–281;MR 2467853 (2010a:11228).Google Scholar