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A database of number fields

  • John W. Jones (a1) and David P. Roberts (a2)
Abstract
Abstract

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.

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References
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2. Belabas K., ‘On quadratic fields with large 3-rank’, Math. Comp. 73 (2004) no. 248, 20612074 (electronic); MR 2059751 (2005c:11132).
3. Bergé A.-M., Martinet J. and Olivier M., ‘The computation of sextic fields with a quadratic subfield’, Math. Comp. 54 (1990) no. 190, 869884; MR 1011438 (90k:11169).
4. Bhargava M., ‘The density of discriminants of quartic rings and fields’, Ann. of Math. (2) 162 (2005) no. 2, 10311063; MR 2183288 (2006m:11163).
5. 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).
6. Bhargava M., ‘The density of discriminants of quintic rings and fields’, Ann. of Math. (2) 172 (2010) no. 3, 15591591; MR 2745272 (2011k:11152).
7. Bordeaux tables of number fields, http://pari.math.u-bordeaux.fr/pub/pari/packages/nftables/.
8. Bosman J., ‘A polynomial with Galois group SL2(F16)’, LMS J. Comput. Math. 10 (2007) 14611570 (electronic); MR 2365691 (2008k:12008).
9. Boston N. and Ellenberg J. S., ‘Random pro-p groups, braid groups, and random tame Galois groups’, Groups Geom. Dyn. 5 (2011) no. 2, 265280; MR 2782173 (2012b:11172).
10. Boston N. and Perry D., ‘Maximal 2-extensions with restricted ramification’, J. Algebra 232 (2000) no. 2, 664672; MR 1792749 (2001k:12005).
11. Butler G. and McKay J., ‘The transitive groups of degree up to eleven’, Comm. Algebra 11 (1983) no. 8, 863911.
12. Dahmen S. R., ‘Classical and modular methods applied to diophantine equations’, PhD Thesis, University of Utrecht, 2008.
13. Driver E. D. and Jones J. W., ‘Minimum discriminants of imprimitive decic fields’, Exp. Math. 19 (2010) no. 4, 475479; MR 2778659 (2012a:11169).
14. Hoelscher J. L., ‘Infinite class field towers’, Math. Ann. 344 (2009) no. 4, 923928; MR 2507631 (2010h:11185).
15. 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).
16. Hunter J., ‘The minimum discriminants of quintic fields’, Proc. Glasgow Math. Assoc. 3 (1957) 5767; MR 0091309 (19,944b).
17. Jones J. W. and Roberts D. P., ‘Artin inline-graphic $L$ -functions with small conductor’, in preparation.
18. 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.
19. Jones J. W. and Roberts D. P., ‘Septic fields with discriminant ± 2 a 3 b ’, Math. Comp. 72 (2003) no. 244, 19751985 (electronic); MR 1986816 (2004e:11119).
20. Jones J. W. and Roberts D. P., ‘A database of local fields’, J. Symbolic Comput. 41 (2006) no. 1, 8097; website: http://math.asu.edu/∼jj/localfields.
21. Jones J. W. and Roberts D. P., ‘Galois number fields with small root discriminant’, J. Number Theory 122 (2007) no. 2, 379407; MR 2292261 (2008e:11140).
22. 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).
23. Jones J. W. and Roberts D. P., ‘The tame–wild principle for discriminant relations for number fields’, Algebra Number Theory 8 (2014) no. 3, 609645.
24. Jones J. W. and Wallington R., ‘Number fields with solvable Galois groups and small Galois root discriminants’, Math. Comp. 81 (2012) no. 277, 555567.
25. Klüners J. and Malle G., ‘A database for field extensions of the rationals’, LMS J. Comput. Math. 4 (2001) 182196 (electronic); MR 2003i:11184.
26. 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).
27. Malle G., ‘On the distribution of Galois groups’, J. Number Theory 92 (2002) no. 2, 315329; MR 1884706 (2002k:12010).
28. Malle G. and Matzat B. H., Inverse Galois theory , Springer Monographs in Mathematics (Springer, Berlin, 1999); MR 1711577 (2000k:12004).
29. 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.
30. 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, 119141; MR 1061762 (91i:11154).
31. 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, 75101; MR 2172333 (2006e:11057).
32. 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, 99117; MR 644904 (83g:12009).
33. Roberts D. P., ‘Chebyshev covers and exceptional number fields’, in preparation. http://facultypages.morris.umn.edu/∼roberts/.
34. Roberts D. P., ‘Wild partitions and number theory’, J. Integer Seq. 10 (2007) no. 6, Article 07.6.6;MR 2335791 (2009b:11206).
35. 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.
36. Schwarz A., Pohst M. and Diaz y Diaz F., ‘A table of quintic number fields’, Math. Comp. 63 (1994) no. 207, 361376; MR 1219705 (94i:11108).
37. The GAP Group, GAP—Groups, Algorithms, and Programming, version 4.4, 2006 (http://www.gap-system.org).
38. The PARI Group, Bordeaux, Pari/gp, version 2.6.2, 2013.
39. Voight J., ‘Tables of totally real number fields’, http://www.math.dartmouth.edu/∼jvoight/nf-tables/index.html.
40. 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).
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