Source of global number field data (reviewed)

Number fields were drawn from the following sources.

• The PARI database from the Bordeaux PARI group [pari.math.u-bordeaux.fr/pub/numberfields/], which in turn, was a combination of work by several authors:
  • M. Olivier for degree 3 fields;
  • J. Buchmann, D. Ford, and M. Pohst for degree 4 fields [MR:1176706];
  • A. Schwarz, M. Pohst, and F. Diaz y Diaz for degree 5 fields [MR:1219705];
  • M. Olivier for degree 6 fields [MR:1106977], [MR:1149805], [MR:1116107], [MR:1096589], [MR:1061760], [MR:1050276], and with A.-M. Bergé and J. Martinet [MR:1011438];
  • P. Létard for degree 7 fields.
• Totally real fields of degrees from 6 to 10 computed by John Voight [arXiv:0802.0194, math.dartmouth.edu/~jvoight/nf-tables/index.html].
• Almost totally real fields computed by John Voight. [arXiv:1408.2001, MR:3384884].
• Octic and nonic fields of small discriminant from Francesco Battistoni [MR:3912944, MR:4174280]
• Fields from John Jones-David Roberts database. [arXiv:1404.0266, MR:3356048, hobbes.la.asu.edu/NFDB].
• Fields from Jürgen Klüners-Gunter Malle database. [arXiv:math/0102232, galoisdb.math.upb.de/].
• Fields of degrees 2 and 3 unramified outside {2,3,5,...,29} from Benjamin Matschke.
• Octic fields with Galois group $Q_8$ from Fabian Gundlach.
• Some fields associated to elliptic curves from Noam Elkies.
• Fields which have arisen because they are connected to other objects in the LMFDB.

Complete lists of global number fields were computed using two techniques, Hunter searches (see Section 9.3 of [MR:1728313, 10.1007/978-1-4419-8489-0]) and class field theory (see Chapter 5 and Section 9.2 of [MR:1728313, 10.1007/978-1-4419-8489-0]). Hunter searches are named for named for John Hunter [MR:0091309], whose technique was refined significantly by M. Pohst [MR:0644904]. In some cases, the Hunter searches are "targeted" in that they focused on specific prime ramification [MR:1986816, 10.1090/S0025-5718-03-01510-2].

Class groups of imaginary quadratic fields with $|D|<2^{40}$ were computed by A. Mosunov and M. J. Jacobson, Jr. [MR:3471116, 10.1090/mcom3050].

Data on whether or not a field is monogenic, its index, and its inessential primes were computed using pair-gp and are based on Gaál [MR:1896601], Gras [MR:0846964], Śliwa [MR:0678997], Nart [MR:0779058], Mushtaq et. al. [MR:3844199], and a classical theorem of Dedekind.