Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

A table of quintic number fields


Authors: A. Schwarz, M. Pohst and F. Diaz y Diaz
Journal: Math. Comp. 63 (1994), 361-376
MSC: Primary 11Y40; Secondary 11R21, 11R32
DOI: https://doi.org/10.1090/S0025-5718-1994-1219705-3
MathSciNet review: 1219705
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: All algebraic number fields F of degree 5 and absolute discriminant less than $ 2 \times {10^7}$ (totally real fields), respectively $ 5 \times {10^6}$ (other signatures) are determined. We describe the methods which we applied and list significant data.


References [Enhancements On Off] (What's this?)

  • [1] E. H. Bareiss, Sylvester's identity and multistep integer preserving Gaussian elimination, Math. Comp. 22 (1968), 565-578. MR 0226829 (37:2416)
  • [2] J. Buchmann and D. J. Ford, On the computation of totally real quartic fields of small discriminant, Math. Comp. 52 (1989), 161-174. MR 946599 (89f:11147)
  • [3] J. Buchmann, D. Ford, and M. Pohst, Enumeration of quartic fields of small discriminant, Math. Comp. 61 (1993), 873-879. MR 1176706 (94a:11164)
  • [4] F. Diaz y Diaz, A table of totally real quintic number fields, Math. Comp. 56 (1991), 801-808. MR 1068820 (91h:11155)
  • [5] D. Ford, Enumeration of totally complex quartic fields of small discriminant, Computational Number Theory, Proc. Colloq. on Comp. Number Theory (Debrecen, Hungary, 1989) (A. Pethö, M. E. Pohst, H. C. Williams, and H. G. Zimmer, eds.), de Gruyter, Berlin and New York, 1991, pp. 129-138. MR 1151860 (93b:11140)
  • [6] J. Hunter, The minimum discriminant of quintic fields, Proc. Glasgow Math. Assoc. 3 (1957), 57-67. MR 0091309 (19:944b)
  • [7] N. Obreschkoff, Verteilung und Berechnung der Nullstellen reeller Polynome, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963. MR 0164003 (29:1302)
  • [8] M. Pohst, Berechnung kleiner Diskriminanten total reeller algebraischer Zahlkörper, J. Reine Angew. Math. 278/279 (1975), 278-300. MR 0387242 (52:8085)
  • [9] -, On the computation of number fields of small discriminants including the minimum discriminants of sixth degree fields, J. Number Theory 14 (1982), 99-117. MR 644904 (83g:12009)
  • [10] -, On computing isomorphisms of equation orders, Math. Comp. 48 (1987), 309-314. MR 866116 (88b:11066)
  • [11] M. Pohst, J. Martinet, and F. Diaz y Diaz, The minimum discriminant of totally real octic fields, J. Number Theory 36 (1991), 145-159. MR 1072461 (91g:11128)
  • [12] M. Pohst and H. Zassenhaus, Algorithmic algebraic number theory, Cambridge Univ. Press, Cambridge, 1989. MR 1033013 (92b:11074)
  • [13] A. Schwarz, Berechnung von Zahlkörpen fünften Grades mit kleiner Diskriminante, Diplomarbeit, Heinrich-Heine-Universität Düsseldorf, 1991.
  • [14] J. Graf von Schmettow, KANT-a tool for computations in algebraic number fields, Computational Number Theory (A. Pethö, M. E. Pohst, H. C. Williams, and H. G. Zimmer, eds.), de Gruyter, Berlin and New York, 1991, pp. 321-330. MR 1151875 (92m:11151)
  • [15] C. L. Siegel, The trace of totally positive and real algebraic integers, Ann. of Math. (2) 46 (1945), 302-312. MR 0012092 (6:257a)
  • [16] R. P. Stauduhar, The determination of Galois groups, Math. Comp. 27 (1973), 981-996. MR 0327712 (48:6054)
  • [17] B. L. van der Waerden, Algebra. I, 8th ed., Heidelberger Taschenbücher 12, Springer-Verlag, Berlin, Heidelberg, New York, 1971.
  • [18] H. Zassenhaus, On the second round of the maximal order program, Applications of Number Theory to Numerical Analysis, Academic Press, New York, 1972, pp. 398-431. MR 0371862 (51:8079)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 11Y40, 11R21, 11R32

Retrieve articles in all journals with MSC: 11Y40, 11R21, 11R32


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1994-1219705-3
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society