|
The density of discriminants of  -sextic number fields
Author(s):
Manjul
Bhargava;
Melanie
Matchett
Wood
Journal:
Proc. Amer. Math. Soc.
136
(2008),
1581-1587.
MSC (2000):
Primary 11R21, 11R45
Posted:
October 12, 2007
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
Abstract:
We prove an asymptotic formula for the number of sextic number fields with Galois group  and absolute discriminant  . In addition, we give an interpretation of the constant in the formula in terms of the asymptotic densities of given local completions among these sextic fields. Our proof gives analogous results when we count  -sextic extensions of any number field, and also when finitely many local completions have been specified for the sextic extensions.
References:
-
- 1.
- K. Belabas, M. Bhargava, and C. Pomerance, Error terms for the Davenport-Heilbronn theorems, preprint.
- 2.
- K. Belabas and E. Fouvry, Discriminants Cubiques et Progressions Arithmétiques, preprint.
- 3.
- M. Bhargava, Higher composition laws III: The parametrization of quartic rings, Ann. of Math. 160 (2004), 1329-1360. MR 2113024 (2005k:11214)
- 4.
- M. Bhargava, The density of discriminants of quartic rings and fields, Ann. of Math. 162 (2005), 1031-1063. MR 2183288 (2006m:11163)
- 5.
- M. Bhargava, The density of discriminants of quintic rings and fields, Ann. of Math., to appear.
- 6.
- M. Bhargava, Mass formulae for extensions of local fields, and conjectures on the density of number field discriminants, Internat. Math. Research Notices, (2007), Vol. 2007, Articke ID rnm052, 20 pp.
- 7.
- H. Cohen, F. Diaz y Diaz, and M. Olivier, Counting discriminants of number fields of degree up to four, in Algorithmic Number Theory (Leiden, 2000), Lecture Notes in Comput. Sci. 1838, 269-283, Springer-Verlag, New York, 2000. MR 1850611 (2002g:11148)
- 8.
- B. Datskovsky and D. J. Wright, Density of discriminants of cubic extensions, J. Reine Angew. Math. 386 (1988), 116-138. MR 0936994 (90b:11112)
- 9.
- H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields II, Proc. Roy. Soc. London Ser. A 322 (1971), no. 1551, 405-420. MR 0491593 (58:10816)
- 10.
- J. W. Jones and D. P. Roberts, A database of local fields, J. Symbolic Comput. 41 (2006), no. 1, 80-97. http://math.asu.edu/
 jj/localfields/ MR 2194887 (2006k:11230) - 11.
- J. Klüners, A counterexample to Malle's conjecture on the asymptotics of discriminants, C. R. Math. Acad. Sci. Paris, Ser. I 340 (2005), 411-414. MR 2135320 (2005m:11214)
- 12.
- J. Klüners and G. Malle, Counting nilpotent Galois extensions, J. reine angew. Math. 572 (2004), 1-26. MR 2076117 (2005f:11259)
- 13.
- S. Mäki, On the density of abelian number fields. Annales Academiae Scientiarum Fennicae. Series A I. Mathematica Dissertationes 54 (1985), 104 pp. MR 791087 (86h:11079)
- 14.
- G. Malle, On the distribution of Galois groups II, Experiment. Math. 13 (2004), 129-135, MR 2068887 (2005g:11216)
- 15.
- D. J. Wright, Distribution of discriminants of abelian extensions, Proc. London Math. Soc. (3) 58 (1989), 17-50. MR 969545 (90b:11115)
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical Society
with MSC
(2000):
11R21, 11R45
Retrieve articles in all Journals with MSC
(2000):
11R21, 11R45
Additional Information:
Manjul
Bhargava
Affiliation:
Department of Mathematics, Princeton University, Princeton, New Jersey 08544
Melanie
Matchett
Wood
Affiliation:
Department of Mathematics, Princeton University, Princeton, New Jersey 08544
DOI:
10.1090/S0002-9939-07-09171-X
PII:
S 0002-9939(07)09171-X
Received by editor(s):
December 19, 2006
Received by editor(s) in revised form:
March 24, 2007
Posted:
October 12, 2007
Communicated by:
Wen-Ching Winnie Li
Copyright of article:
Copyright
2007,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
Comments: webmaster@ams.org