Counting discriminants of number fields

Henri Cohen[1]; Francisco Diaz y Diaz[1]; Michel Olivier[1]

  • [1] Laboratoire A2X, U.M.R. 5465 du C.N.R.S., Université Bordeaux I, 351 Cours de la Libération, 33405 TALENCE Cedex, FRANCE

Journal de Théorie des Nombres de Bordeaux (2006)

  • Volume: 18, Issue: 3, page 573-593
  • ISSN: 1246-7405

Abstract

top
For each transitive permutation group G on n letters with n ≤ 4 , we give without proof results, conjectures, and numerical computations on discriminants of number fields L of degree n over ℚ such that the Galois group of the Galois closure of L is isomorphic to G .

How to cite

top

Cohen, Henri, Diaz y Diaz, Francisco, and Olivier, Michel. "Counting discriminants of number fields." Journal de Théorie des Nombres de Bordeaux 18.3 (2006): 573-593. <http://eudml.org/doc/249645>.

@article{Cohen2006,
abstract = {For each transitive permutation group $G$ on $n$ letters with $n\le 4$, we give without proof results, conjectures, and numerical computations on discriminants of number fields $L$ of degree $n$ over $\mathbb\{Q\}$ such that the Galois group of the Galois closure of $L$ is isomorphic to $G$.},
affiliation = {Laboratoire A2X, U.M.R. 5465 du C.N.R.S., Université Bordeaux I, 351 Cours de la Libération, 33405 TALENCE Cedex, FRANCE; Laboratoire A2X, U.M.R. 5465 du C.N.R.S., Université Bordeaux I, 351 Cours de la Libération, 33405 TALENCE Cedex, FRANCE; Laboratoire A2X, U.M.R. 5465 du C.N.R.S., Université Bordeaux I, 351 Cours de la Libération, 33405 TALENCE Cedex, FRANCE},
author = {Cohen, Henri, Diaz y Diaz, Francisco, Olivier, Michel},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {discriminants of number fields; Galois group; Galois closure},
language = {eng},
number = {3},
pages = {573-593},
publisher = {Université Bordeaux 1},
title = {Counting discriminants of number fields},
url = {http://eudml.org/doc/249645},
volume = {18},
year = {2006},
}

TY - JOUR
AU - Cohen, Henri
AU - Diaz y Diaz, Francisco
AU - Olivier, Michel
TI - Counting discriminants of number fields
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2006
PB - Université Bordeaux 1
VL - 18
IS - 3
SP - 573
EP - 593
AB - For each transitive permutation group $G$ on $n$ letters with $n\le 4$, we give without proof results, conjectures, and numerical computations on discriminants of number fields $L$ of degree $n$ over $\mathbb{Q}$ such that the Galois group of the Galois closure of $L$ is isomorphic to $G$.
LA - eng
KW - discriminants of number fields; Galois group; Galois closure
UR - http://eudml.org/doc/249645
ER -

References

top
  1. A. Baily, On the density of discriminants of quartic fields. J. reine angew. Math. 315 (1980), 190–210. Zbl0421.12007MR564533
  2. K. Belabas, A fast algorithm to compute cubic fields. Math. Comp. 66 (1997), 1213–1237. Zbl0882.11070MR1415795
  3. K. Belabas, On quadratic fields with large 3 -rank. Math. Comp. 73 (2004), 2061–2074. Zbl1051.11055MR2059751
  4. K. Belabas, M. Bhargava, C. Pomerance, Error estimates for the Davenport–Heilbronn theorems. Preprint available at http://www.math.u-bordeaux1.fr/~belabas/pub/ Zbl1227.11114
  5. M. Bhargava, Higher Composition Laws I, II, III, IV. Zbl1072.11078
  6. H. Cohen, A course in computational algebraic number theory (fourth printing). GTM 138, Springer-Verlag, 2000. Zbl0786.11071MR1228206
  7. H. Cohen, Advanced topics in computational number theory. GTM 193, Springer-Verlag, 2000. Zbl0977.11056MR1728313
  8. H. Cohen, Comptage exact de discriminants d’extensions abéliennes. J. Th. Nombres Bordeaux 12 (2000), 379–397. Zbl0976.11055
  9. H. Cohen, Enumerating quartic dihedral extensions of ℚ with signatures. Ann. Institut Fourier 53 (2003) 339–377. Zbl1114.11085MR1990000
  10. H. Cohen, High precision computation of Hardy-Littlewood constants. Preprint available on the author’s web page at the URL http://www.math.u-bordeaux.fr/~cohen. 
  11. H. Cohen, Counting A 4 and S 4 extensions of number fields with given resolvent cubic, in “High primes and misdemeanours: lectures in honour of the 60th birthday of Hugh Cowie Williams”. Fields Institute Comm. 41 (2004), 159–168. Zbl1197.11140
  12. H. Cohen, Constructing and counting number fields. Proceedings ICM 2002 Beijing vol II, Higher Education Press, China (2002), 129–138. Zbl1042.11067MR1957027
  13. H. Cohen, F. Diaz y Diaz, M. Olivier, Enumerating quartic dihedral extensions of ℚ . Compositio Math. 133 (2002), 65–93. Zbl1050.11104MR1918290
  14. H. Cohen, F. Diaz y Diaz, M. Olivier, Construction of tables of quartic fields using Kummer theory. Proceedings ANTS IV, Leiden (2000), Lecture Notes in Computer Science 1838, Springer-Verlag, 257–268. Zbl0987.11079MR1850610
  15. H. Cohen, F. Diaz y Diaz, M. Olivier, Constructing complete tables of quartic fields using Kummer theory. Math. Comp. 72 (2003) 941–951. Zbl1081.11081MR1954977
  16. H. Cohen, F. Diaz y Diaz, M. Olivier, Densité des discriminants des extensions cycliques de degré premier. C. R. Acad. Sci. Paris 330 (2000), 61–66. Zbl0941.11042MR1745187
  17. H. Cohen, F. Diaz y Diaz, M. Olivier, On the density of discriminants of cyclic extensions of prime degree. J. reine angew. Math. 550 (2002), 169–209. Zbl1004.11063MR1925912
  18. H. Cohen, F. Diaz y Diaz, M. Olivier, Cyclotomic extensions of number fields. Indag. Math. (N.S.) 14 (2003), 183–196. Zbl1056.11058MR2026813
  19. H. Cohn, The density of abelian cubic fields. Proc. Amer. Math. Soc. 5 (1954), 476–477. Zbl0055.26901MR64076
  20. B. Datskovsky, D. J. Wright, Density of discriminants of cubic extensions. J. reine angew. Math. 386 (1988), 116–138. Zbl0632.12007MR936994
  21. H. Davenport, H. Heilbronn, On the density of discriminants of cubic fields I. Bull. London Math. Soc. 1 (1969), 345–348. Zbl0211.38602MR254010
  22. H. Davenport, H. Heilbronn, On the density of discriminants of cubic fields II. Proc. Royal. Soc. A 322 (1971), 405–420. Zbl0212.08101MR491593
  23. H. Hasse, Arithmetische Theorie der kubischen Zahlkörper auf klassenkörpertheoretischer Grundlage. Math. Zeitschrift 31 (1930), 565–582. Zbl56.0167.02MR1545136
  24. J. Klüners, A counter-example to Malle’s conjecture on the asymptotics of discriminants. C. R. Acad. Sci. Paris 340 (2005), 411–414. Zbl1083.11069
  25. S. Mäki, On the density of abelian number fields. Thesis, Helsinki, 1985. Zbl0566.12001MR791087
  26. S. Mäki, The conductor density of abelian number fields. J. London Math. Soc. (2) 47 (1993), 18–30. Zbl0727.11041MR1200974
  27. G. Malle, On the distribution of Galois groups. J. Number Th. 92 (2002), 315–329. Zbl1022.11058MR1884706
  28. G. Malle, On the distribution of Galois groups II, Exp. Math. 13 (2004), 129–135. Zbl1099.11065MR2068887
  29. G. Malle, The totally real primitive number fields of discriminant at most 10 9 . Proceedings ANTS VII (Berlin), 2006, Springer Lecture Notes in Computer Science XXX. Zbl1143.11371MR2282919
  30. D. Roberts, Density of cubic field discriminants. Math. Comp. 70 (2001), 1699–1705. Zbl0985.11068MR1836927
  31. G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres. Cours Spécialisés SMF 1, Société Mathématique de France, 1995. Zbl0880.11001MR1366197
  32. D. J. Wright, Distribution of discriminants of Abelian extensions. Proc. London Math. Soc. (3) 58 (1989), 17–50. Zbl0628.12006MR969545
  33. D. J. Wright, personal communication. 
  34. D. J. Wright, A. Yukie, Prehomogeneous vector spaces and field extensions. Invent. Math. 110 (1992), 283–314. Zbl0803.12004MR1185585

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.