Counting discriminants of number fields
Journal de Théorie des Nombres de Bordeaux, Tome 18 (2006) no. 3, pp. 573-593.

Pour tout groupe de permutations transitif sur n lettres G avec n4 nous donnons sans démonstration des résultats, des conjectures et des calculs numériques sur le nombre de discriminants de corps de nombres L de degré n sur tels que le groupe de Galois de la clôture galoisienne de L soit isomorphe à G.

For each transitive permutation group G on n letters with n4, we give without proof results, conjectures, and numerical computations on discriminants of number fields L of degree n over such that the Galois group of the Galois closure of L is isomorphic to G.

Reçu le :
Publié le :
DOI : https://doi.org/10.5802/jtnb.559
@article{JTNB_2006__18_3_573_0,
     author = {Henri Cohen and Francisco Diaz y Diaz and Michel Olivier},
     title = {Counting discriminants of number fields},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {573--593},
     publisher = {Universit\'e Bordeaux 1},
     volume = {18},
     number = {3},
     year = {2006},
     doi = {10.5802/jtnb.559},
     mrnumber = {2330428},
     zbl = {1193.11109},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.559/}
}
Henri Cohen; Francisco Diaz y Diaz; Michel Olivier. Counting discriminants of number fields. Journal de Théorie des Nombres de Bordeaux, Tome 18 (2006) no. 3, pp. 573-593. doi : 10.5802/jtnb.559. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.559/

[1] A. Baily, On the density of discriminants of quartic fields. J. reine angew. Math. 315 (1980), 190–210. | MR 564533 | Zbl 0421.12007

[2] K. Belabas, A fast algorithm to compute cubic fields. Math. Comp. 66 (1997), 1213–1237. | MR 1415795 | Zbl 0882.11070

[3] K. Belabas, On quadratic fields with large 3-rank. Math. Comp. 73 (2004), 2061–2074. | MR 2059751 | Zbl 1051.11055

[4] K. Belabas, M. Bhargava, C. Pomerance, Error estimates for the Davenport–Heilbronn theorems. Preprint available at

[5] M. Bhargava, Higher Composition Laws I, II, III, IV. | Zbl 1072.11078

[6] H. Cohen, A course in computational algebraic number theory (fourth printing). GTM 138, Springer-Verlag, 2000. | MR 1228206 | Zbl 0786.11071

[7] H. Cohen, Advanced topics in computational number theory. GTM 193, Springer-Verlag, 2000. | MR 1728313 | Zbl 0977.11056

[8] H. Cohen, Comptage exact de discriminants d’extensions abéliennes. J. Th. Nombres Bordeaux 12 (2000), 379–397. | Numdam | Zbl 0976.11055

[9] H. Cohen, Enumerating quartic dihedral extensions of with signatures. Ann. Institut Fourier 53 (2003) 339–377. | Numdam | MR 1990000 | Zbl 01940698

[10] H. Cohen, High precision computation of Hardy-Littlewood constants. Preprint available on the author’s web page at the URL .

[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. | Zbl 02154279

[12] H. Cohen, Constructing and counting number fields. Proceedings ICM 2002 Beijing vol II, Higher Education Press, China (2002), 129–138. | MR 1957027 | Zbl 1042.11067

[13] H. Cohen, F. Diaz y Diaz, M. Olivier, Enumerating quartic dihedral extensions of . Compositio Math. 133 (2002), 65–93. | MR 1918290 | Zbl 1050.11104

[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. | MR 1850610 | Zbl 0987.11079

[15] H. Cohen, F. Diaz y Diaz, M. Olivier, Constructing complete tables of quartic fields using Kummer theory. Math. Comp. 72 (2003) 941–951. | MR 1954977 | Zbl 1081.11081

[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. | MR 1745187 | Zbl 0941.11042

[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. | MR 1925912 | Zbl 1004.11063

[18] H. Cohen, F. Diaz y Diaz, M. Olivier, Cyclotomic extensions of number fields. Indag. Math. (N.S.) 14 (2003), 183–196. | MR 2026813 | Zbl 1056.11058

[19] H. Cohn, The density of abelian cubic fields. Proc. Amer. Math. Soc. 5 (1954), 476–477. | MR 64076 | Zbl 0055.26901

[20] B. Datskovsky, D. J. Wright, Density of discriminants of cubic extensions. J. reine angew. Math. 386 (1988), 116–138. | MR 936994 | Zbl 0632.12007

[21] H. Davenport, H. Heilbronn, On the density of discriminants of cubic fields I. Bull. London Math. Soc. 1 (1969), 345–348. | MR 254010 | Zbl 0211.38602

[22] H. Davenport, H. Heilbronn, On the density of discriminants of cubic fields II. Proc. Royal. Soc. A 322 (1971), 405–420. | MR 491593 | Zbl 0212.08101

[23] H. Hasse, Arithmetische Theorie der kubischen Zahlkörper auf klassenkörpertheoretischer Grundlage. Math. Zeitschrift 31 (1930), 565–582. | MR 1545136

[24] J. Klüners, A counter-example to Malle’s conjecture on the asymptotics of discriminants. C. R. Acad. Sci. Paris 340 (2005), 411–414. | Zbl 1083.11069

[25] S. Mäki, On the density of abelian number fields. Thesis, Helsinki, 1985. | MR 791087

[26] S. Mäki, The conductor density of abelian number fields. J. London Math. Soc. (2) 47 (1993), 18–30. | MR 1200974 | Zbl 0727.11041

[27] G. Malle, On the distribution of Galois groups. J. Number Th. 92 (2002), 315–329. | MR 1884706 | Zbl 1022.11058

[28] G. Malle, On the distribution of Galois groups II, Exp. Math. 13 (2004), 129–135. | MR 2068887 | Zbl 1099.11065

[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. | MR 2282919

[30] D. Roberts, Density of cubic field discriminants. Math. Comp. 70 (2001), 1699–1705. | MR 1836927 | Zbl 0985.11068

[31] G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres. Cours Spécialisés SMF 1, Société Mathématique de France, 1995. | MR 1366197 | Zbl 0880.11001

[32] D. J. Wright, Distribution of discriminants of Abelian extensions. Proc. London Math. Soc. (3) 58 (1989), 17–50. | MR 969545 | Zbl 0628.12006

[33] D. J. Wright, personal communication.

[34] D. J. Wright, A. Yukie, Prehomogeneous vector spaces and field extensions. Invent. Math. 110 (1992), 283–314. | MR 1185585 | Zbl 0803.12004