On small discriminants of number fields of degree 8 and 9
Journal de Théorie des Nombres de Bordeaux, Volume 32 (2020) no. 2, pp. 489-501.

We classify all the number fields with signature (4,2), (6,1), (1,4) and (3,3) having discriminant lower than a specific upper bound. This completes the search for minimum discriminants for fields of degree 8 and continues it in the degree 9 case. We recall the theoretical tools and the algorithmic steps upon which our procedure is based, then we focus on the novelties due to a new implementation of this process on the computer algebra system PARI/GP; finally, we make some remarks about the final results, among which the existence of a number field with signature (3,3) and small discriminant which was not previously known.

Nous classifions tous les corps de nombres de signature (4,2), (6,1), (1,4) et (3,3) et discriminant inférieur à une certaine borne spécifique. Ceci achève la recherche du discriminant minimal pour les corps de degré 8 et contribue à l’étude du cas de degré 9. On rappelle les outils théoriques et les étapes algorithmiques sur lesquels repose notre méthode, on se concentre ensuite sur les aspects nouveaux qui proviennent de la nouvelle implémentation de ce processus dans le système de calcul formel PARI/GP ; enfin, on fait quelques remarques sur nos résultats finals, parmi lesquels mentionnons l’existence d’un corps de nombres de signature (3,3) et d’un petit discriminant, inconnu jusqu’à présent.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1131
Classification: 11R21,  11R29,  11Y40
Keywords: Number fields, classification for small discriminant.
Francesco Battistoni 1

1 Dipartimento di Matematica Università di Milano via Saldini 50 20133 Milano, Italy
@article{JTNB_2020__32_2_489_0,
     author = {Francesco Battistoni},
     title = {On small discriminants of number fields of degree~8 and 9},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {489--501},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {32},
     number = {2},
     year = {2020},
     doi = {10.5802/jtnb.1131},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1131/}
}
TY  - JOUR
TI  - On small discriminants of number fields of degree 8 and 9
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2020
DA  - 2020///
SP  - 489
EP  - 501
VL  - 32
IS  - 2
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1131/
UR  - https://doi.org/10.5802/jtnb.1131
DO  - 10.5802/jtnb.1131
LA  - en
ID  - JTNB_2020__32_2_489_0
ER  - 
%0 Journal Article
%T On small discriminants of number fields of degree 8 and 9
%J Journal de Théorie des Nombres de Bordeaux
%D 2020
%P 489-501
%V 32
%N 2
%I Société Arithmétique de Bordeaux
%U https://doi.org/10.5802/jtnb.1131
%R 10.5802/jtnb.1131
%G en
%F JTNB_2020__32_2_489_0
Francesco Battistoni. On small discriminants of number fields of degree 8 and 9. Journal de Théorie des Nombres de Bordeaux, Volume 32 (2020) no. 2, pp. 489-501. doi : 10.5802/jtnb.1131. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1131/

[1] Sergio Astudillo; Francisco Diaz y Diaz; Eduardo Friedman Sharp lower bounds for regulators of small-degree number fields, J. Number Theory, Volume 167 (2016), pp. 232-258 | Article | MR: 3504045 | Zbl: 1415.11158

[2] Francesco Battistoni Tables of Number Fields (available at http://www.mat.unimi.it/users/battistoni/index.html)

[3] Francesco Battistoni The minimum discriminant of number fields of degree 8 and signature (2,3), J. Number Theory, Volume 198 (2019), pp. 386-395 | Article | MR: 3912944 | Zbl: 1442.11143

[4] Karim Belabas A fast algorithm to compute cubic fields, Math. Comput., Volume 66 (1997) no. 219, pp. 1213-1237 | Article | MR: 1415795 | Zbl: 0882.11070

[5] Anne-Marie Bergé; Jacques Martinet; Michel Olivier The computation of sextic fields with a quadratic subfield, Math. Comput., Volume 54 (1990) no. 190, pp. 869-884 | Article | MR: 1011438 | Zbl: 0709.11056

[6] Johannes Buchmann; David Ford; Michael Pohst Enumeration of quartic fields of small discriminant, Math. Comput., Volume 61 (1993) no. 204, pp. 873-879 | Article | MR: 1176706 | Zbl: 0788.11060

[7] Henri Cohen Advanced topics in computational number theory, Graduate Texts in Mathematics, Volume 193, Springer, 2000 | MR: 1728313 | Zbl: 0977.11056

[8] Henri Cohen; Francisco Diaz y Diaz; Michel Olivier Tables of octic fields with a quartic subfield, Math. Comput., Volume 68 (1999) no. 228, pp. 1701-1716 | Article | MR: 1642813 | Zbl: 1036.11066

[9] Harold Davenport; H. Heilbronn On the density of discriminants of cubic fields. II, Proc. R. Soc. Lond., Ser. A, Volume 322 (1971), pp. 405-420 | MR: 491593 | Zbl: 0212.08101

[10] Francisco Diaz y Diaz Tables minorant la racine n-ième du discriminant d’un corps de degré n, Publications Mathématiques d’Orsay, Volume 6, 1980 | MR: 607864 | Zbl: 0482.12003

[11] Francisco Diaz y Diaz Valeurs minima du discriminant des corps de degré 7 ayant une seule place réelle, C. R. Math. Acad. Sci. Paris, Volume 296 (1983), pp. 137-139 | MR: 693185 | Zbl: 0527.12007

[12] Francisco Diaz y Diaz Valeurs minima du discriminant pour certains types de corps de degré 7, Ann. Inst. Fourier, Volume 34 (1984) no. 3, pp. 29-38 | Article | Numdam | MR: 762692 | Zbl: 0546.12004

[13] Francisco Diaz y Diaz Petits discriminants des corps de nombres totalement imaginaires de degré 8, J. Number Theory, Volume 25 (1987), pp. 34-52 | Article | Zbl: 0606.12005

[14] Francisco Diaz y Diaz Discriminant minimal et petits discriminants des corps de nombres de degré 7 avec cinq places réelles, J. Lond. Math. Soc., Volume 38 (1988) no. 1, pp. 33-46 | Article | Zbl: 0653.12003

[15] Francisco Diaz y Diaz; Michel Olivier Imprimitive ninth-degree number fields with small discriminants, Math. Comput., Volume 64 (1995) no. 209, pp. 305-321 | Article | MR: 1260128 | Zbl: 0819.11070

[16] David Ford; Sebastian Pauli; Xavier-François Roblot A fast algorithm for polynomial factorization over p , J. Théor. Nombres Bordeaux, Volume 14 (2002) no. 1, pp. 151-169 | Article | MR: 1925995 | Zbl: 1032.11053

[17] Alexander Hulpke Constructing transitive permutation groups, J. Symb. Comput., Volume 39 (2005) no. 1, pp. 1-30 | Article | MR: 2168238 | Zbl: 1131.20003

[18] Jüurgen Klüners; Gunter Malle A database for number fields (available at http://galoisdb.math.upb.de/home) | Zbl: 1067.11516

[19] Jacques Martinet Méthodes géométriques dans la recherche des petits discriminants, Séminaire de théorie des nombres (Progress in Mathematics) Volume 59, Birkhäuser, 1983, p. 1983-84 | Zbl: 0567.12009

[20] Andrew M. Odlyzko Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results, Sémin. Théor. Nombres Bordx., Sér. II, Volume 2 (1990) no. 1, pp. 119-141 | Article | Numdam | MR: 1061762 | Zbl: 0722.11054

[21] Michel Olivier Corps sextiques contenant un corps quadratique. I, Sémin. Théor. Nombres Bordx., Sér. II, Volume 1 (1989) no. 1, pp. 205-250 | Article | Numdam | MR: 1050276 | Zbl: 0719.11087

[22] Michel Olivier Corps sextiques contenant un corps quadratique. II, Sémin. Théor. Nombres Bordx., Sér. II, Volume 2 (1990) no. 1, pp. 49-102 | Article | Numdam | MR: 1061760 | Zbl: 0719.11088

[23] Michel Olivier Corps sextiques primitifs, Ann. Inst. Fourier, Volume 40 (1990) no. 4, pp. 757-767 | Article | Numdam | MR: 1096589 | Zbl: 0734.11054

[24] Michel Olivier Corps sextiques contenant un corps cubique. III, Sémin. Théor. Nombres Bordx., Sér. II, Volume 3 (1991) no. 1, pp. 201-245 | Article | Numdam | MR: 1116107 | Zbl: 0726.11081

[25] Michel Olivier Corps sextiques primitifs. IV, Sémin. Théor. Nombres Bordx., Sér. II, Volume 3 (1991) no. 2, pp. 381-404 | Article | Numdam | MR: 1149805 | Zbl: 0768.11051

[26] Michel Olivier The computation of sextic fields with a cubic subfield and no quadratic subfield, Math. Comput., Volume 58 (1992) no. 197, pp. 419-432 | Article | MR: 1106977 | Zbl: 0746.11041

[27] Michael Pohst The minimum discriminant of seventh degree totally real algebraic number fields, Number theory and algebra, Academic Press Inc., 1977, pp. 235-240 | Zbl: 0373.12006

[28] Michael Pohst On the computation of number fields of small discriminants including the minimum discriminants of sixth degree fields, J. Number Theory, Volume 14 (1982), pp. 99-117 | Article | MR: 644904 | Zbl: 0478.12005

[29] Michael Pohst; Jacques Martinet; Francisco Diaz y Diaz The minimum discriminant of totally real octic fields, J. Number Theory, Volume 36 (1990) no. 2, pp. 145-159 | Article | MR: 1072461 | Zbl: 0719.11079

[30] Georges Poitou Sur les petits discriminants, Séminaire Delange-Pisot-Poitou, 18e année: (1976/77), Théorie des nombres, Fasc. 1, Secrétariat Mathématique, 1976 | Numdam | Zbl: 0393.12010

[31] A. Schwarz; Michael Pohst; Francisco Diaz y Diaz A table of quintic number fields, Math. Comput., Volume 63 (1994) no. 207, pp. 361-376 | Article | MR: 1219705 | Zbl: 0822.11087

[32] Schehrazad Selmane Non-primitive number fields of degree eight and of signature (2,3), (4,2) and (6,1) with small discriminant, Math. Comput., Volume 68 (1999) no. 225, pp. 333-344 | Article | MR: 1489974 | Zbl: 0958.11082

[33] Schehrazad Selmane Odlyzko–Poitou–Serre lower bounds for discriminants for some number fields, Maghreb Math. Rev., Volume 8 (1999), pp. 151-162 | MR: 1871537

[34] Jean-Pierre Serre Minorations de discriminants, Jean-Pierre Serre, collected papers. Vol. 3, Volume 3, 1975, pp. 240-243 (note of october 1975)

[35] Denis Simon Petits discriminants de polynomes irréductibles (available at https://simond.users.lmno.cnrs.fr/maths/TableSmallDisc.html)

[36] Kisao Takeuchi Totally real algebraic number fields of degree 9 with small discriminant, Saitama Math. J., Volume 17 (2000), pp. 63-85 | MR: 1740248 | Zbl: 0985.11069

[37] The LMFDB Collaboration The L-functions and Modular Forms Database, 2013 (http://www.lmfdb.org)

[38] The PARI Group PARI/GP version 2.11.0, 2018 (available from http://pari.math.u-bordeaux.fr/)

Cited by Sources: