×

Sharp lower bounds for regulators of small-degree number fields. (English) Zbl 1415.11158

Summary: Minimal discriminants of number fields are presently known for 22 signatures. For 20 of these we give the minimal regulator. Except in the totally complex case, in each signature we find that the field with the minimal discriminant has the minimal regulator.

MSC:

11R29 Class numbers, class groups, discriminants
Full Text: DOI

References:

[1] Bergé, A.-M.; Martinet, J.; Olivier, M., The computation of sextic fields with a quadratic subfield, Math. Comp., 54, 869-884 (1990) · Zbl 0709.11056
[2] Bertin, M. J., Sur une conjecture de Pohst, Acta Arith., 74, 347-349 (1996) · Zbl 0860.11064
[3] Boyd, D., Reciprocal polynomials having small measure, Math. Comp., 35, 1361-1377 (1980) · Zbl 0447.12002
[4] Buchmann, J.; Ford, D. J., On the computation of totally real quartic fields of small discriminant, Math. Comp., 52, 161-174 (1989) · Zbl 0668.12001
[5] Buchmann, J.; Ford, D. J.; Pohst, M., Enumeration of quartic fields of small discriminant, Math. Comp., 61, 873-879 (1993) · Zbl 0788.11060
[6] Cassels, J., An Introduction to the Geometry of Numbers (1959), Springer: Springer Berlin · Zbl 0086.26203
[7] Cohen, H., A Course in Computational Algebraic Number Theory (1996), Springer: Springer Berlin
[8] PARI program by H. Cohen, K. Belabas and collaborators, freely available at
[9] Cohen, H.; Diaz y. Diaz, F.; Olivier, M., Constructing complete tables of quartic fields using Kummer theory, Math. Comp., 72, 941-951 (2003) · Zbl 1081.11081
[10] Delone, B. N.; Faddeev, D. K., The Theory of Irrationalities of the Third Degree (1964), Amer. Math. Soc.: Amer. Math. Soc. Providence · Zbl 0133.30202
[11] Diaz y. Diaz, F., Valeurs minima du discriminant des corps de degré 7 ayant une seule place réelle, C. R. Acad. Sci. Paris Sér. I Math., 296, 137-139 (1983) · Zbl 0527.12007
[12] Diaz y. Diaz, F., Valeurs minima du discriminant pour certains types de corps de degré 7, Ann. Inst. Fourier, 34, 29-38 (1984) · Zbl 0546.12004
[13] Diaz y. Diaz, F., Petits discriminants des corps de nombres totalement imaginaires de degré 8, J. Number Theory, 25, 34-52 (1987) · Zbl 0606.12005
[14] Diaz y. Diaz, F., Discriminant minimal et petits discriminants des corps de nombres de degré 7 avec cinq places réelles, J. Lond. Math. Soc. (2), 38, 33-46 (1988) · Zbl 0653.12003
[15] Diaz y. Diaz, F., A table of totally real quintic number fields, Math. Comp., 56, 801-808 (1991) · Zbl 0726.11080
[16] Ford, D. J., Enumeration of totally complex quartic fields of small discriminant, (Pethö; etal., Computational Number Theory, Proc. Colloq. on Comp. Number Theory. Computational Number Theory, Proc. Colloq. on Comp. Number Theory, Debrecen, Hungary, 1989 (1991), de Gruyter: de Gruyter Berlin), 129-138 · Zbl 0729.11051
[17] Friedman, E., Analytic formulas for the regulator of a number field, Invent. Math., 98, 599-622 (1989) · Zbl 0694.12006
[18] Friedman, E., Regulators and total positivity, Proceedings of the Primeras Jornadas de Teoría de Números. Proceedings of the Primeras Jornadas de Teoría de Números, Publ. Mat., 119-130 (2007) · Zbl 1176.11054
[19] Godwin, H. J., Real quartic fields with small discriminant, J. Lond. Math. Soc., 31, 478-485 (1956) · Zbl 0071.03401
[20] Godwin, H. J., On totally complex quartic fields with small discriminants, Proc. Cambridge Philos. Soc., 53, 1-4 (1957) · Zbl 0077.04601
[21] Godwin, H. J., On quartic fields of signature one with small discriminant, Quart. J. Math. Oxford Ser. (2), 8, 214-222 (1957) · Zbl 0079.05704
[22] Hecke, E., Lectures on the Theory of Algebraic Numbers (1981), Springer: Springer Berlin · Zbl 0504.12001
[23] Hunter, J., The minimum discriminants of quintic fields, Proc. Glasg. Math. Assoc., 3, 57-67 (1957) · Zbl 0080.03003
[24] Katok, A.; Katok, S.; Rodriguez Hetz, R., The Fried average entropy and slow entropy for actions of higher rank abelian groups, Geom. Funct. Anal., 24, 1204-1228 (2014) · Zbl 1308.37005
[25] Lenstra, H. W., Euclidean number fields of large degree, Invent. Math., 38, 237-254 (1976) · Zbl 0328.12007
[26] Létard, P., Construction de corps de nombres en degré 7 et 9 (1995), Université Bordeaux 1, unpublished and only available at the mathematics library of the Université de Bordeaux, but its results for degree 7 were incorporated into the tables available at
[27] Malle, G., The totally real primitive number fields of discriminant at most \(10^9\), (Algorithmic Number Theory. Algorithmic Number Theory, Lecture Notes in Comput. Sci., vol. 4076 (2006), Springer: Springer Berlin), 114-123 · Zbl 1143.11371
[28] Odlyzko, A. M., Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results, Sémin. Théor. Nombres Bordeaux, 2, 119-141 (1990) · Zbl 0722.11054
[29] Olivier, M., Corps sextiques primitifs, Ann. Inst. Fourier, 40, 757-767 (1991) · Zbl 0734.11054
[30] Olivier, M., The computation of sextic fields with a cubic subfield and no quadratic subfield, Math. Comp., 58, 419-432 (1992) · Zbl 0746.11041
[31] Pohst, M., Berechnung kleiner Diskriminanten total reeller algebraischer Zahlkörper, J. Reine Angew. Math., 278/279, 278-300 (1975) · Zbl 0314.12014
[32] Pohst, M., Regulatorabschätzungen für total reelle algebraische Zahlkörper, J. Number Theory, 9, 459-492 (1977) · Zbl 0366.12011
[33] Pohst, M., The minimum discriminant of seventh degree totally real algebraic number fields, (Number Theory and Algebra (1977), Academic Press: Academic Press New York), 235-240 · Zbl 0373.12006
[34] Pohst, M., Eine Regulatorabschätzung, Abh. Math. Semin. Univ. Hambg., 47, 95-106 (1978) · Zbl 0381.12006
[35] Pohst, M., On the computation of number fields of small discriminants including the minimum discriminants of sixth degree fields, J. Number Theory, 14, 99-117 (1982) · Zbl 0478.12005
[36] Pohst, M.; Martinet, J.; Diaz y. Diaz, F., The minimum discriminant of totally real octic fields, J. Number Theory, 36, 145-159 (1990) · Zbl 0719.11079
[37] Poitou, G., Sur les petits discriminants, Sémin. Delange-Pisot-Poitou (Théor. Nombres), 9, 6 (1976-1977) · Zbl 0393.12010
[38] Remak, R., Über Grössenbeziehungen zwischen Diskriminante und Regulator eines algebraischen Zahlkörpers, Compos. Math., 10, 245-285 (1952) · Zbl 0047.27202
[39] Rish, D. G., On number fields of degree five, Moscow Univ. Math. Bull., 37, 99-103 (1982) · Zbl 0512.12009
[40] Schwarz, A.; Pohst, M.; Diaz y. Diaz, F., A table of quintic number fields, Math. Comp., 63, 361-376 (1994) · Zbl 0822.11087
[41] Takeuchi, K., Totally real algebraic number fields of degree 9 with small discriminant, Saitama Math. J., 17, 63-85 (1999) · Zbl 0985.11069
[42] Voight, J., Enumeration of totally real number fields of bounded root discriminant, (van der Poorten, A.; Stein, A., Algorithmic Number Theory. Algorithmic Number Theory, ANTS VIII, Banff, 2008. Algorithmic Number Theory. Algorithmic Number Theory, ANTS VIII, Banff, 2008, Lecture Notes in Comput. Sci., vol. 5011 (2008), Springer: Springer Berlin), 268-281 · Zbl 1205.11125
[43] Washington, L., Introduction to Cyclotomic Fields (1982), Springer: Springer Berlin · Zbl 0484.12001
[44] Weil, A., Basic Number Theory (1974), Springer: Springer Berlin · Zbl 0326.12001
[45] Zimmert, R., Ideale kleiner Norm in Idealklassen und eine Regulatorabschätzung, Invent. Math., 62, 367-380 (1981) · Zbl 0456.12003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.