\input amstex \documentstyle{amsppt} \font\tfont=cmbx10 scaled 1200 \font\mone=cmbx10 \def\parag#1{\medskip\medskip\noindent{\bf{#1}}\medskip} \document {\bf Table 1.} Minimal absolute values of discriminants of number fields of degree $n$ with $2r_2$ complex conjugate fields. \medskip \settabs \+mmm&mmmmmmmm&mmmmmmmm&mmmmmmmm&mmmmmm&mmmmmm\cr \+$n$&$r_2$=0&$r_2$=1&$r_2$=2&$r_2$=3&$r_2$=4\cr \smallskip \+1&1\cr \+2&5&3\cr \+3&49&23\cr \+4&725&275&117\cr \+5&14,641&4,511&1,609\cr \+6&300,125&92,779&28,037&9,747\cr \+7&20,134,393&2,306,599&612,233&184,607\cr \+8&282,300,416&?&?&?&1,257,728\cr \medskip {\bf Table 2.} Minimal root-discriminants of number fields of degree $n$ with $2r_2$ complex conjugate fields. \settabs \+mmm&mmmmmmmm&mmmmmmmm&mmmmmmmm&mmmmmm&mmmmmm\cr \+$n$&$r_2$=0&$r_2$=1&$r_2$=2&$r_2$=3&$r_2$=4\cr \smallskip \+1&1\cr \+2&2.236&1.732\cr \+3&3.659&2.844\cr \+4&5.189&4.072&3.289\cr \+5&6.809&5.381&4.378\cr \+6&8.132&6.728&5.512&4.622\cr \+7&11.051&8.110&6.710&5.653\cr \+8&11.385&?&?&?&5.787\cr \medskip {\bf Table 3.} Small root-discriminants of totally complex fields and the best known lower bounds. (For $n\le 8$, the root-discriminants are known to be minimal for each degree.) \medskip \settabs \+mmm&mmmmmmmm&mmmmmmmmmmmmm&mmmmmmmmmmmmmmm\cr \+&&&&unconditional\cr \+$n$&$D^{1/n}$&GRH bound&bound\cr \+$2$&1.732&1.722&1.722\cr \+$4$&3.289&3.263&3.254\cr \+$6$&4.622&4.592&4.557\cr \+$8$&5.787&5.734&5.659\cr \+$10$&6.793&6.726&6.600\cr \+$14$&8.426&8.371&8.122\cr \+$20$&10.438&10.270&9.805\cr \+$32$&13.181&12.912&12.002\cr \+$48$&15.472&15.225&13.772\cr \medskip {\bf Table 4.} Small root-discriminants of totally real fields and the best known lower bounds. (For $n\le 8$, the root-discriminants are known to be minimal for each degree.) \settabs \+mmm&mmmmmmmm&mmmmmmmmmmmmm&mmmmmmmmmmmmmmm\cr \+&&&&unconditional\cr \+$n$&$D^{1/n}$&GRH bound&bound\cr \+$1$&1&0.997&0.997\cr \+$2$&2.236&2.225&2.223\cr \+$3$&3.659&3.630&3.610\cr \+$4$&5.189&5.124&5.067\cr \+$5$&6.809&6.640&6.523\cr \+$6$&8.182&8.143&7.941\cr \+$7$&11.051&9.611&9.301\cr \+$8$&11.385&11.036&10.596\cr \+$9$&12.869&12.410&11.823\cr \medskip {\bf Table 5.} Contributions of prime ideals and zeros to discriminant bounds for some fields. (See Section 6 for detailed explanation.) \medskip \settabs \+mmmmmmm&mmm&mmm&mmmmmm&mmmmm&mmmm&mmmmmmmmmmmmmmmm\cr \+$D$&$n$&$r_2$&\text{deficiency}&\text{ideals}&\text{zeros}&\text{norms}\cr \medskip \+$20134393$&7&0&0.1397&0.0761&0.0636&7,17,23,37,43\cr \+$1257728$&8&4&0.0092&0.0013&0.0079&16\cr \+$1265625$&8&4&0.0100&0.0074&16,16\cr \+$282300416$&8&0&0.0311&0.0183&0.0128&16,41,47,47,47,47,49,49\cr \+$309593125$&8&0&0.0427&0.029&0.0398&3&,41,41\cr \+$9685993193$&9&0&0.0364&0.0143&0.0221&27,31,41,43,67,67,73,73,79,79,79,79\cr \vskip 0.6 in \centerline {\bf {REFERENCES}} \vskip 0.3 in References are grouped by topic, and are arranged approximately chronologically within each group. For most areas only papers published after 1970 are listed. References to earlier ones can be found in the publications listed here, especially in the book of Narkiewicz [A1]. \hbox{} \vskip 0.4 in {\bf A - Books and surveys} \medskip 1. W. NARKIEWICZ, {\sl Elementary and Analytic Theory of Algebraic Numbers}, PWN-Polish Scientific Publishers, Warsaw 1974. MR 50$\#$ 268. (Very complete references for work before 1973.) 2. L. C. WASHINGTON, {\sl Introduction to Cyclotomic Fields}, Springer 1982. MR 85g:11001. \medskip 3. J. MARTINET, {\sl M\'ethodes g\'eom\'etriques dans la recherche des petits discriminants,} pp. 147-179, S\'eminaire Th\'eorie des Nombres, Paris, 1983-84, C. Goldstein \'ed., Birkh\"auser Boston, 1985. MR 88h:11083. \medskip 4. A. M. ODLYZKO, {\sl Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results,} S\'em. de Th\'eorie des Nombres, Bordeaux 2 (1990), 119-141. \bigskip {\bf B - Analytic lower bounds for discriminants} \medskip 1. H.M. STARK, {\sl Some effective cases of the Brauer-Siegel theorem,} Invent. math. 23 (1974), 135-152. MR 49 $\#$ 7218. 2. H.M. STARK, {\sl The analytic theory of algebraic numbers,} Bull. Am. Math. Soc. 81 (1975), 961-972. MR 56 $\#$ 2961. 3. A.M. ODLYZKO, {\sl Some analytic estimates of class numbers and discriminants,} Invent. math. 29 (1975), 275-286. MR 51 $\#$ 12788. 4. A.M. ODLYZKO, {\sl Lower bounds for discriminants of number fields,} Acta Arith. 29 (1976) 275--297. MR 53 $\#$ 5531. 5. A.M. ODLYZKO, {\sl Lower bounds for discriminants of number fields. II.} Tohoku Math. J. 29 (1977), 209-216. MR 56 $\#$ 309. 6. A.M. ODLYZKO, {\sl On conductors and discriminants,} pp. 377-407, {\sl Algebraic Number Fields,} (Proc. 1975 Durham Symp.), A. Fr\"ohlich, ed., Academic Press 1977. MR 56 $\#$11961. 7. J.-P. SERRE, {\sl Minorations de discriminants,} note of October 1975, published on pp. 240-243 in vol. 3 of Jean-Pierre SERRE, Collected Papers, Springer 1986. 8. A.M. ODLYZKO, {\sl Discriminant bounds,} tables dated Nov. 29, 1976 (unpublished). Some of these bounds are included in Ref. B12. 9. G. POITOU, {\sl Minorations de discriminants (d'apr\`es A.M. Odly\-zko),} S\'eminaire Bourbaki, Vol. 1975/76 28\`eme ann\'ee, Exp. No. 479, pp. 136-153, Lecture Notes in Math. \#567, Springer 1977. MR 55 \#7995. 10. G. POITOU, {\sl Sur les petits discriminants,} S\'eminaire Delange-Pisot-Poitou, 18e ann\'ee : (1976/77), Th\'eorie des nombres, Fasc. 1, Exp. No. 6, 18pp., Secr\'etariat Math., Paris, 1977. MR 8li:12007. 11. F. DIAZ Y DIAZ, {\sl Tables minorant la racine n-i\`eme du discriminant d'un corps de degr\'e n,} Publications Math\'ematiques d'Orsay 80.06. Universit\'e de Paris-Sud, D\'epartement de Math\'ematique, Orsay, 1980. 59 pp. MR 82i:12007. (Some of these bounds are included in Ref. B12.) 12. J. MARTINET, {\sl Petits discriminants des corps de nombres,} pp. 151-193, Journ\'ees Arithm\'etiques 1980, J.V. Armitage, ed., Cambridge Univ. Press 1982. MR 84g:12009. See also A3,D9,E4,E6,E10,E16. \bigskip {\bf C. Constructions of fields with small discriminants} \medskip 1. H.W. LENSTRA, {\sl Euclidean number fields of large degree,} Invent. math. 38 (1977), 237-254. MR 55\#2836. 2. J. MARTINET, {\sl Tours de corps de classes et estimations de discri\-minants,} Invent. math. 44 (1978), 65-73. MR 57 $\#$275. 3. J. MARTINET, {\sl Petits discriminants,} Ann. Inst. Fourier (Grenoble) 29, no 1 (1979), 159-170. MR 81h:12006. 4. R. SCHOOF, {\sl Infinite class field towers of quadratic fields}, J. reine angew. Math. 372 (1986), 209--220. MR 88a:11121. See also A3,B12. \bigskip {\bf D. Bounds for class numbers} \medskip 1. J.M. MASLEY, {\sl Odlyzko bounds and class number problems,} pp. 465-474, Algebraic Number Fields (Proc. Durham Symp., 1975), A. Fr\"ohlich, ed., Academic Press 1977. MR 56 \#5493. 2. J.M. MASLEY, {\sl Class numbers of real cyclic number fields with small conductor,} Compositio Math. 37 (1978), 297-319. MR 80e:12005. 3. J.M. MASLEY, {\sl Where are number fields with small class numbers?,} pp. 221-242, Number Theory, Carbondale 1979, Lecture Notes in Math. \#751, Springer, 1979. MR 81f:12004. 4. J. HOFFSTEIN, {\sl Some analytic bounds for zeta functions and class numbers,} Invent. math. 55 (1979), 37-47. MR 80k:12019. 5. J.M. MASLEY, {\sl Class groups of abelian number fields,} pp. 475-497, Proc. Queen's Number Theory Conf. 1979, P. Ribenboim, ed., Queen's Papers in Pure and Applied Mathematics no. 54, Queen's Univ., 1980, MR 83f:12007. 6. J. MARTINET, {\sl Sur la constante de Lenstra des corps de nombres,} S\'em. Th\'eorie des Nombres de Bordeaux 1979-1980, Exp. \#17, 21 pp., UNIV. BORDEAUX 1980. MR 83b:12007. 7. F.J. van der LINDEN, {\sl Class number computations of real abelian number fields,} Math. Comp. 39 (1982), 693-707. MR 84e:12005. 8. A. LEUTBECHER and J. MARTINET, {\sl Lenstra's constant and Euclidean number fields,} Ast\'erisque 94 (1982), 87-131. MR 85b:11090. 9. A. LEUTBECHER, {\sl Euclidean fields having a large Lenstra constant,} Ann. Inst. Fourier (Grenoble) 35, no.2 (1985), 83-106. MR 86j:11107. 10. J. HOFFSTEIN and N. JOCHNOWITZ, {\sl On Artin's conjecture and the class number of certain CM fields,} Duke Math. J., 59 (1989), 553-563. 11. J. HOFFSTEIN and N. JOCHNOWITZ, {\sl On Artin's conjecture and the class number of certain CM fields-II,} Duke Math. J., 59 (1989), 565-584. 12. K. YAMAMURA, {\sl Determination of imaginary abelian number fields with class number one,} Math. Comp. 62 (1994), 899-921. 13. K. YAMAMURA, {\sl The maximal unramified extensions of the imaginary quadratic number fields with class number two,} J. Number Theory 60 (1996), 42-50. 14. K. YAMAMURA, {\sl Maximal unramified extensions of imaginary quadratic number fields of small conductor,} to appear. See also B1,B2,B5. \bigskip {\bf E. Bounds for regulators and norms of ideals in ideal classes} \medskip 1. M. POHST, {\sl Regulatorabsch\"atzungen f\"ur total reelle algebraische Zahlk\"orper}, J. Number Theory 9 (1977) 459-492. MR 57 \#268. 2. G. GRAS and M.-N. GRAS, {\sl Calcul du nombre de classes et des unit\'es des extensions ab\'eliennes r\'eelles de Q,} Bull. Sci. Math. 101 (2) (1977), 97-129. MR 58 \#586. 3. M. POHST, {\sl Eine Regulatorabsch\"atzung,} Abh. Math. Sem. Univ. Hamburg 47 (1978), 95-106. MR 58 \#16596. 4. R. ZIMMERT, {\sl Ideale kleiner Norm in Idealklassen und eine Regulatorabsch\"atzung,} Invent. math. 62 (1981), 367-380. MR 83g:12008. 5. G. POITOU, {\sl Le th\'eor\`eme des classes jumelles de R. Zimmert,} S\'em. de Th\'eorie des Nombres de Bordeaux 1983-1984, Exp. \#86b:11003.) (Listed in MR 86b:11003.) 6. J. OESTERL\'E, {\sl Le th\'eor\`eme des classes jumelles de Zimmert et les formules explicites de Weil}, pp. 181-197, S\'em. Th\'eorie des Nombres, Paris 1983-84, C. Goldstein, ed., Birkh\"auser Boston, 1985. 7. J. SILVERMAN, {\sl An inequality connecting the regulator and the discriminant of a number field}, J. Number Theory 19 (1984), 437-442. MR 86c:11094. 8. T.W. CUSICK, {\sl Lower bounds for regulators,} pp. 63-73 in Number Theory, Noordwijkerhout 1983, H. Jager, ed., Lecture Notes in Math. \# 1068, Springer 1984. MR 85k:11052. 9. A.-M. BERGE and J. MARTINET, {\sl Sur les minorations g\'eom\'e\-tri\-ques des r\'egulateurs,} pp. 23-50, S\'eminaire Th\'eorie des Nombres, Paris 1987-88, C. Goldstein, ed., Birkh\"auser Boston, 1990. 10. E. FRIEDMAN, {\sl Analytic formulas for regulators of number fields,} Invent. math., 98 (1989), 599-622. 11. M. POHST and H. ZASSENHAUS, {\sl Algorithmic Algebraic Number Theory,} Cambridge Univ. Press., 1989. 12. R. SCHOOF and L.C. WASHINGTON, {\sl Quintic polynomials and real cyclotomic fields with large class numbers,} Math. Comp. 50 (1988), 543-556. 13. A.-M. BERG\'E and J. MARTINET, {\sl Notions relatives de r\'egula\-teurs et de hauteurs,} Acta Arith. 54 (1989), 155-170. MR 90m:11167. 14. A.-M. BERG\'E and J. MARTINET, {\sl Minorations de hauteurs et petits r\'egulateurs relatifs,} S\'em. Th\'eorie des Nombres Bordeaux 1987-88, Exp.\#11, Univ. Bordeaux 1988. 15. A. COSTA and E. FRIEDMAN, {Ratios of regulators in totally real extensions of number fields,} to be published. 16. E. FRIEDMAN and N.-P. SKORUPPA, {\sl Explicit formulas for regulators and ratios of regulators of number fields}, manuscript in preparation. \bigskip {\bf F. Determination of minimal discriminants} \medskip 1. P. CARTIER and Y. ROY, {\sl On the enumeration of quintic fields with small discriminants}, J. reine angew. Math 268/269 (1974), 213-215. MR 50 \# 2119. 2. M. POHST, {\sl Berechnung kleiner Diskriminanten total reeller algebraischer Zahlk\"orper.} J. reine angew. Math. 278/279 (1975), 278-300. MR 52 \# 8085. 3. M. POHST, {\sl The minimum discriminant of seventh degree totally real algebraic number fields,} pp. 235-240, Number theory and algebra, H. Zassenhaus, ed., Academic Press 1977. MR 57 \# 5952. 4. J. LIANG and H. ZASSENHAUS, {\sl The minimum discriminant of sixth degree totally complex algebraic number field,} J. Number Theory 9 (1977), 16-35. MR 55 \# 305. 5. M. POHST, {\sl On the computation of number fields of small discriminants including the minimum discriminants of sixth degree fields,} J. Number Theory 14 (1982), 99-117. MR 83g:12009. 6. M. POHST, P. WEILER, and H. ZASSENHAUS, {\sl On effective computation of fundamental units,} Math. Comp. 38 (1982), 293-329. MR 83e:12005b. 7. D.G. RISH, {\sl On algebraic number fields of degree five,} Vestnik Moskov. Univ. Ser. I Mat. Mekh. (1982), no 2, 76-80. English translation in Moscow Univ. Math. Bull. 37 (1982), no. 99-103. MR 83g:12006. 8. F. DIAZ Y DIAZ, {\sl Valeurs minima du discriminant des corps de degr\'e 7 ayant une seule place r\'eelle,} C.R. Acad. Sc. Paris 296 (1983), 137-139. MR 84i:12004. 9. F. DIAZ Y DIAZ, {\sl Valeurs minima du discriminant pour certains types de corps de degr\'e 7,} Ann. Inst. Fourier (Grenoble) 34, no 3 (1984), 29-38. MR 86d:11091. 10. K. TAKEUCHI, {\sl Totally real algebraic number fields of degree 5 and 6 with small discriminant,} Saitama Math. J. 2(1984), 21-32. MR 86i:11060. 11. H.J. GODWIN, {\sl On quartic fields of signature one with small discriminant. II,} Math. Comp. 42 (1984), 707-711. {\sl Corrigendum,} Math. Comp. 43 (1984), 621. MR 85i:11092a, 11092b. 12. F. DIAZ Y DIAZ, {\sl Petits discriminants des corps de nombres totalement imaginaires de degr\'e 8,} J. Number Theory 25 (1987), 34-52. 13. S.-H. KWON and J. MARTINET, {\sl Sur les corps r\'esolubles de degr\'e premier,} J. reine angew. Math. 375/376 (1987),12-23. MR 88g:11080. 14. F. DIAZ Y DIAZ, {\sl Discriminants minima et petits discriminants des corps de nombres de degr\'e 7 avec cinq places r\'eelles,} J. London Math. Soc. (2) 38 (1988), 33-46. 15. J. BUCHMANN and D. FORD, {\sl On the computation of totally real quartic fields of small discriminant,} Math. Comp. 52 (1989), 161-174. 16. S.-H. KWON, {\sl Sur les discriminants minimaux des corps quaternioniens,} preprint 1987. 17. P. LLORENTE and J. QUER, {\sl On totally real cubic fields with discriminant $D<10^7$,} Math. Comp. 50 (1988), 581-594. 18. J. BUCHMANN, M. POHST and J. v. SCHMETTOW, {\sl On the computation of unit groups and class groups of totally real quartic fields,} Math. Comp. 53 (1989), 387-397. 19. A.-M. BERG\'E, J. MARTINET and M. OLIVIER, {\sl The computation of sextic fields with a quadratic subfield,} Math. Comp. 54 (1990), 869-884. 20. F. DIAZ Y DIAZ, {\sl A table of totally real quintic number fields,} Math. Comp. 56 (1991), 801-808. 21. M. OLIVIER, {\sl Corps sextiques contenant un corps quadratique (I),} S\'em. de Th\'eorie des Nombres, Bordeaux 1 (1990), 205-250. 22. M. OLIVIER, {\sl Corps sextiques primitifs,} Ann. Inst. Fourier (Grenoble) 40 (1990), 757-767. 23. J. MARTINET, {\sl Discriminants and permutation groups,} pp. 359-385, Number Theory, R. A. Mollin, ed., De Gruyter, 1990. 24. M. POHST, J. MARTINET and F. DIAZ Y DIAZ, {\sl The minimum discriminant of totally real octic fields,} J. Number Theory 36 (1990), 145-159. 25. D. FORD, {\sl Enumeration of totally complex quartic fields of small discriminant,} pp. 129-138, Computational Number Theory, A. Petho, M. Pohst, H. C. Williams, and H. G. Zimmer, eds., De Gruyter 1991. 26. M. OLIVIER, {\sl The computation of sextic fields with a cubic subfield and no quadratic subfield,} Math. Comp. 58 (1992), 419-432. 27. J. BUCHMANN, D. FORD, and M. POHST, {\sl Enumeration of quartic fields of small discriminant,} Math. Comp. 61 (1993), 873-879. 28. H. FUJITA, {\sl The minimum discriminant of totally real algebraic number field of degree 9 with cubic subfields,} Math. Comp. 60 (1993), 801-810. 29. H. FUJITA, {\sl The minimum discriminant of totally real algebraic number field of degree 9 with cubic subfields. II,} Saitama Math. J. 9 (1991), 9-18. 30. D. FORD and M. POHST, {\sl The totally real A5 extension of degree 6 with minimum discriminant,} Experimental Math. 1 (1992), 231-235. 31. D. FORD and M. POHST, {\sl The totally real A6 extension of degree 6 with minimum discriminant,} Experimental Math. 2 (1993), 231-232. 32. A. SCHWARZ, M. POHST, and F. DIAZ Y DIAZ, {\sl A table of quintic number fields,} Math. Comp. 63 (1994), 361-376. \bigskip {\bf G. Small zeros of Dedekind zeta functions} \medskip 1. J. HOFFSTEIN, {\sl Some results related to minimal discriminants,} pp. 185-194, Number Theory, Carbondale 1979, Lecture Notes in Math. \# 751, Springer 1979, MR 81d:12005. 2. A. NEUGEBAUER, {\sl On zeros of zeta functions in low rectangles in the critical strip} (in Polish), Ph.D. Thesis, A. Mickiewicz University, Poznan, Poland, 1985. 3. A. NEUGEBAUER, {\sl On the zeros of the Dedekind zeta-function near the real axis,} Funct. Approx. Comment. Math. 16 (1988), 165-167. MR 90b:11122. 4. A. NEUGEBAUER, {\sl Every Dedekind zeta-function has a zero in the rectangle $1/2\le \sigma\le 1,0