Maximal unramified extensions of imaginary quadratic number fields of small conductors

Yamamura, Ken

Journal de théorie des nombres de Bordeaux, Tome 9 (1997) no. 2, pp. 405-448.

Nous déterminons la structure du groupe de Galois Gal$(K_{ur}/K)$ de l’extension maximale non ramifiée $K_{ur}$ de chaque corps quadratique imaginaire de conducteur $\leqq 420(\leqq 719$ sous GRH). Pour tous ces corps $K$, l’extension $K_{ur}$ coïncide avec $K$, ou avec le corps de classes de Hilbert de $K$, ou avec le second corps de classes de Hilbert de $K$ ou avec le troisième corps de classes de Hilbert de $K$. Les bornes d’Odlyzko sur les discriminants et les informations sur la structure des groupes de classes obtenues par l’action du groupe de Galois sur les groupes de classes sont ici essentielles. Nous utilisons aussi des relations sur le nombre de classes et un ordinateur pour le calcul du nombre de classes de corps de bas degré pour obtenir le nombre de classes de corps de degré plus élevé. Nous utilisons aussi des résultats sur les tours de corps de classes, ainsi que notre connaissance des $2$-groupes d’ordre $\leqq 2^6$ et des groupes linéaires sur des corps finis.

We determine the structures of the Galois groups Gal$(K_{ur}/K)$ of the maximal unramified extensions $K_{ur}$ of imaginary quadratic number fields $K$ of conductors $\leqq 420(\leqq 719$ under the Generalized Riemann Hypothesis). For all such $K$, $K_{ur}$ is $K$, the Hilbert class field of $K$, the second Hilbert class field of $K$, or the third Hilbert class field of $K$. The use of Odlyzko’s discriminant bounds and information on the structure of class groups obtained by using the action of Galois groups on class groups is essential. We also use class number relations and a computer for calculation of class numbers of fields of low degrees in order to get class numbers of fields of higher degrees. Results on class field towers and the knowledge of the $2$-groups of orders $\leqq 2^6$ and linear groups over finite fields are also used.

1 S. Arno, The imaginary quadratic fields of class number 4, Acta Arith. 60 (1992), no. 4, 321-334; MR 93b:11144. | EuDML | MR | Zbl

2 S. Arno, M.L. Robinson, and F.S. Wheeler, The imaginary quadratic fields of small odd class numbers, preprint, 1993. | MR | Zbl

3 J. Basmaji and I. Kiming, A table of A5-fields, On Artin's conjecture for odd 2-dimensional representations (G. Frey, ed.), Lecture Notes in Math., vol. 1585, Springer-Verlag, New York and Berlin, 1994, pp. 37-46, 122-141; MR 96e:11141. | MR | Zbl

4 E. Benjamin, Remarks concerning the 2-Hilbert class field of imaginary quadratic number fields, Bull Austral. Math. Soc. 48 (1993), no. 3, 379-383; MR 94m:11133; Corrigenda, ibid. 50 (1994), no. 2, 351-352. | MR | Zbl

5 E. Benjamin, F. Lemmermeyer, and C. Snyder, Imaginary quadratic fields k with cyclic Cl2(k1), J. Number Theory 67 (1997), no. 2, 229-245. | MR | Zbl

6 R. Brauer, Beziehung zwischen Klassenzahl von Teilkörpern eines galoisschen Körpers, Math. Nachr. 4 (1951), no. 139, 158-174; MR 12, 593b; reprinted in Collected papers, vol. III, MIT Press, Cambridge, Mass.-London, 1980, pp. 497-513. | MR | Zbl

7 D.A. Buell, Small class number and extreme values of L-functions of quadratic fields, Math. Comp. 31 (1977), no. 139, 786-796; MR 55 #12684. | MR | Zbl

8 R. Carter and P. Fong, The Sylow 2-subgroups of the finite classical groups, J. Algebra 1 (1964), no. 2, 139-151; MR 29 #3548. | MR | Zbl

9 C. Castela, Nombre de classes d'idéaux d'une extension diédrale d'un corps de nombres, C. R. Acad. Sci. Paris Sér. A-B 287 (1978), no. 7, 483-486; MR 80c:12012. | MR | Zbl

10 H. Cohen and H.W. Lenstra, Jr., Heuristics on class groups of number fields, Number Theory, Noordwijkerhout, 1983 (H. Jager, ed.), Lecture Notes in Math., vol. 1068, Springer-Verlag, Berlin and New York, 1984, pp. 33-62; MR 85j:11144. | MR | Zbl

11 H. Cohn, A classical invitation to algebraic numbers and class fields, Universitext, Springer-Verlag, Berlin and New York, 1978; MR 80c:12001. | MR | Zbl

12 F. Diaz Y Diaz, Tables minorant la racine n-ième du discriminant d'un corps de degré n, Publications Mathématiques d'Orsay 80, 6., Université de Paris-Sud, Département de Mathématique, Orsay, 1980; MR 82i:12007. | MR | Zbl

13 H.J. Godwin, On quartic fields of signature one with small discriminant. II, Math. Comp. 42 (1984), no. 166, 707-711; MR 85i:11092a; Corrigendum, ibid. 43 (1984), no. 168, 621; MR 85i:11092b. | MR | Zbl

14 ____, On totally complex quartic fields with small discriminant, Proc. Cambridge Philos. Soc. 53 (1957), 1-4; MR 18, 565c. | MR | Zbl

15 ____, On relations between cubic and quartic fields, Quart. J. Math. Oxford (2) 13 (1962), 206-212; Corrigendum, ibid. (3) 26 (1975), no. 104, 511-512; MR 52 #8078. | MR

16 F. Hajir, Unramified elliptic units, thesis, MIT, 1993.

17 M. Hall, Jr. and J.K. Senior, The groups of order 2n(n ≤ 6), The Macmillan Co., New York, 1964; MR 29 #5889. | Zbl

18 F. Halter-Koch, Einheiten und Divisorenklassen in Galois'schen algebraischen Zahlkörpern mit Diedergruppe der Ordnung 2l für eine ungerade Primzahll, Acta Arith. 33 (1977), no. 4, 355-364; MR 56 #11955. | MR | Zbl

19 F. Halter-Koch et N. Moser, Sur le nombre de classes de certaines extensions métacycliques sur Q ou sur un corps quadratiques imaginaires, J. Math. Soc. Japan 30 (1978), no. 2, 237-248; MR 58 #5587. | MR | Zbl

20 H. Hayashi, On elliptic units and class number of a certain dihedral extension of degree 2l, Acta Arith. 45 (1985), no. 1, 35-45; MR 86m:11081. | MR | Zbl

21 C.S. Herz, Construction of class fields, Seminar on Complex Multiplication, Chap. VII, Lecture Notes in Math., vol. 21, Springer-Verlag, Berlin and New York, 1966. | MR

22 B. Huppert, Endliche Gruppen I, Die Grundlehren der math. Wiss., Bd. 134, Springer-Verlag, Berlin and New York, 1967; MR 37 #302. | MR | Zbl

23 A. Jehanne, Sur les extensions de Q à groupe de Galois S4 et S4, Acta Arith. 70 (1995), no. 3, 259-276; MR 95m:11127. | MR | Zbl

24 G. Kientega and P. Barrucand, On quartic fields with symmetric group, Number theory (R. A. Mollin, ed.), de Gruyter, Berlin, 1990, pp. 287-297; MR 92e:11113. | MR | Zbl

25 H. Kisilevsky, Number fields with class number congruent to 4 mod 8 and Hilbert's Theorem 94, J. Number Theory 8 (1976), no. 3, 271-279; MR 54 #5188. | MR | Zbl

26 T. Kondo, Algebraic number fields with the discriminant equal to that of a quadratic number field, J. Math. Soc. Japan 47 (1995), no. 1, 31-36; MR 95h:11121. | MR | Zbl

27 S. Kuroda, Über die Klassenzahlen algebraischer Zahlkörper, Nagoya Math. J. 1 (1950), 1-10; MR 12, 593a. | MR | Zbl

28 F. Lemmermeyer, Kuroda's class number formula, Acta Arith. 66 (1994), no. 3, 245-260; MR 95f:11090. | MR | Zbl

29 ____, On 2-class field towers of imaginary quadratic number fields, J. Théor. Nombres Bordeaux 6 (1994), no. 2, 261-272; MR 96k:11136. | Numdam | MR

30 ____, On unramified quaternion extension of imaginary quadratic number fields, J. Théor. Nombres Bordeaux 9 (1997), no. 1, 51-68. | Numdam | MR

31 ____, On 2-Class field towers of some imaginary quadratic number fields, Abh. Math. Sem. Univ. Hamburg 67 (1997), 205-214. | MR | Zbl

32 ____, Private communication, 1996.

33 J. Martinet, Corps de nombres de classes 1, Séminaire de Théorie des Nombres 1977-1978, Exp. No. 12, CNRS, Talence, 1978; MR 80k:12009. | MR | Zbl

34 ____, Petits discriminants des corps de nombres, Number theory days, 1980 (Exeter, 1980), London Math. Soc. Lecture Note Ser. 56, Cambridge Univ. Press, Cambridge, New York, 1982, pp. 151-193; MR 84g:12009. | MR

35 J.M. Masley, Class numbers of real cyclic number fields with small conductor, Compositio Math. 37 (1978), no. 3, 297-319; MR 80e:12005. | Numdam | MR | Zbl

36 N. Moser, Unités et nombre de classes d'une extension galoisienne diédrale de Q, Abh. Math. Sem. Univ. Hamburg 48 (1979), 54-75; MR 81h:12009. | MR | Zbl

37 A. Nomura, On the existence of unramified p-extensions, Osaka J. Math. 28 (1991), no. 1, 55-62; MR 92e:11115. | MR | Zbl

38 A.M. Odlyzko, Discriminant bounds, (unpublished tables), Nov. 29th 1976.

39 ____, Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results, Sém. Théor. Nombres Bordeaux (2) 2 (1990), no. 1, 119-141; MR 91i:11154. | Numdam | MR | Zbl

40 J. Oesterlé, Nombres de classes de corps quadratiques imaginaires, Sém. Bourbaki 1983-1984, Exp. 631, 14pp; MR 86k:11064. | Numdam | MR | Zbl

41 M. Olivier, Corps sextique primitifs, Ann. Inst. Fourier (Grenoble) 40 (1990), no. 4, 757-767; MR 92a:11123. | Numdam | MR | Zbl

42 T.W. Sag and J.W. Wamsley, Minimal presentations for groups of order 2n, n ≦ 6, J. Austral. Math. Soc. 15 (1973), 461-469; MR 49 #406. | Zbl

43 R. Schoof, Private communication, 1996.

44 A. Schwarz, M. Pohst and F. Diaz Y Diaz, A table of quintic number fields, Math. Comp. 63 (1994), no. 207, 361-374; MR 94i:11108. | MR | Zbl

45 J.-P. Serre, Modular forms of weight one and Galois representations, Algebraic number fields: L-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975) (A. Fröhlich, ed.), Academic Press, London, 1977, pp. 193-268; MR 56 #8497; repreinted in Collected papres, vol. III, Springer-Verlag, New York and Berlin, 1986, pp. 292-367. | MR | Zbl

46 ____, Topics in Galois theory, Research Notes in Math., vol. 1, Jones and Bartlett Publishers, Boston, MA, 1992; MR 94d:12006. | MR | Zbl

47 M.W. Short, The primitive soluble permutation groups of degree less than 256, Lecture Notes in Math., vol. 1519, Springer-Verlag, Berlin and New York, 1992; MR 93g:20006. | MR | Zbl

48 A.G. Stephens and H.C. Williams, Computation of real quadratic fields with class number one, Math. Comp. 51 (1988), no. 184, 809-824; MR 90b:11106. | MR | Zbl

49 O. Taussky, A remark on the class field tower, J. London Math. Soc. 12 (1937), 82-85. | JFM | Zbl

50 F. Van Der Linden, Class number computations of real abelian number fields, Math. Comp. 39 (1982), no. 160, 693-707; MR 84e:12005. | MR | Zbl

51 H. Wada, On the class number and the unit group of certain algebraic number fields, J. Fac. Sci. Univ. Tokyo Sect. IA 13 (1966), 201-209; MR 35 #5414. | MR | Zbl

52 C. Wagner, Class number 5, 6 and 7, Math. Comp. 65 (1996), no. 214, 785-800; MR 96g:11135. | MR | Zbl

53 L.C. Washington, Introduction to Cyclotomic Fields, Graduate Text in Math., vol. 83, Springer-Verlag, Berlin and New York, 1982; MR 85g:11101. | MR | Zbl

54 Y. Yamamoto, Divisibility by 16 of class numbers of quadratic fields whose 2-class groups are cyclic, Osaka J. Math. 21 (1984), no. 1, 1-22; MR 85g:11092. | MR | Zbl

55 K. Yamamura, On unramified Galois extensions of real quadratic number fields, Osaka J. Math. 23 (1986), no. 2, 471-486; MR 88a:11112. | MR | Zbl

56 ____, Some analogue of Hilbert's irreducibility theorem and the distribution of algebraic number fields, J. Fac. Sci. Univ. Tokyo Sect. IA 38 (1991), no. 1, 99-135; MR 92e:11132. | MR | Zbl

57 ____, The determination of the imaginary abelian number fields with class number one, Math. Comp. 62 (1994), no. 206, 899-921; MR 94g:11096. | MR | Zbl

58 ____, The maximal unramified extensions of the imaginary quadratic number fields with class number two, J. Number Theory 60 (1996), no. 2, 42-50; MR 97g:11119. | MR | Zbl

59 ____, Determination of the non-CM imaginary normal octic number fields with class number one, submitted for publication.

60 ____, Real quadratic number fields with class number one having an unramified An-extension, in preparation.

61 K. Yamazaki, Computations of Galois groups, Proc. Symp. Group theory and its application (T. Kondo, ed.), 1981, pp. 9-57. (Japanese)