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Volumn 9, Issue 2, 1997, Pages 405-448

Maximal unramified extensions of imaginary quadratic number fields of small conductors

Author keywords

Class field tower; Discriminant bounds; Imaginary quadratic number field; Maximal unramified extension

Indexed keywords


EID: 85009932464     PISSN: 12467405     EISSN: None     Source Type: Journal    
DOI: 10.5802/jtnb.211     Document Type: Article
Times cited : (19)

References (61)
  • 1
    • 0000713248 scopus 로고
    • The imaginary quadratic fields of class number 4
    • MR 93b: 11144
    • S. Arno, The imaginary quadratic fields of class number 4, Acta Arith. 60 (1992), no. 4, 321-334; MR 93b: 11144.
    • (1992) Acta Arith. , vol.60 , Issue.4 , pp. 321-334
    • Arno, S.1
  • 3
    • 0002730580 scopus 로고
    • A table of A5-fields, On Artin’s conjecture for odd 2-dimensional representations
    • (G. Frey, ed.),, Springer-Verlag, New York and Berlin, 122-141; MR 96e: 11141
    • 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.
    • (1994) Lecture Notes in Math , vol.1585 , pp. 37-46
    • Basmaji, J.1    Kiming, I.2
  • 4
    • 84971707667 scopus 로고
    • Remarks concerning the 2-Hilbert class field of imaginary quadratic number fields
    • MR 94m: 11133; Corrigenda, ibid. 50 (1994), no. 2, 351-352
    • 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.
    • (1993) Bull Austral. Math. Soc. , vol.48 , Issue.3 , pp. 379-383
    • Benjamin, E.1
  • 5
    • 0037891691 scopus 로고    scopus 로고
    • Imaginary quadratic fields k with cyclic Cl2 (k1)
    • E. Benjamin, F. Lemmermeyer, and C. Snyder, Imaginary quadratic fields k with cyclic Cl2 (k1), J. Number Theory 67 (1997), no. 2, 229-245.
    • (1997) J. Number Theory , vol.67 , Issue.2 , pp. 229-245
    • Benjamin, E.1    Lemmermeyer, F.2    Snyder, C.3
  • 6
    • 84990271089 scopus 로고
    • Beziehung zwischen Klassenzahl von Teilkörpern eines galoisschen Körpers
    • MR 12, 593b; reprinted in Collected papers, vol. III, MIT Press, Cambridge, Mass.-London, 1980, pp. 497-513
    • 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.
    • (1951) Math. Nachr. , vol.4 , Issue.139 , pp. 158-174
    • Brauer, R.1
  • 7
    • 84966212179 scopus 로고
    • Small class number and extreme values of L-functions of quadratic fields
    • MR 55 #12684
    • 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.
    • (1977) Math. Comp. , vol.31 , Issue.139 , pp. 786-796
    • Buell, D.A.1
  • 8
    • 0000603317 scopus 로고
    • The Sylow 2-subgroups of the finite classical groups
    • MR 29 #3548
    • 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.
    • (1964) J. Algebra , vol.1-2 , pp. 139-151
    • Carter, R.1    Fong, P.2
  • 9
    • 0011535138 scopus 로고
    • Nombre de classes d’idéaux d’une extension diédrale d’un corps de nombres
    • MR 80c: 12012
    • 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.
    • (1978) C. R. Acad. Sci. Paris Sér. A-B , vol.287 , Issue.7 , pp. 483-486
    • Castela, C.1
  • 10
    • 0003049266 scopus 로고
    • Heuristics on class groups of number fields, Number Theory
    • H. Jager, ed.), Lecture Notes in Math., vol. 1068, Springer-Verlag, Berlin and New York, 1984, pp. 33-62; MR 85j: 11144
    • 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.
    • (1983) Noordwijkerhout
    • Cohen, H.1    Lenstra, H.W.2
  • 12
    • 85009904318 scopus 로고
    • Publications Mathématiques d’Orsay 80, 6., Université de Paris-Sud, Département de Mathématique, Orsay, MR 82i: 12007
    • 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.
    • (1980) Tables Minorant La Racine n-ième Du Discriminant d’un Corps De degré N
    • Diazdiaz, F.1
  • 13
    • 84966210483 scopus 로고
    • On quartic fields of signature one with small discriminant
    • MR 85i: 11092a; Corrigendum, ibid. 43 (1984), no. 168, 621; MR 85i: 11092b
    • 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.
    • (1984) II, Math. Comp. , vol.42 , Issue.166 , pp. 707-711
    • Godwin, H.J.1
  • 14
    • 84931814893 scopus 로고
    • On totally complex quartic fields with small discriminant
    • MR 18, 565c
    • H. J. Godwin, On totally complex quartic fields with small discriminant, Proc. Cambridge Philos. Soc. 53 (1957), 1-4; MR 18, 565c.
    • (1957) Proc. Cambridge Philos. Soc. , vol.53 , pp. 1-4
    • Godwin, H.J.1
  • 15
    • 84963014996 scopus 로고
    • On relations between cubic and quartic fields, Quart
    • Corrigendum, ibid. (3) 26 (1975), no. 104, 511-512; MR 52 #8078
    • H. J. Godwin, 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.
    • (1962) J. Math. Oxford , vol.2 , Issue.13 , pp. 206-212
    • Godwin, H.J.1
  • 18
    • 0037614340 scopus 로고
    • Einheiten und Divisorenklassen in Galois’schen algebraischen Zahlkörpern mit Diedergruppe der Ordnung 2l für eine ungerade Primzahll
    • MR 56 #11955
    • 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.
    • (1977) Acta Arith , vol.33 , Issue.4 , pp. 355-364
    • Halter-Koch, F.1
  • 19
    • 84972552251 scopus 로고
    • Sur le nombre de classes de certaines extensions métacycliques sur Q ou sur un corps quadratiques imaginaires
    • MR 58 #5587
    • F. Halter-Koch and 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.
    • (1978) J. Math. Soc. Japan , vol.30 , Issue.2 , pp. 237-248
    • Halter-Koch, F.1    Moser, N.2
  • 20
    • 85009892384 scopus 로고
    • On elliptic units and class number of a certain dihedral extension of degree 2l
    • MR 86m: 11081
    • 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.
    • (1985) Acta Arith , vol.45 , Issue.1 , pp. 35-45
    • Hayashi, H.1
  • 21
    • 0002960285 scopus 로고
    • Construction of class fields, Seminar on Complex Multiplication, Chap
    • Springer-Verlag, Berlin and New York
    • 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.
    • (1966) VII, Lecture Notes in Math , vol.21
    • Herz, C.S.1
  • 22
    • 0003871698 scopus 로고
    • Die Grundlehren der math
    • Springer-Verlag, Berlin and New York, MR 37 #302
    • B. Huppert, Endliche Gruppen I, Die Grundlehren der math. Wiss., Bd. 134, Springer-Verlag, Berlin and New York, 1967; MR 37 #302.
    • (1967) Wiss., Bd. , vol.134
    • Huppert, B.1    Endliche Gruppen, I.2
  • 23
    • 0011572844 scopus 로고
    • Sur les extensions de Q à groupe de Galois S4 and S4
    • MR 95m: 11127
    • A. Jehanne, Sur les extensions de Q à groupe de Galois S4 and S4, Acta Arith. 70 (1995), no. 3, 259-276; MR 95m: 11127.
    • (1995) Acta Arith , vol.70 , Issue.3 , pp. 259-276
    • Jehanne, A.1
  • 24
    • 0011661541 scopus 로고
    • On quartic fields with symmetric group, Number theory
    • (R. A. Mollin, ed.), MR 92e: 11113
    • 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.
    • (1990) De Gruyter, Berlin , pp. 287-297
    • Kientega, G.1    Barrucand, P.2
  • 25
    • 0001477991 scopus 로고
    • Number fields with class number congruent to 4 mod 8 and Hilbert’s Theorem 94, J
    • MR 54 #5188
    • 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.
    • (1976) Number Theory , vol.8 , Issue.3 , pp. 271-279
    • Kisilevsky, H.1
  • 26
    • 84972561061 scopus 로고
    • Algebraic number fields with the discriminant equal to that of a quadratic number field, J
    • MR 95h: 11121
    • 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.
    • (1995) Math. Soc. Japan , vol.47 , Issue.1 , pp. 31-36
    • Kondo, T.1
  • 27
    • 84961492978 scopus 로고
    • Über die Klassenzahlen algebraischer Zahlkörper
    • MR 12, 593a
    • S. Kuroda, Über die Klassenzahlen algebraischer Zahlkörper, Nagoya Math. J. 1 (1950), 1-10; MR 12, 593a.
    • (1950) Nagoya Math. J. , vol.1 , pp. 1-10
    • Kuroda, S.1
  • 28
    • 0000012131 scopus 로고
    • Kuroda’s class number formula
    • MR 95f: 11090
    • F. Lemmermeyer, Kuroda’s class number formula, Acta Arith. 66 (1994), no. 3, 245-260; MR 95f: 11090.
    • (1994) Acta Arith , vol.66 , Issue.3 , pp. 245-260
    • Lemmermeyer, F.1
  • 29
    • 0039991461 scopus 로고
    • On 2-class field towers of imaginary quadratic number fields, J
    • MR 96k: 11136
    • F. Lemmermeyer, On 2-class field towers of imaginary quadratic number fields, J. Théor. Nombres Bordeaux 6 (1994), no. 2, 261-272; MR 96k: 11136.
    • (1994) Théor. Nombres Bordeaux , vol.6 , Issue.2 , pp. 261-272
    • Lemmermeyer, F.1
  • 30
    • 85010007306 scopus 로고    scopus 로고
    • On unramified quaternion extension of imaginary quadratic number fields
    • F. Lemmermeyer, On unramified quaternion extension of imaginary quadratic number fields, J. Théor. Nombres Bordeaux 9 (1997), no. 1, 51-68.
    • (1997) J. Théor. Nombres Bordeaux , vol.9 , Issue.1 , pp. 51-68
    • Lemmermeyer, F.1
  • 31
    • 0002341247 scopus 로고    scopus 로고
    • On 2-Class field towers of some imaginary quadratic number fields
    • F. Lemmermeyer, On 2-Class field towers of some imaginary quadratic number fields, Abh. Math. Sem. Univ. Hamburg 67 (1997), 205-214.
    • (1997) Abh. Math. Sem. Univ. Hamburg , vol.67 , pp. 205-214
    • Lemmermeyer, F.1
  • 33
    • 85009913364 scopus 로고
    • Corps de nombres de classes 1, Séminaire de Théorie des Nombres 1977-1978
    • MR 80k: 12009
    • J. Martinet, Corps de nombres de classes 1, Séminaire de Théorie des Nombres 1977-1978, Exp. No. 12, CNRS, Talence, 1978; MR 80k: 12009.
    • (1978) Exp. No. 12, CNRS, Talence
    • Martinet, J.1
  • 34
    • 0011664893 scopus 로고
    • Petits discriminants des corps de nombres, Number theory days, 1980 (Exeter, 1980), London Math
    • Cambridge Univ. Press, Cambridge, New York, MR 84g: 12009
    • J. Martinet, 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.
    • (1982) Soc. Lecture Note Ser. , vol.56 , pp. 151-193
    • Martinet, J.1
  • 35
    • 0001271995 scopus 로고
    • Class numbers of real cyclic number fields with small conductor
    • MR 80e: 12005
    • J. M. Masley, Class numbers of real cyclic number fields with small conductor, Compositio Math. 37 (1978), no. 3, 297-319; MR 80e: 12005.
    • (1978) Compositio Math , vol.37 , Issue.3 , pp. 297-319
    • Masley, J.M.1
  • 36
    • 0011593811 scopus 로고
    • Unités and nombre de classes d’une extension galoisienne diédrale de Q, Abh
    • MR 81h: 12009
    • N. Moser, Unités and nombre de classes d’une extension galoisienne diédrale de Q, Abh. Math. Sem. Univ. Hamburg 48 (1979), 54-75; MR 81h: 12009.
    • (1979) Math. Sem. Univ. Hamburg , vol.48 , pp. 54-75
    • Moser, N.1
  • 37
    • 84972566427 scopus 로고
    • On the existence of unramified p-extensions
    • MR 92e: 11115
    • A. Nomura, On the existence of unramified p-extensions, Osaka J. Math. 28 (1991), no. 1, 55-62; MR 92e: 11115.
    • (1991) Osaka J. Math. , vol.28 , Issue.1 , pp. 55-62
    • Nomura, A.1
  • 39
    • 0001749152 scopus 로고
    • Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: A survey of recent results
    • 1, 119-141; MR 91i: 11154
    • A. M. Odlyzko, 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.
    • (1990) Sém. Théor. Nombres Bordeaux , vol.2 , Issue.2
    • Odlyzko, A.M.1
  • 40
    • 0000468761 scopus 로고    scopus 로고
    • Nombres de classes de corps quadratiques imaginaires
    • MR 86k: 11064
    • J. Oesterlé, Nombres de classes de corps quadratiques imaginaires, Sém. Bourbaki 1983-1984, Exp. 631, 14pp; MR 86k: 11064.
    • Sém. Bourbaki 1983-1984, Exp. , vol.631
    • Oesterlé, J.1
  • 41
    • 0010075593 scopus 로고
    • Corps sextique primitifs, Ann
    • MR 92a: 11123
    • M. Olivier, Corps sextique primitifs, Ann. Inst. Fourier (Grenoble) 40 (1990), no. 4, 757-767; MR 92a: 11123.
    • (1990) Inst. Fourier (Grenoble) , vol.40 , Issue.4 , pp. 757-767
    • Olivier, M.1
  • 42
    • 0039399392 scopus 로고
    • Minimal presentations for groups of order 2n, n ~ 6, J
    • MR 49 #406
    • 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.
    • (1973) Austral. Math. Soc. , vol.15 , pp. 461-469
    • Sag, T.W.1    Wamsley, J.W.2
  • 44
    • 84968468593 scopus 로고
    • Diaz, A table of quintic number fields
    • MR 94i: 11108
    • 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.
    • (1994) Math. Comp. , vol.63 , Issue.207 , pp. 361-374
    • Schwarz, A.1    Pohst, M.2    Diaz, F.3
  • 48
    • 84968475641 scopus 로고
    • Computation of real quadratic fields with class number one
    • MR 90b: 11106
    • 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.
    • (1988) Math. Comp. , vol.51 , Issue.184 , pp. 809-824
    • Stephens, A.G.1    Williams, H.C.2
  • 49
    • 84963018577 scopus 로고
    • A remark on the class field tower
    • O. Taussky, A remark on the class field tower, J. London Math. Soc. 12 (1937), 82-85.
    • (1937) J. London Math. Soc , vol.12 , pp. 82-85
    • Taussky, O.1
  • 50
    • 84966206153 scopus 로고
    • Class number computations of real abelian number fields
    • MR 84e: 12005
    • F. van der Linden, Class number computations of real abelian number fields, Math. Comp. 39 (1982), no. 160, 693-707; MR 84e: 12005.
    • (1982) Math. Comp. , vol.39 , Issue.160 , pp. 693-707
    • Van Der Linden, F.1
  • 51
    • 0000766026 scopus 로고
    • On the class number and the unit group of certain algebraic number fields
    • MR 35 #5414
    • 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.
    • (1966) J. Fac. Sci. Univ. Tokyo Sect. IA , vol.13 , pp. 201-209
    • Wada, H.1
  • 52
    • 0030372815 scopus 로고    scopus 로고
    • Class number 5, 6 and 7
    • MR 96g: 11135
    • C. Wagner, Class number 5, 6 and 7, Math. Comp. 65 (1996), no. 214, 785-800; MR 96g: 11135.
    • (1996) Math. Comp. , vol.65 , Issue.214 , pp. 785-800
    • Wagner, C.1
  • 53
    • 0003254134 scopus 로고    scopus 로고
    • Introduction to Cyclotomic Fields
    • Springer-Verlag, Berlin and New York, 1982; MR 85g: 11101
    • L. C. Washington, Introduction to Cyclotomic Fields, Graduate Text in Math., vol. 83, Springer-Verlag, Berlin and New York, 1982; MR 85g: 11101.
    • Graduate Text in Math , vol.83
    • Washington, L.C.1
  • 54
    • 84972551814 scopus 로고
    • Divisibility by 16 of class numbers of quadratic fields whose 2-class groups are cyclic
    • MR 85g: 11092
    • 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.
    • (1984) Osaka J. Math. , vol.21 , Issue.1 , pp. 1-22
    • Yamamoto, Y.1
  • 55
    • 84972562450 scopus 로고
    • On unramified Galois extensions of real quadratic number fields
    • MR 88a: 11112
    • K. Yamamura, On unramified Galois extensions of real quadratic number fields, Osaka J. Math. 23 (1986), no. 2, 471-486; MR 88a: 11112.
    • (1986) Osaka J. Math. , vol.23 , Issue.2 , pp. 471-486
    • Yamamura, K.1
  • 56
    • 0011532806 scopus 로고
    • Some analogue of Hilbert’s irreducibility theorem and the distribution of algebraic number fields
    • MR 92e: 11132
    • K. Yamamura, 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.
    • (1991) J. Fac. Sci. Univ. Tokyo Sect. IA , vol.38 , Issue.1 , pp. 99-135
    • Yamamura, K.1
  • 57
    • 84968482167 scopus 로고
    • The determination of the imaginary abelian number fields with class number one
    • MR 94g: 11096
    • K. Yamamura, The determination of the imaginary abelian number fields with class number one, Math. Comp. 62 (1994), no. 206, 899-921; MR 94g: 11096.
    • (1994) Math. Comp , vol.62 , Issue.206 , pp. 899-921
    • Yamamura, K.1
  • 58
    • 0030240852 scopus 로고    scopus 로고
    • The maximal unramified extensions of the imaginary quadratic number fields with class number two
    • MR 97g: 11119
    • K. Yamamura, 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.
    • (1996) J. Number Theory , vol.60 , Issue.2 , pp. 42-50
    • Yamamura, K.1
  • 61
    • 84972546720 scopus 로고
    • Computations of Galois groups, Proc
    • (T. Kondo, ed.), (Japanese)
    • K. Yamazaki, Computations of Galois groups, Proc. Symp. Group theory and its application (T. Kondo, ed.), 1981, pp. 9-57. (Japanese)
    • (1981) Symp. Group Theory and Its Application , pp. 9-57
    • Yamazaki, K.1


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