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The class number one problem for the normal CM-fields of degree 32

Author(s): Sun-Mi Park; Hee-Sun Yang; Soun-Hi Kwon
Journal: Trans. Amer. Math. Soc. 359 (2007), 5057-5089.
MSC (2000): Primary 11R29; Secondary 11R21
Posted: April 16, 2007
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Abstract: We prove that there are exactly six normal CM-fields of degree 32 with class number one. Five of them are composita of two normal CM-fields of degree 16 with the same maximal totally real octic field.

References:

[BLS]
E. Benjamin, F. Lemmermeyer, and C. Snyder, Real quadratic fields with abelian $ 2$-class field tower, J. Number Theory, 73 (1998), 182-194. MR 1658015 (2000c:11179)

[CK]
K.-Y.Chang and S.-H. Kwon, Class numbers of imaginary abelian number fields, Proc. Amer. Math. Soc., Vol. 128, No. 9 (2000), 2517-2528. MR 1707511 (2000m:11108)

[Co]
H.Cohen, Advanced topics in computational number theory, Springer-Verlag Graduate Texts in Math. 193, 2000. MR 1728313 (2000k:11144)

[G]
M.-N. Gras, Classes et unites des extensions cycliques reelles de degre $ 4$ de $ \mathbb{Q}$, Ann. Inst. Fourier, 29 no 1, xiv (1979), 102-124. MR 526779 (81f:12003)

[H]
H. Hasse, Uber die Klassenzahl abelscher Zahlkorper, Akademie-Verlag : Berlin, 1952. Reprinted with an introduction by J. Martinet : Springer-Verlag : Berlin, 1985. MR 842666 (87j:11122a)

[He]
E. Hecke, Lectures on the theory of algebraic numbers, Springer-Verlag Graduate Texts in Math. 77, 1981. MR 638719 (83m:12001)

[JL]
G. James and M. Liebeck, Representations and characters of groups, Cambridge University Press, 1993. MR 1237401 (94h:20007)

[K]
S.-H. Kwon, Sur les discriminants minimaux des corps quaternioniens, Arch. Math., 67 (1996), 119-125. MR 1399827 (97d:11165)

[KT]
M. Daberkow, F. Fieker, J. Kluners, M. Pohst, K. Roegner and K. Wildanger, KANT $ V_4$, J. Symbolic Computation, 24 (1997), 267-283. MR 1484479 (99g:11150)

[Lef]
Y. Lefeuvre, Corps diédraux à multiplication complexe principaux, Ann. Inst. Fourier, 50.1 (2000), 67-103. MR 1762338 (2001g:11166)

[Lem1]
F. Lemmermeyer, Ideal class groups of cyclotomic number fields I, Acta Arith., 72.4 (1995), 347-359. MR 1348202 (96h:11111)

[Lem2]
F. Lemmermeyer, Unramified quaternion extensions of quadratic number fields, J. de Théorie des Nombres de Bordeaux, 9 (1997), 51-68. MR 1469661 (98j:11090)

[LO1]
S. Louboutin and R. Okazaki, Determination of all non-normal quartic CM-fields and of all non-abelian normal octic CM-fields with class number one, Acta Arith., 67 (1994), 47-62. MR 1292520 (95g:11107)

[LO2]
S. Louboutin and R. Okazaki, The class number one problem for some non-abelian normal CM-fields of $ 2$-power degree, Proc. London Math. Soc., 76 part 3 (1998), 523-548. MR 1616805 (99c:11138)

[LOO]
S. Louboutin, R. Okazaki, and M. Olivier, The class number one problem for some non-abelian normal CM-fields, Trans. of the A.M.S., Vol.349, No.9 (1997), 3657-3678. MR 1390044 (97k:11149)

[Lou1]
S. Louboutin, Lower bounds for relative class numbers of CM-fields, Proc. Amer. Math. Soc., Vol 120 (1994), 425-434. MR 1169041 (94d:11089)

[Lou2]
S. Louboutin, Determination of all quaternion octic CM-fields with class number $ 2$, J. London Math. Soc., (2) 54 (1996), 227-238. MR 1405052 (97g:11122)

[Lou3]
S. Louboutin, The class number one problem for the non-abelian normal CM-fields of degree $ 16$, Acta Arith., 82.2 (1997), 173-196. MR 1477509 (98j:11097)

[Lou4]
S. Louboutin, Majoration explicites du résidu au point 1 des fonctions $ z\hat{e}ta$ de certains corps de nombres, J. Math. Soc. Japan, Vol. 50 (1998), 57-69. MR 1484611 (99a:11131)

[Lou5]
S. Louboutin, Upper bounds on $ \vert L(1,\chi)\vert$ and applications, Canada J. Math. Vol. 50(4) (1998), 794-815. MR 1638619 (99f:11149)

[Lou6]
S. Louboutin, The class number one problem for the dihedral and dicyclic CM-fields, Colloquium Mathematicum, Vol. 80, No. 2 (1999), 259-265. MR 1703822 (2000e:11140)

[Lou7]
S. Louboutin, Computation of relative class numbers of CM-fields by using Hecke L-functions, Math. Comp., Vol.69 (2000), 371-393. MR 1648395 (2000i:11172)

[Lou8]
S. Louboutin, Computation of $ L(0,\chi)$ and of relative class numbers of CM-fields, Nagoya Math. J., Vol 161 (2001), 171-191. MR 1820217 (2002e:11152)

[Lou9]
S. Louboutin, Explicit upper bounds for residues of Dedekind zeta functions and values of $ L$-functions at $ s=1$, and explicit lower bounds for relative class numbers of CM-fields, Can. J. Math. 53 (2001), 1194-1222. MR 1863848 (2003d:11167)

[Lou10]
S. Louboutin, Explicit lower bounds for residues at $ s=1$ of Dedekind zeta functions and relative class numbers of CM-fields, Trans. Amer. Math. Soc., 355 (2003), 3079-3098. MR 1974676 (2004f:11134)

[Mar]
J. Martinet, Une introduction $ \grave{a}$ la théorie du corps de classes (notes de M. Olivier), Ecole dotorale de Mathématiques de Bordeaux (1991).

[Mas]
J. M. Masley, Class numbers of real cyclic number fields with small conductor, Compositio Mathematica, Vol. 37. Fasc. 3 (1978), 297-319. MR 511747 (80e:12005)

[N]
J. Neukirch, Class field theory, A Series of Comprehensive studies in Mathematics 280, Springer-Verlag (1986). MR 819231 (87i:11005)

[Ok]
R. Okazaki, Inclusion of CM-fields and divisibility of relative class numbers, Acta Arith., 112.4 (2000), 319-338. MR 1760241 (2001h:11138)

[P]
C. Batut, K. Belabas, D. Bernardi, H. Cohen, M. Olivier, PARI-GP version 2.0.11.

[PK]
Y.-H. Park and S.-H. Kwon, Determination of all non-quadratic imaginary cyclic number fields of $ 2$-power degree with relative class number $ \leq 20$, Acta Arith., 83.3 (1998), 211-223. MR 1611205 (99a:11125)

[R1]
O. Ramar$ \acute{e}$, Approximate formulae for $ L(1,\chi)$, Acta, Arith., C.3 (2001), 245-266. MR 1865385 (2002k:11144)

[R2]
O. Ramar$ \acute{e}$, Approximate formulae for $ L(1,\chi)$, II, Acta, Arith., 112.2 (2004), 141-149. MR 2051374 (2005g:11156)

[S]
J.-P. Serre, Corps lacaux, Hermann, $ 3^e$édition, 1980.

[TW]
A.D. Thomas and G.V. Wood, Group Tables, Shiva Mathematics series 2, 1980. MR 572793 (81d:20002)

[W]
L.C. Washington, Introduction to Cyclotomic Fields, Springer-Verlag, Grad. Texts Math. Second Edition, 1997. MR 1421575 (97h:11130)

[YK]
H.-S. Yang and S.-H. Kwon, The non-normal quartic CM-fields and the octic dihedral CM-fields with relative class number two, J. Number Theory, 79 (1999), 175-193. MR 1728146 (2000h:11117)

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Additional Information:

Sun-Mi Park
Affiliation: Department of Mathematics, Korea University, 136-701, Seoul, Korea
Email: smpark@korea.ac.kr

Hee-Sun Yang
Affiliation: Department of Mathematics, Korea University, 136-701, Seoul, Korea
Address at time of publication: Korea Minting and Security Printing Corporation, 54, Gwahakro, Yusong-Gu, 305-713 Daejon, Korea
Email: yanghs@komsco.com

Soun-Hi Kwon
Affiliation: Department of Mathematics Education, Korea University, 136-701, Seoul, Korea
Email: sounhikwon@korea.ac.kr

DOI: 10.1090/S0002-9947-07-04219-5
PII: S 0002-9947(07)04219-5
Received by editor(s): May 6, 2004
Received by editor(s) in revised form: September 30, 2005
Posted: April 16, 2007
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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