The class number one problem for the normal CM-fields of degree 32
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- by Sun-Mi Park, Hee-Sun Yang and Soun-Hi Kwon PDF
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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
- Elliot Benjamin, Franz Lemmermeyer, and C. Snyder, Real quadratic fields with abelian $2$-class field tower, J. Number Theory 73 (1998), no. 2, 182–194. MR 1658015, DOI 10.1006/jnth.1998.2291
- Ku-Young Chang and Soun-Hi Kwon, Class numbers of imaginary abelian number fields, Proc. Amer. Math. Soc. 128 (2000), no. 9, 2517–2528. MR 1707511, DOI 10.1090/S0002-9939-00-05555-6
- Henri Cohen, Advanced topics in computational number theory, Graduate Texts in Mathematics, vol. 193, Springer-Verlag, New York, 2000. MR 1728313, DOI 10.1007/978-1-4419-8489-0
- Marie-Nicole Gras, Classes et unités des extensions cycliques réelles de degré $4$ de $\textbf {Q}$, Ann. Inst. Fourier (Grenoble) 29 (1979), no. 1, xiv, 107–124 (French, with English summary). MR 526779
- Helmut Hasse, Über die Klassenzahl abelscher Zahlkörper, 1st ed., Springer-Verlag, Berlin, 1985 (German). With an introduction to the reprint edition by Jacques Martinet. MR 842666, DOI 10.1007/978-3-642-69886-6
- Erich Hecke, Lectures on the theory of algebraic numbers, Graduate Texts in Mathematics, vol. 77, Springer-Verlag, New York-Berlin, 1981. Translated from the German by George U. Brauer, Jay R. Goldman and R. Kotzen. MR 638719, DOI 10.1007/978-1-4757-4092-9
- Gordon James and Martin Liebeck, Representations and characters of groups, Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, 1993. MR 1237401
- S.-H. Kwon, Sur les discriminants minimaux des corps quaternioniens, Arch. Math. (Basel) 67 (1996), no. 2, 119–125 (French). MR 1399827, DOI 10.1007/BF01268925
- M. Daberkow, C. Fieker, J. Klüners, M. Pohst, K. Roegner, M. Schörnig, and K. Wildanger, KANT V4, J. Symbolic Comput. 24 (1997), no. 3-4, 267–283. Computational algebra and number theory (London, 1993). MR 1484479, DOI 10.1006/jsco.1996.0126
- Yann Lefeuvre, Corps diédraux à multiplication complexe principaux, Ann. Inst. Fourier (Grenoble) 50 (2000), no. 1, 67–103 (French, with English and French summaries). MR 1762338, DOI 10.5802/aif.1747
- Franz Lemmermeyer, Ideal class groups of cyclotomic number fields. I, Acta Arith. 72 (1995), no. 4, 347–359. MR 1348202, DOI 10.4064/aa-72-4-347-359
- Franz Lemmermeyer, Unramified quaternion extensions of quadratic number fields, J. Théor. Nombres Bordeaux 9 (1997), no. 1, 51–68 (English, with English and French summaries). MR 1469661, DOI 10.5802/jtnb.189
- Stéphane Louboutin and Ryotaro 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), no. 1, 47–62. MR 1292520, DOI 10.4064/aa-67-1-47-62
- Stéphane Louboutin and Ryotaro Okazaki, The class number one problem for some non-abelian normal CM-fields of $2$-power degrees, Proc. London Math. Soc. (3) 76 (1998), no. 3, 523–548. MR 1616805, DOI 10.1112/S0024611598000318
- Stéphane Louboutin, Ryotaro Okazaki, and Michel Olivier, The class number one problem for some non-abelian normal CM-fields, Trans. Amer. Math. Soc. 349 (1997), no. 9, 3657–3678. MR 1390044, DOI 10.1090/S0002-9947-97-01768-6
- Stéphane Louboutin, Lower bounds for relative class numbers of CM-fields, Proc. Amer. Math. Soc. 120 (1994), no. 2, 425–434. MR 1169041, DOI 10.1090/S0002-9939-1994-1169041-0
- Stéphane Louboutin, Determination of all quaternion octic CM-fields with class number $2$, J. London Math. Soc. (2) 54 (1996), no. 2, 227–238. MR 1405052, DOI 10.1112/jlms/54.2.227
- Stéphane Louboutin, The class number one problem for the non-abelian normal CM-fields of degree $16$, Acta Arith. 82 (1997), no. 2, 173–196. MR 1477509, DOI 10.4064/aa-82-2-173-196
- Stéphane Louboutin, Majorations explicites du résidu au point 1 des fonctions zêta de certains corps de nombres, J. Math. Soc. Japan 50 (1998), no. 1, 57–69 (French). MR 1484611, DOI 10.2969/jmsj/05010057
- Stéphane Louboutin, Upper bounds on $|L(1,\chi )|$ and applications, Canad. J. Math. 50 (1998), no. 4, 794–815. MR 1638619, DOI 10.4153/CJM-1998-042-2
- Stéphane Louboutin, The class number one problem for the dihedral and dicyclic CM-fields, Colloq. Math. 80 (1999), no. 2, 259–265. MR 1703822, DOI 10.4064/cm-80-2-259-265
- Stéphane Louboutin, Computation of relative class numbers of CM-fields by using Hecke $L$-functions, Math. Comp. 69 (2000), no. 229, 371–393. MR 1648395, DOI 10.1090/S0025-5718-99-01096-0
- Stéphane Louboutin, Computation of $L(0,\chi )$ and of relative class numbers of CM-fields, Nagoya Math. J. 161 (2001), 171–191. MR 1820217, DOI 10.1017/S0027763000022170
- Stéphane 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, Canad. J. Math. 53 (2001), no. 6, 1194–1222. MR 1863848, DOI 10.4153/CJM-2001-045-5
- Stéphane 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), no. 8, 3079–3098. MR 1974676, DOI 10.1090/S0002-9947-03-03313-0
- J. Martinet, Une introduction $\grave {a}$ la théorie du corps de classes (notes de M. Olivier), Ecole dotorale de Mathématiques de Bordeaux (1991).
- John Myron Masley, Class numbers of real cyclic number fields with small conductor, Compositio Math. 37 (1978), no. 3, 297–319. MR 511747
- Jürgen Neukirch, Class field theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 280, Springer-Verlag, Berlin, 1986. MR 819231, DOI 10.1007/978-3-642-82465-4
- Ryotaro Okazaki, Inclusion of CM-fields and divisibility of relative class numbers, Acta Arith. 92 (2000), no. 4, 319–338. MR 1760241, DOI 10.4064/aa-92-4-319-338
- C. Batut, K. Belabas, D. Bernardi, H. Cohen, M. Olivier, PARI-GP version 2.0.11.
- Young-Ho Park and Soun-Hi Kwon, Determination of all non-quadratic imaginary cyclic number fields of $2$-power degree with relative class number $\le 20$, Acta Arith. 83 (1998), no. 3, 211–223. MR 1611205, DOI 10.4064/aa-83-3-211-223
- Olivier Ramaré, Approximate formulae for $L(1,\chi )$, Acta Arith. 100 (2001), no. 3, 245–266. MR 1865385, DOI 10.4064/aa100-3-2
- Olivier Ramaré, Approximate formulae for $L(1,\chi )$. II, Acta Arith. 112 (2004), no. 2, 141–149. MR 2051374, DOI 10.4064/aa112-2-4
- J.-P. Serre, Corps lacaux, Hermann, $3^e$édition, 1980.
- A. D. Thomas and G. V. Wood, Group tables, Shiva Mathematics Series, vol. 2, Shiva Publishing Ltd., Nantwich; distributed by Birkhäuser Boston, Inc., Cambridge, Mass., 1980. MR 572793
- Lawrence C. Washington, Introduction to cyclotomic fields, 2nd ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1997. MR 1421575, DOI 10.1007/978-1-4612-1934-7
- Hee-Sun Yang and Soun-Hi Kwon, The non-normal quartic CM-fields and the octic dihedral CM-fields with relative class number two, J. Number Theory 79 (1999), no. 2, 175–193. MR 1728146, DOI 10.1006/jnth.1999.2421
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
- Received by editor(s): May 6, 2004
- Received by editor(s) in revised form: September 30, 2005
- Published electronically: April 16, 2007
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 359 (2007), 5057-5089
- MSC (2000): Primary 11R29; Secondary 11R21
- DOI: https://doi.org/10.1090/S0002-9947-07-04219-5
- MathSciNet review: 2320660