The class number one problem for the dihedral and dicyclic CM-fields

• Autorzy: Stéphane Louboutin
• Języki publikacji: EN

Abstrakty

EN

We recall the determination of all the dihedral CM-fields with relative class number one, and prove that dicyclic CM-fields have relative class numbers greater than one.

Słowa kluczowe

EN

relative class number   CM-field   dihedral group   dicyclic group  

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 1999
• Tom: 80
• Numer: 2
• Strony: 259-265

Daty

wydano

1999

otrzymano

1998-12-14

Twórcy

autor

Stéphane Louboutin

• Département de Mathématiques, Université de Caen, Campus 2, BP 5186, 14032 Caen Cedex, France

Bibliografia

• [Hof] J. Hoffstein, Some analytic bounds for zeta functions and class numbers, Invent. Math. 55 (1979), 37-47.
• [Lef] Y. Lefeuvre, Corps diédraux à multiplication complexe principaux, preprint, Univ. Caen, 1998.
• [LL] Y. Lefeuvre and S. Louboutin, The class number one problem for the dihedral CM-fields, in: Proc. Conf. on Algebraic Number Theory and Diophantine Analysis, Graz, August-September 1998, to appear.
• [LLO] F. Lemmermeyer, S. Louboutin and R. Okazaki, The class number one problem for some non-abelian normal CM-fields of degree 24, J. Théor. Nombres Bordeaux, to appear.
• [Lou1] S. Louboutin, Determination of all quaternion octic CM-fields with class number 2, J. London Math. Soc. 54 (1996), 227-238.
• [Lou2] S. Louboutin, CM-fields with cyclic ideal class groups of 2-power orders, J. Number Theory 67 (1997), 1-10.
• [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.
• [LO2] S. Louboutin and R. Okazaki, The class number one problem for some non-abelian normal CM-fields of 2-power degrees, Proc. London Math. Soc. (3) 76 (1998), 523-548.
• [LOO] S. Louboutin, R. Okazaki and M. Olivier, The class number one problem for some non-abelian normal CM-fields, Trans. Amer. Math. Soc. 349 (1997), 3657-3678.
• [Mar] J. Martinet, Sur l'arithmétique des extensions galoisiennes à groupe de Galois diédral d'ordre 2p, Ann. Inst. Fourier (Grenoble) 19 (1969), no. 1, 1-80.
• [Odl] A. Odlyzko, Some analytic estimates of class numbers and discriminants, Invent. Math. 29 (1975), 275-286.
• [TW] A. D. Thomas and G. V. Wood, Group Tables, Shiva Publ. Kent, 1980.
• [Wa] L. C. Washington, Introduction to Cyclotomic Fields, Grad. Texts in Math. 83, Springer, 1982; 2nd ed., 1997.
• [Yam] K. Yamamura, The determination of the imaginary abelian number fields with class-number one, Math. Comp. 206 (1994), 899-921.