Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Generating functions for the numbers of abelian extensions of a local field


Author: Artur Travesa
Journal: Proc. Amer. Math. Soc. 108 (1990), 331-339
MSC: Primary 11S15; Secondary 11S40
DOI: https://doi.org/10.1090/S0002-9939-1990-1007513-6
MathSciNet review: 1007513
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The aim of this paper is to give an explicit formula for the numbers of abelian extensions of a $ {\mathbf{p}}$-adic number field and to study the generating function of these numbers. More precisely, we give the number of abelian extensions with given degree and ramification index, and the number of abelian extensions with given degree of any local field of characteristic zero. Moreover, we give a concrete expression of a generating function for these last numbers.


References [Enhancements On Off] (What's this?)

  • [Ap1] Tom M. Apostol, Introduction to analytic number theory, Springer-Verlag, New York-Heidelberg, 1976. Undergraduate Texts in Mathematics. MR 0434929
  • [Ha1] H. Hasse, Number theory, GMW 229 Springer-Verlag, Berlin-Heidelberg, 1970.
  • [Kr1] M. Krasner, Nombre des extensions d'un degré donné d'un corps $ p$-adique, C.R. Acad. Sc. Paris 254 (1962), 3470-3472; C. R. Acad. Sc. Paris 255 (1962), 224-226, 1682-1684, 2342-2344, 3095-3097, See also Les tendances géométriques en algèbre et théorie des nombres. Colloques internationaux du C.N.R.S. 143 (1966), pp. 143-169..
  • [La1] Serge Lang, Algebraic number theory, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London-Don Mills, Ont., 1970. MR 0282947
  • [Ne1] Jürgen Neukirch, Class field theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 280, Springer-Verlag, Berlin, 1986. MR 819231
  • [Se1] Jean-Pierre Serre, Une “formule de masse” pour les extensions totalement ramifiées de degré donné d’un corps local, C. R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 22, A1031–A1036 (French, with English summary). MR 500361
  • [Tr1] A. Travesa, Nombres d'extensions abelianes i les seves funcions generatrius, Doctoral Thesis, Universitat de Barcelona, 1987.
  • [Tr2] -, Sobre el número de extensiones de grado dado de un cuerpo local, Actas de las X Jornadas Hispano-Lusas de Matemáticas, Sección I: Álgebra y Fundamentos, Murcia, 1985, pp. 235-243.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 11S15, 11S40

Retrieve articles in all journals with MSC: 11S15, 11S40


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1990-1007513-6
Keywords: Abelian extension, local field, generating function, Dirichlet series
Article copyright: © Copyright 1990 American Mathematical Society