Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Restricted ramification for imaginary quadratic number fields and a multiplicator free group

Author: Stephen B. Watt
Journal: Trans. Amer. Math. Soc. 288 (1985), 851-859
MSC: Primary 11R32; Secondary 11R11
MathSciNet review: 776409
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ K$ be an imaginary quadratic number field with unit group $ {E_K}$ and let $ \ell $ be a rational prime such that $ \ell \nmid \left\vert {{E_K}} \right\vert$. Let $ S$ be any finite set of finite primes of $ K$ and let $ K(\ell ,S)$ denote the maximal $ \ell $-extension of $ K$ (inside a fixed algebraic closure of $ K$) which is nonramified at the finite primes of $ K$ outside $ S$. We show that the finitely generated pro-$ \ell $-group $ \Omega (\ell ,S) = \operatorname{Gal}(K(\ell ,S)/K)$ has the property that a complete set of defining relations for $ \Omega (\ell ,S)$ as a pro-$ \ell $-group can be obtained by lifting the nontrivial abelian or torsion relations in the maximal abelian quotient group $ \Omega {(\ell ,S)^{{\text{ab}}}}$. In addition we use the key idea of the proof to derive some interesting results on towers of fields over $ K$ with restricted ramification.

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

  • [1] Algebraic number theory, Proceedings of an instructional conference organized by the London Mathematical Society (a NATO Advanced Study Institute) with the support of the Inter national Mathematical Union. Edited by J. W. S. Cassels and A. Fröhlich, Academic Press, London; Thompson Book Co., Inc., Washington, D.C., 1967. MR 0215665
  • [2] A. Fröhlich, On fields of class two, Proc. London Math. Soc. (3) 4 (1954), 235–256. MR 0063404
  • [3] -, Algebraic number fields, Academic Press, London, 1977.
  • [4] Albrecht Fröhlich, Central extensions, Galois groups, and ideal class groups of number fields, Contemporary Mathematics, vol. 24, American Mathematical Society, Providence, RI, 1983. MR 720859
  • [5] H. Koch, Galoissche Theorie der 𝑝-Erweiterungen, Springer-Verlag, Berlin-New York; VEB Deutscher Verlag der Wissenschaften, Berlin, 1970 (German). Mit einem Geleitwort von I. R. Šafarevič. MR 0291139
  • [6] Serge Lang, Algebraic number theory, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London-Don Mills, Ont., 1970. MR 0282947
  • [7] Stephen S. Shatz, Profinite groups, arithmetic, and geometry, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972. Annals of Mathematics Studies, No. 67. MR 0347778

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 11R32, 11R11

Retrieve articles in all journals with MSC: 11R32, 11R11

Additional Information

Article copyright: © Copyright 1985 American Mathematical Society