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)

 
 

 

Kida's theorem for a class of nonnormal extensions


Authors: Robert Gold and Manohar Madan
Journal: Proc. Amer. Math. Soc. 104 (1988), 55-60
MSC: Primary 11R23
DOI: https://doi.org/10.1090/S0002-9939-1988-0958043-X
MathSciNet review: 958043
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ E,F$ be $ {{\mathbf{Z}}_p}$-fields of CM-type such that $ E/F$ is an extension of degree $ p$. Let $ L$, the normal closure of $ E/F$, be such that $ \operatorname{Gal} (L/F)$ has a normal subgroup of order $ p$. Denote the fixed field of this group by $ K$. We prove a Kida type formula which describes the minus part of the Iwasawa lambda invariant of $ E$ in terms of the lambda invariants of $ F$ and $ K$.


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

  • [DM] J. D'Mello and M. Madan, Class group rank relations in $ {{\mathbf{Z}}_l}$-extensions, Manuscripta Math. 41 (1983), 75-107. MR 689133 (84m:12016)
  • [GM] R. Gold and M. Madan, Galois representations of Iwasawa modules, Acta Arith. 46 (1986), 243-255. MR 864260 (88c:11061)
  • [I] K. Iwasawa, On the $ \mu $-invariants of $ {{\mathbf{Z}}_l}$-extensions, Number Theory, Algebraic Geometry and Commutative Algebra, Kinokuniya, Tokyo, 1973. MR 0357371 (50:9839)
  • [K] Y. Kida, $ l$-extensions of CM-fields and cyclotomic invariants, J. Number Theory 12 (1980), 519-528. MR 599821 (82c:12006)
  • [R] H.-G. Rück, Hass-Witt-invariants and dihedral extensions, Math. Z. 191 (1986), 513-517. MR 832808 (87j:11128)
  • [Sa] I. R. Safarevic, On $ p$-extensions, Amer. Math. Soc. Transl. (2) 4 (1954).
  • [S] W. Sinnott, On $ p$-adic $ L$-functions and the Riemann-Hurwitz genus formula, Compositio Math. 53 (1984), 3-17. MR 762305 (86e:11103)
  • 1. [W L. Washington, Introduction to cyclotomic fields, Springer-Verlag, Berlin and New York, 1982. MR 718674 (85g:11001)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 11R23

Retrieve articles in all journals with MSC: 11R23


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0958043-X
Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society