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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)



Class groups in cyclic $ \ell$-extensions: Comments on a paper by G. Cornell

Author: Michael Rosen
Journal: Proc. Amer. Math. Soc. 142 (2014), 21-28
MSC (2010): Primary 11R29
Published electronically: September 10, 2013
MathSciNet review: 3119177
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Abstract: Let $ E/F$ be a cyclic extension of number fields of degree $ \ell ^{n}$, where $ \ell $ is a prime. It is proved that the $ \ell $-rank of the class group of $ E$ is bounded by $ \ell ^{n}(t-1+ \mathrm {rk}_\ell Cl_{F})$, where $ t$ is the number of primes of $ F$, including infinite primes, which ramify in $ E$ and $ Cl_{F}$ is the class group of $ F$. This generalizes a result of G. Cornell which applies when $ n=1$ and $ \ell $ is odd. A similar result is shown to hold when the Galois group is an abelian $ \ell $-group and the Hasse norm theorem is valid for $ E/F$.

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Michael Rosen
Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912

Keywords: $\ell$-rank, class groups, genus field, central class field
Received by editor(s): March 9, 2011
Received by editor(s) in revised form: December 21, 2011, and February 23, 2012
Published electronically: September 10, 2013
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.