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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 numbers of imaginary abelian number fields


Authors: Ku-Young Chang and Soun-Hi Kwon
Journal: Proc. Amer. Math. Soc. 128 (2000), 2517-2528
MSC (1991): Primary 11R29; Secondary 11R20
Published electronically: April 27, 2000
MathSciNet review: 1707511
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Abstract:

Let $N$ be an imaginary abelian number field. We know that $h_{N}^{-}$, the relative class number of $N$, goes to infinity as $f_N$, the conductor of $N$, approaches infinity, so that there are only finitely many imaginary abelian number fields with given relative class number. First of all, we have found all imaginary abelian number fields with relative class number one: there are exactly 302 such fields. It is known that there are only finitely many CM-fields $N$ with cyclic ideal class groups of 2-power orders such that the complex conjugation is the square of some automorphism of $N$. Second, we have proved in this paper that there are exactly 48 such fields.


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Additional Information

Ku-Young Chang
Affiliation: Department of Mathematics, Korea University, 136-701, Seoul, Korea
Email: jang@semi.korea.ac.kr

Soun-Hi Kwon
Affiliation: Department of Mathematics Education, Korea University, 136-701, Seoul, Korea
Email: shkwon@semi.korea.ac.kr

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05555-6
PII: S 0002-9939(00)05555-6
Received by editor(s): May 1, 1998
Published electronically: April 27, 2000
Additional Notes: This research was supported by Grant BSRI-97-1408 from the Ministry of Education of Korea.
Communicated by: David E. Rohrlich
Article copyright: © Copyright 2000 American Mathematical Society