Class numbers of imaginary abelian number fields

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...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2000-09, Vol.128 (9), p.2517-2528
Main Authors: Chang, Ku-Young, Kwon, Soun-Hi
Format: Article
Language:English
Subjects:
Automorphisms
Infinity
Integers
Mathematical theorems
Mathematics education
Number theory
Numbers
Prime numbers
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Summary: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.
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-00-05555-6