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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

On the ideal class groups of imaginary abelian fields with small conductor


Authors: Kuniaki Horie and Hiroko Ogura
Journal: Trans. Amer. Math. Soc. 347 (1995), 2517-2532
MSC: Primary 11R29; Secondary 11R20
MathSciNet review: 1297529
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Abstract: Let $ k$ be any imaginary abelian field with conductor not exceeding 100, where an abelian field means a finite abelian extension over the rational field $ {\mathbf{Q}}$ contained in the complex field. Let $ C(k)$ denote the ideal class group of $ k$, $ {C^ - }(k)$ the kernel of the norm map from $ C(k)$ to the ideal class group of the maximal real subfield of $ k$, and $ f(k)$ the conductor of $ k;f(k) \leqslant 100$. Proving a preliminary result on $ 2$-ranks of ideal class groups of certain imaginary abelian fields, this paper determines the structure of the abelian group $ {C^ - }(k)$ and, under the condition that either $ [k:{\mathbf{Q}}] \leqslant 23$ or $ f(k)$ is not a prime $ \geqslant 71$, determines the structure of $ C(k)$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1995-1297529-6
PII: S 0002-9947(1995)1297529-6
Article copyright: © Copyright 1995 American Mathematical Society