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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the Stickelberger ideal and the relative class number

Authors: Tatsuo Kimura and Kuniaki Horie
Journal: Trans. Amer. Math. Soc. 302 (1987), 727-739
MSC: Primary 11R18; Secondary 11R29
MathSciNet review: 891643
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Abstract: Let $ k$ be any imaginary abelian field, $ R$ the integral group ring of $ G = {\text{Gal}}(k/\mathbb{Q})$, and $ S$ the Stickelberger ideal of $ k$. Roughly speaking, the relative class number $ {h^ - }$ of $ k$ is expressed as the index of $ S$ in a certain ideal $ A$ of $ R$ described by means of $ G$ and the complex conjugation of $ k;{c^ - }{h^ - } = [A:S]$, with a rational number $ {c^ - }$ in $ \frac{1} {2}\mathbb{N} = \{ n/2;n \in \mathbb{N}\} $, which can be described without $ {h^ - }$ and is of lower than $ {h^ - }$ if the conductor of $ k$ is sufficiently large (cf. [6, 9, 10]; see also [5]). We shall prove that $ 2{c^ - }$, a natural number, divides $ 2{([k:\mathbb{Q}]/2)^{[k:\mathbb{Q}]/2}}$. In particular, if $ k$ varies through a sequence of imaginary abelian fields of degrees bounded, then $ {c^ - }$ takes only a finite number of values. On the other hand, it will be shown that $ {c^ - }$ can take any value in $ \frac{1} {2}\mathbb{N}$ when $ k$ ranges over all imaginary abelian fields. In this connection, we shall also make a simple remark on the divisibility for the relative class number of cyclotomic fields.

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Keywords: (Imaginary) abelian field, Stickelberger ideal, relative class number, analytic class number formula
Article copyright: © Copyright 1987 American Mathematical Society