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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On the Stickelberger ideal and the relative class number
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by Tatsuo Kimura and Kuniaki Horie PDF
Trans. Amer. Math. Soc. 302 (1987), 727-739 Request permission

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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Additional Information
  • © Copyright 1987 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 302 (1987), 727-739
  • MSC: Primary 11R18; Secondary 11R29
  • DOI: https://doi.org/10.1090/S0002-9947-1987-0891643-8
  • MathSciNet review: 891643