Triviality of Stickelberger Ideals of Conductor $p$ | 東京大学大学院数理科学研究科理学部数学科

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Triviality of Stickelberger Ideals of Conductor $p$

J. Math. Sci. Univ. Tokyo
Vol. 13 (2006), No. 4, Page 617--628.

ICHIMURA, Humio
Triviality of Stickelberger Ideals of Conductor $p$
[Full Article (PDF)] [MathSciNet Review (HTML)] [MathSciNet Review (PDF)]

Abstract:
Let

$p$ be an odd prime number,

$G={\mathbb F}_p^\times,$ and

${\mathcal S}_G$ the classical Stickelberger ideal of the group ring

${\mathbb Z}[G]$ . For each subgroup

$H$ of

$G$ , we defined in [4] a Stickelberger ideal

${\mathcal S}_H$ of

${\mathbb Z}[H]$ as a

$H$ -part of

${\mathcal S}_G$ . We prove that if

${\mathcal S}_H$ is \lq\lq nontrivial", then the relative class number

$h^-_p$ of the

$p$ -cyclotomic field is divisible \lq\lq too often" by some prime number. This implies that

${\mathcal S}_H$ is nontrivial quite rarely. We also give an application of the triviality of

${\mathcal S}_H$ for a normal integral basis problem.

Mathematics Subject Classification (2000): Primary 11R18; Secondary 11R33.

Mathematical Reviews Number: MR2306221

Received: 2006-06-27

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