Triviality of Stickelberger Ideals of Conductor $p$

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

Triviality of Stickelberger Ideals of Conductor $p$
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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