Tohoku Mathematical Journal
2007
June
SECOND SERIES VOL. 59, NO. 2

Tohoku Math. J.
59 (2007), 259-291

Title CERTAIN PRIMARY COMPONENTS OF THE IDEAL CLASS GROUP OF THE $\boldsymbol{Z}_p$-EXTENSION OVER THE RATIONALS

Author Kuniaki Horie

(Received December 14, 2005, revised November 21, 2006)
Abstract. We study, for any prime number $p$, the triviality of certain primary components of the ideal class group of the $\boldsymbol{Z}_p$-extension over the rational field. Among others, we prove that if $p$ is $2$ or $3$ and $l$ is a prime number not congruent to $1$ or $-1$ modulo $2p^2$, then $l$ does not divide the class number of the cyclotomic field of $p^u$th roots of unity for any positive integer $u$.

2000 Mathematics Subject Classification. Primary 11R29; Secondary 11R18, 11R20, 11R23.
Key words and phrases. Ideal class group, $\boldsymbol{Z}_p$-extension, cyclotomic field, class number formula, decomposition field.

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