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

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 $u$.