Primary components of the ideal class group of the $\mathbf{Z}_p$-extension over $\mathbf{Q}$ for typical inert primes

Abstract

Let $p$ be an odd prime, $\mathbf Z_p$ the ring of $p$-adic integers, and $l$ a prime number different from $p$. We have shown in [1] that, if $l$ is a sufficiently large primitive root modulo $p^2$, then the $l$-class group of the $\mathbf Z_p$-extension over the rational field is trivial. We shall modify part of the proof of the above result and see, in the case $p\leq 7$, that the result holds without assuming $l$ to be sufficiently large.