The $l$-class group of the Z$_p$-extension over the rational field

Abstract

Let $p$ be an odd prime, and let B$_\infty$ denote the Z$_p$-extension over the rational field. Let $l$ be an odd prime different from $p$. The question whether the $l$-class group of B$_\infty$ is trivial has been considered in our previous papers mainly for the case where $l$ varies with $p$ fixed. We give a criterion, for checking the triviality of the $l$-class group of B$_\infty$, which enables us to discuss the triviality when $p$ varies with $l$ fixed. As a consequence, we find that, if $l$ does not exceed 13 and $p$ does not exceed 101, then the $l$-class group of B$_\infty$ is trivial.