Arithmetic of certain $\ell$-extensions ramified at three places. IV

Abstract

Let $\ell$ be an odd regular prime, $k$ be the $\ell$th cyclotomic field, and $K=k(\sqrt[\ell]{a})$, where $a$ is a natural number that has exactly three distinct prime divisors. Assuming that there are exactly three places ramified in $K_\infty/k_\infty$, we study the $\ell$-component of the class group of the field $K$. For $\ell>3$, we prove that there always exists an unramified extension $\mathcal{N}/K$ such that $G(\mathcal{N}/K)\cong (\mathbb Z/\ell\mathbb Z)^3$, and all places over $\ell$ split completely in $\mathcal{N}/K$. If $\ell=3$ and $a$ is of the form $a=p^rq^s$, we give a complete description of the possible structure of the $\ell$-component of the class group of $K$. Some other results are also obtained.