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

Abstract

Let $\ell$ be a regular odd prime, $K$ the $\ell$ th cyclotomic field and$K=k(\sqrt[\ell]{a} )$, where $a$ is a positive integer. Under theassumption that there are exactly three places ramified in the extension$K_\infty/k_\infty$, we study the $\ell$-component of the class group of thefield $K$. We prove that in the case $\ell>3$ there always is an unramifiedextension $\mathcal{N}/K$ such that $G(\mathcal{N}/K)\cong (\mathbbZ/\ell\mathbb Z)^2$ and all places over $\ell$ split completely in theextension $\mathcal{N}/K$. In the case $\ell=3$ we give a completedescription of the situation. Some other results are obtained.