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

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 exactlythree distinct prime divisors. Assuming that there are exactly three placesramified in $K_\infty/k_\infty$, we study the $\ell$-component of the classgroup 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 $\ell$-componentof the class group of $K$.Some other results are also obtained.

Iwasawa theory, Tate module, extension with given ramification, the Riemann–Hurwitz formula