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