On a family of algebraic number fields with finite 3-class field tower

Let $\ell=3$, $k=\mathbb Q(\sqrt{-3})$ and $K=k(\sqrt[3]{a})$, where $a$ is a natural number such that $a^2\equiv 1\pmod 9$. Under the assumption that there are exactly three places not over $\ell$ that ramify in the extension $K_\infty/k_\infty$, where $k_\infty$ and $K_\infty$ are cyclotomic $\mathbb Z_3$-extensions of the fields $k$ and $K$, respectively, we study 3-class field towers for intermediate fields $K_n$ of the extension $K_\infty/K$.
It is shown that for each $K_n$ the 3-class field tower of the field $K_n$ terminates already at the first step, which means that the Galois group of the extension $\mathbf H_\ell(K_n)/K_n$, where $\mathbf H_\ell(K_n)$ is the maximal unramified $\ell$-extension of the field $K_n$, is Abelian.
Bibliography: 7 titles.