On the $p$-class Tower of a $\textbf{Z}_{\textrm{p}}$-extension

Abstract

For a number field $k$ and a prime number $p$, let $k_{\infty}$ be a $\textbf{Z}_{\textrm{p}}$-extension of $k$ and $X_{\infty}(k)$ the Galois group over $k_{\infty}$ of the maximal abelian unramified $p$-extension of $k_{\infty}$. We first give a sufficient condition, bearing on the norm index of units in the layers of $k_{\infty}$, for $X_{\infty}(k)$ to be finite. When the prime $p$ is 2 and $X_{\infty}(k)\simeq \textbf{Z}/2\textbf{Z}\times \textbf{Z}/2\textbf{Z}$, we study the structure of the Galois group of the maximal unramified $p$-extension of $k_{\infty}$, improving on some previous results in the case of quadratic fields.