  Let $K$ be the $N$th layer in the cyclotomic $\widehat{\mathbb{Z}}$-extension
$\widehat{\mathbb{Q}}$. Many authors (Aoki, Fukuda, Horie, Ichimura, Inatomi,
Komatsu, Miller, Morisawa, Nakajima, Okazaki, Washington,\,$\ldots$) analyse
the behavior of the $p$-class groups ${\mathcal C}_K$. We revisit this problem,
in a more conceptual form, since computations show that the $p$-torsion group
${\mathcal T}_K$ of the Galois groups of the maximal abelian $p$-ramified
pro-$p$-extension of $K$ (Tate--Shafarevich group of $K$) is often non-trivial;
this raises questions since $\# {\mathcal T}_K = \# {\mathcal C}_K\, \#
{\mathcal R}_K$ where ${\mathcal R}_K$ is the normalized $p$-adic regulator. We
give a new method testing ${\mathcal T}_K \ne 1$ (Theorem 4.6, Table 6.2) and
characterize the $p$-extensions $F$ of $K$ in $\widehat{\mathbb{Q}}$ with
${\mathcal C}_F \ne 1$ (Theorem 7.5 and Corollary 7.6). We publish easy to use
programs, justifying again the eight known examples, and allowing further
extensive computations.
