The Hilbert's class field and the p-class group of the cyclotomic fields

Let $p$ be an irregular prime and $K=\Q(\zeta)$ the $p$-cyclotomic field. Let $\sigma$ be a $\Q$-isomorphism of $K$ generating $Gal(K/\Q)$. Let $S/K$ be a cyclic unramified extension of degree $p$, defined by $S= K(A^{1/p})$ where $A\in K\backslash K^p$, $A\Z_K=\mk a^p$ with $\mk a$ non-principal ideal of $\Z_K$, $A^{\sigma-\mu}\in K^p$ and $\mu\in{\bf F}_p$. We compute explicitly the decomposition of the prime $p$ in the subfields $M$ of $S$ of degree $[M:\Q]=p$.