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

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