  Let $p$ be an irregular prime. Let $K=\Q(\zeta)$ be the $p$-cyclotomic field.
  From Kummer and class field theory, there exist Galois extensions $S/\Q$ of
degree $p(p-1)$ such that $S/K$ is a cyclic unramified extension of degree
$[S:K]=p$. We give an algebraic construction of the subfields $M$ of $S$ with
degree $[M:\Q]=p$ and an explicit formula for the prime decomposition and
ramification of the prime number $p$ in the extensions $S/K$, $M/\Q$ and $S/M$.
In the last section, we examine the consequences of these results for the
Vandiver's conjecture. This article is at elementary level on Classical
Algebraic Number Theory.
