  Let p be an odd prime. Let F_p^* be the no-null part of the finite field of p
elements. Let K = Q(zeta) be the p-cyclotomic field and let O_K be the ring of
integers of K. Let pi be the prime ideal of K lying over p. An integer B \in
O_K is said singular if B^{1/p} not \in K and if B O_K = b^p where b is an
ideal of O_K. An integer B \in O_K is said semi-primary if B = beta mod pi^2
where the natural beta is coprime with p. Let sigma be a Q-isomorphism of the
field K generating the Galois group Gal(K/Q). When p is irregular, there exists
at least one subgroup Gamma of order p of the class group of K annihilated by a
polynomial sigma - mu with mu \in F_p^*. We prove the existence, for each
Gamma, of singular semi-primary integers B where B O_K= b^p with class Cl(b)
\in Gamma and B^{sigma-mu} \in K^p and we describe their pi-adic expansion.
This paper is at a strictly elementary level.
