  Let p be an odd prime. Let K = Q(zeta) be the p-cyclotomic field. Let v be
any primitive root mod p. Let sigma be a Q-isomorphism of K. Let P(sigma) =
sigma^{p-2}v^{-(p-2)}+ ... + sigma v^{-1} +1 \in Z[G] where 1 \leq v^n \leq p-1
is a notation mod p. We apply a Kummer and Stickelberger relation of K to some
singular not primary numbers A of K connected to p-class group C_p of K and
prove they verify the congruence A^P(sigma) = 1 mod p^2.
  This p-adic method on singular numbers A allows us to prove: in a
straightforward way the connection between relative p-class group C_p^- and the
solutions of some explicit congruences mod p in Z[X]:
  \sum_{i=1}^{p-2} ((v^{-(i-1)} - v^{-i} v) /p) X^{i-1} \equiv 0 mod p and that
if (p-1)/2 is odd then the Bernoulli Number B_((p+1)/2) not = 0 mod p.
  In this version some congruences deduced of Stickelberger relation for prime
ideals Q of K of inertial degree f > 1 are added.
