  Let K/Q be Galois and let eta in K* be such that the multiplicative
Z[G]-module generated by eta is of Z-rank n.We define the local
theta-regulators Delta\_p^theta(eta) in F\_p for the Q\_p-irreducible
characters theta of G=Gal(K/Q). Let V\_theta be the theta-irreducible
representation. A linear representation L^theta=delta.V\_theta is associated
withDelta\_p^theta(eta) whose nullity is equivalent to delta$\ge$1 (Theorem
3.9). Each Delta\_p^theta(eta) yields Reg\_p^theta(eta) modulo p in the
factorization $\prod$\_theta (Reg\_p^theta(eta))^phi(1) of Reg\_p^G(eta) :=
Reg\_p(eta)/p^[K : Q] (normalized p-adic regulator), where phi divides theta is
absolutely irreducible.From the probability Prob(Delta\_p^theta(eta) = 0 \&
L^theta=delta.V\_theta)$\le$p^(-f.delta^2) (f= residue degree of p in the field
of values of phi) and the Borel--Cantelli heuristic, we conjecture that, for p
large enough, Reg\_p^G(eta) is a p-adic unit or that p^phi(1) divides exactly
Reg\_p^G(eta) (existence of a single theta with f=delta=1); this obstruction
may be lifted assuming the existence of a binomial probability law (Sec. 7)
confirmed through numerical studies (groups C\_3, C\_5, D\_6). This conjecture
would imply that, for all p large enough, Fermat quotients of rationals
andnormalized p-adic regulators are p-adic units (Theorem 1.1), whence the fact
that number fields are p-rational for p\textgreater{}\textgreater{}0. We recall
\S8.7 some deep cohomological results, which may strengthen such conjectures.
