  Preprint of a paper to appear in "Communications in Advanced Mathematical
Sciences". Let K be a real abelian extension of Q. Let p be a prime number, S
the set of p-places of K and G\_K,S the Galois group of the maximal S-ramified
pro-p-extension of K (i.e., unramified outside p and infinity). We revisit the
problem of annihilation of the p-torsion group T\_K:=tor\_Z\_p(G\_K,S^ab)
initiated by us and Oriat then systematized in our paper on the construction of
p-adic L-functions in which we obtained a canonical ideal annihilator of T\_K
in full generality (1978--1981). Afterwards (1992--2014) some annihilators,
using cyclotomic units, were proposed by Solomon, Belliard--Nguyen Quang Do,
Nguyen Quang Do--Nicolas, All, Belliard--Martin.In this text, we improve our
original papers and show that, in general, the Solomon elements are not optimal
and/or partly degenerated. We obtain, whatever K and p, an universal
non-degenerated annihilator in terms of p-adic logarithms of cyclotomic numbers
related to L\_p-functions at s=1 of its primitive characters of K (Theorem
9.4). Some computations are given with PARI programs; the case p=2 is analyzed
and illustrated in degrees 2, 3, 4 to test a conjecture.
