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Preprint of a paper to appear in "Communications in Advanced MathematicalSciences". Let K be a real abelian extension of Q. Let p be a prime number, Sthe set of p-places of K and G\_K,S the Galois group of the maximal S-ramifiedpro-p-extension of K (i.e., unramified outside p and infinity). We revisit theproblem 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 ofp-adic L-functions in which we obtained a canonical ideal annihilator of T\_Kin 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 ouroriginal papers and show that, in general, the Solomon elements are not optimaland/or partly degenerated. We obtain, whatever K and p, an universalnon-degenerated annihilator in terms of p-adic logarithms of cyclotomic numbersrelated to L\_p-functions at s=1 of its primitive characters of K (Theorem9.4). Some computations are given with PARI programs; the case p=2 is analyzedand illustrated in degrees 2, 3, 4 to test a conjecture.

Annihilation of tor\_p(G\_S^ab) for real abelian extensions