To a connected reductive group \(G\) over a field \(K\) of characteristic 0 and any integer \(i\geq-1\) the author associates the \(i\)th Abelian cohomology group \(H^i_{ab}(K,G)\) and shows that it depends only on the algebraic fundamental group \(\pi_1(G)\). It is related to the usual Galois cohomology by the abelianization map \(ab^i:H^i_{ab}(K,G)\to H^i(K,G)\). It is announced that the map \(ab^1\) is surjective when \(K\) is a number field. When \(K\) is a local field, this was proved by R. Kottwitz [Math. Ann. 275, 365-399 (1986; Zbl 0591.10020)].
As a corollary, there is given a description of \(H^1(K,G)\) in terms of Abelian cohomology and real cohomology. This generalizes results of J.-J. Sansuc [J. Reine Angew. Math. 327, 12-80 (1981; Zbl 0468.14007)] and J. S. Milne and K.-Y. Shih [Lect. Notes Math. 900, 280-356 (1982; Zbl 0478.14029)]. In particular, \(H^1(K,G)\) turns out to coincide with \(H^1_{ab}(K,G)\) when \(K\) is totally imaginary. The proofs will be published in Mem. Am. Math. Soc.