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Galois cohomology Let K/k be a finite Galois extension with the Galois group G(K/k). Suppose, further, that G(K/k) acts on some Abelian group A. The Galois cohomology groups Hn(G(K/k), A) ≡ Hn(K/k,A), n ≥ 0 are then the cohomology groups defined by the (cochain) complex (F n, d), with Fn consisting of all mappings G(K/k)n → A and d designating the coboundary operator (see cohomology groups). When the extension K/k is of an infinite degree, one also requires that the Galois topological group acts continuously on the discrete group A and the mappings for the cochains in Fn are also continuous.