Nonabelian H^1 and the descent slogan
When the coefficient module is a nonabelian group on which G acts, H^1(G, A) is no longer a group — just a pointed set (cocycles are still f(gh) = f(g)·g(f(h)), modulo f(g) ~ a^{−1} f(g) g(a)). The price of nonabelianness is losing the group structure above degree 1, but the payoff is enormous: H^1(Gal(K̄/K), Aut(X₀)) classifies the K-forms of an object X₀ — everything that becomes isomorphic to X₀ over the algebraic closure but might not be isomorphic over K. This is the descent / twisting principle.
TWISTING SLOGAN: { K-forms of X_0 } / iso <--> H^1( Gal(Kbar/K), Aut_Kbar(X_0) ).
Example A (vanishing). X_0 = the K-vector space K^n. Aut over Kbar is GL(n, Kbar).
H^1(Gal, GL(n, Kbar)) = * (one point): a form of GL_n is again GL_n.
This is Theorem 90 generalized (n=1 recovers H^1(G, Kbar*) = 0). Moral: vector spaces
have no twists -- a finite-dim space over K is determined by its dimension.
Example B (nontrivial). X_0 = the split quadratic form x1^2 + ... + xn^2 over K = Q.
Aut over Kbar is O(n, Kbar). H^1(Gal, O(n)) classifies n-dim QUADRATIC FORMS over Q
up to isometry that become equivalent over Qbar -- i.e. all nondegenerate ones.
Diagonal entries (the d_i in <d_1,...,d_n>) mod squares are exactly the cocycle data.
The twists are real: x^2 + y^2 and x^2 - y^2 are NOT isometric over Q (different signature),
yet both diagonalize to <1,1> over C. H^1 sees the difference; the algebraic closure does not.H^2 again: the Brauer group
Just as nonabelian H^1 classifies twisted forms, H^2(Gal(K̄/K), K̄*) classifies something of degree 2 — and it is the [[brauer-group|Brauer group]] Br(K). Its elements are central simple algebras over K up to Morita equivalence; the cohomology class is, once more, a factor set, now valued in K̄*. The Hamilton quaternions over R represent the nontrivial element of Br(R) = Z/2 — a central simple R-algebra that is not a matrix algebra precisely because its class in H^2 is nonzero. The H^2-classifies-extensions story of guide 3 and the H^2-as-Brauer-group story are the same cohomology in two costumes.
To navigate between fields, two functorial maps recur. The [[restriction-map|restriction map]] res : H^n(G, A) → H^n(H, A) passes to a subgroup H ≤ G (shrinking the symmetry, e.g. to a larger field). The [[inflation-map|inflation map]] inf : H^n(G/N, A^N) → H^n(G, A) pulls classes up from a quotient. Together with the transgression they assemble the inflation–restriction exact sequence, the low-degree shadow of the Lyndon–Hochschild–Serre spectral sequence — the tool for computing cohomology of a group from a normal subgroup and the quotient.