Galois cohomology
Galois cohomology is group cohomology specialized to the most important groups in arithmetic: Galois groups. When you study a field by passing to a larger field where things become simpler, the Galois group measures the difference between the two, and it acts on every arithmetic object — units, roots, points of varieties — that lives upstairs. Galois cohomology records what is lost in descending back down, packaging questions about solving equations over the base field as cohomology classes.
Concretely, if L/K is a Galois extension with group G = Gal(L/K), and A is a G-module (a discrete module on which G acts continuously, such as L^* or the points of an algebraic group over L), then the Galois cohomology groups are the continuous group cohomology H^n(G, A). For infinite extensions G is profinite and one uses continuous cochains, equivalently the colimit of cohomology over finite subextensions. The absolute Galois group Gal(K-sep / K) and its cohomology encode deep arithmetic of K.
Two cornerstone results organize the subject in low degree. Hilbert's Theorem 90 gives H^1(G, L^*) = 0, which underlies Kummer theory and the description of cyclic extensions. The Brauer group Br(K) is identified with H^2(Gal(K-sep/K), (K-sep)^*) and classifies central simple algebras over K. Higher Galois cohomology feeds into class field theory, the cohomological formulation of duality (Tate-Poitou), and the étale cohomology used throughout modern number theory and arithmetic geometry.
For K = R, the absolute Galois group is Gal(C/R) = Z/2Z. Then H^2(Z/2Z, C^*) = Z/2Z = Br(R), whose nonzero class is the Hamilton quaternions H — the unique nontrivial central division algebra over the reals.
The Brauer group of R is Z/2Z, realized by the quaternions.
The continuity requirement is essential: for an infinite Galois group one must use cohomology computed with continuous cochains, where the module is given the discrete topology and stabilizers are open. Without it the answers would be wrong and functoriality would break.