 Galois cohomology

Galois cohomology is continuous group cohomology of a Galois group acting on a discrete Galois module. For a field $K$, the usual group is its absolute Galois group $G_K=\mathrm{Gal}(K_s/K)$. In degree zero it gives the invariant subgroup. In degree one, continuous cocycles satisfyand two cocycles have the same class when their difference is $\sigma \mapsto \sigma m-m$. If the action is trivial, these are continuous homomorphisms modulo the zero subgroup of coboundaries.

Continuity matters for the profinite group $G_K$: it connects the cohomology to finite Galois subextensions. Hilbert's theorem 90 makes the first multiplicative cohomology of a finite Galois extension vanish, and the Kummer exact sequence turns that vanishing into the power-class description in Kummer theory.

Galois cohomology is a branch of mathematics that studies objects known as "cohomology groups" in the context of Galois theory, which is a part of algebra concerned with the symmetries of polynomial equations. To understand Galois cohomology, we start with a few key ideas: 1. **Galois Groups**: A Galois group is a group associated with a field extension, representing the symmetries of the roots of polynomials.