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.
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Galois cohomology is continuous group cohomology of a Galois group acting on a discrete Galois module. For a field , the usual group is its absolute Galois group . In degree zero it gives the invariant subgroup. In degree one, continuous cocycles satisfyand two cocycles have the same class when their difference is . If the action is trivial, these are continuous homomorphisms modulo the zero subgroup of coboundaries.
Continuity matters for the profinite group : 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.