Galois cohomology and descent data

Vector bundles can be classified via transition functions up to change of trivialisation of vector bundles. In this video, we look at an analagous setup, where we use the Grothendieck topology arising from the Galois theory of a field as opposed to the usual topology one uses to study vector bundles. We describe Galois descent data via 1-cocycles of the Galois group. These play the role of transition functions. We look at the case of twisted forms of matrix algebras and see how these 1-cocycles can be used to twist the standard Galois action on matrix algebras to a new one whose invariant ring recovers the twisted form. Finally, we introduce an equivalence relation on 1-cocycles which is analagous to change of vector bundle trivialisation. We thus arrive at the first Galois cohomology set which can be used to classify twisted forms.