Cohomology
The theorem can be stated in terms of group cohomology: if L× is the multiplicative group of any (not necessarily finite) Galois extension L of a field K with corresponding Galois group G, then
- H1(G, L×) = {1}.
A further generalization using non-abelian group cohomology states that if H is either the general or special linear group over L, then
- H1(G,H) = {1}.
This is a generalization since L× = GL1(L).
Another generalization is for X a scheme, and another one to Milnor K-theory plays a role in Voevodsky's proof of the Milnor conjecture.
Read more about this topic: Hilbert's Theorem 90