# Galois cohomology

In the mathematical branch of number theory, Galois cohomology is the study of the group cohomology of Galois groups .

Is L | K is a field extension and A a Galoismodul , so a module of the Galois group Gal ( L | K ), we write

${\ displaystyle \, H ^ {*} (L | K, A) = H ^ {*} (\ mathrm {Gal} (L | K), A)}$(for notation see the article group cohomology )

If especially L = K sep is a separable closure of K , then one also writes

${\ displaystyle \, H ^ {*} (K, A) = H ^ {*} (G_ {K}, A) = H ^ {*} (\ mathrm {Gal} (K ^ {\ mathrm {sep} } | K), A).}$

One of the first results of Galois cohomology is Hilbert's Theorem 90 , which says:

${\ displaystyle H ^ {1} (K, (K ^ {\ mathrm {sep}}) ^ {\ times}) = 0}$.

Especially in class field theory , the relationship between Galois cohomology and brewer group is important:

${\ displaystyle H ^ {2} (K, (K ^ {\ mathrm {sep}}) ^ {\ times}) = \ mathrm {Br} (K)}$.