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:

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

literature

• Jean Pierre Serre: Galois cohomology . Springer, Berlin 2002, ISBN 3-540-42192-0 .