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Questions tagged [galois-cohomology]

For questions on Galois cohomology, the study of the group cohomology of Galois modules.

In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group G associated to a field extension L/K acts in a natural way on some abelian groups, for example those constructed directly from L, but also through other Galois representations that may be derived by more abstract means. Galois cohomology accounts for the way in which taking Galois-invariant elements fails to be an exact functor.

221 questions
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Computing $H^1$ in Galois cohomology

I'm trying to learn the basics of Galois cohomology. If $k$ is a perfect field and $k^s$ is a separable closure, and $A$ is a finite group, how do you compute $H^1(\text{Gal}(k^s/k),A)=H^1(k,A)$?
Not a grad student
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Unramified Galois representation and cohomology

I wish to understand a remark made by Rubin in his book "Euler Systems" (just after definition 5.1) about the first cohomology group of an unramified Galois representation. Let $K$ be a number field, $v$ a finite place of $K$ above the rational…
Yoël
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Galois cohomology example

Let G be the Galois group of Q-bar over Q. I have a good idea of what $H^1(G,GL_1)$ is when $G$ acts trivially on $GL_1$: it is the group of homomorphisms from $G$ to $GL_1$. In particular it factors through the abelian quotient of $G$. In fact…
Class F. Student
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$H^1(G,GL_n(K))$ is trivial.

Suppose $K/F$ is a Galois extension with group $G=\text{Gal}(K/F)$, how to prove that $H^1(G,GL_n(K))$ is trivial with Galois descent? Thanks.
Ryze
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