# Questions tagged [galois-cohomology]

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

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### On prime ideal and irreducible ideal in R[X].

For a noetherian domain $R$, an irreducible ideal $I$ implies $\sqrt{I}$ is a prime ideal. Irreducible implies primary, but not always vice versa. That said, I would like to ask whether the following ...
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### Galois Cohomology and Loop Groups

I am trying to understand problem 8.5 in Kac's Infinite dimensional Lie algebras. It goes as follows. Let $G$ be a semisimple algebraic group, let $\alpha$ be an automorphism of $G$ of order $m$, and ...
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### Exactness of Inflation restriction sequence, Galois Cohomology

I am trying to prove the following. Let $K/k$ be a finite Galois extension, $G= G(K/k)$, $k \subset F \subset K$ with $K/k$ normal and $H=G(K/F)$. Then: $\rho : C^{2} (G,A) \rightarrow C^{2} (H,A)$ ...
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### Exact sequence for Galois cohomology groups

Let $K/k$ be a finite Galois extension and let $G=\mbox{Gal}(K/k)$. Let $F$ be a subfield of $K$ such that $k\subseteq F\subseteq K$ and $F/k$ is normal. Let $H=\mbox{Gal}(K/F)\trianglelefteq G$. ...
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### Why are central simple algebras classified by cohomology?

In their article on the Brauer group Wikipedia writes: Since all central simple algebras over a field $K$ become isomorphic to the matrix algebra over a separable closure of $K$, the set of ...
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### Finite Galois Cohomology of Abelian Variety

Let $l/k$ be a finite galois extension let $A$ be an abelian variety over $k$. Then $A(l)$ is a $Gal(l/k)$-module. Hence it makes sense to study $H^1(Gal(l/k),A(l))$. I know that for $A=\mathbb{G}_m$, ...
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### Prove $A^\times$ is an inner form of $GL_n$

Question Let $A$ be a central simple algebra over $F$ where $F$ is a field and $dim_FA=n^2$. $A^\times$ can be seen as an algebraic group over $F$ and let $G=A^\times$. Prove that $G$ is an ...
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### Vanishing of a Galois cohomology of a pro-$p$ group

Suppose $G$ is a pro-$p$ group and $M$ is a discrete $p$-primary module not necessarily finite. Suppose the $p$-cohomological dimension of $G$ is $n$. Can one conclude $$H^n(G,M) \neq 0?$$ This is ...
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### $p$ cohomological dimension of profinite groups

Suppose $p$ is an odd prime and $S$ is any finite set containing the primes above $p$ and the Archimedean primes. Does there exist any number field $K$ such that $\textrm{Gal}(K_S/ K_{cyc})$ has $p$-...
### Calculating the Galois cohomology of $U(n)$
I would like to know know what the Galois cohomology $H^1(Gal( \Bbb{C}/\Bbb{R}),U(n)(\Bbb{C}) )$ is. The unitary group $U(n)$ can be defined as the real form of $Gl_{n,\Bbb{C}}$ relative to the ...