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Results tagged with galois-cohomology
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user 51164
7
votes
1
answer
494
views
A vanishing condition for cup products in Galois cohomology
Let $k$ be a field of characteristic $\neq 2$. For a non-zero element $a \in k^*$, let us write $[a] \in H^1(k,\mathbb{Z}/2)$ for the Galois cohomology class corresponding to the quadratic extension $ …
- 9,491
2
votes
A vanishing condition for cup products in Galois cohomology
After realizing that the claim is wrong (thanks to the answers above), I managed to find a weaker statement, that turned out to be what I needed anyway. So just in case someone else happens to be inte …
- 9,491
2
votes
Accepted
A question on the injectivity of a canonical map between galois cohomology groups
I believe this map is always injective. Here is a quick argument: first note that $K'$ is normal over $k$ (because it is invariant under any automorphism of the algebraic closure of $k$ which preserve …
- 9,491
9
votes
Accepted
Nonabelian $H^2$ and Galois descent
Let me elaborate more on the remark above. Let $k$ be a perfect field. Let $\mathrm{Field}_k$ denote the category of finite extensions of $k$, i.e., the objects of $\mathrm{Field}_k$ are fields $k'$ e …
- 9,491