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One argument I love is the following: let $L/K$ be a Galois extension with group $G$ and let $n\geq1$. One can show very straightforwardly that $H^1(G,\mathrm{GL}(n, L))$ classifies $K$-vector spaces $V$ such that $L\otimes_KV$ is isomorphic as an $L$-vector space to $L\otimes K^n$, up to $K$-linear isomorphisms; Serre does it in chapter X, §2, of his Corps Locaux. Now, linear algebra tells us that all such $V$'s are in fact isomorphic to $K^n$ as $K$-vector spaces, so we conclude that $H^1(G,\mathrm{GL}(n, L))$ is trivial.

This is, in fact, the same argument that Brian gave. Yet it is nice that the theorem becomes essentially a statement saying that all vector spaces of the same dimension are isomorphic :)

Also, other somewhat mystifying statements, like $«H^1(G,\mathrm{Sp}(n, L))=0»$ can be proved by exactly the same argument.

2 added 127 characters in body

One argument I love is the following: let $L/K$ be a Galois extension with group $G$ and let $n\geq1$. One can show very straightforwardly that $H^1(G,\mathrm{GL}(n, L))$ classifies $K$-vector spaces $V$ such that $L\otimes_KV$ is isomorphic as an $L$-vector space to $L\otimes K^n$, up to $K$-linear isomorphisms; Serre does it in chapter X, §2, of his Corps Locaux. Now, linear algebra tells us that all such $V$'s are in fact isomorphic to $K^n$ as $K$-vector spaces, so we conclude that $H^1(G,\mathrm{GL}(n, L))$.

This is, in fact, the same argument that Brian gave. Yet it is nice that the theorem becomes essentially a statement saying that all vector spaces of the same dimension are isomorphic :)

Also, other somewhat mystifying statements, like $«H^1(G,\mathrm{Sp}(n, L))=0»$ can be proved by exactly the same argument.

1

One argument I love is the following: let $L/K$ be a Galois extension with group $G$ and let $n\geq1$. One can show very straightforwardly that $H^1(G,\mathrm{GL}(n, L))$ classifies $K$-vector spaces $V$ such that $L\otimes_KV$ is isomorphic as an $L$-vector space to $L\otimes K^n$, up to $K$-linear isomorphisms; Serre does it in chapter X, §2, of his Corps Locaux. Now, linear algebra tells us that all such $V$'s are in fact isomorphic to $K^n$ as $K$-vector spaces, so we conclude that $H^1(G,\mathrm{GL}(n, L))$.

This is, in fact, the same argument that Brian gave. Yet it is nice that the theorem becomes essentially a statement saying that all vector spaces of the same dimension are isomorphic :)