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Suppose I have a finite Galois extension of fields $K/k$, as well as a finite group $G$ with a surjection $f: G \rightarrow \mathrm{Gal}(K/k)$.

Finally, suppose I have an action $\sigma$ of $G$ on a vector space $V$ over $K$ by twisted endomorphisms, i.e. $\sigma(av) = f(\sigma)(a)\sigma(v)$.

How can I detect whether $G$ descends to a representation over $k$, i.e. when does there exist an untwisted representation of $G$ on a vector space $W$ such that $V = W \otimes_{k} K$ with the diagonal action?

I guess this is some group/Galois cohomology obstruction, but I wasn't sure how to set up the problem.

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  • $\begingroup$ It always descends to the space of invariants in $V$. $\endgroup$ – Angelo Sep 15 '13 at 11:57
  • $\begingroup$ @Angelo: Consider $G = {\rm{Gal}}(K/k) \times H$ for a finite group $H$ with $f = {\rm{pr}}_1$, and $V= K \otimes_k W$ with $(\gamma, h) \in G$ acting by $c \otimes w \mapsto \gamma(c) \otimes h(w)$. Then $V^G=W^H$ but $K \otimes_k V^G = K \otimes_k W^H$ is typically a proper $K$-subspace of $V$, so what do you mean by "It always descends to the space of invariants in $V$"? Invariants under what group? $\endgroup$ – Marguax Sep 15 '13 at 14:08
  • $\begingroup$ Sorry, I had not read the question properly, I thought that $G$ was the Galois group. $\endgroup$ – Angelo Sep 15 '13 at 20:16

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