Question

I have a question about fixed points of Galois group actions. I am hoping that this is easy for the experts.

Let $k$ be a field of characteristic $0$. Let $K$ be a finite Galois extension of $k$ with Galois group $G$.

Supose that $A$ is any finite dimensional $G$-representation over $k$. Then $G$ acts diagonally on $A\otimes K$. The question is to determine $\dim_{k}(A\otimes K)^{G}$. I am hoping the answer is $\dim_{k}(A)$.

Any ideas on how to attack this problem are more than welcomed.

Answer 1

This follows from the fact that $K$ is isomorphic as $G$ module to the free module $k[G]$. (use the existence of a basis of $K$ over $k$ consisting of Galois conjugates.