Dimension of fixed points of Galois group actions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T01:41:07Z http://mathoverflow.net/feeds/question/82509 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/82509/dimension-of-fixed-points-of-galois-group-actions Dimension of fixed points of Galois group actions José Manuel Gómez 2011-12-02T21:18:10Z 2011-12-02T22:18:17Z <p>I have a question about fixed points of Galois group actions. I am hoping that this is easy for the experts.</p> <p>Let $k$ be a field of characteristic $0$. Let $K$ be a finite Galois extension of $k$ with Galois group $G$. </p> <p>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 $\dim_{k}(A)$. </p> <p>Any ideas on how to attack this problem are more than welcomed.</p> http://mathoverflow.net/questions/82509/dimension-of-fixed-points-of-galois-group-actions/82513#82513 Answer by kassabov for Dimension of fixed points of Galois group actions kassabov 2011-12-02T22:18:17Z 2011-12-02T22:18:17Z <p>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.</p>