Analogon to Brauer characters, if K not algebraically closed - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T09:59:12Z http://mathoverflow.net/feeds/question/103459 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103459/analogon-to-brauer-characters-if-k-not-algebraically-closed Analogon to Brauer characters, if K not algebraically closed Bill 2012-07-29T17:29:52Z 2012-07-29T18:02:35Z <p>Hello, is there a theory for characters of a finite group over a field $K$ with prime characteristic, if $K$ is not algebraically closed? For algebraically closed fields $K$ for example Brauer found some interesting theorems.</p> <p>Any hints and suggested literature is very welcome.</p> <p>Regards, Bill</p> http://mathoverflow.net/questions/103459/analogon-to-brauer-characters-if-k-not-algebraically-closed/103463#103463 Answer by Florian Eisele for Analogon to Brauer characters, if K not algebraically closed Florian Eisele 2012-07-29T18:02:35Z 2012-07-29T18:02:35Z <p>There really isn't any need for character theory over non-algebraically closed fields. The Grothendieck group of $KG$ embeds into the Grothendieck group of $\bar K G$ (i.e. of the group ring over the algebraic closure). Therefore all information about a $KG$-module $M$ you could possibly hope to recover from any type of "character of $M$" is already contained in the Brauer character of the $\bar K G$-module $\bar K \otimes M$. </p> <p>Also, the values of the Brauer character of $M$ will lie in the ring of Witt-vectors over $K$. </p>