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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.

Any hints and suggested literature is very welcome.

Regards, Bill

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It isn't a question of working over an algebraically closed field, but instead a splitting field. Otherwise you have to compare what happens over a given field and a larger one. Brauer theory deals with a splitting field of prime characteristic dividing the group order; then the usual traces are not so useful and are replaced by corresponding sums of roots of unity in a field of characteristic 0. There is a range of helpful textbook literature: Curtis-Reiner, Serre, etc. – Jim Humphreys Jul 29 '12 at 20:45

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 $\bar K \otimes M$.

Also, the values of the Brauer character of $M$ will lie in the ring of Witt-vectors over $K$.

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Or, put another way, it is relatively easy to understand the structure of simple modules over a general field once you understand the algebraically closed field. However, there are issues of Clifford theory and Galois theory which need to be analysed sometimes. – Geoff Robinson Jul 29 '12 at 19:45

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