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In studying Linear Representation theory of finite groups, many authors state the important theorems over the field $\mathbb{C}$ or algebraically closed field, which is much stronger assumption. I am interested in representations over general fields. I know that we have, atleast to consider cases where characteristic of field is zero or prime to $|G|$ (ordinary representations) and characteristic dividing $|G|$ (modular representations).

I would like to see the important concepts such as "Schur's Lemma, Maschke's theorem, McKay irreducibility criteria, orthogonality relations, number of irreducible $F$-representations and number of some type of conjugacy classes of group etc (in ordinary case at least).

Can someone suggest some good reference for the subject which states these results with weaker hypothesis (means not always consider algebraically closed field)?

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up vote 5 down vote accepted

The book by Curtis-Reiner, "Representation Theory of Finite Groups and Associative Algebras" might be what you are looking for.

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This would have been my recommendation, too. They cover modular rep theory, splitting fields and such. I'm sure many other authors do the same, but I was brought up with C & R, so... –  Jyrki Lahtonen Oct 8 '11 at 13:08

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