Let $G$ be a finite group, and let $K$ be a finite field whose characteristic does not divide $|G|$. I am interested in the theory of finitely generated modules over $K[G]$. Of course many problems are not present here because $K[G]$ is semisimple and all modules are projective. My case is partly covered by Section 15.5 of Serre's book "Linear Representations of Finite Groups". However, Serre likes to assume that $K$ is "sufficiently large", meaning that it has a primitive $m$'th root of unity, where $m$ is the least common multiple of the orders of the elements of $G$. I do not want to assume this, so some Galois theory of finite extensions of $K$ will come into play. I do not think that anything desperately complicated happens, but it would be convenient if I could refer to the literature rather than having to write it out myself. Is there a good source for this?


[UPDATED]:


In particular, I would like to be able to control the dimensions over $K$ of the simple $K[G]$-modules. As pointed out in Alex Bartel's answer, these need not divide the order of $G$. I am willing to assume that $G$ is a $p$-group for some prime $p\neq\text{char}(K)$.


[UPDATED AGAIN]:


OK, here is a sharper question. Put $m=|K|$ (which is a power of a prime different from $p$) and let $t$ be the order of $m$ in $(\mathbb{Z}/p)^\times$. Let $L$ be a finite extension of $K$, let $G$ be a finite abelian $p$-group, and let $\rho:G\to L^\times$ be a homomorphism that does not factor through the unit group of any proper subfield containing $K$. Then $\rho$ makes $L$ into an irreducible $K$-linear representation of $G$, and every irreducible arises in this way. If I've got this straight, we see that the possible degrees of nontrivial irreducible $K$-linear representations of abelian $p$-groups are the numbers $tp^k$ for $k\geq 0$. I ask: if we let $G$ be a nonabelian $p$-group, does the set of possible degrees get any bigger? 



Modular representations with unequal characteristic - reference request
Your last statement is not true in general. Let $G=C\_3$ and take your favourite finite field that does not contain the cube roots of unity, e.g. $K=\mathbb{F}\_5$. Then the two non-trivial one-dimensional representations over $\bar{K}$ are not defined over $K$, but their sum is, since it's the regular representation minus the trivial.


In general, $K[G]$-modules for $K$ finite of characteristic co-prime to $|G|$ behave pretty much like modules over characteristic zero fields (the simple $K[G]$-modules are just sums over Galois orbits of the absolutely simple ones), the major simplification being that there are no Schur indices. I am not sure that there is much more to say about this. The second volume of Curtis and Reiner contains a whole chapter on rationality questions, i.e. fields that are not "sufficiently large" in the sense of Serre. Most of it is for characteristic zero, but as I say, a lot of it carries over.


**Edit** Re updated question: if $G$ is a finite $p$-group and $K$ is a finite field of characteristic different from $p$, then it is indeed true that any irreducible representation of $G$ over $K$ has dimension $tp^k$, for some $k$ and some $t\;|\;(p-1)$. Indeed, the absolutely irreducible representations have dimension a power of $p$, and the field of definition of any absolutely irreducible representation is $K$ adjoin $p^r$-th roots of unity for some $r$, so the Galois orbit of a representation has size dividing $(p-1)p^{r-1}$ for some $r$.


