Irreducible characters of k[G] when k is not algebraically closed and char k divides order of G.

Question

Let k[G] be the group algebra where char(k) divides |G| with G being a finite group. Assume k is not algebraically closed. How can one show that the characters associated with the irreducible representations are linearly independent and non-zero over k? Is there any argument that doesn't involve Galois theory or is that the only line of attack? Thank you in advance for any assistance! :)

Answer 1

I’m not sure how it translates to characters, but the simple modules of $k[G]$ are the same as the simple modules of $k[G]/J(k[G])$, where $J(-)$ refers to the Jacobson radical.

So you should be able to compute the simple modules from $k[G]/J(k[G])$ and extract their characters.