Group ring of finite group over field of characteristic not dividing its order is semisimple Artinian

Statement
Suppose $$G$$ is a finite group and $$K$$ is a field whose characteristic does not divide the order of $$G$$. Then, the group ring $$K[G]$$ (i.e., the group ring of $$G$$ over $$K$$ is a semisimple Artinian ring. In particular, it is a finite-dimensional semisimple algebra over $$K$$.

Related facts

 * Artin-Wedderburn theorem that tells us further that the group ring must therefore be a direct sum of matrix rings over division rings.
 * Group ring over splitting field is direct sum of matrix rings for each irreducible representation: In the case of a splitting field, the group ring is a direct sum of matrix rings over the field $$K$$ itself, with the sizes of the matrices corresponding to the degrees of irreducible representations.