Group ring over splitting field is direct sum of matrix rings for each irreducible representation

Statement
Suppose $$G$$ is a finite group and $$K$$ is a splitting field for $$G$$. Suppose the irreducible representations of $$G$$ are $$\varphi_1, \varphi_2, \dots, \varphi_r$$ and the degrees of irreducible representations are $$d_1,d_2,\dots,d_r$$ respectively. Then, the group ring $$K[G]$$ is a semisimple Artinian ring and is expressible as a direct sum of matrix rings over $$K$$ as follows:

$$K[G] \cong M_{d_1}(K) \oplus M_{d_2}(K) \oplus \dots \oplus M_{d_r}(K)$$

where $$M_n(K)$$ is the ring of $$n \times n$$ matrices over $$K$$. Note that each $$M_n(K)$$ is a simple ring and therefore corresponds to a minimal two-sided ideal in the decomposition.

Over non-splitting fields
Even if $$K$$ is not a splitting field, it is true that if the characteristic of $$K$$ does not divide the order of $$G$$, $$K[G]$$ is a semisimple Artinian ring. By the Artin-Wedderburn theorem, it splits as a direct sum of matrix rings over division rings (in an essentially unique fashion), where each division ring has $$K$$ in its center. If the division rings that we need to use are not fields, then we have the Schur index phenomenon.

Alternative formulations

 * Sum of squares of degrees of irreducible representations equals order of group
 * Regular representation over splitting field has multiplicity of each irreducible representation equal to its degree