What are the groups $G$ and fields $\Bbb K$ for which $\Bbb K[G]\cong\Bbb K^{G}$ holds?
For example $\Bbb R[\Bbb F_2^n]\cong\Bbb R^{2^n}$ holds.
What are the groups $G$ and fields $\Bbb K$ for which $\Bbb K[G]\cong\Bbb K^{G}$ holds? For example $\Bbb R[\Bbb F_2^n]\cong\Bbb R^{2^n}$ holds. 


This is true iff $G$ is finite and abelian, the characteristic of $K$ does not divide $G$, and $K$ has all $n^{th}$ roots of unity whenever $G$ has an element of order $n$. Hopefully it is clear why $G$ must be finite and abelian. The characteristic and root of unity conditions follow from writing $G$ as a product of cyclic groups and applying the Chinese remainder theorem. 

