Let, $C_3=\langle\sigma\mid\sigma^3=1\rangle$, $a=\frac{1}{3}( 1+ \sigma +\sigma^2)$, $b=\frac{1}{3}(1+w \sigma+ w^2 \sigma^2)$ and $c=\frac{1}{3}(1+w^2\sigma+ w\sigma^2)$ where $w$ is the primitive cube root of 1.
Characteristic is $0$. Need to decompose the group ring $KG$ (where $K$ is a field) into $KG \cong aK \oplus bK \oplus cK$. Personally, I don't see why there are two decomposition from $KG$ into $aK \oplus bK \oplus cK$(when char $K=0$) and $KG \cong K[x]/(x^3)$ (when char $K=3$). Seems do arbitrary.
In the notes it says you need to calculuate $a^2=a$, e.t.c show they are all idempotents.
However, it says that you need to show all crossproduct of $a,b,c$ are $0$.
However, when I do $ab=\frac{1}{3}(1+w+w^2+(1+w+w^2)\sigma+(1+w+w^2)\sigma^2)$. This doesn't equal $0$ when you take $w=1$.