This is a question from one of our 'ancient' qualification exams. I am not familiar with representation theory, so the second question baffles me a lot.
Let $G=\langle g: g^3=1\rangle$ be the cyclic group of order $3$.
- Show that $\mathbb{C}[G]$ is isomorphic to $C\times C\times C$ as $C$-algebra. What about $\mathbb{Q}[G]$ and $\mathbb{F}_2[G]$
- Classify all simple modules of $\mathbb{C}[G],\mathbb{Q}[G],\mathbb{F}_2[G]$ and $\mathbb{F}_3[G]$ respectively.
My attempts
Let $\zeta$ be the root of unity that $\zeta^3=1$. According the Chinese remainder theorem, we have $$\mathbb{C}[G]\simeq \mathbb{C}[x]/(x^3-1)\simeq\mathbb{C}[x]/(x-\zeta)\times \mathbb{C}[x]/(x-\zeta)\times \mathbb{C}[x]/(x-\zeta)\simeq \mathbb{C}^3.$$
For $F=\mathbb{Q},\mathbb{F}_2$, we have $$F[G]\simeq F\times F[x]/(x^2+x+1).$$
Therefore, $$F[G]/F\simeq F[x]/(x^2+x+1),$$which is a field. Hence, a similar result does not hold for the other two cases.
For the second question, I know that $\mathbb{F}_3[G]$ will be pretty different from the others, since $\gcd(\text{char}(F), |G|)=3$ is non-trivial. Any ideas on how to come up with all simple modules? THX:)