How to compute the Wedderburn-Artin decomposition for a group algebra I have a homework question that asks me to

Compute the Wedderburn-Artin decomposition for $F[G]$ where $G$ is the
  cyclic group of order $3$ and $F$ is one of complex numbers or reals
  or rationals.

I'm clueless about how to begin. I'll be glad if someone can provide with an answer to one of the fields and hopefully I'll figure out what to do for the other fields.
The group is abelian, so I suppose the factors will be matrix rings with dimension $1$?
Thank you
 A: Hint: for a general cyclic group $G$ of order $n$ and $F$ a field of characteristic zero, show that $F[G]$ and $F[X]/\langle X^{n}-1 \rangle$ are isomorphic as $F$-algebras. You can see then how things will change for $F = \mathbb{Q}, \mathbb{R}$ and $\mathbb{C}$, since this will affect how $X^{n}-1$ factors over $F$. Namely, over both $\mathbb{Q}$ and $\mathbb{R}$, $X^{3}-1$ factors as $(X-1)(X^{2}+X+1)$, so by the Chinese Remainder theorem, we get
$$\mathbb{Q}[G] \cong \mathbb{Q} \oplus \mathbb{Q}(\zeta_{3})$$
$$\mathbb{R}[G] \cong \mathbb{R} \oplus \mathbb{R}(\zeta_{3}) \cong \mathbb{R} \oplus \mathbb{C}$$
Finally, since $X^{3}-1$ splits into distinct linear factors over $\mathbb{C}$, we get 
$$\mathbb{C}[G] \cong \mathbb{C} \oplus \mathbb{C} \oplus \mathbb{C}$$
Note that in the last case, this agrees with the classical results from representation theory over $\mathbb{C}$: since $G$ is an abelian group of order $3$, there should be one irreducible representation of $G$ of dimension $1$ for each element of $G$, hence one factor of $\mathbb{C}$ in the Wedderburn decompostion of $\mathbb{C}[G]$ for each element of $G$ - and this is exactly what we find.  
