A question in representations theory My question is about irreducible representations of groups over the field $\mathbb{Q}$.
Let $G$ be a cyclic or an abelian group. I want to check that under what conditions we have a $\mathbb{Q}G$-module that is irreducible.
Because $\operatorname{char} \mathbb{Q}=0$, by Maschke 's theorem , we know that every $\mathbb{Q}$-representation of G is com­pletely  reducible. Although, because $\mathbb{Q}$ is not algebraically closed, we can not say that every irreducible $\mathbb{Q}$-representations of an abelian group has degree 1.
Specially, I want to find a faithful irreducible $\mathbb{Q}$-representation of degree $\phi(n)$ (where $\phi$ is Euler's function) for a cyclic group of order $n$. I know that I should use $n^{th}$ roots of unity but I don't know how I can move this representation to $\mathbb{Q}$. 
Thanks a lot.
 A: Hint: $\Bbb Q[\zeta_n]$ is a quotient ring of $\Bbb Q[C_n]$.
(All quotient rings are modules over the original ring...)
A: You can classify the irreducible representations of a finite cyclic group over an arbitrary field as follows. The group algebra of the cyclic group $C_n$ of order $n$ over an arbitrary field $k$ is $k[x]/(x^n - 1)$. If $x^n - 1 = \prod_k f_k(x)^{m_k}$ is the factorization of $x^n - 1$ into irreducibles over $k$, then by the Chinese remainder theorem $k[x]$ factors (as a $k$-algebra) into a product
$$\prod_k k[x]/f_k(x)^{m_k}$$
and determining the simple modules of $k[x]/(x^n - 1)$ (which is equivalent to determining the irreducible representations of $C_n$ over $k$) from here is straightforward: they are precisely the quotients $k[x]/f_k(x)$, with a generator of $C_n$ acting as multiplication by $x$. Explicitly, $x$ acts on the basis $1, x, \dots x^{\deg f_k - 1}$ by the companion matrix of $f_k$. 
Example. If $k = \mathbb{C}$, then $x^n - 1$ factors as 
$$\prod_{k=0}^{n-1} (x - \zeta_n^k)$$ 
where $\zeta_n$ is a primitive $n^{th}$ root of unity. This recovers the familiar classification of irreducible representations of $C_n$ over the complex numbers.
Example. If $k = \mathbb{Q}$, then $x^n - 1$ factors as
$$\prod_{d | n} \Phi_d(x)$$
where $\Phi_d(x)$ is the $d^{th}$ cyclotomic polynomial. Hence there are $\sigma_0(n)$ irreducible representations, one for each divisor $d$ of $n$, which have dimension $\varphi(d)$ and are given explicitly in matrix form by the companion matrices of $\Phi_d$.  
Example. Now suppose that $n = p$ is prime and that the characteristic of $k$ is also $p$. Then $x^p - 1$ factors as
$$(x - 1)^p.$$
Hence there is exactly one irreducible representation of $C_p$ over a field of characteristic $p$, namely the trivial representation. 
