This is not really an answer, but is too long for a comment. The proof given by Moonface above is given in more or less that form in the 1962 book of Curtis and Reiner. As far as I know, it is still open whether all irreducible representations of a finite group $G$ can be realized over
$\mathbb{Z}[\omega]$, where $\omega$ is a complex primitive $|G|$-th roots of unity, though I think the paper of Cliff,Ritter and Weiss settles the questions for finite solvable groups. The paper
of Serre ( the three letters to Feit) give counterexamples to a slightly different question:
they show (among other things) that a representation of a finite group can be realised over
some number fields, but might not be able to be realised over the ring of algebraic integers
of that field. Brauer's characterization of characters/Brauer's induction theorem show that
all representations of the finite group $G$ may be realised over $\mathbb{Q}[\omega]$ for
$\omega$ as above ( $|G|$ can be replaced by the exponent of $G$ if desired). As I said,
realizability over $\mathbb{Z}[\omega]$ is a different matter.