$\newcommand{\Q}{\mathbb Q}$ Let $F=\mathbb Q^{ab} \subset \mathbb C$, i.e. the algebraic numbers. Let $G$ be a finite group of order $n$ and let $\phi: G \rightarrow GL_m(F)$ be a representation. Denote the extension of $\Q$ generated by the entries of $\phi(g)$ for each $g \in G$ by $\Q(\phi)$. Note this is a finite extension. My question is whether or not $\Q(\phi)$ is contained in a cyclotomic extension of $\Q$ or if we can find a change of basis such that $\Q(\phi)$ is wrt to this basis.
Clearly this is the case if $\phi$ is a sum of $1$-dimensional representations, because then the $\phi(g)$ are simultaneously diagonalizable. Since each $\phi(g)$ has only roots of unity for eigenvalues it follows. Other than this rather simple case, I'm at somewhat of a loss. Mainly I'm unsure of how to show something is not contained in a cyclotomic extension. I would guess we would want to show $\Q(\phi)$ has some $\alpha$ in it such that the splitting field of $\alpha$'s minimum polynomial has non-abelian galois group.