Call a square matrix **Galois-irreducible** if all its eigenvalues are Galois conjugates of each other.

Let $M$ be an integer $n\times n$ matrix which is not Galois-irreducible. Is it always possible to find an integer matrix $S$ such that $S^{-1}MS=diag(A_1,...,A_k)$ is a block diagonal matrix with Galois-irreducible matrices $A_i$ ?

Intuitively this should be true, but I have no idea how to construct such a matrix $S$.

Note that this is not true when replacing everywhere "Galois-irreducibility" by "irreducibility of the characteristic polynomial" because any non-trivial Jordan blocks like $\begin{pmatrix}\lambda&1\\ 0&\lambda\\ \end{pmatrix}$ for, say, $\lambda\in\mathbb Z$, would yield a counterexample.

In a similar vein:

Suppose $M$ is this time Galois-irreducible, but with each eigenvalue of multiplicity $k$ and trivial Jordan blocks. Does there exist $S$ as above, maybe even such that $A_1=\cdots=A_k $?

(It is clear that for non-trivial Jordan blocks the latter isn't possible.)