I'm trying to prove the following: Given a matrix $M \in \mathbb{Z}^{n\times n}$ with an irreducible characteristic polynomial $f$ (irreducible over $\mathbb{Z}$ or $\mathbb{Q}$). If I'm not mistaken, this implies that $\mathbb{Q}(f)$ is Galois over $\mathbb{Q}$, and furthermore $$f(X) = \prod_{\sigma_i \in G} (X-\sigma_i(\lambda)),$$ where $G = \text{Gal}(\mathbb{Q}(f)/\mathbb{Q})$ and $\lambda \in \mathcal{O}_{\mathbb{Q}(f)}$.
Since $f$ is separable, the matrix $M$ is diagonalizable, i.e. $$ M = P^{-1}D P \text{ with } D = \begin{pmatrix} \sigma_1(\lambda) & 0 & \dots \\ 0 & \sigma_2(\lambda) & \dots \\ \vdots & \vdots & \ddots \end{pmatrix}.$$ It can easily been seen that $x = \prod_{\sigma_i\in G} \sigma_i(\lambda)$ lies in $\mathbb{Z}$, because this number is invariant under all automorphisms of $G$ and $\lambda$ is an algebraic integer. Consider $x_k = \prod_{i\neq k, \sigma_i\in G} \sigma_i(\lambda) = \frac{x}{\sigma_k(\lambda)}$. I wish to show that the following matrix is integral (in $\mathbb{Z}^{n\times n}$): $$ A = P^{-1}\begin{pmatrix} x_1 & 0 & \dots \\ 0 & x_2 & \dots \\ \vdots & \vdots & \ddots \end{pmatrix}P.$$ The only idea I had so far was to show that $x_i = m_1 + m_2\sigma_i(\lambda)$ for some integers $m_1$ and $m_2$. (Notice that clearly if those numbers exist, they must be the same for all $x_i$ by applying $\sigma \in G$ to the expression.) However, I have no idea how to prove this claim.
Suppose the claim hold, then $A$ clearly equals $P^{-1}(m_1\text{Id} + m_2 D)P = m_1\mathbb{1} + m_2M$. Hence, this surely suffices to prove the result.
If $\text{Adj}(A)$ is the adjugate of $A$, then we find $\text{Adj}(M) = P^{-1}\text{Adj}(D)P$. Note that $\text{Adj}(D) = x D^{-1}$, so $\text{Adj}(M)$ is precisely $A$. Since the adjugate of an integral matrix is clearly integral, that proves the claim above.
What happens in the following case:
$$ M = P^{-1}D P \text{ with } D = \begin{pmatrix} \sigma_1(\lambda) \text{Id}_k & 0 & \dots \\ 0 & \sigma_2(\lambda)\text{Id}_k & \dots \\ \vdots & \vdots & \ddots \end{pmatrix};$$
$$A = P^{-1}\begin{pmatrix} x_1\text{Id}_k & 0 & \dots \\ 0 & x_2\text{Id}_k & \dots \\ \vdots & \vdots & \ddots \end{pmatrix}P,$$
i.e. the diagonal matrices are block diagonal in structure.
Does it hold that $A$ is integral? Equivalently, does it hold that $x^{k-1}\mid \text{Adj}(M)$?