Recover an integer matrix from its complex diagonalization

Question

Suppose that $A$ is an $n \times n$ integer matrix, and that we have no knowledge of its entries. However, we do know that it is diagonalizable over $\mathbb C$ - in fact, we have the diagonalization (in my case, I am able to obtain this diagonalization because $A$ is of finite order, and I know the characters of its powers). How can we obtain a matrix similar over $\mathbb Z$ to $A$?

Answer 1

Write the characteristic polynomial of $A$ in the form $p(x) = p_1(x) p_2(x) \cdots p_k(x)$, where each polynomial $p_j$ has non-repeating roots and integer coefficients. Then because your $A$ is diagonalizable, $A$ is similar (over $\Bbb Z$) to the block-diagonal matrix $$ \pmatrix{C(p_1) \\ & \ddots \\ && C(p_k)}, $$ where $C(p)$ is the companion matrix associated with $p$.